An ETKF approach for initial state and parameter estimation in ice sheet modelling

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1 An ETKF approach for initial state and parameter estimation in ice sheet modelling Bertrand Bonan University of Reading DARC Seminar, Reading October 30, 2013 DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

2 Context Work supervised by Maëlle Nodet 1,2 and Catherine Ritz 3. Joined work also with Vincent Peyaud 3. 1 : Inria, Laboratoire Jean Kuntzmann (Grenoble) 2 : Université Joseph Fourier Grenoble 1 3 : CNRS, Laboratoire de Glaciologie et Géophysique de l Environnement (Grenoble) Part of project ADAGe funded by French Agence Nationale de la Recherche. DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

3 Outline 1 Problem description 2 Large-scale ice sheet model 3 Data assimilation Twin experiment ETKF and LETKF Initial ensemble Results DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

4 Problem description Outline 1 Problem description 2 Large-scale ice sheet model 3 Data assimilation Twin experiment ETKF and LETKF Initial ensemble Results DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

Problem description Sea level change: Antarctica & Greenland contribution Motivations: Computation of the ice discharge of Antarctica and Greenland in the near future, thanks to simulations of polar

5 Problem description Sea level change: Antarctica & Greenland contribution Motivations: Computation of the ice discharge of Antarctica and Greenland in the near future, thanks to simulations of polar ice sheet model. Ice discharge: narrow outlets, closely linked to ice velocities, highly sensitive to basal friction parameters, highly sensitive to bedrock topography. Surface ice velocities [Rignot et al. 2011] DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

6 Problem description Basal parameters Typically 2 types of basal parameters are mandatory for simulation: bedrock topography B soc (x) basal friction parameters in basal shear stress relationship { C(x, t) ub (x, t) τ b (x, t) = m(x,t) 1 u b (x, t) if T b (x, t) = T melt 0 if T b (x, t) < T melt with u b basal velocity, T b basal ice temperature, C > 0 and m basal friction parameters. Simplified relationship { β(x)ub (x, t) if T τ b (x, t) = b (x, t) = T melt 0 if T b (x, t) < T melt with β > 0. DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

7 Problem description What is behind β due to sliding of ice over the bed deformation of the bed effective pressure = ice pressure - water pressure (ex: thin water layer between ice and bedrock, water in cavities,...) rock debris sediments... DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

2013] Bedmap2 estimation of bed topography DARC Seminar (Reading)

8 Problem description Poorly known basal parameters... Illustration for bedrock topography with Bedmap2 [Fretwell et al. 2013] Bedmap2 estimation of bed topography DARC Seminar (Reading) Estimated uncertainty of bed topography ETKF for ice sheet initialisation October 30, / 50

9 Problem description... Fortunately surface observations Ice velocity Surface elevation (ex.: [Bamber et al, 2009]) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

10 Problem description Overview What we want: Build estimation of actual basal parameters: bedrock topography B soc sliding coefficient β = 10 α Give an estimation of the uncertainty of our estimation. What we have: 20 years of observations of surface elevation and surface velocities Sparse observations of bedrock topography Time dependent ice sheet model What we need: efficient data assimilation system To develop the method: Synthetic experiments along a flowline DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

11 Large-scale ice sheet model Outline 1 Problem description 2 Large-scale ice sheet model 3 Data assimilation Twin experiment ETKF and LETKF Initial ensemble Results DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

12 Large-scale ice sheet model Ice dynamics processes DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

13 Large-scale ice sheet model Simplified physics DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

14 Large-scale ice sheet model Model equations: mass balance Flowline SIA model (1D + time) with a sliding law Mass balance equation: H t = ḃm (UH) x H t=0 = H 0 with x latitude, t time H(x, t) ice thickness, H 0 (x) initial ice thickness U(x, t) ice velocity averaged over ice thickness ḃ m (x, t) surface mass balance rate DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

15 Large-scale ice sheet model Model equations: dynamics (1) Vertically averaged ice velocity is a diagnostic variable = no partial derivative in time involved, computed from geometry at each time step with S(x, t) surface elevation U = u def + u slid S = B soc + H H 0 B soc (x, t) bedrock topography u def ice velocity component due to ice deformation u slid ice velocity component due to sliding DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

16 Large-scale ice sheet model Model equations: dynamics (2) u def is the averaged velocity due to ice deformation. Under SIA approximation S H 2 ( ) S 3 u def = a 1 x 3 a H 4 2 x 5 with a 1, a 2 given coefficients u slid is the velocity component due to basal sliding with β > 0 basal friction coefficient. u slid = 1 β ρ i gh S x DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

17 Large-scale ice sheet model Model equations: mass balance rate ḃ m (x, t) = Acc(x, t) + Abl(x, t) Acc(x, t) 0 ice accumulation rate (mainly snow precipitation) Abl(x, t) 0 ice ablation rate (snow melting) Both are function of surface temperature T s T s (x, t) = F clim (t) + λ x + γs(x, t) (en m/an) Accumulation Ablation Bilan de masse Température en surface (en ) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

