CITES-2019 Bayesian approach to the data assimilation problem based - PowerPoint PPT Presentation

CITES-2019 Bayesian approach to the data assimilation problem based on the use of ensembles of forecasts and observations Ekaterina Klimova ICT SB RAS

Introduction • The task of data assimilation is usually understood as the time-sequential estimation of an unknown quantity from observational data. • The purpose of data assimilation - both the preparation of initial fields for subsequent forecasting and the more general one - the description of the behavior of the studied fields over time, the study of climate, the estimation of parameters, etc.

Bayesian approach to the data assimilation problem    1 η k k k 1, ( ) x f x The time change of the estimated quantity:  k k   ε k k k ( ) y h x The observations: k ε k η k - random errors of forecast and observations , The Bayesian approach consists of applying the Bayes theorem to obtain an optimal estimate from observational data and a forecast: ( | ) ( ) p y x p x  ( | ) p x y ( ) p y

Bayesian approach to the data assimilation problem There are various options for assessing the state of data and forecast :  ( | ), - forecast, p x y k l ,1 l k ( | ) - filtration, p x y ,1 k k ( | ) - smoothing, p x y ,0 ,1 k k   { , , , } { , , } where x x x x y y y , . k  ,0 1 0 ,1 1 k k k k

Bayesian approach to the data assimilation problem: the ensemble Kalman filter   t t η t ( ) x f x Consider a nonlinear dynamic system   1 1 k k k   t ε t ( ) An observation equation y h x k k k     ε t   ε η η t t t 0, 0 and are Gaussian random variables: E E  k  1 k k 1 k     T T   ε ε R η η Q t t t t t t [ ] , [ ] E E    1 1 1 k k k k k k  f n , { , 1, , } x n N The ensemble Kalman filter consists of an ensemble of forecasts k   , , η f n a n n ( ) x f x   k k 1 k 1     a n , f n , n ε n x f n ,  a n , ( ( )) x x K y h { , 1, , } x n N and an ensemble of analyses k k k k k k k

Bayesian approach to the data assimilation problem: the ensemble Kalman filter    f T f T 1 K is a matrix of the form ( ) K P H H P H R k k k k k k k k     1 N 1 N   T T , , ε ε f f n f n n n , P dx dx R   k k k k k k 1 1 N N   n 1 n 1 1 N     - an ensemble of forecast errors f n , f n , f n ,  { , 1, , } dx x x f n , f n , n N x x k k k k k N  n 1  ε n { , 1, , } - an ensemble of observation errors k n N    η n T { , 1, , }  n N η η Q n n - an ensemble of model noise [ ] E     1 k 1 1 1 k k k   , H ε H is the linearized operator f n f ( ) ( ) h x h x k k k k k   N 1   T , , a a n a n The analysis error covariance matrix P dx dx k k k 1 N  1 n    , , , a n a n a n { , 1, , } dx x x n N k k k N 1   , , a n a n x x k k N  n 1

Bayesian approach to the data assimilation problem 1. Given a large sample of realizations for each of the prior pdfs, the joint pdfs can be evaluated by integration of each individual realization forward in time using stochastic model equation. 2. The prior pdfs do not need to be Gaussian distributed . 3. The analysis step of EnKF consists of the updates performed on each of the model state ensemble members. 4. For the case with a linear dynamical model a Gaussian prior pdfs the variance minimizing analysis equals the maximum likelihood estimate. 5. For a nonlinear dynamical model the pdfs for the model evolution will become non-Gaussian. In this case analysis will provide only an approximate solution.

Approaches to the implementation of ensemble Kalman filter “True value”     t t t « Stochastic filter » ( ) x f x « Deterministic filter » 1 k k k k (EnKF) Ensemble of forecasts (ESRF, ETKF, LETKF)     ( ) ( ) ( ) f i a i i ( ) x f x 1 k k k k Ensemble of analysis   ( ) ( ) a i a a i , x x dx    ( ) ( ) 0( ) ( ) a i f i i f i ( ) x x K y H x k k k k k k  ( ) ( ) a i a i T a dx dx P Estimation error (skill) The transformation of ensembles of forecast       ( ) ( ) a t a i f t f i { }, { } x x x x  ( ) ( ) a i f i dx A dx k k k Ensemble spread     ( ) ( ) ( ) ( ) a a i a i f f i f i { }, { } X x x X x x

Practical implementation of ensemble algorithms 1. Algorithms with the transformation of forecast ensembles. 2. Local algorithms. 3. Methods to increase ensemble spread.

