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

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

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.

Citation: Samira Amraoui, Didier Auroux, Jacques Blum, Emmanuel Cosme. Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022058
References:
[1]

C. Afri, V. Andrieu, L. Bako and P. Dufour, State and parameter estimation: A nonlinear Luenberger observer approach, IEEE Trans. Automat. Control, 62 (2017), 973–980. https://hal.archives-ouvertes.fr/hal-01232747. doi: 10.1109/TAC.2016.2566804.

[2]

S. Amraoui, Data Assimilation for External Geophysics: The Back-and-Forth Nudging Method, PhD thesis, University of Nice Sophia Antipolis, France, 2019.

[3]

A. Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part Ⅰ, Journal of Computational Physics, 1 (1966), 119–143. http://www.sciencedirect.com/science/article/pii/0021999166900155.

[4]

R. Asselin, Frequency filter for time integrations, Monthly Weather Review, 100 (1972), 487-490.  doi: 10.1175/1520-0493(1972)100<0487:FFFTI>2.3.CO;2.

[5]

D. Auroux, P. Bansart and J. Blum, An easy-to-implement and efficient data assimilation method for the identification of the initial condition: The Back and Forth Nudging (BFN) algorithm, in Proc. Int. Conf. Inverse Problems in Engineering, vol. 135, J. Phys.: Conf. Ser., 2008. doi: 10.1088/1742-6596/135/1/012011.

[6]

D. Auroux and J. Blum, Back and forth nudging algorithm for data assimilation problems, C. R. Math. Acad. Sci. Paris, 340 (2005), 873-878.  doi: 10.1016/j.crma.2005.05.006.

[7]

D. Auroux and J. Blum, A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm, Nonlinear Processes in Geophysics, 15 (2008), 305–319. https://hal.archives-ouvertes.fr/hal-00331117/document. doi: 10.5194/npg-15-305-2008.

[8]

D. AurouxJ. Blum and M. Nodet, Diffusive back and forth nudging algorithm for data assimilation, C. R. Math. Acad. Sci. Paris, 349 (2011), 849-854.  doi: 10.1016/j.crma.2011.07.004.

[9]

D. Auroux and M. Nodet, The back and forth nudging algorithm for data assimilation problems: Theoretical results on transport equations, ESAIM Control Optim. Calc. Var., 18 (2012), 318-342.  doi: 10.1051/cocv/2011004.

[10]

P. Bernard and V. Andrieu, Luenberger observers for non autonomous nonlinear systems, IEEE Trans. Automat. Control, 64 (2019), 270-281.  doi: 10.1109/TAC.2018.2872202.

[11]

N. Bof, R. Carli and L. Schenato, Lyapunov theory for discrete time systems, 2018.

[12] B. Bolin, The atmosphere and the sea in motion: Scientific contributions to the rossby memorial volume, Rockefeller Univ. Press, 1959. 
[13]

A.-C. Boulanger, P. Moireau, B. Perthame and J. Sainte-Marie, Data assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description, Commun. Math. Sci., 13 (2015), 587 – 622. https://hal.archives-ouvertes.fr/hal-00924559. doi: 10.4310/CMS.2015.v13.n3.a1.

[14]

L. Brivadis, V. Andrieu and U. Serres, Luenberger observers for discrete-time nonlinear systems, in 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, 2019, 3435–3440. https://hal.archives-ouvertes.fr/hal-02467958. doi: 10.1109/CDC40024.2019.9029220.

[15]

W. CastaingsD. DartusF.-X. Le Dimet and G.-M. Saulnier, Sensitivity analysis and parameter estimation for distributed hydrological modeling: Potential of variational methods, Hydrology and Earth System Sciences, 13 (2009), 503-517.  doi: 10.5194/hess-13-503-2009.

[16]

D. B. CheltonR. A. deSzoekeM. G. SchlaxK. El Naggar and N. Siwertz, Geographical variability of the first baroclinic rossby radius of deformation, Journal of Physical Oceanography, 28 (1998), 433-460.  doi: 10.1175/1520-0485(1998)028<0433:GVOTFB>2.0.CO;2.