18 Outline 1 Problem description 2 Large-scale ice sheet model 3 Data assimilation Twin experiment ETKF and LETKF Initial ensemble Results DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

19 Twin experiment Outline 1 Problem description 2 Large-scale ice sheet model 3 Data assimilation Twin experiment ETKF and LETKF Initial ensemble Results DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

20 Twin experiment Twin experiment What we control: at each grid point Ice thickness H Bedrock topography B soc Basal parameter β What we observe: each year during 20 years (20 observation times) Surface elevation S at each grid point (σ S = 2 m) Surface ice velocity u s at each grid point (σ u = 3 m/yr) Bedrock topography B soc at few grid points (σ B = 20 m) DA system: ETKF [Wang et al., 2004], [Ott et al. 2004] LETKF [Hunt et al., 2007] DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

21 Twin experiment Background and reference states (1) altitude (in m) reference bedrock observed surface backround bedrock x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

22 Twin experiment Background and reference states (2) ! reference! backround! x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

23 Twin experiment Background and reference states (3) velocity (in m/yr) sliding component deformation component surface velocity x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

24 ETKF and LETKF Outline 1 Problem description 2 Large-scale ice sheet model 3 Data assimilation Twin experiment ETKF and LETKF Initial ensemble Results DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

25 ETKF and LETKF Basis of Ensemble Kalman Filter (EnKF) Ensemble formulation: Let x k ensemble of N ens state estimations of model at time t k (in our case, ice thickness, sliding coefficient and bedrock topography) x (i) k ith member of ensemble, i = 1,..., N ens x k ensemble mean P e,k ensemble covariance matrix 2 step algorithm: Forecast: Predict x f k from xa k 1 using the model. Analysis: Correct x f k with observations yo k (in our case, surface velocities, surface elevation and sparse bedrock topography) to produce x a k. DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

26 Data assimilation ETKF and LETKF Sequential data assimilation: EnKF sequence state x f k 2 yk 1 o y o k 2 y o k k 2 k 1 k t DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

27 Data assimilation ETKF and LETKF Sequential data assimilation: EnKF sequence state x f k 2 yk 1 o x a k 2 y o k 2 y o k k 2 k 1 k t DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

28 Data assimilation ETKF and LETKF Sequential data assimilation: EnKF sequence state x f k 2 yk 1 o x a k 2 y o k 2 y x f k o k 1 k 2 k 1 k t DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

29 Data assimilation ETKF and LETKF Sequential data assimilation: EnKF sequence state x f k 2 yk 1 o x a k 2 x a k 1 y o k 2 y x f k o k 1 k 2 k 1 k t DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

30 Data assimilation ETKF and LETKF Sequential data assimilation: EnKF sequence state x f k 2 yk 1 o x a k 2 x a k 1 x f k y o k 2 y x f k o k 1 k 2 k 1 k t DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

31 Data assimilation ETKF and LETKF x Sequential data assimilation: EnKF sequence state x f k 2 yk 1 o y o k 2 x a k 2 x a k 1 x f k 1 x f k k 2 k 1 k t y o k a k DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

32 ETKF and LETKF Ensemble Transform Kalman Filter (ETKF) Detailled in [Hunt et al. 2007], [Harlim and Hunt 2007]. Analysis step } Build an ensemble {x a(i), i = 1... N ens such as x a minimum of J (x) = 1 2 ( x x f ) T P f e 1 ( x x f ) (yo H(x)) T R 1 (y o H(x)) and P a e P a. Attention: rank(p f e) N ens 1 by construction so P f e not invertible. DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

33 ETKF and LETKF First hypothesis Transform Filter w R Nens Assume x = x f + X f w with X f matrix whose ith column is x f (i) x f Why? w R Nens, x R nx and minimise J (x f + X f w) and N ens < n x (generally). ( x x f ) T P f 1 ( e x x f ) = w T X f T P f 1 e X f w (background term) }) if V = Vect ({x f (i) x f, i = 1... N ens, P f e = 1 N 1 Xf X f T invertible as a restriction to V. DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

34 ETKF and LETKF Where are we? J (x f + X f w) = N 1 ( ) w T X f T X f X f T 1 X f w ( y o H(x f + X w)) ( T ) f R 1 y o H(x f + X f w) 2 Attention: J invariant on Ker(X f ) and dim(ker(x f )) 1. = non unicity of w for minimal J. DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

35 ETKF and LETKF Gauge-fixing term Minimise under constraints To avoid this invariance we can minimise J over the w that have a null orthogonal projection on the kernel of X f ( ( ) ) I Nens X f T X f X f T 1 X f w = 0 It is equivalent to add a gauge-fixing term G into J G (w) = N 1 ( ( ) w T I Nens X f T X f X f T 1 X f 2 and minimise the new cost function J. ) w DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

36 ETKF and LETKF Where are we? J (w) = N 1 w T w+ 1 ( y o H(x f + X w)) ( T ) f R 1 y o H(x f + X f w) 2 2 DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

