Polynomial Chaos Expansions for QuantifyingUncertainty in Ocean Models

Mohamed Iskandarani Ashwanth Srinivasan Carlisle Thacker,University of Miami

Omar Knio,Duke University

Alen Alexandrian Justin Winokur Ihab Sraj,The Johns Hopkins University

Funding: Office of Naval Research

January 11, 2012

Polynomial Expansions for Quantifying Uncertainties

Outline

Objective: Explore the use of Polynomial Chaos (PC)methods to quantify uncertainties in oceanic forecastsApplications to date

Nesting BC uncertainties in the Gulf of MexicoUncertainties in Deep Water Horizon oil spill modelKPP-Parametric Uncertainties and their impact on SSTresponse to hurricane forcing

Hurricane Ivan 2004Typhoons Fanapi & Malakas 2010

Polynomial Expansions for Quantifying Uncertainties

Ocean Modeling Uncertainties

Surface ForcingMomentum flux (wind-stress)Heat fluxFresh water flux

Initial Conditions: Observation sparse in space-timeLateral Boundary Conditions in Regional ModelsParameterization of small scale processes

mixed layer physicsbottom boundary layerbulk formula for air-sea fluxes

Quantify uncertainties in ocean forecast given inputuncertainties

Polynomial Expansions for Quantifying Uncertainties

What is Polynomial Chaos

Series Representation

u(x , t , ξ) =P∑

k=0

uk (x , t)ψk (ξ) (1)

u(x , t , ξ): a model output (aka observable)uk (x , t): series coefficientsψk (ξ): basis functionsξ: input characterized by its PDF ρ(ξ)e.g. IC uncertainty: u(x ,0, ξ) = u(x) + ξ δu

Basic QuestionsHow to choose the basis functions ψk?How to determine the coefficients uk?Where to truncate the series, P ?

Different PC flavors depending on choices

Polynomial Expansions for Quantifying Uncertainties

Polynomial Chaos Basis

Basis functions are orthonormal polynomials w.r.t. PDF ρ(ξ)

〈ψj , ψk

〉=

∫ψk (ξ)ψj(ξ)ρ(ξ)dξ = δi,j (2)

ρ(ξ) ψk (ξ)

Gaussian Hermite polynomialsGamma Laguerre polynomials

Beta Jacobi polynomialsUniform Legendre polynomialsGeneral Wiener-Askey polynomials

Note that ψ0 = 1

Polynomial Expansions for Quantifying Uncertainties

How do we determine PC coefficients

Series: u(x , t , ξ) =∑P

k=0 uk (x , t)ψk (ξ)Non Intrusive Spectral Projection: minimime l2-norm

uk (x , t) = 〈u, ψk 〉 =∫

u(x , t , ξ)ψk (ξ)ρ(ξ)dξ

Approximaxte integral with numerical Quadrature

uk (x , t) ≈Q∑

q=1

u(x , t , ξq)ψk (ξq)ωq

ξq/ωq quadrature points/weightsQuadrature requires an ensemble run at ξq

Polynomial Expansions for Quantifying Uncertainties

Polynomial Chaos: Computing statistics

mean:

E [u] = 〈u, ψ0〉 =P∑

k=0

uk (x , t) 〈ψk , ψ0〉 = u0(x , t)

Variance:

E[(u − E [u])2

]=

P∑k=1

u2k (x , t)

Covariance:

E [ (u − E [u]) (v − E [v ]) ] =P∑

k=1

uk (x)vk (x , t)

Orthogonality simplifies computations of statisticalquantities

Polynomial Expansions for Quantifying Uncertainties

How many terms to retain

Monitor variance

E[(u − E [u])2

]=

P∑k=1

u2k (x , t)

Power in high modes indicate if series has convergedsufficiently

Polynomial Expansions for Quantifying Uncertainties

Benefits of Polynomial Chaos Expansions

Combination of statistical and approximation frameworksCan quantify error and “convergence” to solutionNo a-priori restriction/assumption on output statisticsApproach robust to model non-linearity and modeldifferentiabilityCan be done non-intrusively via ensembles.Multiple independent stochastic variables can be handledby multi-dimensonal tensorization of 1D basis functionsand quadratures.Series can act as surrogate for the model, e.g. a PDF canbe generated without running model

Polynomial Expansions for Quantifying Uncertainties

Challenges of PC expansions

Works best for smooth observable (adapt ψk otherwise)Curse of dimensionality: number of unknowns increasesexponentially with number of stochastic dimensions, N,and truncation P = (N+P)!N!P! .

