Introduction to Data Assimilation Ma¨ elle Nodet team.inria.fr/moise/maelle Universit´ e de Grenoble, INRIA, LJK GdT Couplage, Montpellier 6 mars 2012
Intoduction What is data assimilation? What is data assimilation? Combine at best different sources of information to estimate the state of a system: model equations observations, data background, a priori information statistics Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 2 / 57
Intoduction What is data assimilation? Data only data reference (1 every 25 gripoint) Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 3 / 57
Intoduction What is data assimilation? Model only reference model (after 4 months) Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 4 / 57
Intoduction What is data assimilation? Model-Data coupling reference after data assimilation Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 5 / 57
Intoduction What is data assimilation? What is data assimilation for? Historically: meteorology. Later, oceanography. Today, many other fields glaciology, seismology, nuclear fusion, medicine, agronomy, etc. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 6 / 57
Intoduction What is data assimilation? What is data assimilation for? Historically: initial state estimation, for weather forecasting. Today, many other applications: initial conditions for predictions, calibration and validation, observing system design, monitoring and assessment, reanalysis, better understanding (model errors, data errors, physical process interactions, parameters, etc), etc. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 7 / 57
Intoduction Problem position Example from ocean forecasting Short term ocean forecasting (a few weeks) of currents, temperature, salinity, for fisheries, coastal economy, sailing... We aim at producing an estimate x a of the true state x t = ( u , v , T , S ) of the ocean at initial time, to initialize forecasts. We are given: a background estimate x b = ( u b , v b , T b , S b ), which is either a previous forecast or comes from climatology, partial observations y o = H ( x t ) + ǫ o , distributed over a given time window, e.g. temperatures from buoys, sea surface elevation from satellites, currents from moorings, . . . Observation operator H contains the dynamical model mapping the initial state of the ocean to the actual temperature, currents, sea surface height at given points in space and time. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 8 / 57
Intoduction Problem position Example from glaciology Short term ice dynamics forecasting (100 years), to estimate Antarctica and Greenland contribution to sea level change. We aim at producing an estimate x a of the true input parameters x t = β of the basal drag coefficient (function of space, constant over time) at the bottom of the ice cap. We are given: a background estimate x b = β b , which is roughly inferred from surface velocities, partial observations y o = H ( x t ) + ǫ o , e.g. surface velocities, ice surface elevation, approximate bedrock topography . . . Observation operator H contains the dynamical model mapping the basal drag of the ice cap to the surface variables at given points in space and time. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 9 / 57
Intoduction Problem position Outline Stochastic data assimilation 1 Best linear unbiased estimator (BLUE) Kalman filter algorithm Variational Data Assimilation 2 Principle of variational methods Gradient-based optimization Variational algorithms Implementation issues 3 Non linearities High dimensional problems Gradient computation: adjoint method Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 10 / 57
Stochastic data assimilation Best linear unbiased estimator (BLUE) Errors statistics Mean: E ( x ) = < x > scalar , E ( x ) = ( E ( x 1 ) , E ( x 2 ) , ..., E ( x n )) vector-valued Variance, covariance ( x , y scalar): Var ( x ) = E (( x − E ( x )) 2 ) , Cov ( x , y ) = E (( x − E ( x ))( y − E ( y ))) We say that errors are: unbiased if E ( ǫ ) = 0 ; uncorrelated if E ( ǫ 1 ǫ T 2 ) = 0 ; non trivial if Cov ( ǫ ) is positive-definite ; Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 13 / 57