[17]

A. Donovan, M. Mirrahimi and P. Rouchon, Back and forth nudging for quantum state reconstruction, in 2010 4th International Symposium on Communications, Control and Signal Processing (ISCCSP), 2010, 1–5. doi: 10.1109/ISCCSP.2010.5463439.

[18]

D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Texts in Applied Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3081-4.

[19]

D. Esteban-Fernandez, Swot Project: Mission Performance and Error Budget, Technical Report JPL D-79084, NASA/JPL, 2017. https://swot.jpl.nasa.gov/system/documents/files/2178_2178_SWOT_D-79084_v10Y_FINAL_REVA__06082017.pdf.

[20]

L.-L. Fu and C. Ubelmann, On the transition from profile altimeter to swath altimeter for observing global ocean surface topography, J. Atmos. Oceanic Technol., 31 (2014), 560-568.  doi: 10.1175/JTECH-D-13-00109.1.

[21]

L. GaultierC. Ubelmann and L.-L. Fu, The challenge of using future SWOT data for oceanic field reconstruction, J. Atmos. Oceanic Technol., 33 (2016), 119-126.  doi: 10.1175/JTECH-D-15-0160.1.

[22] J.-P. Gauthier and I. Kupka, Deterministic Observation Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546648.
[23]

J. E. Hoke and R. A. Anthes, The initialization of numerical models by a dynamic-initialization technique, Monthly Weather Review, 104 (1976), 1551-1556.  doi: 10.1175/1520-0493(1976)104<1551:TIONMB>2.0.CO;2.

[24]

A. H. Jazwinski, Stochastic Processes and Filtering Theory, Courier Corporation, 1970.

[25]

E. Kazantsev, Local Lyapunov exponents of the quasi-geostrophic ocean dynamics, Appl. Math. Comput., 104 (1999), 217-257.  doi: 10.1016/S0096-3003(98)10078-4.

[26]

N. Kazantzis and C. Kravaris, Nonlinear observer design using Lyapunov's auxiliary theorem, Systems Control Lett., 34 (1998), 241-247.  doi: 10.1016/S0167-6911(98)00017-6.

[27]

H. K. Khalil, Nonlinear Systems; 3rd Ed., Prentice-Hall, Upper Saddle River, NJ, 2002.

[28]

F. Le Guillou, S. Metref, E. Cosme, C. Ubelmann, M. Ballarotta, J. Verron and J. Le Sommer, Mapping altimetry in the forthcoming SWOT era by back-and-forth nudging a one-layer quasi-geostrophic model, 2020, https://hal.archives-ouvertes.fr/hal-03084218, Preprint.

[29]

L. LeiD. R. Stauffer and A. Deng, A hybrid nudging-ensemble kalman filter approach to data assimilation in wrf/dart, Quarterly Journal of the Royal Meteorological Society, 138 (2012), 2066-2078.  doi: 10.1002/qj.1939.

[30]

A. C. LorencN. E. BowlerA. M. ClaytonS. R. Pring and D. Fairbairn, Comparison of hybrid-4denvar and hybrid-4dvar data assimilation methods for global NWP, Monthly Weather Review, 143 (2014), 212-229.  doi: 10.1175/MWR-D-14-00195.1.

[31]

D. Luenberger, Observers for multivariable systems, IEEE Transactions on Automatic Control, 11 (1966), 190-197.  doi: 10.1109/TAC.1966.1098323.

[32]

J. MarshallA. AdcroftC. HillL. Perelman and C. Heisey, A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers, J. of Geophys. Res., 102 (1997), 5753-5766. 

[33]

R. Morrow, L.-L. Fu, F. Ardhuin, M. Benkiran, B. Chapron, E. Cosme, F. d'Ovidio, J. T. Farrar, S. T. Gille, G. Lapeyre, P.-Y. Le Traon, A. Pascual, A. Ponte, B. Qiu, N. Rascle, C. Ubelmann, J. Wang and E. D. Zaron, Global observations of fine-scale ocean surface topography with the surface water and ocean topography (SWOT) mission, Frontiers in Marine Science, 6. doi: 10.3389/fmars.2019.00232.

