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doi: 10.3934/dcdss.2022059
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Physics informed model error for data assimilation

1. 

Univ Bretagne - Sud, CNRS UMR 6205, LMBA, F-56000 Vannes, France

2. 

Univ Brest, CNRS UMR 6205, Laboratoire de Mathematiques de Bretagne Atlantique, France

*Corresponding author: Jules Guillot

Received  January 2022 Early access March 2022

Data assimilation consists in combining a dynamical model with noisy observations to estimate the latent true state of a system. The dynamical model is generally misspecified and this generates a model error which is usually treated using a random noise. The aim of this paper is to suggest a new treatment for the model error that further takes into account the physics of the system: the physics informed model error. This model error treatment is a noisy stationary solution of the true dynamical model. It is embedded in the ensemble Kalman filter (EnKF), which is a usual method for data assimilation. The proposed strategy is then applied to study the heat diffusion in a bar when the external heat source is unknown. It is compared to usual methods to quantify the model error. The numerical results show that our method is more accurate, in particular when the observations are available at a low temporal resolution.

Citation: Jules Guillot, Emmanuel Frénod, Pierre Ailliot. Physics informed model error for data assimilation. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022059
References:
[1]

P. Ailliot, A. Cuzol, G. Durrieu, H. Flourent, E. Frénod, J. Guillot, J.-P. Lucas and F. Septier, Synthèse des questions mathématiques soulevées par la mise en oeuvre de jumeaux numériques pour le suivi et le pilotage de systèmes dynamiques en entreprises, hal-03167416.

[2]

J. L. Anderson and S. L. Anderson, A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Monthly Weather Review, 127 (1999), 2741-2758.  doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

[3]

J. BernerU. AchatzL. BatteL. BengtssonA. De La CamaraH. M. ChristensenM. ColangeliD. R. ColemanD. Crommelin and S. I. Dolaptchiev, Stochastic parameterization: Toward a new view of weather and climate models, Bulletin of the American Meteorological Society, 98 (2017), 565-588.  doi: 10.1175/BAMS-D-15-00268.1.

[4]

J. Brajard, A. Carrassi, M. Bocquet and L. Bertino, Combining data assimilation and machine learning to infer unresolved scale parametrization, Philos. Trans. Roy. Soc. A, 379 (2021), Paper No. 20200086, 16 pp. doi: 10.1098/rsta.2020.0086.

[5]

G. DesroziersL. BerreB. Chapnik and P. Poli, Diagnosis of observation, background and analysis-error statistics in observation space, Quarterly Journal of the Royal Meteorological Society: A Journal of the Atmospheric Sciences, Applied Meteorology and Physical Oceanography, 131 (2005), 3385-3396.  doi: 10.1256/qj.05.108.

[6]

G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, Journal of Geophysical Research: Oceans, 99 (1994), 10143-10162.  doi: 10.1029/94JC00572.

[7]

A. FarchiP. LaloyauxM. Bonavita and M. Bocquet, Using machine learning to correct model error in data assimilation and forecast applications, Quarterly Journal of the Royal Meteorological Society, 147 (2021), 3067-3084.  doi: 10.1002/qj.4116.

[8]

M. Ghil and P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography, Advances in Geophysics, 33 (1991), 141-266.  doi: 10.1016/S0065-2687(08)60442-2.

[9]

R. LguensatP. TandeoP. AilliotM. Pulido and R. Fablet, The analog data assimilation, Monthly Weather Review, 145 (2017), 4093-4107.  doi: 10.1175/MWR-D-16-0441.1.

[10]

T. N. Palmer, R. Buizza, F. Doblas-Reyes, T. Jung, M. Leutbecher, G. J. Shutts, M. Steinheimer and A. Weisheimer, Stochastic parametrization and model uncertainty, ECMWF Technical Memoranda.

