introduction to data assimilation - cirm · introduction to data assimilation sebastian reich ()...

Introduction to data assimilation

Sebastian Reich (www.sfb1294.de)

Universität Potsdam/ University of Reading

CIRM2018, October 26, 2018

Universität Potsdam/ University of Reading 1

Ensemble prediction I

Source: The quiet revolution of numerical weather prediction, Nature, 2015


Ensemble prediction II

Ensemble prediction system with M members:d

dtZit

= f (Zit) , Zi0 ∼ π0 , i = 1, . . . ,M .

Source: ECMWF


Data assimilation I



Data assimilation II



Data assimilation III

StateState StateModel Model

Data

Assimilation

Data

Assimilation

Data

Assimilation

time

Model:dZt = f (Zt)dt + γ1/2dWt

Data: ytn at discrete times tn


Data assimilation IV

Discrete–time observations:

ytn = h(Ztn) +R1/2 Ξtn , n = 1, . . . ,N .

Likelihood function:

L(z[0,T]) := exp

�

−1

2

∑

n

‖ytn − h(ztn)‖2R

�

.

Bayes:dbP[0,T]

dP[0,T](Z[0,T]) :=

L(Z[0,T])

P[0,T][L].


Data assimilation V

prior P[9:21] in yellow; posterior bP[9:21] in green


References

É Daley, R., Atmospheric Data Analysis, Cambridge University Press,1991.

É Bain, A. and Crisan, D., Fundamentals of Stochastic Filtering,Springer, 2009.

É SR, Cotter, J., Probabilistic Forecasting and Bayesian DataAssimilation, Cambridge University Press, 2015.

É Law, K. et al., Data Assimilation – A Mathematical Introduction,Springer, 2015.

É P. Bauer et al, The quiet revolution of numerical weather prediction,Nature, 2015


Sequential processing of data I

StateState StateModel Model

Data

Assimilation

Data

Assimilation

Data

Assimilation

time

Notation:

Prediction/Forecast: π(bzn+1 |y1:n)

Filtering/Analysis: π(zn+1 |y1:n+1)


Sequential processing of data II

Classic bootstrap particle filter (sequential MC):

1) propagate M particles zin

under the model dynamics to yield forecastsbzin+1, i = 1, . . . ,M.

2) data likelihood assigns importance weights win+1 to each

propagated particle bzin+1

3) resample to obtain equally weighted particles zin+1 from the filtering

distribution at tn+1.

This approach is not applicable to geophysical/meteorologicalapplications because of:

i) small sample sizes M ∼ 102

ii) high dimensionality of models (discretised PDEs) D ∼ 107

iii) medium number of observations N ∼ 105


Sequential processing of data III

Remedies used in data assimilation:

1) Formulate assimilation of data as an optimisation problem (4DVar,3DVar)

2) Replace importance–resampling step by low variance ensembletransformations

zjn+1 =

M∑

i=1

bzin+1 t

∗ij

3) Localise assimilation of data

zjn+1(x) =

M∑

i=1

zin+1(x) t∗

ij(x)

x ∈ R3 grid point beingupdated.

State variable on grid points

Observations outside localization radius

Gridpoint being updated

Observations inside localization radius


Ensemble Kalman filter (EnKF)

Forward observation model: yobsn+1 = h(bzn+1) +R1/2Ξn+1

Best linear unbiased estimator (BLUE): zn+1 = Ayobsn+1 + b

Empirical BLUE/EnKF:

A = PR−1

bj = bzjn+1 − A

�

h(bzjn+1) +R1/2

bΞjn+1

�

with

P =1

M− 1

M∑

i=1

bzin+1(h(bzi

n+1)− hn+1)T , bΞin+1 ∼ N(0, I) .

EnKF induced particle transformation:

zjn+1 =

M∑

i=1

bzin+1t

∗ij.


Ensemble transform particle filter

Importance weights: win+1 ∝ exp

�

−1

2‖h(bzi

n+1)− yn+1)‖2R

�

5 10 15 200

0.02

0.04

0.06

0.08

0.1

Impo

rta

nce

we

igth

Particle Number

⇒OptimalCoupling

T∗⇒

5 10 15 200

0.02

0.04

0.06

0.08

0.1

Impo

rta

nce

we

igth

Particle Number

T∗ minimiser of J({tij}) =M∑

i,j=1

tij‖bzin+1 − bzjn+1‖

2

subject to:tij ≥ 0,

∑

i

tij = 1,∑

j

tij = wiM

T∗ induces linear transformation: zjn+1 =

M∑

i=1

bzin+1t

∗ij




�

−1

2‖h(bzi

n+1)− yn+1)‖2R

�

5 10 15 200

0.02

0.04

0.06

0.08

0.1

Imp

ort

an

ce

we

igth

Particle Number

⇒OptimalCoupling

T∗⇒

5 10 15 200

0.02

0.04

0.06

0.08

0.1

Impo

rta

nce

we

igth

Particle Number


i,j=1


2


∑

i

tij = 1,∑

j

tij = wiM


M∑

i=1

bzin+1t

∗ij




�

−1

2‖h(bzi

n+1)− yn+1)‖2R

�

5 10 15 200

0.02

0.04

0.06

0.08

0.1

Imp

ort

an

ce

we

igth

Particle Number

⇒OptimalCoupling

T∗⇒

5 10 15 200

0.02

0.04

0.06

0.08

0.1

Imp

ort

an

ce

we

igth

Particle Number


i,j=1


2


∑

i

tij = 1,∑

j

tij = wiM


M∑

i=1

bzin+1t

∗ij


Numerical example I

Lorenz-63 model, first component observed infrequently (∆t = 0.12) andwith large measurement noise (R = 8):

Figure: RMSEs for various second-order accurate LETFs compared to the ETPF, theESRF, and the SIR PF as a function of the sample size, M.