N \ P 1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9 102 3 6 10 15 21 28 36 45 553 4 10 20 35 56 84 120 165 2204 5 15 35 70 126 210 330 495 7155 6 21 56 126 252 462 792 1287 20026 7 28 84 210 462 924 1716 3003 50057 8 36 120 330 792 1716 3432 6435 11440

Ensemble size to compute all term1 Tensorized Gauss quadrature: (P + 1)N (GoM simulation)2 Nested Sparse Smolyak quadrature (Hurricane simulations)3 adaptive quadrature (on-going research)

Polynomial Expansions for Quantifying Uncertainties

The Ocean Model

HYbrid Coordinate Ocean Model (HYCOM)Solves the hydrostatic Navier-Stokes equations

Horizontal momentum equation solved for ~uhContinuity equation solved for w and Sea Surface HeightAdvection-Diffusion equations for T and SHydrosatic pressure and density are diagnosed

Horizontal grids: Structured Finite VolumeVertical grid: ALE-type hybrid vertical coordinate

isopycnal in ocean interior to eliminate numerical diapycnalmixingisobaric near surface to resolve shearterrain-following near shelves to resolve BBL dynamics

Operationally used by NOAA and NRL for near real timeocean prediction

Polynomial Expansions for Quantifying Uncertainties

Uncertainty in Nesting Boundary Conditions

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

GOM − Model Domain and Bathymetry (m)

Latitude

Long

itude

1000 2000 3000 4000 5000 6000 7000

1/25◦ GoM nested in 1/12◦ North Atlantic HYCOMExchange at south and east boundaries (red lines)3-hourly surface fluxes from 1/2◦ COAMPSSouth Open BC: spatially varying random field

Polynomial Expansions for Quantifying Uncertainties

Stochastic southern boundary data

α = α(~x , t) +√λ1α1ξ1 +

√λ2α2ξ2 (3)

α: Stochastic boundary inputα: reference boundary input(λk , αk ): are eigenvalues/eigenvectors of covariance matrixInitial Conditions obtained from 5-year climatological run(ξ1, ξ2) are independent normally distributed stochasticvariablesPC expansion with orthonormal basis Ψn:u(x , t , ξ1, ξ2) =

∑Pn=0 ûnΨn(ξ1, ξ2)

NISP approach to find ûn = 〈u,Ψn〉Ψn tensorized 2D Hermite polynomials, P = 28 in 2DEnsemble of 49 realizations for Hermite quadrature

Polynomial Expansions for Quantifying Uncertainties

Eigenvalues

0 5 10 15 20 250

200

400

600

800

1000

1200

1400

non−

dim

ensi

onal

eig

enva

lues

modes

0 5 10 15 20 250

200

400

600

800

1000

1200

1400

Sin

gula

r va

lue

Mode

Figure: Eigenvalues of correlation matrix of the boundary climatology,the first 4 account for ≈ 1/2 variance.

Polynomial Expansions for Quantifying Uncertainties

0 100 200 300 400

−4

−2

0

2

4

Temporal Patterns

day

ampl

itude

mode 1

mode 2

87W 86W 85W 84W

0

200

400

600

800

Mean Meridional Velocity

pres

sure

(db

ar)

longitude

87W 86W 85W 84W

0

200

400

600

800

Meridional Velocity Mode 1

pres

sure

(db

ar)

longitude87W 86W 85W 84W

0

200

400

600

800

Meridional Velocity Mode 2

pres

sure

(db

ar)

longitude

cm/sec−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Figure: Space-Time structures of Nesting BC EOFs

!1

! 2−4

−2

0

2

4

−4 −2 0 2 4

!

! ! !

! !

! ! !

! ! !

! !

! !

! !

! ! !

! ! !

! !

! !

! !

! !

! !

! ! !

! ! ! !

! !

! !

! ! ! !

Figure 4: Circles enclose regions of 90%, 99%, . . ., 99.9999% probability. Dots marklocations of the quadrature points, with red dots corresponding to relatively likely, blueless likely, green unlikely, and magenta highly unlikely boundary conditions.