Local Ensemble Transform Kalman Filter- LETKF (Hunt et al, 2007)   ( ) ( ) a a i a i { } X x x - the ensembles of analysis and   ( ) ( ) f f i f i { } forecasts X x x  1         1 a f T f f f 1) ( 1) / ( ) , P k I Y R Y Y HX   1/2     LETKF: a a 2) ( 1) W k P    - “inflation     1 a a f f T 3) ( ), ( ) w P C y Hx C Y R factor”. o     a i ( ) f f a i ( ) a i ( ) 4) , x x X w w i th row of W   T    a f f a a f a f 5) 6) x x X w P X P X

Ensemble π -algorithm The ensemble π -algorithm is a stochastic filter in which the analysis step is performed only for the ensemble mean. D is a matrix the columns of which are vectors The ensemble of analysis errors  n { , 1, , } dx k n N 1 1       Π T C I I      T Π T 1 B T T T 1 Ε C C ( 0,25 ) 2 0,5 , ( ) , ( ) . D I C F H R HB  1 2 1 N B is a matrix with columns {    n f n , f n , n , 1, , } b k n N b x x k k k Ε ε n is a matrix with columns - the ensemble of observation errors k Klimova E. A suboptimal data assimilation algorithm based on the ensemble Kalman filter. Quarterly Journal of the Royal Meteorological Society. 2012. DOI:10.1002/qj.1941 . Klimova E.G. The Kalman stochastic ensemble filter with transformation of perturbation ensemble. Siberian. J. Nun, Math. /Sib. Branch of Russ. Acad. of Sci. – Novosibirsk, 2019. Vol. 22 N 1. P.27-40.

Classical particle filter ( ) l b Ensemble of states representing the prior probability distribution p x k at time k t . The analysis step at time k t : calculation of new weights   ( ) ( ) ( ) a l l b l ( ) ( | ) ( ) w p x cp y x p x ( , ) k l k k k

Gaussian particle filter The Gaussian particle filter treats each particle as a Gaussian probability distribution L   ( , ) b l ( | ) ( , ) p x Y N x B   k k 1 k 1  l 1 L   ( , ) a l ( | ) ( , ) p x Y N x B k k k  1 l ( , ) a l The analysis ensembles are calculated by treating each particle as x an individual Gaussian distribution :    ( , ) a l ( , ) b l ( , ) b l ( ) x x K y H x k k k   T ( ) K B H H B H R   1 1 k k k k k k   ( ) B I K H B  1 k k k k

Nonlinear ensemble filter (T.Bengtsson, C.Snyder, D.Nychka J. of Geoph. Res. V. 108 No D24 2003)   L  f f f Suppose that ( | ) ( , ) p x Y N x P  1 , , , k k k l k l k l  1 l    a f f ( ) x x K y H x L k l , k l , k l , k k k l ,    a a a ( | ) ( , ) p x Y N x P   , , , k k k l k l k l f f T ( ) K P H H P H R  1 l , , k k l k k k l k   a f ( ) P I K H P , , k l k k k l  f w   , k l l a , k l L   f w , k j j  1 j  1/2        f T f T f T 1 f ( ) exp[ 1/ 2( ) ( ) ( )] w H P H R y H x H P H R y H x , , , , l k k l k k k k l k k l k k k k l      a f f ( ) Ensemble of analysis x x K y H x k j , k j , k j , k j k k j , ( | ) p x Y Sampling according to k k

The behavior of ensemble spread in the ensemble Kalman filter (stochastic filter) The stochastic ensemble Kalman filter can be written in the following form:        η K y ε n n n n n ( ) ( ) ( ) x I K H x f     k k k k 1 k 1 k k k The optimal estimate in the ensemble Kalman filter is the ensemble mean value n x k   n n n Deviation from the mean (spread) simulates the estimate error dx x x k k k      η K ε n n n n n ( )( ( ) ( ) ) dx I K H f x f x    1 1 1 k k k k k k k k   A ‘theoretical’ estimation error ( skill) t t n dx x x k k k      η K ε t t n t t ( )( ( ) ( ) ) dx I K H x x f f    k k k k 1 k 1 k 1 k k