Stochastic data assimilation Best linear unbiased estimator (BLUE) Covariance matrix Covariance matrix ( x vector-valued): E (( x − E ( x ))( x − E ( x )) T ) Cov ( x ) = ( Cov ( x )) i , j = Cov ( x i , x j ) = E (( x i − E ( x i ))( x j − E ( x j ))) E.g. for x = ( x 1 , x 2 , x 3 ): Var ( x 1 ) Cov ( x 1 , x 2 ) Cov ( x 1 , x 3 ) Cov ( x ) = Cov ( x 1 , x 2 ) Var ( x 2 ) Cov ( x 2 x 3 ) Cov ( x 1 , x 3 ) Cov ( x 2 , x 3 ) Var ( x 3 ) Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 14 / 57
Stochastic data assimilation Best linear unbiased estimator (BLUE) Notations State x state vector or input parameters x t true state (unknown) x b background state (a priori information), background error ǫ b = x b − x t , covariance matrix B x a analyzed state (result of the assimilation process), analysis error ǫ a = x a − x t , covariance matrix A Observations observation vector y o observation operator H , mapping state space to observation space: y o = H ( x t ) + ǫ o observation error ǫ o , covariance matrix R Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 15 / 57
Stochastic data assimilation Best linear unbiased estimator (BLUE) Problem position: what we have We aim at producing an estimate x a of the true input parameters x t of the system. We are given: a background estimate x b , whose error ǫ b are assumed unbiased and non trivial, with covariance matrix B given, partial observations y o = H ( x t ) + ǫ o , where ǫ o are unbiased and non trivial, with covariance matrix R given. Observation operator H maps the input parameters to the observation variables (can contain complex laws, PDEs, non linear physics, . . . ). We also assume that: H = H is a linear operator, ǫ o and ǫ b are not correlated. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 16 / 57
Stochastic data assimilation Best linear unbiased estimator (BLUE) Problem position: what we look for We aim at producing an estimate x a of the true state x t of the system. The best estimate is searched for as a linear combination of the background estimate and the observation: x a = L x b + K y o Optimality criterium We look for an unbiased estimate x a , with minimal variance tr( A ). Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 17 / 57
Stochastic data assimilation Best linear unbiased estimator (BLUE) Best linear unbiased estimator, or least squares analysis BLUE analysis: 1 � x a = ( I − KH ) x b + K y o = x b + K ( y o – H ( x b )) K = BH T ( HBH T + R ) –1 K : gain, or weight matrix, y o – H ( x b ) innovation. Analysis covariance matrix: A = ( I – KH ) B 2 Equivalent variational optimization problem: (optimal least squares) 3 � x a = arg min J J ( x ) = ( x – x b ) T B –1 ( x – x b ) + ( y o – H ( x )) T R –1 ( y o – H ( x )) J : cost function. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 18 / 57
Stochastic data assimilation Best linear unbiased estimator (BLUE) Data assimilation methods Two types of methods: Direct computation of the BLUE, and the gain matrix K . 1 Main algorithm: Kalman filter − → stochastic data assimilation, this section. Minimization of the cost function J using optimization and adjoint methods. 2 Main algorithm: 4D-Var − → variational data assimilation, next section. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 19 / 57
b b u b b b u Stochastic data assimilation Kalman filter algorithm Time-dependant problems: the Kalman filter sequence state y o x f x f x a x a b x a u y o y o x f t k − 2 k − 1 k Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 21 / 57
Stochastic data assimilation Kalman filter algorithm Back to notations Vectors: k time index x f k forecast state (background), forecast error covariance matrix P f k x a k analyzed state (result of the assimilation process), analysis error covariance matrix P a k Operators: model operator x t k +1 = M k , k +1 ( x t k ) + η k , k +1 , model error η k , k +1 , covariance matrix Q k observation operator y o k = H k ( x t ) + ǫ o k , observation error ǫ o k , covariance matrix R k Kalman’s hypotheses, schematically: Model and observations operators M k , k +1 and H k are linear ; Errors are unbiased, gaussian, independant and white in time. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 22 / 57
Stochastic data assimilation Kalman filter algorithm Kalman’s hypotheses Schematically: Model and observations operators are linear, denoted M k , k +1 and H k ; Errors are unbiased, gaussian, independant and white in time. Ma¨ elle Nodet (Grenoble) Introduction to Data Assimilation 6 mars 2012 23 / 57
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