Ensemble methods for ice sheet init.

Ensemble methods for ice sheet model initialisation Bertrand Bonan 1 Maëlle Nodet 1,2 Catherine Ritz 3 : INRIA Laboratoire Jean Kuntzmann (Grenoble) 2 3 1 : Université Joseph Fourier (Grenoble) : CNRS Laboratoire de Glaciologie et Géophysique de l Environnement (Grenoble) International Conference on Ensemble Methods in Geophysical Sciences Toulouse November 13, 2012 Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 1 / 25

Outline 1 Problem presentation 2 Large-scale ice sheet model 3 Local Ensemble Transform Kalman Filter (LETKF) 4 Data assimilation experiments Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 2 / 25

Problem presentation Outline 1 Problem presentation 2 Large-scale ice sheet model 3 Local Ensemble Transform Kalman Filter (LETKF) 4 Data assimilation experiments Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 3 / 25

Problem presentation 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: governed by a couple of ice streams, closely linked to ice velocities, highly sensitive to basal friction parameters, highly sensitive to bedrock topography, = Data assimilation to estimate the best basal parameters (basal drag + bedrock topography) Balance velocities Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 4 / 25

Problem presentation Poorly known basal parameters Basal drag and bedrock topography are crucial to perform accurate simulations of ice sheets. Uncertainties of bedrock topography measures But: basal drag is unknown (impossible measures, not representative lab experiments, geothermal flux impacting basal temperature not well known) bedrock topography is measured along tracks = up to 400-500 meters uncertainties on central regions of Greenland Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 5 / 25

Large-scale ice sheet model Outline 1 Problem presentation 2 Large-scale ice sheet model 3 Local Ensemble Transform Kalman Filter (LETKF) 4 Data assimilation experiments Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 6 / 25

Large-scale ice sheet model Ice dynamics processes Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 7 / 25

Large-scale ice sheet model Simplified physics Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 8 / 25

Large-scale ice sheet model Model equations: mass balance Large time and space scales: flowline shallow ice model (1D + time) 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) Euler velocity averaged over ice thickness ḃ m (x, t) surface mass balance rate Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 9 / 25

Large-scale ice sheet model Model equations: dynamics Vertically averaged ice velocity is a diagnostic variable = no partial derivative in time involved, computed from geometry at each time step with S H 2 U = u B a 1 x 3 a 2 S = B + H H 0 u B (x, t) basal velocity (here u B (x, t) = 0) S(x, t) surface elevation B(x, t) bedrock topography a 1, a 2 given coefficients ( ) S 3 H 4 x 5 Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 10 / 25

Large-scale ice sheet model Model equations: surface mass balance Surface mass balance is given as a function of ice surface temperature: ḃ m = Acc + Abl Acc = f (T s ) Abl = g(t s ) with Acc(x, t) accumulation rate (snow falls) Abl(x, t) ablation rate (snow melting) f, g given function T s (x, t) surface temperature: T s (x, t) = T clim (t) + b x c S(x, t) with b, c given coefficients (b, c > 0) strong retroaction with dynamics throught surface elevation S. Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 11 / 25

Local Ensemble Transform Kalman Filter (LETKF) Outline 1 Problem presentation 2 Large-scale ice sheet model 3 Local Ensemble Transform Kalman Filter (LETKF) 4 Data assimilation experiments Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 12 / 25

Local Ensemble Transform Kalman Filter (LETKF) Ensemble Transform Kalman Filter (ETKF) Detailled in [Hunt et al. 2007]. Forecast step Analysis step ( x f (i) k = Mk x a (i)) k 1, i = 1,..., N ens x a k (i) = x f k + Xf k (wa k + ith column of Wa k ) ] X f k [x = f (1) k x f k,..., x f (N ens) k x f k ( ) [ ] yk f (i) = Hk x f (i) k and Yk f = yk f (1) y f k,..., yk f (N ens) y f k ( ) P a k = (N ens 1) I + Yk f T 1 ( Rk Yk f and W a k = (N ens 1) P ) 1/2 a k w a k = P a k Yf T ( k R 1 k y o k y f k) Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 13 / 25

Local Ensemble Transform Kalman Filter (LETKF) Local Ensemble Transform Kalman Filter (LETKF) Localisation based on observations [Ott et al. 2004] For each grid point (red dot), perform ETKF analysis step with only all observations (purple diamonds) within a local region. Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 14 / 25

Data assimilation experiments Outline 1 Problem presentation 2 Large-scale ice sheet model 3 Local Ensemble Transform Kalman Filter (LETKF) 4 Data assimilation experiments Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 15 / 25

Data assimilation experiments Context What we want: control variables We search the actual state of our model governed entirely by B and H Twin experiments Simulate data thanks to the model and do data assimilation to retrieve the state variables and/or parameters you choose to simulate data. Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 16 / 25

Data assimilation experiments Context What we have: surface observations We observe each year during 10 years surface elevation (σ S = 10 m) surface velocity (σ u = 3 m/yr) Obs. taken at each grid point except in the centre every 2 grid points. Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 17 / 25

Data assimilation experiments Initial ensemble How to build initial ensemble? In our case, well-known surface elevation S = B + H But poor information on B and H. We use these two facts. Assume B back a priori information on bedrock topography. For member (i), we compute [B (i), H (i) ]: B (i) = B back + b (i) with b (i) N (0, Cov B ) with a good length scale for space correlation. S (i) = S obs + s (i) with s (i) N (0, σ 2 S I). H (i) = S (i) B (i) then run the model for 10 years and take resulting ice thickness for H (i) (which is more physical). Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 18 / 25

Data assimilation experiments Initial ensemble Example of initial ensemble for ensemble size N ens = 10 Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 19 / 25

Data assimilation experiments First results Influence of ensemble size N ens on results quality for ETKF Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 20 / 25

Data assimilation experiments First results Inflation and localisation on results quality for LETKF Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 21 / 25

Data assimilation experiments First results LETKF for optimal inflation and localisation N ens = 10 Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 22 / 25

Data assimilation experiments First results LETKF for optimal inflation and localisation N ens = 10 Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 23 / 25

Conclusion Ontgoing and future works Add more physics (basal sliding law using Shallow Shelves Approximation) Add basal drag control and perform joint control of bedrock topography and basal drag Perform more realistic experiments Compare with variationnal approaches LETKF can product negative ice thickness at some grid points (edges of ice sheet). How to deal with efficiently? Implement both methods into a 2D+time model Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 24 / 25

Conclusion Thank you for your attention! Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 25 / 25

To complete Influence of ensemble size N ens on results quality for ETKF Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 26 / 25

To complete Influence of inflation on results quality for ETKF Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 27 / 25

To complete Influence of inflation on results quality for ETKF Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 28 / 25

To complete Influence of inflation on results quality for ETKF (N ens = 10) Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 29 / 25

To complete Influence of inflation on results quality for ETKF (N ens = 20) Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 30 / 25

To complete Influence of inflation on results quality for ETKF (N ens = 30) Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 31 / 25

To complete Influence of inflation on results quality for ETKF (N ens = 40) Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 32 / 25

To complete Influence of inflation and localisation on results quality for LETKF Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 33 / 25

To complete ETKF with N ens = 40 Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 34 / 25

To complete ETKF with N ens = 40 Bonan et al. (Grenoble) Ensemble methods for ice sheet init. November 13, 2012 35 / 25