Stochastic and doubly stochastic spatio-temporal field ... · are spatio-temporal random ﬁelds by...

Stochastic and doubly stochastic spatio-temporal fieldmodeling for data assimilation

Michael Tsyrulnikov, Dmitry Gayfulin and Alexander Rakitko

HydroMetCentre of Russia

Melbourne, 5 December 2016

Michael Tsyrulnikov, Dmitry Gayfulin and Alexander Rakitko (HMC)Stochastic and doubly stochastic spatio-temporal field modeling for data assimilationMelbourne, 5 December 2016 0 / 32

Outline

1 A limited area Stochastic Pattern Generator (SPG) for model errorsimulation

I Model errors: a brief introductionI The SPG

2 Doubly stochastic models for simulation of “truth” in testing DAmethodologies

I A one-variable modelI A stochastic advection-diffusion-decay model on the circle


Model-error modeling


Model (tendency) errors

Consider a forecast equation:

dxdt

= F(x) (1)

Due to imperfections in F, the truth satisfies

dxdt

= F(x)− 𝜉 (2)

where 𝜉 is the model error.


Model error models

1 Non-stochasticI Multi-modelI Multi-physicsI Multi-parameter

2 StochasticI Additive perturbations, like Stochastic Kinetic Energy Backscatter

(SKEB).I Multiplicative perturbations, like Stochastic Perturbations of Physics

Tendencies (SPPT)I Stochastically Perturbed Parameterizations (SPP)

All the existing stochastic model-error models require a stochasticspatio-temporal pattern generator.


The SPG: motivation


Space-time interactions and proportionality of scales

The existing pattern generators for ensemble applications produceseparable space-time correlations:

C (t, s) = Ct(t) · Cs(s)

But: no space-time interactions.

In reality, larger spatial scales ‘live longer’ than smaller spatial scales,which ‘die out’ quicker. For large k ,

𝜏k ∼1k

This ‘proportionality of scales’ is widespread in geophysical fields(Tsyroulnikov QJRMS 2001) and other media.


Illustration:Separable vs. Proportional-scales model error models

Experiment

Simulate model error fields on a 1D spatial domain with:I Separable correlations: exp(−|Δx |/L) · exp(−U|Δt|/L)I Proportional-scales correlations: exp(−

√︀(Δx)2 + (UΔt)2/L)

Note that both fields have exactly the same spatial length scales andexactly the same temporal length scales.

Setup:

The domain: 100 km × 3 h.The grid: 100× 100 points.L05 = 50 km and U = 20 m/s.


Separable vs. Proportional-scales fields: model errors

20 40 60 80 1000.0

0.5

1.0

1.5

2.0

2.5

Model error: Separable

Space, km

Tim

e, h

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

20 40 60 80 1000.0

0.5

1.0

1.5

2.0

2.5

Model error: Proportional Scales

Space, km

Tim

e, h

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0


Illustration: impact on forecast errors

For small model errors and small time periods, the forecast model operatorF can be linearized, so that the forecast error 𝛿x due to the model error 𝜉satisfies

d𝛿xdt

= A𝛿x− 𝜉

so that

𝛿x(t) = eAt∫︁ t

0e−A𝜏𝜉(𝜏) d𝜏 =

∫︁ t

0𝜉(𝜏) d𝜏 + o(t2) ≈

∫︁ t

0𝜉(𝜏) d𝜏

Integrate both fields in time, getting, approximately, forecast errors.

Examine the spatial length scales of the forecast errors as functions oftime.


Separable vs. Proportional-scales fields: forecast errors

0 20 40 60 80 100−1.

5−

1.0

−0.

50.

00.

51.

0

Model error field

Space, km

SeparableProportional scales

0 20 40 60 80 100

−50

050

100

Time integrated model error

Space, km



0.0 0.5 1.0 1.5 2.0 2.5

010

2030

40

Spatial MICRO SCALE: forecast error

Time, h

Mic

ro s

cale

, km


Dynamical relevance of space-time interactions:

Separability ⇒ the spatial length scale Lx of the forecast errorfield is constant.

Proportionality of scales ⇒ Lx grows with t. Lx =s.d. 𝛿

s.d. 𝜕𝛿/𝜕x

NB: The forecast errors length scale is a very important attribute of thebackground-error covariances.


The SPG: design


Formulation of the SPGRequirement: The generated fields should have “proportional scales”.

Approach: Linear evolutionary stochastic partial differential equations.

Starting point:𝜕𝜉(t, s)𝜕t

+ A 𝜉(t, s) = 𝛼(t, s)

= t is time, s = (x , y , z) is the spatial vector= 𝛼 is the white driving noise= A is the spatial operator.= 𝜉 is the output random field

∙ To get the spatial isotropy, we specify A = P(−Δ) = 𝜇(1− 𝜆2Δ)q,where Δ is the spatial Laplacian and P is the polynomial or order q.

∙ We impose the “proportionality of scales’ (𝜏k ∼ 1/k as k →∞) ⇒

q =12


Formulation of the SPG

Model: (︂𝜕

𝜕t+ 𝜇√︀

1− 𝜆2Δ

)︂3

𝜉(t, s) = 𝜎 𝛼(t, s)

𝜎 controls the variance𝜆 controls the spatial scale𝜇 controls the temporal scale

Computational domain: the cube with cyclic boundary conditions.