[11]

P. TandeoP. AilliotM. BocquetA. CarrassiT. MiyoshiM. Pulido and Y. Zhen, A review of innovation-based methods to jointly estimate model and observation error covariance matrices in ensemble data assimilation, Monthly Weather Review, 148 (2020), 3973-3994. 

show all references

References:
[1]

P. Ailliot, A. Cuzol, G. Durrieu, H. Flourent, E. Frénod, J. Guillot, J.-P. Lucas and F. Septier, Synthèse des questions mathématiques soulevées par la mise en oeuvre de jumeaux numériques pour le suivi et le pilotage de systèmes dynamiques en entreprises, hal-03167416.

[2]

J. L. Anderson and S. L. Anderson, A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Monthly Weather Review, 127 (1999), 2741-2758.  doi: 10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.

[3]

J. BernerU. AchatzL. BatteL. BengtssonA. De La CamaraH. M. ChristensenM. ColangeliD. R. ColemanD. Crommelin and S. I. Dolaptchiev, Stochastic parameterization: Toward a new view of weather and climate models, Bulletin of the American Meteorological Society, 98 (2017), 565-588.  doi: 10.1175/BAMS-D-15-00268.1.

[4]

J. Brajard, A. Carrassi, M. Bocquet and L. Bertino, Combining data assimilation and machine learning to infer unresolved scale parametrization, Philos. Trans. Roy. Soc. A, 379 (2021), Paper No. 20200086, 16 pp. doi: 10.1098/rsta.2020.0086.

[5]

G. DesroziersL. BerreB. Chapnik and P. Poli, Diagnosis of observation, background and analysis-error statistics in observation space, Quarterly Journal of the Royal Meteorological Society: A Journal of the Atmospheric Sciences, Applied Meteorology and Physical Oceanography, 131 (2005), 3385-3396.  doi: 10.1256/qj.05.108.

[6]

G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, Journal of Geophysical Research: Oceans, 99 (1994), 10143-10162.  doi: 10.1029/94JC00572.

[7]

A. FarchiP. LaloyauxM. Bonavita and M. Bocquet, Using machine learning to correct model error in data assimilation and forecast applications, Quarterly Journal of the Royal Meteorological Society, 147 (2021), 3067-3084.  doi: 10.1002/qj.4116.

[8]

M. Ghil and P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography, Advances in Geophysics, 33 (1991), 141-266.  doi: 10.1016/S0065-2687(08)60442-2.

[9]

R. LguensatP. TandeoP. AilliotM. Pulido and R. Fablet, The analog data assimilation, Monthly Weather Review, 145 (2017), 4093-4107.  doi: 10.1175/MWR-D-16-0441.1.

[10]

T. N. Palmer, R. Buizza, F. Doblas-Reyes, T. Jung, M. Leutbecher, G. J. Shutts, M. Steinheimer and A. Weisheimer, Stochastic parametrization and model uncertainty, ECMWF Technical Memoranda.

[11]

P. TandeoP. AilliotM. BocquetA. CarrassiT. MiyoshiM. Pulido and Y. Zhen, A review of innovation-based methods to jointly estimate model and observation error covariance matrices in ensemble data assimilation, Monthly Weather Review, 148 (2020), 3973-3994. 

