4. DATA ASSIMILATION FUNDAMENTALS

Transcription

1 4. DATA ASSIMILATION FUNDAMENTALS... [the atmosphere] "is a chaotic system in which errors introduced into the system can grow with time... As a consequence, data assimilation is a struggle between chaotic destruction of knowledge and its restoration by new observations." Leith (1993) ATMOSPHERIC STATE OBSERVATIONS ASSIMILATED MODEL REALITY TIME Ross Bannister, EO and DA, QUEST ES Page 3 of 51

2 The vector notation for fields and data and the need for an a- priori The 'state vector', x The 'observation vector', y u v θ p q λ1 λ n φ 1 φ m l 1 l L u zonal wind field v meridional wind field θ potential temperature p pressure q specific humidity λ longitude φ latitude l vertical level NWP models: > 10 7 elements (5 n m L) y 1 y y 3 y N NWP models: ~10 6 elements The 'forward model', h y = h [x ] + ε No. of obs. << No. of (unknown) elements in x. This is an under-constrained (and inexact) inverse problem. Need to fill-in the missing information with prior knowledge. An 'a-priori' state (a.k.a. 'first guess', 'background', 'forecast') is needed to make the assimilation problem well posed. Ross Bannister, EO and DA, QUEST ES Page 4 of 51

3 Vectors and matrices Vector/matrix notation is a powerful and compact way of dealing with large volumes of data. A matrix operator acts on an input vector to give an output vector, e.g. ( ) x () ( 1 A 11 A 1 A x (1) 1 1N x () (1) x () A = Ax = 1 A A N x (1) x () A N1 A N A NN N x (1) N )( ) Matrix products do not commute in general, e.g. x (3) = ABCx (1) (1) CBAx Some matrices can be inverted (must be 'square' and non-singluar), e.g. ( ) x (1) ( 1 (A 1 ) 11 (A 1 ) 1 (A 1 ) 1 1N x (1) (A = 1 ) 1 (A 1 ) (A 1 ) N x (1) (A 1 ) N N1 (A 1 ) N (A 1 ) NN N ) ( ) x () x () x () (matrices are complicated to invert, ie (A 1 ) ij A 1 ij.) The matrix transpose make rows into columns and columns into rows (also for vectors), e.g. ( ( A 11 A 1 A 1N A 11 A 1 A N1 A A = 1 A A N ), A T A = 1 A A N A N1 A N A NN A 1N A N A ( ) NN x 1 x x =, x T = (x 1, x, x N ) x N ) The inner product ('scalar' or 'dot' product), e.g. () x T x (1) = x () 1 x (1) 1 + x () x (1) + +x () The outer product (a matrix), e.g. x () x (1) T = ( N x (1) N ) x () 1 x (1) 1 x () 1 x (1) x () x (1) 1 x () x (1) x () 1 x (1) N x (1) N x () N x (1) x () x () N x (1) 1 x () N x (1) N = scalar Ross Bannister, EO and DA, QUEST ES Page 5 of 51

4 Early data assimilation ("objective analysis") The method of "successive corrections" Bergthorsson & Doos (1955), Cressman (1959) Analysis obs. (obs-rich regions). Analysis a-priori (obs-poor regions). Simple scheme to develop. Computationally cheap. Use a-priori in absence of observations. obs. grid-point Analysis is a linear combination of nearby observations, and an a-priori. Poor account of error statistics of obs and a-priori. Direct observations only. Anomalous spreading of obs. information. No multivariate relations (e.g. geostrophy). Ross Bannister, EO and DA, QUEST ES Page 6 of 51

5 'Optimal' Interpolation (OI) Introduced in the 1970s - a more powerful formulation of data assimilation xa = xb + K (y h [x B]) K is a rectangular matrix operator (the 'gain matrix'). K depends upon the error covariance matrices B and R, and linearization H. K = BH T (HBH T + R) 1. The Best Linear Unbiased Estimator (BLUE). R : observation error covariance matrix. Describes the error statistics of the observations (see later). B : background error covariance matrix. Describes the error statistics of the a-priori state (see later). H : linearized observation operator. h [x B + δx ] h [x B] + Hδx. Account taken of a-priori and obs. error statistics. Allows assimilation of some indirect obs. Use a-priori in absence of obs. Works as an inverse model. Too expensive for single global solution. Difficult to know B. No consistency with the equations of motion. Ross Bannister, EO and DA, QUEST ES Page 7 of 51

6 Types of errors 1. Random errors Data Arising from Obs. Noise Forecast Stochastic processes in model, init. conds. Assim. Input data E.g. repeated measurement of temperature: truth mean spread (ref. 'error statistics'). Systematic errors Data Arising from Obs. E.g. reading errors Forecast Model formulation, init. conds. Assim. Input data, formulation E.g. As before but with biased thermometer: truth mean spread Biases should be corrected where possible. 3. Representativeness error Data Arising from Assim. Unresolved variability E.g. Interpolation of model grid values to location of observation (forward operator h [x ]): V 1 obs V V 11 V 1 Ross Bannister, EO and DA, QUEST ES Page 8 of 51

