Climate, Assimilation of Data and Models - When Data Fail Us · 2010-06-29 · UQG Group Focus...

Climate Assimilation of Data and ModelsWhen Data Fail Us

JUAN M RESTREPOGroup Leader

Uncertainty Quantification Group

University of Arizona

June 2010

JUAN M RESTREPO Group Leader Uncertainty Quantification Group ( University of Arizona )Climate Assimilation of Data and Models June 2010 1 88

UQG Group


FacultyJMR (Math Physics Atmospheric Sciences)Shankar Venkataramani Kevin Lin Hermann Flaschka (Math)Shlomo Neuman Larry Winter (Hydrology)Rabi Bhattacharya Walt Piegorsch (Math Statistics)Kobus Barnard Alon Efrat (Computer Science)

Post-DocsP Krause (estimation)J Ramırez (probability)P Dostert (scientific computing)T-T Shieh (variational methods)

Graduate Students 15 (Brad Weir Darin Comeau Suz Tolwinski)Undergraduate Students 2 (Jason Ditmann)

UQGQG


UQG Group

Focus ProblemsClimateWeather Hydrogeology Computer Vision

Complex Problems Problems in which progress will result from acombination of statistical methods and deterministic and stochasticmathematics

UQGQG


UQG Group



UQGQG


UQG Group



UQGQG


UQG Group

Earthrsquos Climate ForcedDissipativeThermodynamicSystem

TIME

SP

AC

E

WEATHER

300 years1 day 1 season

Climate

10 years

continent

country

city


UQG Group

The Meteorology vs Climate Problem

Climate poor andor sparse data Very large spatio-temporal ranges(oceans) Challenging interations oceanrsquos interaction with theatmosphere and ice

Aim is to discern variability and understanding dynamicsBayesian data assimilation is viable strategyNeed to make more use of models need to develop parameterizationsNeed to focus more on non-Gaussian issues as well as the use of dataassimilation combined with time of transit dynamics

Meteorology lots of data very nonlinear very stiffSocietal imperative forecastsEnsemble forecastingReduced state space representationLarge deviation theory could be explored here


UQG Group






UQG Group






UQG Group






UQG Group






UQG Group

What Models tell Us about Data the Non-Gaussian Issue

When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group


When data fool ussame data zoomed in

0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group


use our understanding of the dynamics

dx = 4x(1minus x2)dt +κdWt

x(0) = x0

minus3 minus2 minus1 0 1 2 30

01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD


Time Dependent Estimation

Data Assimilation in Climate Studies

Combine information derived from data and models

Bayes Theorem

P(X|Y) prop likelihoodtimesprior


Time Dependent Estimation LinearGaussian Estimation

Least Squares and Kalman FilterSmoother

W(m)xminusV = Θ

ΘsimN (0σ)

Find x mean such that E(θgtθ) is minimizedFind the uncertainty U = E[(xminus x)(xminus x)gt]

Alternatively can find x and U using a sequential approach the KalmanFilterSmoother (RTS Algorithm)

Review in C Wunsch The Ocean Inverse Circulation Problem Cambridge U Press



Kalman Filter

Forecast

Xlowast = MX(t)+BP(t) t = 01

Ulowast = MU(t)Mgt

Analysis

X(t +1) = Xlowast+K(t +1)[Y(t +1)minusH(t +1)Xlowast]U(t +1) = UlowastK(t +1)H(t +1)Ulowast

where the Kalman Gain Matrix is

K(t +1) = UlowastH(t +1)gt[H(t +1)UlowastH(t +1)gt]minus1

X(0) and U(0) are known




Nonlinear Non-Gaussian Problems

Forecast not much of a problem

X(t +1) = N(X(t)BP(t))

But not clear how to propagate uncertainty U(t +1)

Extended Kalman Filter used extensively on nonlinear problems linearizeabout X(t) and use closure ideas for moments


Nonlinear Non-Gaussian Estimation

A Nonlinear Non-Gaussian Example

Let x(t) isinR1 0 lt t le T

Model


x(0) = x0 known distrib

Data

y(tm) = x(tm)+η(tm)E(ηmηl) = Rδtmtl

m = 12 M

minus2 minus15 minus1 minus05 0 05 1 15 2minus2

0

2

4

6

8

10

12



Double-Well Stationary Distribution


The Stationary distribution for the double-well problem (κ = 05)


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12

A single realization of a random process with known statistics



Goals of Traditional Data Assimilation

Find mean history xS(t) conditioned on observations E(x(t)|y1 yM)and the uncertainty Cs(t) = E[(x(t)minus xs(t))(x(t)minus xs(t))T |y1 yM]xs(t) should minimize the tr(Cs)Guarantee statistical convergence of moments



The EKF Results1

Figure 10 uncertainty ∆t = 1



1R Miller M Ghil P Gauthiez Advanced data assimilation in strongly nonlineardynamical systems J Atmo Sci 51 1037-1056 (1994)