37 ETKF and LETKF Second hypothesis Linear approximation for observation operator ( ) H x f + X f w y f + Y f w with y f (i) = H (x f (i)) y f = 1 N N ens y f (i) i=1 matrix Y f whose ith column is y f (i) y f Finally, we search to minimise a quadratic cost function J (w) = N 1 w T w + 1 ( y o y f Y w) ( T ) f R 1 y o y f Y f w 2 2 DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

38 ETKF and LETKF ETKF analysis step Direct computation of: w a = P ( ) a Y f T R 1 y o y f (minimum of J ) ( ) P a = (N ens 1) I Nens + Y f T 1 ( 1 R 1 Y f (= Hess J (w )) a ) ( ) 1/2 W a = (N ens 1) P a (symetric square root matrix) } Analysis ensemble {x a(i), i = 1... N ens is defined as x a = x f + X f w a (mean) X a = X f W a (anomalies matrix) x a(i) = x a + X a i (with X a i ith column of X a ) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

39 Initial ensemble Outline 1 Problem description 2 Large-scale ice sheet model 3 Data assimilation Twin experiment ETKF and LETKF Initial ensemble Results DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

40 Initial ensemble Initial ensemble: Cooking recipe (1) Generate an ensemble of α = log(β) following a Gaussian law N (α b, B α ) with [B α ] ij = σα 2 exp ( (x i x j ) 2 ) L Generate an ensemble of B soc following a Gaussian law N (B b soc, B b ) Generate an ensemble of S following a Gaussian law N (S obs, σ 2 S I) Compute H = S B soc and correct to avoid negative ice thicknesses. For each member, run the model 1 year in order to obtain more physically balanced ice sheets. Rescale the produced ensemble so that its mean remains equal to the background. Warning: we use observations to produce initial ensemble! DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

41 Initial ensemble Initial ensemble: Cooking recipe (2) B b = ΣCΣ with: Σ square root of diagonal matrix of variances C correlation matrix standart deviation (in m) x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

42 Initial ensemble Initial ensemble: Cooking recipe (3) [C] ij = c 1 exp ( (x i x j ) 2 ) + c 2 exp ( (x i x j ) 2 ) L 1 L 2 1 large scale component short scale component correlation function x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

43 Initial ensemble Initial ensemble 4000 Initial ice sheet ensemble (Ne = 50) altitude (in m) x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

44 Results Outline 1 Problem description 2 Large-scale ice sheet model 3 Data assimilation Twin experiment ETKF and LETKF Initial ensemble Results DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

45 Results ETKF (N ens = 1000) 3000 ETKF result for ice sheet (ensemble size = 1000) altitude (in m) reference background final analysis observed surface x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

46 Results ETKF (N ens = 1000) 800 Standard deviation for bedrock topography (ensemble size = 1000) background final analysis Standard deviation (in m) x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

47 Results ETKF (N ens = 1000) ETKF result for! (ensemble size = 1000) ! reference background final analysis x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

48 Results ETKF (N ens = 1000) 3000 ETKF result for sliding velocity (ensemble size = 1000) velocity (in m/yr) reference background final analysis x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

49 Results ETKF (N ens = 1000) 2500 Standard deviation for sliding velocity (ensemble size = 1000) background final analysis 2000 standard deviation (in m/yr) x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

50 Results LETKF (N ens = 30, 50, 100) LETKF results for ice sheet (ensemble size = 100, 50, 30) reference background final anal. (100 m.) final anal. (50 m.) final anal. (30 m.) observed surface altitude (in m) x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

51 Results LETKF (N ens = 30, 50, 100) 800 Standard deviation for bedrock topography (ensemble size = 100, 50, 30) background final anal. (100 m.) final anal. (50 m.) final anal. (30 m.) standard deviation (in m) x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

52 Results LETKF (N ens = 30, 50, 100) 2500 LETKF results for sliding velocity (ensemble size = 100, 50, 30) sliding velocity (in m/yr) reference background final anal. (100 m.) 2000 final anal. (50 m.) final anal. (30 m.) x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

53 Results LETKF (N ens = 30, 50, 100) Standard deviation for sliding velocity (ensemble size = 100, 50, 30) background final anal. (100 m.) final anal. (50 m) final anal. (30 m.) standard deviation (in m/yr) x axis (in km) DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

54 Conclusion Ontgoing and future works Compare with variational approaches used in glaciology Add more physics: ice shelves, criterion on basal temperature for example. Perform experiments on a real flow line Implement this method into a 2D ice sheet model. DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

55 Conclusion Thank you for your attention! DARC Seminar (Reading) ETKF for ice sheet initialisation October 30, / 50

Asynchronous data assimilation

Ensemble Kalman Filter, lecture 2 Asynchronous data assimilation Pavel Sakov Nansen Environmental and Remote Sensing Center, Norway This talk has been prepared in the course of evita-enkf project funded