7 Probability distribution functions Error statistics are described by a probability density function (PDF). PDFs of random and representativeness errors are often expressed together. They are often approximated by the normal (Gaussian) distribution. A scalar (ie a single piece of information), y P (y) A vector (multiple pieces of information), x x P ( x ) σ x x 1 xn ( ) x = y (y y ) P (y) exp σ Mean : y Variance : σ ( Std. dev. : σ) y ( P (x ) exp 1 (x x ) B 1 (x x )) Mean Covariance : x : B = ( x 1 ) δx 1 δx 1 δx δx 1 δx N δx δx 1 δx δx δx N δx N δx 1 δx N δx δx N = δx δx T where δx = x x Ross Bannister, EO and DA, QUEST ES Page 9 of 51

8 Example of covariances: forecast as the a-priori xb is a forecast and so the equations of motion will influence strongly the covariance patterns. Example: Geostrophic error covariances Geostrophic balance: v = 1 ρf p x u = 1 ρf p y Courtesy, Univ. of Washington Pressure-pressure covariances assumption: δp i δp j = σ exp r ij L L 750 km Ross Bannister, EO and DA, QUEST ES Page 30 of 51

9 Variational data assimilation The 'method of least squares' - simple version J (x ) = (x xb) + (y h [x ]) J : cost function (a scalar) xb : a-priori (background) state y : observations h [x ] : observation operator (forward model) x : variable x A = x min J = Carl Fredrich Gauss "analysis" The 'method of least squares' - considering error statistics J (x ) = 1 (x 1 (y x B) T B 1 (x h [x ]) T R 1 (y xb) + h [x ]) B : background error covariance matrix R : observation error covariance matrix This is the form used in operational weather forecasting, deriving satellite retrievals, etc. Non-Euclidean L norm. Assumes perfect forward model, unbiased data. This is consistent with a Gaussian model of error statistics (next slide). 'Var.' is efficient enough to solve the global problem. Ross Bannister, EO and DA, QUEST ES Page 31 of 51

10 The Bayesian view of data assimilation Bayes' Theorem P(y, x) = P(x y)p(y ) P(x P(x, y) = P(y x)p(x ) y) = P(y x)p(x ) P(y ) P (x ) P (y x) Rev. Thomas Bayes P (x ( y) exp 1 (x x B) T B 1 (x B)) x exp ( 1 (h [x ] y) T R 1 (h [x ] y )) ( exp 1 (x x B) T B 1 (x xb) + 1 (h [x ] y) T R 1 (h )) [x ] y Maximum likelihood Minimum penalty, J J [x ] = 1 (x x B) T B 1 (x xb) + 1 (h [x ] y) T R 1 (h [x ] y) xa = x min J = "analysis" Ross Bannister, EO and DA, QUEST ES Page 3 of 51

11 Minimising the cost function The problem reduces to a (badly conditioned) optimisation problem in dimensional phase space. J [x ] = 1 (x x B) T B 1 (x xb) + 1 (h [x ] y) T R 1 (h [x ] y) x xb xa x 1 Descent algorithms minimize J iteratively. They need the local gradient, x J of the cost function at each iteration. The adjoint method is used to compute the adjoint. 1 The curvature (a.k.a. inverse Hessian, ) at x indicates the error statistics of the analysis. A very badly conditioned problem. ( xj) 1 Ross Bannister, EO and DA, QUEST ES Page 33 of 51 A

12 Algebraic minimization of the cost function Under simplified conditions the cost function can be minimized algebraically. Assume that the linearization of the forward model is reasonable h [x ] h [x B] + H (x xb) J [x ] = 1 (x x B) T B 1 (x xb) + 1 (H (x x B) (y h [x B]) ) T R 1 (H (x xb) (y h [x B]) ) 1. Calculate the gradient vector ( x J = ) J / x 1 J / x = J / x N B 1 (x xb) + H T R 1 (h [x ] y). The special x that has zero gradient minimizes J (this cost function is quadratic and convex) xa x J xa = 0 = xb + (B 1 + H T R 1 H) 1 H T R 1 (y h [x B ]) = xb + BH T (R + HBH T ) 1 (y h [x B ]) This is the OI formula with the BLUE! Ross Bannister, EO and DA, QUEST ES Page 34 of 51