The EKF Results1







The EKF Results1







The EKF Results1







Other Approaches on NonlinearNon-Gaussian Problems

Optimal (variance-minimizer) KSP (Kushner Stratonovich Pardoux) early 60rsquos

4D-VarAdjoint (Maximum Likelihood) (Wunsch McLaughlin Courtier late 80rsquos)

ensemble KF (Evensen rsquo97)

Mean Field Variational (Eyink Restrepo rsquo01)

Parametrized Resampling Particle Filter (Kim Eyink Restrepo Alexander Johnson rsquo02)

Path Integral Monte Carlo (Restrepo rsquo07 Alexander Eyink amp Restrepo rsquo05)

Diffusion Kernel Filter (Krause Restrepo rsquo09)

Langevin Sampler (A Stuart rsquo05)














KSP Optimalso why not use it

Filter between tm and tm+1 solve

parttP = minuspartx[f (x)P]+12

κ2partxxP

P(x0) = Ps(x)

At measurement times tm the probability jumps

P(x t+) =1N

e1

R2 [ymminus x2m2 ]P(x tminus)



Smoother between tm+1 and tm solve

parttA = minuspartx[f (x)]A +12

κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1

R2 [ymminus x2m2 ]A (x t+)

Finally P(x t|ymm = 12 M) = A (x t)P(x t)



KSP Filter and Smoother Results

Figure KSP Filter

Figure KSP Smoother



enKF Most Favored in Practice

The enKF (rdquostate-of-the-artrdquo)

Use model for forecast

Update the uncertainty using Monte Carlo

Pros and Cons

Can handle legacy code easily

Linear (Gaussian) analysis

Requires full model runs

Ad-hoc

G Evensen Sequential data assimilation with a nonlinear quasigeostrophic model using Monte Carlo methods to forecast errorstatistics J Geophys Res 99 10143-10162


Nonlinear Non-Gaussian Estimation The Path Integral Monte Carlo (PIMC)

PIMC The Path Integral Monte Carlo

Optimal on the discretized model

Simple to implement but very subtle

Can handle legacy code

Relies on sampling

Can yield a variety of different estimators

J Restrepo A Path Integral Method for Data Assimilation Physica D 2007F Alexander G Eyink J Restrepo Accelerated Monte-Carlo for Optimal Estimation of Time Series J Stat Phys 2005



Bayesian Statement

P(x|y) prop LikelihoodtimesPrior

Use data for likelihood

Use model for prior

P(x|y) prop eminusAmodeleminusAdata = eminusA (x)



Amodel

dx = f (x t)dt +[2D(x t)]12dW

is discretized

xn+1 = xn +∆tf (xn tn)+ [2D(xn tn)]12[Wn+1minusWn]

n = 01 Tminus1

Amodel asympT

sumn=1

[(xn+1minus xnminus∆tf (xn tn))gtD(xn tn)minus1 (xn+1minus xnminus∆tf (xn tn))]

if Prob(∆W) prop exp(minus∆W2D)



Adata

ym = H(xm)+ [2R[xm tm)]12ηm

m = 12 M

Adata =M

summ=1

[(ymminusH(xm))gtR(xn tn)minus1 (ymminusH(xm))]

if Prob(η) prop exp(minusη2R)


Nonlinear Non-Gaussian Estimation Samplers

MCMC Samplers


The Path Integral Monte Carlo practicality depends on fast sampling

Multigrid (UMC)

Langevin Sampler (LS)

Hybrid Monte Carlo (HMC)

Shadow Hybrid MC (sHMC)

Riemannian Manifold Hamiltonian Monte Carlo (RM-HMC)

generalized Hybrid Monte Carlo (gHMC)



(HMC) Hybrid Markov Chain Monte Carlo

Proposals generated bysolving Hamiltoniansystem in fictitious timeτ

Acceptreject viaMetropolis Hastings



HMC Algorithm

Let qn(τ = 0) = xn

To each qn a conjugate generalized momemtum pn is assigned

The momenta pn give rise to a kinetic contributionK = sum

Tn=1 pgtn Mminus1pn2

The Hamiltonian of the system H = A (q)+K(p)The dynamics are

partqn

partτ= Mminus1pn

partpn

partτ= Fn where Fn =minusgrad(A (q))

Solve using Verlet integrator (detailed balance)

AcceptReject MetropolisHastings



Why does HMC work What are good HMC properties

Write probability Π(q) = 1ZΠ

eminusA (q)

Sampling π(qp) = 1Zπ

eminusH (qp) sim 1Z eminusA (q) samples Π(q)

Gradient dynamics makes system search through configuration spacemore efficiently

Moves in qn are linear in pn ie partqpartτ

= Mminus1p

A (q) and grad(A (q)) should be easily evaluated



Sampler Efficiency Estimates

Sampler Efficiency key to choosing and tuning sampler

Computational Cost O(NT)r nmethod(pL)p =lt Pacc gt=lt min1exp[minus∆H ]gtprop erfc

(12 δτm(NT)12

)

c(L) =lt H (0)H (0+L) gt Depends on problem dimension and statespace characteristics