Numerical scheme: spectral in space and finite-difference in time.


Time

distance

cov

Spatio−temporal covariances

Ranges: t=0...12 h, r=0...750 km

0.1

0.2

0.3

0.4

0.5

0.6


Spatial field (xy)


Space-time plot (xt)


Application with the COSMO model


Conclusions on the SPG

The SPG produces Gaussian pseudo-random fields on a 2D or 3D limitedarea domain with non-separable spatio-temporal correlations.

= The Fortran code of the SPG is freely available fromgithub.com/gayfulin/SPG.

= Tsyrulnikov M. and Gayfulin D. A limited-area spatio-temporalstochastic pattern generator for ensemble prediction and ensemble dataassimilation. – Meteorol. Zeitschrift, 2016 (accepted). (A copy can bedownloaded from ResearchGate.)


Doubly stochastic models of “truth”


Simple models of “truth”

Goal: in-depth investigation of DA methodologies in a simplifiedsetting with synthetic and thus known “truth”.

Examples: nonlinear one-variable models (the logistic map, the Henonmap) and multi-variable models (the Lorenz models, the Burgers’equation on the circle).

A step forward: we wish to know not only the true state but also thetrue statistics.

Bishop and Satterfield (MWR 2013, Part 1) computed the true EnKFerror statistics with a deterministic model of truth and a stochasticforecast model.

In our doubly stochastic models, the truth is stochastic whereas theforecast model is deterministic.


Building a 1D doubly stochastic model of “truth”

We start with the simplest time discrete stochastic equation

xk = Fkxk−1 − 𝜎k𝜀k

If Fk = const and 𝜎k = const , then the solution xk is a stationary randomprocess.

In reality, the dynamics and statistics of the truth vary in time (andspace).


Building a 1D doubly stochastic model of “truth” (2)

To mimic this variability, we specify Fk and 𝜎k to be random processes bythemselves:

Fk = F̄ + 𝜇(Fk−1 − F̄ ) + 𝜎F𝜀Fk

𝜎k = exp(Σk)

Σk = κΣk−1 + 𝜎Σ𝜀Σk

xk = Fkxk−1 + 𝜎k𝜀k


The doubly stochastic advection-diffusion-decay modelNon-stochastic:

𝜕x

𝜕t+ U𝜕x

𝜕s+ 𝜌 x − 𝜈 𝜕

2x

𝜕s2= 0

(t is time, s is the spatial coordinate, U is the advection velocity, 𝜌 is the decaycoefficient, and 𝜈 is the diffusion coefficient).Singly stochastic:

𝜕x

𝜕t+ U𝜕x

𝜕s+ 𝜌 x − 𝜈 𝜕

2x

𝜕s2 = 𝜎 · 𝛼(t, s) (*)

(𝛼 is the driving noise and 𝜎 its intensity).Doubly stochastic:

𝜕x

𝜕t+ U(t, s)

𝜕x

𝜕s+ 𝜌(t, s) x − 𝜈(t, s)𝜕

2x

𝜕s2 = eΣ(t,s) 𝛼(t, s)

where the coefficients,U(t, s), 𝜌(t, s), 𝜈(t, s), and Σ(t, s) (or some of them),are spatio-temporal random fields by themselves postulated to satisfythe singly stochastic advection-diffusion-decay model (*).


A space-time x(t, s) plot for our model of truth


A space-time x(t, s) plot for the Lorenz-96 model


Hierarchical simulation

xk = Fkxk−1 + 𝜎k𝜀k

1 Specify parameters of the processes Fk and 𝜎k . This defines a“universe”. All worlds in a “universe” share the same “climate”.

2 Generate {Fk} and {𝜎k}. This pair of sequences defines a “galaxy”.All worlds in a “galaxy” share the same “statistics of the day”,e.g. Var xk for each k.

3 Given the sequences {Fk} and {𝜎k}, generate {𝜀k} and thus {xk}getting “worlds” in the “galaxy”. Var xk can be estimated byaveraging over worlds within the galaxy.In the same way, true error covariances, say Bk , can be estimated forany filter in question.


Space-time plots for the variance of the “truth” Var x(t, s)The 4 plots correspond to the 4 different variances of the fields 𝜌(t, s) and Σ(t, s).


Examples of applications of thedoubly stochastic models of truth


Stochastic 1D EnKF with the optimally tuned inflation 1.01

0 5 10 15 20 25

−4

−2

02

4

B_true_enkf

Bia

s in

S_e

nkf

EnKF: Bias in S vs B_tru


Stochastic 64-D EnKF:Errors in (localized) sample correlations, Si,i+2 − Bi,i+2, vs. true Bi,i+2


Conclusions on the doubly stochastic models of “truth”

Are capable of generating complicated spatio-temporal variability.

Provide the “true” time and space specific spatio-temporal errorstatistics (instantaneous variances, length scales, etc.)

Allow the use of the exact KF (as a benchmark).

Thank you!


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