Figure 1.  Optimization of $ \sigma_{PIME} $, $ \sigma_{QD} $ and $ \sigma_{QSS} $
Figure 2.  Comparison of the evolution of the heat diffusion for the analysis of each algorithm (the more the color is red, the more the temperature is close to zero)
Figure 3.  Comparison of the values of $ X^f_{4} $ and $ X^a_{4} $ for the different algorithms
Figure 4.  Temporal evolution of the estimated temperature of the middle point for each method
Figure 5.  Temporal evolution of the estimated temperature of the middle point for each method with $ dt = 1.5 $
Figure 6.  Global RMSE of each algorithm according to the value of $ dt $
Figure 7.  Confidence interval for the temporal evolution of the RMSE of PIME
Table 1.  EnKF with the physics informed model error.
$\underline {{Initialization}}$: for $ i = 1, \ldots, N $
generate $ w_{1}^{i} $
$ X_{1}^{a, i} = X_{0}+w_{1}^i $
For $ k \geq 2 $
$\underline{Forecast}$: for $ i = 1, \ldots, N $
generate $ w_{k}^{i} $
$ X_{k}^{f,i} = M[X^{a,i}_{k-1}]+w_{k}^i $
$ X_{k}^{f} = \frac{1}{N} \sum_{i = 1}^{N} X_{k}^{f, i} $
$ P_{k}^{f} = \frac{1}{N-1}\sum_{i = 1}^{N}(X_{k}^{f, i}-X_{k}^{f})(X_{k}^{f, i}-X_{k}^{f})^{T} $
$\underline{Analysis}:$: for $ i = 1, \ldots, N $
generate $ \varepsilon_{k}^{i} \sim \mathcal{N}(0,R) $
$ K_{k} = P_{k}^{f} H^{T}(H P_{k}^{f} H^{T}+R)^{-1} $
$ d^{i}_{k} = Y_{k}+\varepsilon^{i}_{k}-HX^{f,i}_{k} $
$ X^{a,i}_{k} = X^{f,i}_{k}+K_{k}d^{i}_{k} $
$ X_{k}^{a} = \frac{1}{N} \sum_{i = 1}^{N} X_{k}^{a, i} $
$\underline {{Initialization}}$: for $ i = 1, \ldots, N $
generate $ w_{1}^{i} $
$ X_{1}^{a, i} = X_{0}+w_{1}^i $
For $ k \geq 2 $
$\underline{Forecast}$: for $ i = 1, \ldots, N $
generate $ w_{k}^{i} $
$ X_{k}^{f,i} = M[X^{a,i}_{k-1}]+w_{k}^i $
$ X_{k}^{f} = \frac{1}{N} \sum_{i = 1}^{N} X_{k}^{f, i} $
$ P_{k}^{f} = \frac{1}{N-1}\sum_{i = 1}^{N}(X_{k}^{f, i}-X_{k}^{f})(X_{k}^{f, i}-X_{k}^{f})^{T} $
$\underline{Analysis}:$: for $ i = 1, \ldots, N $
generate $ \varepsilon_{k}^{i} \sim \mathcal{N}(0,R) $
$ K_{k} = P_{k}^{f} H^{T}(H P_{k}^{f} H^{T}+R)^{-1} $
$ d^{i}_{k} = Y_{k}+\varepsilon^{i}_{k}-HX^{f,i}_{k} $
$ X^{a,i}_{k} = X^{f,i}_{k}+K_{k}d^{i}_{k} $
$ X_{k}^{a} = \frac{1}{N} \sum_{i = 1}^{N} X_{k}^{a, i} $
Table 2.  Parameters values
Parameter Value
$ n $ 100
$ p $ 50
$ N $ 30
$ R $ $ 0.01I_{n} $
$ K_{final} $ 30
$ \alpha $ 0.05
$ dt $ 1
Parameter Value
$ n $ 100
$ p $ 50
$ N $ 30
$ R $ $ 0.01I_{n} $
$ K_{final} $ 30
$ \alpha $ 0.05
$ dt $ 1
Table 3.  Optimal values for $ \sigma_{PIME} $, $ \sigma_{QD} $ and $ \sigma_{QSS} $
$ \sigma_{PIME} $ 0.016
$ \sigma_{QD} $ 0.001
$ \sigma_{QSS} $ 0.050
$ \sigma_{PIME} $ 0.016
$ \sigma_{QD} $ 0.001
$ \sigma_{QSS} $ 0.050
Table 4.  Global RMSE of each algorithm
Algorithm Global RMSE
PIME 0.017
$ Q $D 0.048
$ Q $SS 0.025
Algorithm Global RMSE
PIME 0.017
$ Q $D 0.048
$ Q $SS 0.025
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