RM-HMC Algorithm2

Hamiltonian replaced by

H = A (q)+12

pgtG(q)minus1p

where the non-degenerate Fisher information matrix G = EnablaA nablaA gt

Challenges

find a time-reversiblevolume-preserving discrete integrator forHamiltonian problem

optimize its computational efficiency

2Girolami Calderhead Chin preprint 2009JUAN M RESTREPO Group Leader Uncertainty Quantification Group ( University of Arizona )Climate Assimilation of Data and Models June 2010 32 88


Decrease decorrelation length L gHMC Algorithm

Hamiltonian dynamics replaced by

partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)

where C isinRTtimesT matrix

Challenge find C that leads to a significant reduction in the sampledecorrelation length

We used the circulant matrix C = circ(1eminusα eminus2α eminusTα)



Sampler Efficiency Comparison

Table T is the number of time steps (middot) is the standard deviation on the number ofsamples [α] used in C J is the number of τ time steps

T +1 HMC (J=1) HMC (J=8) UMC gHMC (J=1)8 900(125) 170(7) 800(40) 40(8) [020]16 5300(1600) 560(20) 1040(60) 60(10) [010]32 13300(8300) 2700 (140) 1430(100) 200(30) [005]64 30000(7800) 2800(400) 1570(100) 420(70) [00245]


The DKF

The Diffusion Kernel Filter

Collaborator

Paul Krause UQG U Arizona

A particle-filter based method

p(x|ym tm) =p(ym|x) p(x tm)

p(ym)

SequentialSimple to implement provided an adjoint existsThe Clustered cDKF is competitive with EKFCan handle nonlinearnon-Gaussian problems

P Krause and J Restrepo The Diffusion Kernel Filter Applied to Lagrangian Data Assimilation Mon Wea Rev 2009P Krause and J Restrepo Sequential Estimation with Deterministic Models and Noisy Data SIAM J Sci Comp 2009


The DKF

The Diffusion Kernel

Φ(middot) = φ(middot)+Φprime(middot)

Φprime(middot) is obtained by applyingDuhamelrsquos principle andexpectation projections

Φprime(ξ (tm i) t)sim

int t

tmG dw(sminus tm)

where

G(ξ (tm i) ts) = nablaφ(ξ (tm i) t) g(Φ(ξ (tm i)s))

is the Diffusion Kernel


The DKF

The Uncertainty Norm

The diffusion kernel is


The uncertainty norm ||G||(t) for branches of prediction bounds the sup-normof the covariance matrix of Φprime(ξ (tm i) t)

1N||cov(Φprime)||infin le ||G||2 tm le t le tm+1

The key observation is that ||cov(Φprime)||infin a measure of entropy of the branchwill be small whenever ||G|| is small


Application Lagrangian Data Assimilation

The Oceanic Lagrangian Setting

Figure 2 CONTROL run layer thickness displacement (h hrest) superimposed to the

velocity field in each layer (L1L2L3) at initial time (DAY0) and at the end (DAY120) of

the simulation Contour intervals are 25m and the maximum velocity field is approximately

50cms in L1 10cms in L2 5cms in L3

Figure 3 Left column shows successive drifters positions (one point each half day) in

each layer (L1L2L3) Right column shows the corresponding Lagrangian autocovariance

functions (in cm2s2) versus time lag (in days) The solid (dashed) line shows the meridional

(zonal) component

Figure Synthetic Eulerian Flow LagrangianTracks From Molcard et al J Atmos OceanTech (2005)

Figure Four MLFIIrsquos Ready forHurricane Isidore after Puget SoundTesting in July 2002 From EDrsquoAsarorsquos (Washington) Web Page



2D VortexDrifter Problem

Measure the tracks of Np passive tracers ζi(t) = vm(t)+ iwm(t) give anoptimal estimate of tracks and of Nv point vortices zm(t) = xm(t)+ iym(t)

Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl

zlowastmminus zlowastl+η

V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl

ζ lowastn minus zlowastl+η

P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y

particlesvortexvortex

L Kusnetsov and K Ide and CKRT Jones A method for assimilation of Lagrangian data Mon Wea Rev 131 (2003)



EKF Lagrangian Estimation

Kusnetsov et al proposed using Constrained EKF

for reasonable data uncertainties andor frequent enough observationsEKF works reasonably well based on the computation of distancebetween the rdquotruerdquo path and the EKF estimate

if pushed too hard the EKF would fail saddles in the orbit paths wouldcause divergences in the forecast stage

reasonably efficient robust



Comparison of Benchmark DKF and EKF

minus2 minus15 minus1 minus05 0 05 1 15 2

minus15

minus1

minus05

0

05

1

15

X

Y

Mean drifter path (v1w1) as estimated by DKF and benchmark and EKF(jagged dashed) The true path is superimposed (black)



Comparison of Bootstrap DKF and EKF

minus2 minus1 0 1 2minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y

Mean drifter path (v1w1) as estimated by DKF and particle filter and EKF(jagged dashed) The true path is superimposed (black)