13 Types of data assimilation Sequential data assimilation methods Data assimilation performed at each batch of observations. Model forecast made between batches (for background). E.g. OI, KF*, EnKF*, etc. Explicit formula used for analysis. Very expensive. 1d-Var Data assimilation performed for vertical profile only, where satellite makes observations. Used as a 'pre-main-assimilation' step to produce vertical profiles of model quantities (retrievals) from satellite radiances. * Kalman Filter (KF) and Ensemble Kalman Filter (EnKF). Ross Bannister, EO and DA, QUEST ES Page 35 of 51

14 3d-Var Data assimilation performed every 6 hours. 6 hour model forecast between analysis times (for background). Adequate for re-analysis. Relatively cheap. Observations within ±3 hours are not at analysis time. No dynamical constraint used. 3d-Var. time window 4d-Var Data assimilation performed every 6 (1) hours. 6 (1) hour model forecast between analysis times (for background). Used (e.g.) by ECMWF and Met Office for operational weather forecasting. Model used as a dynamical constraint. Observations are compared to the model trajectory at the correct time. Perfect model assumption. Expensive, but not unfeasible. 4d-Var. time window t = 3 t = 0 t = +3 t = +6 t = 3 t = 0 t = +3 t = +6 3dFGAT '3d First Guess at Time' is half-way between 3d and 4d-Var. 'Strong constraint' - it is possible to use 4d-Var with a model under the 'weak-constraint' formulation. Ross Bannister, EO and DA, QUEST ES Page 36 of 51

15 The 4d-Var cost function 4d-Var. time window yt = 1 yt = yt = 3 J [x 0] = 1 (x 0 xb) T B 1 (x 0 xb) + 1 t (h t [x t] yt) T R 1 t (h t [x t] yt) t = 3 t = 0 t = +3 t = +6 The observation vector comprises subvectors, y for time t. The observation operator ht acts on model state xt. Vary x0 in the minimization - the state at the start of the 4d-Var. cycle. Future states in the cycle are computed with the forecast model, xt = M x. Forward model is the composite operator h. Important issues: t t t M 0 t 0 0 Tangent linear model (and its adjoint) needed and can be difficult to find. Forecast model can be highly non-linear (e.g. sensitive dependence in model's convection scheme - on/off 'switches'). Ross Bannister, EO and DA, QUEST ES Page 37 of 51

16 Assimilation of sequences of satellite images in 4d-Var (Courtesy Samantha Pullen, Met Office) Sequence of observed brightness temperatures Sequence of simulated brightness temperatures Ross Bannister, EO and DA, QUEST ES Page 38 of 51

17 '4d-Var.' demonstration with a double pendulum θ1 m 1 ( x = ) θ θ 1 θ1 l 1 m l θ θ l 3 L = T V d d t ( L θ i) = L θi m 3 V = gm 1 l 1 cos θ1 gm l cos θ1 gm 3 (l cos θ1 + l 3 cos θ) T = ½ m 1 (x1 + y1) + ½ m (x + y) + ½ m 3 (x3 + y3) Demonstrate '4d-Var' with an OSSE - 'Observation System Simulation Experiment'. Also known as a 'twin experiment'. Choose a set of initial conditions and run the model (truth). Add random noise to generate pseudo-observations. Forget the truth and try to recover it by assimilating the observations. Use observations of and only (no observations of and ). θ1 θ θ 1 θ Ross Bannister, EO and DA, QUEST ES Page 39 of 51

18 OSSE demonstration (double pendulum) - truth run 4 3 θ1 Angle θ 1 Rate of change angle time 'Truth' Observations Ross Bannister, EO and DA, QUEST ES Page 40 of 51

19 OSSE demonstration (double pendulum) - '4d-Var.' run 4 3 θ1 Angle θ 1 Rate of change angle time 'Truth' Observations '4d-Var.' analysis Ross Bannister, EO and DA, QUEST ES Page 41 of 51

20 OSSE demonstration (double pendulum) - 'obs insertion' run 4 3 θ1 Angle θ 1 Rate of change angle time 'Truth' Observations 'Insertion' run Ross Bannister, EO and DA, QUEST ES Page 4 of 51

21 Issues with data assimilation Data assimilation is a computer intensive process. For one cycle, 4d-Var. can use up to 100 times more computer power than the forecast. The B-matrix (forecast error covariance matrix in Var.) is difficult to deal with. Assimilation process is very sensitive to B. Least well-known part of data assimilation. In operational data assimilation, B is a matrix. Need to model the B-matrix - use technique of 'control variable transforms'. In reality B is flow dependent. Practically, B is quasi-static. Data assimilation replies on optimality. Issues of suboptimality arise if: Actual error distributions are non-gaussian, B or R are inappropriate. Forward models are inaccurate or are non-linear. Data have biases. Cost function has not converged adequately (in Var.). Assimilation can introduce undesirable imbalances. Quantities not constrained by observations can be poor (e.g. diagnosed quantities): Precipitation. Vertical velocity, etc. Ross Bannister, EO and DA, QUEST ES Page 43 of 51