[34]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1992.

[35]

A. J. Robert, The integration of a low order spectral form of the primitive meteorological equations, Journal of the Meteorological Society of Japan. Ser. Ⅱ, 44 (1966), 237-245. 

[36]

G. A. Ruggiero, Y. Ourmières, E. Cosme, J. Blum, D. Auroux and J. Verron, Data assimilation experiments using diffusive back-and-forth nudging for the NEMO ocean model, Nonlinear Processes in Geophysics, 22 (2015), 233–248. https://hal-amu.archives-ouvertes.fr/hal-01232425/document.

[37]

R. E. SchlesingerL. W. Uccellini and D. R. Johnson, The effects of the Asselin time filter on numerical solutions to the linearized shallow-water wave equations, Monthly Weather Review, 111 (1983), 455-467.  doi: 10.1175/1520-0493(1983)111<0455:TEOTAT>2.0.CO;2.

[38]

D. R. Stauffer and N. L. Seaman, Use of four-dimensional data assimilation in a limited-area mesoscale model. Part Ⅰ: Experiments with Synoptic-Scale data, Monthly Weather Review, 118 (1990), 1250-1277.  doi: 10.1175/1520-0493(1990)118<1250:UOFDDA>2.0.CO;2.

[39]

C. Ubelmann, P. Klein and L.-L. Fu, Dynamic interpolation of sea surface height and potential applications for future high-resolution altimetry mapping, Journal of Atmospheric and Oceanic Technology, 32 (2014), 177–184, http://journals.ametsoc.org/doi/abs/10.1175/JTECH-D-14-00152.1.

[40]

B. ZhouG.-B. Cai and G.-R. Duan, Stabilisation of time-varying linear systems via lyapunov differential equations, Internat. J. Control, 86 (2013), 332-347.  doi: 10.1080/00207179.2012.728008.

show all references

References:
[1]

C. Afri, V. Andrieu, L. Bako and P. Dufour, State and parameter estimation: A nonlinear Luenberger observer approach, IEEE Trans. Automat. Control, 62 (2017), 973–980. https://hal.archives-ouvertes.fr/hal-01232747. doi: 10.1109/TAC.2016.2566804.

[2]

S. Amraoui, Data Assimilation for External Geophysics: The Back-and-Forth Nudging Method, PhD thesis, University of Nice Sophia Antipolis, France, 2019.

[3]

A. Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part Ⅰ, Journal of Computational Physics, 1 (1966), 119–143. http://www.sciencedirect.com/science/article/pii/0021999166900155.

[4]

R. Asselin, Frequency filter for time integrations, Monthly Weather Review, 100 (1972), 487-490.  doi: 10.1175/1520-0493(1972)100<0487:FFFTI>2.3.CO;2.

[5]

D. Auroux, P. Bansart and J. Blum, An easy-to-implement and efficient data assimilation method for the identification of the initial condition: The Back and Forth Nudging (BFN) algorithm, in Proc. Int. Conf. Inverse Problems in Engineering, vol. 135, J. Phys.: Conf. Ser., 2008. doi: 10.1088/1742-6596/135/1/012011.

[6]

D. Auroux and J. Blum, Back and forth nudging algorithm for data assimilation problems, C. R. Math. Acad. Sci. Paris, 340 (2005), 873-878.  doi: 10.1016/j.crma.2005.05.006.

[7]

D. Auroux and J. Blum, A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm, Nonlinear Processes in Geophysics, 15 (2008), 305–319. https://hal.archives-ouvertes.fr/hal-00331117/document. doi: 10.5194/npg-15-305-2008.

[8]

D. AurouxJ. Blum and M. Nodet, Diffusive back and forth nudging algorithm for data assimilation, C. R. Math. Acad. Sci. Paris, 349 (2011), 849-854.  doi: 10.1016/j.crma.2011.07.004.