(Short-time) Behavior of Bootstrap DKF and EKF

minus1 minus05 0 05 1 15minus05

0

05

1

15

2

t=0t1

DKF

EKF

Mean drifter path (v1w1) as estimated by DKF and particle filter and EKF(red dashed) rdquoTruthrdquo path (black) Measurements (green dashed)



Comparison of Benchmark DKF and enKF


minus15

minus1

minus05

0

05

1

15

X

Y

Figure Mean drifter path (v1w1) as estimated by DKF and benchmark and enKF(jagged dashed) The true path is superimposed (black)



SECOND MOMENT Comparison of Benchmark DKFenKF and EKF

(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt

Comparison of DKF benchmark filter for v1(t) Second moment (a) EKF (b)enKF



THIRD MOMENT Comparison of Benchmark DKF andenKF

0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt

Comparison of enKF DKF and benchmark filter moment estimates for v1(t)Third moment EKF does not generate a third moment


Application Lagrangian Data Assimilation Computational Cost EKF vs Particle Filter vs DKF

Computational Efficiency from tk to tk+1

Bootstrap Filter Cost

CβαTtimesN2x times I

I sample paths (5times105 in examples)

Nx is the dimension of state variable

T is the number of deterministic time steps

αT is the number of times steps taken in the stochastic differential equation integrator α 1 β gt 1 (SODE time step in examples10minus4 T = 10minus2)

DKF Cost

Ttimes (CprimeNx +CN3x )times Ik asymp TtimesCN3

x times Ik

Conditions whereby the cost of the bootstrap filter exceeds that of the DKF

Nx lt αβ IIk

Reflects the impact of non-Gaussianity in the cost of the DKF the less Gaussian the closer IIk is to 1

EKF vs DKF Cost

In the prediction step both methods are comparable

In the analysis step EKF is minN3x N3

y for DKF it is Ik timesN2x


Application Lagrangian Data Assimilation Computational Cost Clustered DKF




I sample paths (5times105 in examples)Nx is the dimension of state variableT is the number of deterministic time stepsαT is the number of times steps taken in the stochastic differential equation integrator α 1 β gt 1 (SODE time step in examples10minus4 T = 10minus2)

DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost

Can use Ik le 10minusr I r ge 1

In examples r = 2 Conditions whereby the cost of bootstrap exceeds that of rDKF

Nx lt αβ10r


New Projects

In nonlinear problems there is no unique best predictor

Shown average entropy average likelihood average

from P Krause J M Restrepo to be submitted SIAM J Sci Comp 2010


New Projects

Lagrangian Data Blending

Data Blending Contour Dynamics

23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1

ClassicCont ourAnalysis

ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2

A competing approach B Beechler J Weiss G Duane J Tribbia to appear in J Atmos Sci 2010


New Projects


the data assimilation problem will remain dimensionally-challenging

Develop statistically-convergent methods

Increase the availability and quality of the dataReduction of the state space

Exploit vastly different degrees of uncertaintyDevelop closures and stochastic parameterizations

all the while being clear about what it is that we are doing


State-Space Reduction

The Ensemble Bred Vectors

GOAL

Propose an alternative and robust methodology for sensitivity analysisand reduced-space representation

Provide a mathematical basis for the Bred Vector (BV)

COLLABORATORSNusret Balci (IMAUMN) Anna Mazzucato (PSU) George Sell (UMN)




GOAL






The Bred Vectors Defined

Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0

Y(tn)asymp y(t = tn) while Y0 asymp y0

BVs computed as follows

δYn+1 = M(Yn +δYn tn)minusM(Yn tn)δYn+1 = RδYn+1

R =δY0δYn+1

the normalization factor

See Toth and Kalnay 1993 Bull Amer Met Soc Toth and Kalnay Mon Wea Rev 1997



The Lyapunov Vectors Defined

Let δΘ be the solution of the initial value problem

δΘ(tn +δ tn) = LnδΘ(tn) n = 012

δΘ(0) = δΘ0

Take nrarr infinAt each time step the Tangent Linear Model is

Ln = L(Yn tn) =partG(y t)

party

∣∣∣∣y=Ynt=tn

LV make sense in asymptotic time (the BV is local in time)



The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt

= x(rminus z)minus y

dzdt

= xyminusbz t gt 0 (1)

At t = 0 x = 08001 y = 04314 z = 09106 Here we set r = 28b = 83σ = 10



The Cahn Hilliard (Cessi-Young 92 Model)

partSpart t

= αpart 2

part z2 [f (z)+ microS(Sminus sin(z))2 +Sminus γpart 2Spart z2 ] t gt 0z isin [minusππ]

S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2

Zminus3 minus2 minus1 0 1 2 3

minus02

minus015

minus01

minus005

0

005

01

015

02

Z

Forcing function and perturbation

time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV

Solution of CY92 and LV vs BVJUAN M RESTREPO Group Leader Uncertainty Quantification Group ( University of Arizona )Climate Assimilation of Data and Models June 2010 56 88


Numerical Instability of Bred Vectors

dxdt

= Ax

where A is non-normal constant (Jordan) matrix of dim(20)

