Estimation of positive sum-to-one constrained parameters with ensemble-based Kalman filters: application to an ocean ecosystem model

Transcription

1 Estimation of positive sum-to-one constrained parameters with ensemble-based Kalman filters: application to an ocean ecosystem model Ehouarn Simon 1, Annette Samuelsen 1, Laurent Bertino 1 Dany Dumont 2 1 Nansen Environmental and Remote Sensing Center, Norway 2 Institut des sciences de la mer, Université du Québec à Rimouski, Canada 16 November 212

2 Outline 1 Estimation of positive sum-to-one constrained parameters with ensemble-based Kalman filters 2 Numerical results in GOTM-NORWECOM

3 3 Assimilation of ocean color data Time evolution of the errors CHLA: forecast mean ( ) CHLA: observations ( ) RMS error and standard deviations (mg/m 3 ) RMS Ensemble STD Ensemble 1 Jan 27 1 Jan 28 1 Jan 29 1 Jan 21 Difficulties Nonlinear dynamics and positive variables (tracer concentration). Numerous poorly known parameters. Seasonal lack of observations (Arctic) and large error (satellite), few in-situ data. Diet of the herbivorous / predators? Zooplankton grazing preferences (relative diet). No prior knowledge, one set of values for the whole domain. Adaptation of zooplankton to their local environment?

4 Non-Gaussianity and parameter estimation with the EnKF Truncations of negative values (d!1 ) Potential depletion of components of the ensemble Large corrections on parameters in erroneous directions. Grazing efficiency f (d!1 ) Grazing efficiency f Gaussian anamorphosis extension of ensemble-based Kalman filters Analysis with transformed variables and observations (Bertino et al., 23) Applicability of the methods in large systems Effectiveness of the Gaussian anamorphosis extension of the EnKF to estimate parameters in nonlinear frameworks of positive variables.

5 Outline 1 Estimation of positive sum-to-one constrained parameters with ensemble-based Kalman filters 2 Numerical results in GOTM-NORWECOM

6 Positive sum-to-one constrained parameters (π i ) i NN Estimation Constraints: i = 1 : N, π i and NX π i = 1 i=1 Cannot be handled by the EnKF in that form. Dirichlet distribution of order N i = 1 : N, π i = φ i with φ i Γ(θ i, 1) NX φ k k=1 Gelman s formulation (1995) i = 1 : N, π i = eφ i NX k=1 e φ k with φ i N (θ i, Σ i )

7 7 Positive sum-to-one constrained parameters (π i ) i NN Hyperspherical coordinates: N 1 parameters 8 π 1 = cos 2 ( π 2 φ 1) >< >: i = 2 : N 1, π i = π N = i 1 Y sin 2 ( π 2 φ k) cos 2 ( π 2 φ i ) k=1 N 2 Y k=1 sin 2 ( π 2 φ k) sin 2 ( π 2 φ N 1) with (φ i ) i=1=n 1 distributed on the segment line [, 1] Inversion of the hyperspherical coordinate system Recursively. Information on the distribution of the (π i ) i=1:n or available samples: Prior values for the (φ i ) i=1:n 1.

8 8 Expected values and variances of the (π i ) i=1:n Tuning of the parameters of the distributions of the (φ i ) i=1:n 1 i = 1 : N 1, E[π i ] = m i, var(π i ) = σ 2 i Assumption: (φ i ) i=1:n 1 (D i ([, 1], Θ i )) i=1:n 1 are independent. Expected values of the (π i ) i=1:n (E[ejπφ 1 ] + E[e jπφ 1 ]) = m >< >: i = 2 : N 1, 1 4 (E[ejπφ i ] + E[e jπφ m i i ]) = Xi 1 1 m k Variances of the (π i ) i=1:n (E[e2jπφ 1 ] + E[e 2jπφ 1 ]) = σ2 1 + m (E[ejπφ 1 ] + E[e jπφ 1 ]) >< >: 1 i = 2 : N 1, 16 (E[e2jπφ i ] + E[e 2jπφ i ]) = (E[ejπφ i ] + E[e jπφ i ]) σi 2 + mi 2 + ix k 1 ( 2) k 1 (σi k 2 + m2 i k ) Y 1 4 (E[ejπφ i l ] + E[e jπφ i l ]) k=1 l=1 k=1 1 2

