A surrogate-based approach to nonlinear, non-Gaussian joint state-parameter data assimilation

John Maclean; Elaine T. Spiller

doi:10.3934/fods.2021019

Article Contents

2021, Volume 3, Issue 3: 589-614. Doi: 10.3934/fods.2021019

This issue Previous Article Mean field limit of Ensemble Square Root filters - discrete and continuous time Next Article Analysis of the feedback particle filter with diffusion map based approximation of the gain

A surrogate-based approach to nonlinear, non-Gaussian joint state-parameter data assimilation

John Maclean^1, , and
Elaine T. Spiller^2,

1.
School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
2.
Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, WI 53201, USA

^* Corresponding author: John Maclean
^* Corresponding author: John Maclean

Received: October 2020

Revised: June 2021

Early access: August 2021

Published: September 2021

The first author is supported by the ARC grant DP180100050, and acknowledges past support from ONR grant N00014-18-1-2204. The second author is supported by NSF grant DMS-1821338

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Abstract

Many recent advances in sequential assimilation of data into nonlinear high-dimensional models are modifications to particle filters which employ efficient searches of a high-dimensional state space. In this work, we present a complementary strategy that combines statistical emulators and particle filters. The emulators are used to learn and offer a computationally cheap approximation to the forward dynamic mapping. This emulator-particle filter (Emu-PF) approach requires a modest number of forward-model runs, but yields well-resolved posterior distributions even in non-Gaussian cases. We explore several modifications to the Emu-PF that utilize mechanisms for dimension reduction to efficiently fit the statistical emulator, and present a series of simulation experiments on an atypical Lorenz-96 system to demonstrate their performance. We conclude with a discussion on how the Emu-PF can be paired with modern particle filtering algorithms.

Keywords:

Mathematics Subject Classification: Primary: 62R07, 62M20, 62C12; Secondary: 62M05, 62G07, 86A10.

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Figure 1. Schematic for state dependence on parameters: we plot the state at eight different samples (1a), then apply a variety of interpolating schemes (1b) and lastly a statistical surrogate (1c). The shaded region in the rightmost plot shows one standard deviation in uncertainty. The second and third plot allow for the state to be estimated at a variety of parameter values

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Figure 2. Here we demonstrate how the GP example mapping from parameter space to state space (as in Figure 1) can be used in a particle filter update step. (A) The same GP mapping from parameter to state space is plotted (black line) along with the design points (blue dots along black line) used to fit that mapping and an observation (red dot and line) in state space along the left axis. A bi-modal prior distribution is plotted (light blue) along with samples from that distribution ($ 10^3 $ in light blue and 8 in black) along the horizontal axis. (B) Stem plots of the eight-sample PF posterior along with $ 10^3 $-sample normalized posterior histogram of parameters taking into account the likelihood of GP mapped prior samples given the observation in (A). Plotted behind the emuPF histogram is the equivalent (and nearly identical) histogram using the true mapping instead of the GP mapping for each of the $ 10^3 $ samples.

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Figure 3. Overview of the novel synthesis of Gaussian process emulators with Data Assimilation methods

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Figure 4. Visualisation of the internal Emu-PF mechanisms over one assimilation step. Left column shows components of dimension $ n_D = 100. $ Right column shows components of dimension $ N_F = 10,000 $. (a): parameter ensembles at time $ t_j $. (b): distribution of one state variable as a function of parameters. (c): parameter ensembles at time $ t_{j+1} $. Full details for this $ 8 $ state, $ 2 $ parameter experiment are given in section 4

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Figure 5. Long term error statistics for the implementation of Emu-PF from fig. 4, compared to: a "coarse" PF that employs $ n_D = 100 $ model runs (as in the Emu-PF), and a "fine" PF that employs $ N_F = 10,000 $ model runs, equal to the number of samples in the Emu-PF emulator. Performance of Emu-PF is markedly better than the coarse PF

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Figure 6. Error statistics for Experiment One, $ m = 8 $ observations at each observation time, of accuracy $ \sigma_0 = 1 $. In this (and every) plot, only every $ 20 $th data point is shown. For this mildly difficult filtering problem, we observe that the $ \Gamma = -1 $ implementation of section 3.2, that uses no state variables at all as emulator inputs, is stable and reasonably accurate