0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time

standard BV computation

0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time

Well-conditioned computation



The Ensemble Bred Vector (EBV) Defined

The Ensemble Bred VectorPick a norm and its size for an ensemble of initial conditions

Advance the ensemble of initial conditions using the BV

find the element from the ensemble that grew the most and use it toresize the whole ensemble

repeat



The Lorenz63 using EBV

minus01minus005000501

minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z

Superposition of an ensemble of 1500 runs over all times from 0 to 50 (Weshow 150 of these) normalized to the initial condition 2-norm set to 01



The Cahn Hilliard CY92 using EBV

BVrsquos

minus3 minus2 minus1 0 1 2 3minus03

minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z



The Non-Normal Linear Problem using EBV

dxdt

= Ax

where A is non-normal and dim(20)

time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects

Bred Vectors Some of our Conclusions

BV are attractive because they can be applied to legacy code

Established that BV is a nonlinear generalization of the LV though it isnot clear how a finite-time quantity is to be compared to anasymptotically-defined quantity unless we are discussing a steady state

Established that BV is norm dependent

In applications the norm matters a great deal when considering what is tobe analyzed

But do they provide any practical information not contained in singularvectors or LV


Stochastic Parameterization

Dimension Reduction via Stochastic Parameterization

dx = f (x)dt + sochastic term

x(0) = x0




GOAL

Produce stochastically-based subscale parameterization

Make greater contact with data (and data assimilation)

COLLABORATORSDarin Comeau (U Arizona)Brad Weir (U Arizona) Jorge Ramırez (U Nacional de Colombia) J CMcWilliams (UCLA) M Banner (UNSW)




GOAL






WAVE BREAKING DISSIPATION

Geophysical GoalsHow does dissipation at wave scales manifest itself at longer time scales

What is the correct stochastic parameterization

Can we parameterize dissipation in such a way that we use field datamore efficiently



Scale Range of the Model

10 secs-months

100m-basin scale

Speed waves gt currents

kH sim 1Applications

climate dynamics (transport)erodible bed dynamicsriver plume evolutionalgalplankton bloomsOil spillspollution

J McWilliams and J M Restrepo The Wave-driven OceanCirculation J Phys Oceanogr (1999)J Restrepo Wave-Current Interactions and Shore-connected Bars J Estuarine Sci (2001)J McWilliams J M Restrepo E Lane An asymptotic Theory for the Interaction of Waves and Currents in Coastal Waters J Fluid Mechanics(2004)E Lane J M Restrepo J McWilliams Wave-Current Interaction A comparison of radiation-stress and vortex-force representations J PhysOceanogr (2007)



Lagrangian Motion

dX = Vdt

Figure Deterministic



Lagrangian Motion Under White Capping

dXt = V(X t)dt +dWt(X t)

Figure Stochastic



What Stochastic Model


Experiments are needed to determine the right model

minus02 minus015 minus01 minus005 0 005 01 015 02 025minus02

minus01

0

01

X

Z

Figure dW standardwhite noise


minus01

0

01

X

Z

Figure dW with addedmean-reverting process


minus01

0

01

X

Z

Figure dW with addedjump process


Stochastic Parameterization ProjectionFiltering Strategy

LagrangianEulerian Projections in Multiscale Setting

BASIC STRATEGYMultiscale projection of systems of (mostly) hyperbolic equations betweenthe Eulerian and Lagrangian frames


Stochastic Parameterization Model

CurrentWavesBreaking Model

The momentum with breaking-generated stresses and diffusion

partvc

partT= VtimesZminusnablaΦ+ 〈(btimesZ)+btimes (nablatimesb)+ [Vtimesnablatimesb]minus 1

2nabla|b|2〉+nabla middotR

where V = vc +ustokes Z = nablatimesv+2Ω and Φ = p0 + 12 |V|

2

The ustokes is the wave contributionThe break-stresses modify the vortex force and Bernoulli headThe diffusion accounts for mixed-layer boundary-layer effects

The tracers obeypartCpartT

+V middotnablaC =minusb middotnablaC+nabla middotQ

J M Restrepo Wave Breaking Dissipation in the Wave-Driven Ocean Circulation J Phys Oceanogr (2007)JM Restrepo J M Ramırez JC McWilliams amp M Banner Wave Breaking Dissipation and Diffusion in Waves and Currents J PhysOceanogr 2010



Wave Breaking Diffusion

The observation is that wave breaking increases the size of the mixing layerand this layer will then create a great deal of dissipation

Rv asymp νpartvh

part z Rh asymp νnablavh

Qv asymp κpartCpart z

Qh asymp κnablaC

We assume thatν sim 〈`b

∣∣wb∣∣〉 κ sim 〈`θ

∣∣wb∣∣〉

wb is the vertical component of the velocity associated with breaking and themixing length is

`b = γη(x t) `θ = αη(x t)