[9]

D. Auroux and M. Nodet, The back and forth nudging algorithm for data assimilation problems: Theoretical results on transport equations, ESAIM Control Optim. Calc. Var., 18 (2012), 318-342.  doi: 10.1051/cocv/2011004.

[10]

P. Bernard and V. Andrieu, Luenberger observers for non autonomous nonlinear systems, IEEE Trans. Automat. Control, 64 (2019), 270-281.  doi: 10.1109/TAC.2018.2872202.

[11]

N. Bof, R. Carli and L. Schenato, Lyapunov theory for discrete time systems, 2018.

[12] B. Bolin, The atmosphere and the sea in motion: Scientific contributions to the rossby memorial volume, Rockefeller Univ. Press, 1959. 
[13]

A.-C. Boulanger, P. Moireau, B. Perthame and J. Sainte-Marie, Data assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description, Commun. Math. Sci., 13 (2015), 587 – 622. https://hal.archives-ouvertes.fr/hal-00924559. doi: 10.4310/CMS.2015.v13.n3.a1.

[14]

L. Brivadis, V. Andrieu and U. Serres, Luenberger observers for discrete-time nonlinear systems, in 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, 2019, 3435–3440. https://hal.archives-ouvertes.fr/hal-02467958. doi: 10.1109/CDC40024.2019.9029220.

[15]

W. CastaingsD. DartusF.-X. Le Dimet and G.-M. Saulnier, Sensitivity analysis and parameter estimation for distributed hydrological modeling: Potential of variational methods, Hydrology and Earth System Sciences, 13 (2009), 503-517.  doi: 10.5194/hess-13-503-2009.

[16]

D. B. CheltonR. A. deSzoekeM. G. SchlaxK. El Naggar and N. Siwertz, Geographical variability of the first baroclinic rossby radius of deformation, Journal of Physical Oceanography, 28 (1998), 433-460.  doi: 10.1175/1520-0485(1998)028<0433:GVOTFB>2.0.CO;2.

[17]

A. Donovan, M. Mirrahimi and P. Rouchon, Back and forth nudging for quantum state reconstruction, in 2010 4th International Symposium on Communications, Control and Signal Processing (ISCCSP), 2010, 1–5. doi: 10.1109/ISCCSP.2010.5463439.

[18]

D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Texts in Applied Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3081-4.

[19]

D. Esteban-Fernandez, Swot Project: Mission Performance and Error Budget, Technical Report JPL D-79084, NASA/JPL, 2017. https://swot.jpl.nasa.gov/system/documents/files/2178_2178_SWOT_D-79084_v10Y_FINAL_REVA__06082017.pdf.

[20]

L.-L. Fu and C. Ubelmann, On the transition from profile altimeter to swath altimeter for observing global ocean surface topography, J. Atmos. Oceanic Technol., 31 (2014), 560-568.  doi: 10.1175/JTECH-D-13-00109.1.

[21]

L. GaultierC. Ubelmann and L.-L. Fu, The challenge of using future SWOT data for oceanic field reconstruction, J. Atmos. Oceanic Technol., 33 (2016), 119-126.  doi: 10.1175/JTECH-D-15-0160.1.

[22] J.-P. Gauthier and I. Kupka, Deterministic Observation Theory and Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546648.
[23]

J. E. Hoke and R. A. Anthes, The initialization of numerical models by a dynamic-initialization technique, Monthly Weather Review, 104 (1976), 1551-1556.  doi: 10.1175/1520-0493(1976)104<1551:TIONMB>2.0.CO;2.

[24]

A. H. Jazwinski, Stochastic Processes and Filtering Theory, Courier Corporation, 1970.

[25]

E. Kazantsev, Local Lyapunov exponents of the quasi-geostrophic ocean dynamics, Appl. Math. Comput., 104 (1999), 217-257.  doi: 10.1016/S0096-3003(98)10078-4.

[26]

N. Kazantzis and C. Kravaris, Nonlinear observer design using Lyapunov's auxiliary theorem, Systems Control Lett., 34 (1998), 241-247.  doi: 10.1016/S0167-6911(98)00017-6.