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Figure 7. Error statistics for Experiment Two, $ m = 2 $ observations at each observation time, of accuracy $ \sigma_0 = 1 $. The $ \Gamma = -1 $ Emu-PF and fine PF both under-perform compared to their mean behaviour; the Emu-PF employing PCA is stable and accurate

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Figure 8. Error statistics for Experiment Three, $ m = 4 $ observations at each observation time, of accuracy $ \sigma_0 = 0.5 $. In this case the $ \Gamma = 1 $ Emu-PF performs only as well as the coarse PF. However the Emu-PF employing PCA is still competitive with the, much more expensive, fine PF

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Figure 9. Summary statistics for Experiment Four, long-time state estimation with $ m = 4 $ observations of accuracy $ \sigma_o = 1 $. The median RMSE for EnKF and fine PF are similar; however the EnKF error occasionally spikes. The sliced Emu-PF of section 3.2 is stable, with no large error spikes, and performs close to the fine PF in accuracy

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Figure 10. RMSE against time for Experiment Five: dashed red lines plot the Fine PF (formulated under the Optimal Proposal), and solid blue lines plot the best-performing Emu-PF according to table 5. There is a clear improvement in skill in parameter estimation. State estimates are similar in skill (and, importantly, do possess some skill: the state RMSE is well below $ 5 $, the approximate long-term or climatic mean RMSE of forecasting with no DA)

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Table 1. Summary statistics for twenty repetitions of experiment One. The 'Resampling' column counts how many resampling steps, out of a thousand, were performed by each algorithm.

	RMSE ($ \theta $)	Var ($ \theta $)	RMSE ($ {{\mathbf x}} $)	Var ($ {{\mathbf x}} $)	Resampling
Fine PF	0.066	0.0035	0.34	0.15	226
Coarse PF	0.79	0.0015	2.1	0.16	663
EnKF	0.048	0.0018	0.32	0.12	-
Emu-PF ($ \Gamma=-1 $)	0.13	0.00026	2.4	5.1	483

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Table 2. Summary statistics for twenty repetitions of experiment Two.

	RMSE ($ \theta $)	Var ($ \theta $)	RMSE ($ {{\mathbf x}} $)	Var ($ {{\mathbf x}} $)	Resampling
Fine PF	0.074	0.0043	0.7	0.61	173
Coarse PF	0.49	0.0016	4.9	0.13	312
EnKF	0.065	0.0027	0.78	0.66	-
Emu-PF ($ \Gamma=-1 $)	0.38	0.00085	3.8	6.1	526
Emu-PF (PCA)	0.27	0.00051	3.1	0.58	339

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Table 3. Summary statistics for twenty repetitions of experiment Three.

	RMSE ($ \theta $)	Var ($ \theta $)	RMSE ($ {{\mathbf x}} $)	Var ($ {{\mathbf x}} $)	Resampling
Fine PF	0.062	0.0032	0.28	0.13	243
Coarse PF	1	0.0012	3.4	0.11	739
EnKF	0.045	0.0017	0.25	0.1	-
Emu-PF ($ \Gamma=-1 $)	0.15	0.00034	2.4	5.1	590
Emu-PF (PCA)	0.084	0.00075	1.5	0.085	334

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Table 4. Summary statistics for Experiment Four

	RMSE ($ {{\mathbf x}} $)	Var ($ {{\mathbf x}} $)	Resampling
Fine PF	0.47	0.15	1706
Coarse PF	5.1	0.16	9917
EnKF	1	0.096	-
Emu-PF (Localized)	0.83	0.31	3566

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Table 5. Summary statistics for twenty repetitions of experiment Five.

	RMSE ($ \theta $)	Var ($ \theta $)	RMSE ($ {{\mathbf x}} $)	Var ($ {{\mathbf x}} $)	Resampling
Fine OP-PF	1.2	0.0075	1.9	3.3	226
Coarse OP-PF	1.2	0.004	2.0	2.8	205
EnKF	1.1	0.00042	1.5	1.8	-
Emu-PF ($ \Gamma=-1 $)	0.5	0.0017	2.6	3.5	243
Emu-PF ($ \Gamma=+2 $)	0.75	0.0035	2.0	3.7	232
Emu-PF (PCA)	1.1	0.061	2.0	2.8	238

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