Boundary Conditions

The surface boundary conditions at z = η(xh t) are the following

w =Dη

Dt p = gρ0η +pa ν

partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)

wc = nabla middotMminuswb pc = ηc +pa0minusP

ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T

where the wave-induced adjustments (at z = 0) are

Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b

How to determine b(XT)

Thus far we have answered the question

if breaking occurs how do waves and currents get modified at theselarge spatio-temporal scales


Determining b


How is the breaking velocity b determined

it can be determined by field dataor


Determining b





Determining b

Steps to determine b(XT)

Ingredients in determining b using averaging and breaking kinematics

individual breaker b velocity are obtained parametrically from data

breaking events need to be upscaled to yield b

kinematics of their occurrence can be tied to wave-group dynamics

breaking events should be energetically consistent

breaking events should yield a Poisson process in space-time

Details in JM Restrepo J M Ramırez JC McWilliams amp M Banner Wave Breaking Dissipation and Diffusion in Waves and Currents JPhys Oceanogr 2010B Weir amp J M Restrepo Stability of the Langmuir Circulation in the Presence of Wave Breaking in preparation


Determining b

Individual breaker velocity b are obtained parametrically

Individual breaker velocity b given by

0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300

parttb+ b middotnablab =1

Re∆b+A

nabla middot b = 0

where

A = kbX (x)Y (y)Z (z)T (t)a

cf Sullivan McWilliams Melville JFM 507 (2004)


Determining b

Breaking events need to be upscaled to yield b

Model b = B+bprime as the random sum

b(xhzT) = sum(Xhτ)isinΦ

bE(Xhτ)(xhminusXhz)δ (τminusT)

wherebE(xhminusXhz) =

1τE

intτE

0b(xhminusXhz t)dt

The ensemble average at (xhzT) of some functional F of the field b is

〈F (b)〉(xh zT)dT =

langsum

(Xh τ)isinΦ

F (bE(Xh τ)(xhminusXh z))δ (Tminus τ)

rang

=intR

intxhminusΩE

F (bE(xhminusXh z))Λ(dXhdT)p(E)dE

B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b

Kinematics of their occurrence can be tied to wave-groupdynamics

Waves expressed in terms of a carrier and an envelope

η(Xh t) = Re

ei(khmiddotxh+σ t)ρ(xh t)eiθ(xht)

where

P(ρ(xh t) isin dρ) =ρ

2π〈η2〉expminus ρ2

2〈η2〉

a Rayleigh distribution


Determining b

Mean growth rate of wave group energy

If δ = 1〈σ〉

Dmicro

Dt gt δ lowast asymp 14times10minus3 a breaking wave group

where the material derivative is taken following the wave group

Local wave energylowast micro(t) = η2(xmax t)k2(xmax t)

xmax(t) is maximum of crest point in the group

(k〈σ〉) wavenumber mean frequency

lowast see Song and Banner J Phys Oceanogr 2002


Determining b

2D Example

(Loading breakmovie)

minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 100minus100

minus80

minus60

minus40

minus20

0

20

40

60

80

100Trajectories of group maxima and breaking

y (m

)

x (m)

Space-time evolution of breakingevents shown in movie


breakmovqt
Media File (videoquicktime)

Determining b

Breaking events should be energetically consistent

Recall

partt b+ b middotnablab =1

Re∆b+ kbX (x)Y (y)Z (z)T (t)a

nabla middot b = 0

kb found by equating the energy E(Tw) from the break velocity field with thetotal energy change of the surface ρ0g∆η2

E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2

If ∆η2 is total change in η2(xmax middot) during the breaking event then Thus

kb = 182k2∆η2


Determining b

Breaking events should be Poisson process

0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)

Histograms of wave group meanwavenumber klowast and breaking strength

kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r

Poissonfit theoretical indicator function vs

computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008

Initial Current with top speed 01 msec

Current after 64 hours (01 ms contours) with waves and breaking due to 15 ms winds

(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(a) No Stokes drift (b) With Stokes ustokes = (0260) ms


Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05

(o) Current Stokes and No Breaking(minus) Current Stokes and Breaking(o) Current No Stokes and No Breaking(minus) Current No Stokes and Breaking

(o) No Current Stokes and No Breaking(minus) No Current No Stokes and Breaking

time (hrs)

rate

(m

s)

AVERAGE DISPERSION OF PASSIVE TRACERS


Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3

(minus) No Current Stokes and Breaking

(minus) No Current No Stokes and Breaking

(minus) Current No Stokes and Breaking

(minus) Current Stokes and Breaking

(o) Current Stokes and No Breaking

TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments

Data assimilation with sparse data forces us to put emphasis on climatemodel development

The rdquodimensional-curserdquo is unavoidable In the long term convergentmethods allow one to know how much a computational gain is possibleat the expense of optimality

Closure methods and stochastic parameterizations can lead to robust andefficient models and hence dimension reductionit exploits deepknowledge of the system dynamics

Taking advantage of different degrees of uncertainty among the degreesof freedom can be a productive alternative to reduced representations ofbackground error fields