9 9 Example with the triangular distribution Triangular distribution: φ i T (, 1, c i ) Tuning of the mode c i. Characteristic function: t R, E[e jtφ i ] = 2 (1 c i ) e jc i t + c i e jt π 2 c i (1 c i ) Equal preferences: E[π i ] = 1 N A system of N 1 nonlinear equations: i = 1 : N 1, cos(πc i ) + 2c i 1 π 2 c i (1 c i ) + N i 1 2(N i + 1) =. N > 3: non existence of solutions.

10 Outline 1 Estimation of positive sum-to-one constrained parameters with ensemble-based Kalman filters 2 Numerical results in GOTM-NORWECOM

11 11 GOTM-NORWECOM-ASSIM Assimilation Deterministic ensemble Kalman filter (DEnKF, Sakov and Oke, 28). State vector: biogeochemical state vector and parameters. 1 members. Gaussian anamorphosis (Bertino et al., 23) for the biogeochemical state variables (and parameters for the spherical formulation). Reference solution x t Deterministic simulation with different preferences: Mesozooplankton: π dia =.6, π mic =.15, π den =.25. Microzooplankton: π fla =.6, π den =.15, π dia =.25. Chlorophyll Nitrate

12 The data assimilation system Configuration of the experiments One year to warm up the ensemble, then four years with assimilation. Observations: y = x t 1 P G, with G Γ( σo 2, σo 2 ), σo =.3. Surface chlorophyll (2 first layers) every seven days. Truncated-Gaussian perturbations in the whole water column every twelve hours: phytoplankton and zooplankton only. Robustness of the estimation: experiments repeated 2 times. Chlorophyll concentration (mg/m 3 ) Layer 1 Reference Reference (observation time) Observations Chlorophyll concentration (mg/m 3 ) Layer 2 Jan Jan1 Jan2 Jan3 Jan4 Jan Jan1 Jan2 Jan3 Jan4

13 13 Preferences: Gelman s formulation π i = e φ i P N k=1 eφ k MesoZ π dia MesoZooplankton: diatoms MesoZ π mic MesoZooplankton: MicroZooplankton MesoZ π den MesoZooplankton: detritus (N) !.2 Jan Jan1 Jan2 Jan3 Jan4!.2 Jan Jan1 Jan2 Jan3 Jan4!.2 Jan Jan1 Jan2 Jan3 Jan4 MicroZ π fla MicroZooplankton: flagellates MicroZ π den MicroZooplankton: detritus (N) MicroZ π dia MicroZooplankton: diatoms !.2 Jan Jan1 Jan2 Jan3 Jan4!.2 Jan Jan1 Jan2 Jan3 Jan4!.2 Jan Jan1 Jan2 Jan3 Jan4

14 14 Preferences: spherical formulation MesoZ π dia MesoZooplankton: diatoms MesoZ π mic MesoZooplankton: MicroZooplankton MesoZ π den MesoZooplankton: detritus (N) !.2 Jan Jan1 Jan2 Jan3 Jan4!.2 Jan Jan1 Jan2 Jan3 Jan4!.2 Jan Jan1 Jan2 Jan3 Jan4 MicroZ π fla MicroZooplankton: flagellates MicroZ π den MicroZooplankton: detritus (N) MicroZ π dia MicroZooplankton: diatoms !.2 Jan Jan1 Jan2 Jan3 Jan4!.2 Jan Jan1 Jan2 Jan3 Jan4!.2 Jan Jan1 Jan2 Jan3 Jan4