Hermite quadrature points in each direction.Figure 4 shows the locations of the 49 quadrature points relative to

contours of the bi-variate normal density function. There is a 90% prob-ability that an open southern boundary conditions corresponds to points(ξ1, ξ2) within the smallest circle. The next larger circle encloses an addi-tional 9% of the possible boundary conditions, and each larger circle adds asmaller fraction, leaving only 0.0001% outside the largest circle. Note thatmany of the quadrature points correspond to boundary conditions that arehighly unlikely. Thus, the ensemble of HYCOM runs providing values atthe quadrature points includes what might be regarded as quite extremeevents. Monte Carlo methods would require an ensemble of 1,000,000 ran-domly drawn boundary conditions to have a reasonable chance of samplingbeyond the largest circle where the much smaller quadrature ensemble hasfour points. Note however that each of these remote cases has a quadratureweight of only 3.0074 × 10−7.18

Each of the 49 quadrature points provides a different specification of the

18For practical purposes these weights might be taken to be zero and the simulationscorresponding to the four most unlikely members of the ensemble could be avoided. Thetwo-dimensional array of quadrature points does not appear to be optimal and otherapproaches to two- and higher-dimensional quadrature might be more cost-effective.

17

Figure: Circles enclose regions of 90%, 99%, ..., 99.9999%probability. Dots mark locations of the quadrature points, with reddots corresponding to relatively likely, blue less likely, green unlikely,and magenta highly unlikely boundary conditions.

Figure: 17 cm SSH contour for the 49 realizations

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day015

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day150

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day300

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day450

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day600

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day750

Latitude

Long

itude

−0.2 0 0.2 0.4 0.6

Figure: PC estimated mean SSH at day 15, 150, 300, 450, 600 and750

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −015

Latitude

Long

itude

0.01 0.02 0.03 0.04 0.05

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −150

Latitude

Long

itude

0.05 0.1 0.15 0.2 0.25

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −300

Latitude

Long

itude

0.05 0.1 0.15 0.2 0.25 0.3 0.35

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −450

Latitude

Long

itude

0 0.05 0.1 0.15 0.2 0.25 0.3

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −600

Latitude

Long

itude

0.05 0.1 0.15 0.2 0.25 0.3

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −750

Latitude

Long

itude

0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure: PC-estimated SSH standard deviations at day 15, 150, 300,450, 600 and 750

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0 0.1 0.2 0.3 0.4

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0 0.1 0.2 0.3 0.4

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0 0.1 0.2 0.3 0.4

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0 0.1 0.2 0.3 0.4

Figure: PC-convergence at day 90 with decreasing PC-order (2,3,4,5)

Series Convergence

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN day 750 degree 5

latit

ude

0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW

day 750 degree 6

longitude

0 0.2 0.4 0.6

Figure: Fraction of variance of surface elevation at day 750 due to theretained polynomial terms of highest degree: (left) contribution of the6 5th-degree terms relative to the total contributed by the 21 terms ofdegree less than 6; (right) contribution of the 7 6th-degree termsrelative to the total contributed by the 28 terms of degree less than 7.

Polynomial Expansions for Quantifying Uncertainties

*

18oN

21oN

24oN

27oN

30oN day 15

*

day 150

*

day 300

*

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN day 450

*

96oW 92oW 88oW 84oW 80oW

day 600

*

96oW 92oW 88oW 84oW 80oW

day 750

(m2)−0.02 0 0.02 0.04

Figure: Covariance of SSH in GOM with SSH @ 86E,24.1N (whitestar) derived from the Polynomial Chaos Expansions.

Polynomial Expansions for Quantifying Uncertainties

surface elevation (m)

kern

el d

ensi

ty fu

nctio

n (c

m−1

) and

frac

tion

of c

ount

s pe

r 1

cm b

in

0

10

20

30

40

day 15

0.0 0.2 0.4 0.6

day 150

day 300

0

10

20

30

40

day 450

0

10

20

30

40

0.0 0.2 0.4 0.6

day 600 day 750

Conclusions

SummaryPC-approach looks promising in quantifying uncertaintiesPC expansion can be mined as posteriori for valuablestatistical information at little extra cost

ChallengesCurse of dimensionalityTime-dependent stochastic uncertaintyHow do we validate our approach?How well do we know our input uncertainties?Can be sharpened with Bayesian inference andobservation

Polynomial Expansions for Quantifying Uncertainties

Ocean Modeling UncertaintiesWhat is Polynomial ChaosUQ-Boundary Conditions