Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations

Samira Amraoui; Didier Auroux; Jacques Blum; Emmanuel Cosme

doi:10.3934/dcdss.2022058

Article Contents

2023, Volume 16, Issue 2: 197-219. Doi: 10.3934/dcdss.2022058

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Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations

1.
Université Côte d'Azur, CNRS, LJAD, France
2.
Université Grenoble Alpes, CNRS, IRD, G-INP, IGE, France

^* Corresponding author: Didier Auroux
^* Corresponding author: Didier Auroux

This paper is dedicated to the memory of François-Xavier Le Dimet who left us in March 2021. He was an expert in data assimilation and at the origin of the 4D-VAR algorithm, especially in meteorology.

Received: June 2021

Revised: February 2022

Early access: March 2022

Published: February 2023

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Abstract

In data assimilation for geophysical problems, the increasing amount of satellite data to analyze makes it more and more challenging to guarantee near real time forecasting. Thus, low time and memory consuming data assimilation methods become very attractive. The back-and-forth nudging (BFN) method is a non-classical data assimilation method that can be seen as a deterministic and smoothing version of the Kalman filter. From a practical point of view, the BFN method is very valuable for its simplicity of implementation (no optimization, no differentiation, ...) and its rapidity of convergence. Under observability conditions, we prove the mathematical convergence of BFN at deep layers for a multi-layer quasi-geostrophic (MQG) ocean circulation model using an infinite dimensional variant of LaSalle's invariance principle. We also extend the BFN to the problem of joint state-parameter identification. The numerical experiments, performed on 120km large swath sea surface height (SSH) simulated data of the Surface Water Ocean Topography (SWOT) satellite, show the high robustness of the algorithm to uncertainties and the few iterations needed to reach convergence, whereas some problems remain due to non-reversibility properties in time. We also give a strategy to improve geophysical model accuracy, considering the large number of uncertain parameters inherent to models and their impacts on state estimation performance. We propose here a joint state-parameter estimation, tested on the baroclinic wavenumber as an unobserved parameter.

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Mathematics Subject Classification: Primary: 86A05, 86A22, 93D05; Secondary: 35Q86, 35R30, 37N10, 65M32, 93B53.

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Figure 2. Comparison of initial time sea-surface height

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Figure 1. SWOT satellite SSH data coverage after 5 days, 10 days or 21 days

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Figure 3. Lyapunov function versus time, during 10 back-and-forth successive iterations (forward integrations in blue, backward integrations in red), while assimilating time-sampled and space-complete data: one observation every 10 time steps (left) and every 150 time steps (right)

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Figure 4. Lyapunov function during 10 back-and-forth successive iterations (forward integration in blue and backward integration in red) while assimilating different sets of data

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Figure 5. Sea-Surface Height (exact or assimilated by BFN) in the spatial region $ \Omega $ at different times : t = 0 (first column) and t = 21 days (second column)

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Figure 6. Sea-Surface Height spatial error between assimilated and exact maps in the spatial region $ \Omega $ at different times : t = 0 (first column) and t = 21 days (second column)

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Figure 7. Evolution of the Lyapunov functions $ V $ (a-b-c) during the assimilation and forecast windows, and of $ W $ (d) during the assimilation window: (a) exact speed parameter $ c = 2.5 $ used during assimilation and forecast; (b) wrong speed parameter $ c = 1.0 $ used during assimilation and forecast; (c) initially wrong speed parameter $ c = 1.0 $ that is corrected during data assimilation; (d) decrease of $ W $ during the assimilation window in the third case (c)

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Figure 8. Forecasted SSH at 84 days with different scenarios during the full process (data assimilation of 21 days then forecast of $ 3 \times 21 $ days): (a) exact speed parameter $ c = 2.5 $; (b) wrong speed parameter $ c = 1.0 $; (c) initially wrong speed parameter $ c = 1.0 $ corrected with data assimilation

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Table 1. Percentage of error decay at initial time ($ t_0 = 0 $ day), at final time ($ t_f = 21 $ days) and in average over time on $ [t_0,t_f] $ after 10 iterations of BFN for different types of data

Data type	Error decay at $ t_0 $	Error decay at $ t_f $	Error decay in average
Complete data	97.39%	99.99%	99.41%
Time-sampled complete data	88.58%	99.63 %	96.71%
SWOT-like perfect data	77.77%	95.52 %	91.20%
SWOT-like noisy data	77.29%	95.34 %	90.96%

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