15 Preferences: ternary plots of the final estimates Gelman s formulation Diatoms Mesozooplankton Detritus Mesozooplankton True Initial DEnKF MicroZ Flagellates Microzooplankton Diatoms Microzooplankton Detritus Spherical formulation Detritus Diatoms Diatoms MicroZ Flagellates Detritus

16 Spherical formulation: asymmetry of the transformation Robustness of the results to the matching preferences/transformed parameters Same 2 experiments (observation, prior ensembles) Microzooplankton: permutation in the assignment preferences/transformed parameters every four experiments. 8 >< >: Mesozooplankton π DIA = cos 2 ( π 2 φ 1) π MIC = sin 2 ( π 2 φ 1) cos 2 ( π 2 φ 2) π DET = sin 2 ( π 2 φ 1) sin 2 ( π 2 φ 2) 8 >< >: Microzooplankton π DET = cos 2 ( π 2 φ 1) π FLA = sin 2 ( π 2 φ 1) cos 2 ( π 2 φ 2) π DIA = sin 2 ( π 2 φ 1) sin 2 ( π 2 φ 2) Microzooplankton: marginal improvement of the estimates of the preferences. Mesozooplankton: slight degradation of the estimation. Mesozooplankton RMS error (Mesozooplankton.) Detritus Initial DEnKF True 16 Diet Diatoms MicroZ Detritus Prior (%) Gelman Spherical Spher. + perm Diatoms MicroZ

17 17 Conclusion and perspectives Two approaches for estimating positive sum-to-one constrained parameters. Gelman s formulation: Gaussian parameters, symmetry of the transformation, N transformed parameters. Spherical formulation: N 1 parameters to estimate, asymmetry of the transformation, distribution of the transformed parameters? Preliminary experiments indicate that the zooplankton grazing preferences could be estimated with both approaches. Quite simple framework. Further investigations have to be done. Distribution. Additional parameters to estimate. Real observations (Mike station).

18 18 Thank you!

19 Bibliography Bertino L., Evensen G. and Wackernagel H.: Sequential Data Assimilation Techniques in Oceanography, International Statistical Reviews, 71, , 23. Bocquet M., Pires C.A. and Wu L.: Beyond Gaussian statistical modeling in geophysical data assimilation, Monthly Weather Review, 138, 8, , 21. Doron M., Brasseur P. and Brankart J.-M.: Stochastic estimation of biogeochemical parameters of a 3D ocean coupled physical-biogeochemical model: Twin experiments. Journal of Marine Systems, 87 (3-4), , 211. Gelman A.: Method of Moments Using Monte Carlo Simulation. Journal of Computational and Graphical Statistics, 4, 1, 36-54, Gelman A., Bois F. and Jiang J.: Physiological Pharmacokinetic Analysis Using Population Modeling and Informative Prior Distributions. Journal of the American Statistical Association, 91, 436, , Simon E. and Bertino L: Gaussian anamorphosis extension of the DEnKF for combined state and parameter estimation: application to a 1D ocean ecosystem model, Journal of Marine Systems, 89, 1-18, 212. Simon E., Samuelsen A., Bertino L. and Dumont D.: Estimation of positive sum-to-one constrained zooplankton grazing preferences with the DEnKF: a twin experiments, Ocean Science, 8, , 212.

20 Construction of a monovariate anamorphosis: Z(x) = ψ(y (x)) Empirical anamorphosis Empirical marginal distribution of Z(x): sample (z i ) i=1:n. Cumulative distribution function of Y (x): G. ψ(y) = NX i=1 z i 1 ]G 1 ( i 1 N ),G 1 ( N i )](y) 1- Empirical anamorphosis Interpolation 3- Definition of the tails 3 Physical values (mg/m 3 ) Physical values (mg/m 3 ) Physical values (mg/m 3 ) !1!5 5 Gaussian values!1!5 5 Gaussian values!1! Gaussian values 2 Major issues Choice of the physical data set: specification of the likely and unlikely values. Dependance on a reference model run (model bias, extreme events).