Further Information

Further Information

Juan M Restrepohttpwwwphysicsarizonaedusimrestrepo

Uncertainty Quantification Grouphttpwwwphysicsarizonaedusimtolwinski

UQGQG


UQG Group


LinearGaussian Estimation


The Path Integral Monte Carlo (PIMC)

Samplers

The DKF


Computational Cost EKF vs Particle Filter vs DKF

Computational Cost Clustered DKF

New Projects



ProjectionFiltering Strategy

Model

Determining b

Results

Closing

Further Information

UQG Group


FacultyJMR (Math Physics Atmospheric Sciences)Shankar Venkataramani Kevin Lin Hermann Flaschka (Math)Shlomo Neuman Larry Winter (Hydrology)Rabi Bhattacharya Walt Piegorsch (Math Statistics)Kobus Barnard Alon Efrat (Computer Science)

Post-DocsP Krause (estimation)J Ramırez (probability)P Dostert (scientific computing)T-T Shieh (variational methods)

Graduate Students 15 (Brad Weir Darin Comeau Suz Tolwinski)Undergraduate Students 2 (Jason Ditmann)

UQGQG


UQG Group



UQGQG


UQG Group



UQGQG


UQG Group



UQGQG


UQG Group


TIME

SP

AC

E

WEATHER


Climate

10 years

continent

country

city


UQG Group






UQG Group






UQG Group






UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group



UQGQG


UQG Group



UQGQG


UQG Group



UQGQG


UQG Group


TIME

SP

AC

E

WEATHER


Climate

10 years

continent

country

city


UQG Group






UQG Group






UQG Group






UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group



UQGQG


UQG Group



UQGQG


UQG Group


TIME

SP

AC

E

WEATHER


Climate

10 years

continent

country

city


UQG Group






UQG Group






UQG Group






UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group



UQGQG


UQG Group


TIME

SP

AC

E

WEATHER


Climate

10 years

continent

country

city


UQG Group






UQG Group






UQG Group






UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group


TIME

SP

AC

E

WEATHER


Climate

10 years

continent

country

city


UQG Group






UQG Group






UQG Group






UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group






UQG Group






UQG Group






UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group






UQG Group






UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group






UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group






UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group






UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group


When data fool us

0 100 200 300 400 500 600minus2

minus1

0

1

2

time


UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group



0 5 10 15 20minus2

minus1

0

1

2

time


UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

UQG Group




x(0) = x0


01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD





Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




Bayes Theorem





W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



W(m)xminusV = Θ

ΘsimN (0σ)






Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


Kalman Filter

Forecast


Ulowast = MU(t)Mgt

Analysis










X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




X(t +1) = N(X(t)BP(t))







Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




Model



Data


m = 12 M


0

2

4

6

8

10

12







01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information






01

02

03

04

05

06

07

08SOLUTION

X

HOT

COLD



The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


The Observations

0 1 2 3 4 5 6 7minus15

minus1

minus05

0

05

1

15

time

minus2

minus1

5minus

1minus

05

00

51

15

2minus

202468

10

12








The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information






The EKF Results1







The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


The EKF Results1







The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


The EKF Results1







The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


The EKF Results1
































κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



























κ2partxxP

P(x0) = Ps(x)


P(x t+) =1N

e1






κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




κ2partxxA

A (x tf ) = 1


A (x tminus) =1N

e1






Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



Figure KSP Filter

Figure KSP Smoother







Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information






Pros and Cons




Ad-hoc








Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information






Relies on sampling





Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


Bayesian Statement



Use model for prior




Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


Amodel


is discretized


n = 01 Tminus1

Amodel asympT

sumn=1





Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


Adata


m = 12 M

Adata =M

summ=1





MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


MCMC Samplers



Multigrid (UMC)













HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information







HMC Algorithm

Let qn(τ = 0) = xn





partqn

partτ= Mminus1pn

partpn








eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




eminusA (q)





= Mminus1p







(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information





(12 δτm(NT)12

)




RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


RM-HMC Algorithm2


H = A (q)+12

pgtG(q)minus1p


Challenges







partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




partqn

partτ= CMminus1pn

partpn

partτ= CgtF(qn)










The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information






The DKF


Collaborator




p(ym)




The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

The DKF





int t

tmG dw(sminus tm)

where




The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

The DKF

















(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information










(zonal) component







Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




Vortices

dzm

dt=

i2π

Nv

suml=1lm

Γl


V(t)

Drifters

dζn

dt=

i2π

Np

suml=1

Γl


P(t)

minus2 minus1 0 1 2

minus2

minus15

minus1

minus05

0

05

1

15

2

X

Y














minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information











minus15

minus1

minus05

0

05

1

15

X

Y






minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




minus15

minus1

minus05

0

05

1

15

2

X

Y






0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




0

05

1

15

2

t=0t1

DKF

EKF






minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




minus15

minus1

minus05

0

05

1

15

X

Y





(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



(a)0 2 4 6 8 10

0

001

002

003

004

005

006

007

time

2n

d m

om

en

t

(b)0 2 4 6 8 10

0

005

01

time2

nd

mo

me

nt





0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



0 2 4 6 8 10minus002

minus001

0

001

002

003

004

time

3rd

mo

me

nt











DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information









DKF Cost


x times Ik


Nx lt αβ IIk


EKF vs DKF Cost










DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information






DKF Cost


x times Ik


Nx lt αβ IIk


cDKF Cost



Nx lt αβ10r


New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

New Projects





New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

New Projects



23

4

1

23

23

4

1

2

3

2

1

23

4

2

3

23

1


ψ ψ

ψ ψ

(a) (b)