[27]

H. K. Khalil, Nonlinear Systems; 3rd Ed., Prentice-Hall, Upper Saddle River, NJ, 2002.

[28]

F. Le Guillou, S. Metref, E. Cosme, C. Ubelmann, M. Ballarotta, J. Verron and J. Le Sommer, Mapping altimetry in the forthcoming SWOT era by back-and-forth nudging a one-layer quasi-geostrophic model, 2020, https://hal.archives-ouvertes.fr/hal-03084218, Preprint.

[29]

L. LeiD. R. Stauffer and A. Deng, A hybrid nudging-ensemble kalman filter approach to data assimilation in wrf/dart, Quarterly Journal of the Royal Meteorological Society, 138 (2012), 2066-2078.  doi: 10.1002/qj.1939.

[30]

A. C. LorencN. E. BowlerA. M. ClaytonS. R. Pring and D. Fairbairn, Comparison of hybrid-4denvar and hybrid-4dvar data assimilation methods for global NWP, Monthly Weather Review, 143 (2014), 212-229.  doi: 10.1175/MWR-D-14-00195.1.

[31]

D. Luenberger, Observers for multivariable systems, IEEE Transactions on Automatic Control, 11 (1966), 190-197.  doi: 10.1109/TAC.1966.1098323.

[32]

J. MarshallA. AdcroftC. HillL. Perelman and C. Heisey, A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers, J. of Geophys. Res., 102 (1997), 5753-5766. 

[33]

R. Morrow, L.-L. Fu, F. Ardhuin, M. Benkiran, B. Chapron, E. Cosme, F. d'Ovidio, J. T. Farrar, S. T. Gille, G. Lapeyre, P.-Y. Le Traon, A. Pascual, A. Ponte, B. Qiu, N. Rascle, C. Ubelmann, J. Wang and E. D. Zaron, Global observations of fine-scale ocean surface topography with the surface water and ocean topography (SWOT) mission, Frontiers in Marine Science, 6. doi: 10.3389/fmars.2019.00232.

[34]

J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1992.

[35]

A. J. Robert, The integration of a low order spectral form of the primitive meteorological equations, Journal of the Meteorological Society of Japan. Ser. Ⅱ, 44 (1966), 237-245. 

[36]

G. A. Ruggiero, Y. Ourmières, E. Cosme, J. Blum, D. Auroux and J. Verron, Data assimilation experiments using diffusive back-and-forth nudging for the NEMO ocean model, Nonlinear Processes in Geophysics, 22 (2015), 233–248. https://hal-amu.archives-ouvertes.fr/hal-01232425/document.

[37]

R. E. SchlesingerL. W. Uccellini and D. R. Johnson, The effects of the Asselin time filter on numerical solutions to the linearized shallow-water wave equations, Monthly Weather Review, 111 (1983), 455-467.  doi: 10.1175/1520-0493(1983)111<0455:TEOTAT>2.0.CO;2.

[38]

D. R. Stauffer and N. L. Seaman, Use of four-dimensional data assimilation in a limited-area mesoscale model. Part Ⅰ: Experiments with Synoptic-Scale data, Monthly Weather Review, 118 (1990), 1250-1277.  doi: 10.1175/1520-0493(1990)118<1250:UOFDDA>2.0.CO;2.

[39]

C. Ubelmann, P. Klein and L.-L. Fu, Dynamic interpolation of sea surface height and potential applications for future high-resolution altimetry mapping, Journal of Atmospheric and Oceanic Technology, 32 (2014), 177–184, http://journals.ametsoc.org/doi/abs/10.1175/JTECH-D-14-00152.1.

[40]

B. ZhouG.-B. Cai and G.-R. Duan, Stabilisation of time-varying linear systems via lyapunov differential equations, Internat. J. Control, 86 (2013), 332-347.  doi: 10.1080/00207179.2012.728008.

Figure 2.  Comparison of initial time sea-surface height
Figure 1.  SWOT satellite SSH data coverage after 5 days, 10 days or 21 days
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)
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
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)
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)
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)
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
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%
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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