Numerical example II

Hybrid filter: P := PESRF(α) PETPF(1− α).

Figure: RMSEs for hybrid ESRF (α = 0) and 2nd-order corrected NETF/ETPF (α = 1)as a function of the sample size, M.


References

É Evensen, G., Data assimilation – the ensemble Kalman filter,Springer, 2009.

É SR, A nonparametric ensemble transform method for Bayesianinference, SIAM J. Sci. Comput., 2013.

É SR, Cotter, J., Probabilistic Forecasting and Bayesian DataAssimilation, Cambridge University Press, 2015.

É Tödter J., Ahrens, B., A second–order exact ensemble square rootfilter for nonlinear data assimilation, Month. Weather Rev., 2015.

É Chustagulprom, N. et al., A hybrid ensemble transform particle filterfor nonlinear and spatially extended systems, SIAM/ASA J. UQ, 2016.

É Acevedo, W., et al., Second–order accurate ensemble transformparticle filters, SIAM J. Sci. Comput., 2017.

É Fearnhead, P., Künsch, H.R., Particle filters and data assimilation,Annual Review of Statistics and Its Application, 2018.


Continuous–time data assimilation I

Observation model:

dyt = h(Zt)dt +R1/2dVt

Linear SDEs and observations:

dZt = (AZt + b) dt + γ1/2dWt, dyt = HZtdt +R1/2dVt .

Kalman-Bucy equations for the mean

dμt = (Aμt + b)dt − KtdIt, dIt := (Hμtdt − dyt)

and the covariance matrix

d

dtPt = APt + PtA

T + γI− KtHPt .

with Kalman gain matrix Kt = PtHTR−1.


Continuous–time data assimilation II

Controlled mean–field SDE:

dZt = (AZt + b)dt + γ1/2dWt + dut(Zt), dut := −KtdIt

with innovation It

dIt =1

2(HZt +Hμt)dt − dyt

ordIt = HZtdt +R1/2dUt − dyt

and Kalman gain Kt = PtHTR−1.

Extension to nonlinear SDEs:

Ensemble Kalman-Bucy filter(EnKBF)

dtZt = f (Zt)dt + γ1/2dWt − KtdIt .


Continuous–time data assimilation III

Feedback particle filter:

dZt = f (Zt)dt + γ1/2dWt − Kt ◦ dIt

with Kalman gain

Kt := ∇zψtR−1, ∇z · (πt∇zψt) = −πt(h− h)

and innovations

dIt =1

2(h(Zt) + ht)dt − dyt

ordIt = h(Zt)dt +R1/2dUt − dyt .


Continuous–time data assimilation IV

Particle approximation to elliptic PDE:

∇zψt(zjt) ≈

M∑

i=1

zitaijt

with zit∼ πt, i = 1, . . . ,M and coefficients a

ijt ∈ R appropriately defined.

Interacting particle system:

dzjt = f (zjt)dt + γ1/2dWj

t −

M∑

i=1

zitaijt

!

R−1 ◦ dI jt

with innovationdI jt = h(z

jt)dt +R1/2dUjt − dyt .


Parameter estimation I

Parameter–dependent model:

dZt = f (Zt, θ)dt + γ1/2dWt − Kt ◦ dIt

Marginal likelihood (evidence of the data Yt = y[0,t] under θ):

dπt(Yt |θ) = πt(Yt |θ) htR−1dyt .

Example.SPDE model:

ut = θ∆ut + Wt

Wt space–time white noise.

Observation:dyt = ut(x0) dt +R1/2dVt

for fixed spatial location x0.


Parameter estimation II

D = 80 grid points, state estimation with EnKBF, M = 20 particles, andlocalisation, observation interval [0,20], measurement error R = 0.001

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8parameter

-32

-30

-28

-26

-24

-22

-20

-18

-16

-14

-12ne

gativ

e lo

glik

elih

ood

Figure: negative loglikelihood for θ ∈ [0.2,1.8], true value θ∗ = 1.Universität Potsdam/ University of Reading 23

References

É Bergemann, K, SR, An ensemble Kalman–Bucy filter forcontinuous–time data assimilation, Meteorolog. Zeitschrift, 2012.

É Yang, T. et al., Feedback particle filter, IEEE Trans. Autom. Control,2013.

É Taghvaei, A. et al., Kalman filter and its modern extensions for thecontinuous–time nonlinear filtering problem, J. Dyn. Sys., Meas., andControl, 2018.

É de Wiljes, et al., Long–time stability and accuracy of the ensembleKalman–Bucy filter for fully observed processes and smallmeasurement noise, SIAM J. Appl. Dyn. Sys., 2018.

É SR, Data assimilation, Acta Numerica, 2019, preliminary version onarXiv 1807.08351.


Summary

É Data assimilation has become a workhorse for day–to–day weatherforecasting

É Key are a "smart" choice of biased and robust approximations to thesequential filtering problem

É Data assimilation has applications to many areas outside weatherprediction such as medicine, geosciences, biology, etc.

É Current topics of research: combined state–parameter estimation forPDEs, theoretical understanding of approximations such as lineartransformations & localisation

É Continuous–in–time DA naturally leads to an attractive unifyingframework in the form of mean–field SDEs.


The End

(source: J.B. Handelsman, New Yorker, 05/31/1969)


Collaborators

É Walter Acevedo

É Kay Bergemann

É Yuan Cheng

É Nawinda Chustagulprom

É Colin Cotter

É Jana de Wiljes

É Prashant Mehta

É Wilhelm Stannat

É Amari Taghvaei


introduction to data assimilation - cirm · introduction to data assimilation sebastian reich ()...

Documents