(c) (d)

1 2



New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

New Projects










GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



GOAL







GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



GOAL







Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



Given a model

Y(tn +δ tn) = M(Y(tn) tn) n = 012

Y(0) = Y0




R =δY0δYn+1








δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information





δΘ(0) = δΘ0



party





The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


The Lorenz63

0 5 10 15 20 25 30 35 40 45 50minus2

0

2

4

X

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Y

0 5 10 15 20 25 30 35 40 45 50minus5

0

5

Z

Time

dxdt

= σ(yminus x)

dydt


dzdt






partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



partSpart t

= αpart 2


S(z0) = S0(z)


minus8

minus6

minus4

minus2

0

2


minus02

minus015

minus01

minus005

0

005

01

015

02

Z


time

Z

0 10 20 30 40 50 60minus3

minus2

minus1

0

1

2

3

minus1

minus05

0

05


minus04

minus02

0

02

04

Z

LVBV




dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



dxdt

= Ax


0 5 10 15 20 25 300

005

01

015

02

025

03

035

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

Time


0 5 10 15 20 25 300

005

01

015

02

025

Time

Bred Vectors Norm=2

0 5 10 15 20 25 300

5

10

15

20

25

Time








repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information






repeat





minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




minus01

0

01

minus01

minus005

0

005

01

XY

Z

minus010

01


minus01

minus005

0

005

01

YX

Z





BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



BVrsquos


minus02

minus01

0

01

02

03

Z

EBVrsquos


minus004

minus002

0

002

004

006

Z




dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



dxdt

= Ax


time

N

0 10 20 30 40 50 60

2

4

6

8

10

12

14

16

18

20

minus18

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

5 10 15 20minus2

minus15

minus1

minus05

0

N


New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

New Projects











x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




x(0) = x0




GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



GOAL







GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information



GOAL













10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information









10 secs-months

100m-basin scale







Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


Lagrangian Motion

dX = Vdt






Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




Figure Stochastic







minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information






minus01

0

01

X

Z



minus01

0

01

X

Z



minus01

0

01

X

Z










partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information








partvc




2









Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information




Rv asymp νpartvh



Qh asymp κnablaC



∣∣wb∣∣〉





Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information


Boundary Conditions


w =Dη


partqpart z

=1ρ0

τ κpartCpart z

= T

Lead to (at z = 0)


ν

(partvc

part z+S)

= τminusνpartbc

part z κ

partCpart z

= T


Mequivlang

uwη

wrang Pequiv

langpw

z ηwrang Sequiv

langpart 2uw

part z2 ηwrang


Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b





Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b





Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b





Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b










Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b



0 02 04 06 08 1minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

50

100

150

200

250

300


Re∆b+A

nabla middot b = 0

where




Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b






1τE

intτE

0b(xhminusXhz t)dt



langsum

(Xh τ)isinΦ


rang

=intR

intxhminusΩE


B(z) =intR

[intΩE

bE(XhzT)dXhdT]

λ p(E)dE


Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b



η(Xh t) = Re


where



2〈η2〉



Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b


If δ = 1〈σ〉

Dmicro








Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b

2D Example



minus80

minus60

minus40

minus20

0

20

40

60

80


y (m

)

x (m)



breakmovqt

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b


Recall



nabla middot b = 0


E(Tw) =ρ0c

TwΩb

intΩb[0Tw]

Adxds =055gρ0kb

k2


kb = 182k2∆η2


Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Determining b


0 02 04 06 08 1 12 140

05

1

15

2

25

3

kb

002 004 006 008 01 0120

10

20

30

40

50

60

70

k (m1)


kb

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

r (m)

l(r)r


computational l(r)


Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Results

X (km)

Y(k

m)

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

002

003

004

005

006

007

008



(a)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)

(b)0 10 20 30 40 50

0

10

20

30

40

50

X (km)

Y(k

m)



Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Results

0 05 1 15 2 25 30

005

01

015

02

025

03

035

04

045

05



time (hrs)

rate

(m

s)



Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Results

0 05 1 15 2 25 302

04

06

08

1

12

14

16

x 10minus3






TRACER DISPERSION

time (hrs)

dis

per

sio

n |nabla

θ| (

mminus

1)


Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Closing

Closing Comments






Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Further Information

Further Information



UQGQG


UQG Group





Samplers

The DKF




New Projects




Model

Determining b

Results

Closing

Further Information

Climate, Assimilation of Data and Models - When Data Fail Us · 2010-06-29 · UQG Group Focus...

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Transcript of Climate, Assimilation of Data and Models - When Data Fail Us · 2010-06-29 · UQG Group Focus...