A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling
Data assimilation methods are an invaluable tool for operational ocean models. These methods are often based on a variational approach and require the knowledge of the spatial covariances of the background errors (differences between the numerical model and the true values) and the observation error...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-12-01
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| Series: | Journal of Marine Science and Engineering |
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| Online Access: | https://www.mdpi.com/2077-1312/9/12/1461 |
| _version_ | 1797503247896805376 |
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| author | Jose M. Gonzalez-Ondina Lewis Sampson Georgy I. Shapiro |
| author_facet | Jose M. Gonzalez-Ondina Lewis Sampson Georgy I. Shapiro |
| author_sort | Jose M. Gonzalez-Ondina |
| collection | DOAJ |
| description | Data assimilation methods are an invaluable tool for operational ocean models. These methods are often based on a variational approach and require the knowledge of the spatial covariances of the background errors (differences between the numerical model and the true values) and the observation errors (differences between true and measured values). Since the true values are never known in practice, the error covariance matrices containing values of the covariance functions at different locations, are estimated approximately. Several methods have been devised to compute these matrices, one of the most widely used is the one developed by Hollingsworth and Lönnberg (H-L). This method requires to bin (combine) the data points separated by similar distances, compute covariances in each bin and then to find a best fit covariance function. While being a helpful tool, the H-L method has its limitations. We have developed a new mathematical method for computing the background and observation error covariance functions and therefore the error covariance matrices. The method uses functional analysis which allows to overcome some shortcomings of the H-L method, for example, the assumption of statistical isotropy. It also eliminates the intermediate steps used in the H-L method such as binning the innovations (differences between observations and the model), and the computation of innovation covariances for each bin, before the best-fit curve can be found. We show that the new method works in situations where the standard H-L method experiences difficulties, especially when observations are scarce. It gives a better estimate than the H-L in a synthetic idealised case where the true covariance function is known. We also demonstrate that in many cases the new method allows to use the separable convolution mathematical algorithm to increase the computational speed significantly, up to an order of magnitude. The Projection Method (PROM) also allows computing 2D and 3D covariance functions in addition to the standard 1D case. |
| first_indexed | 2024-03-10T03:47:50Z |
| format | Article |
| id | doaj.art-632fcdf4c78f441a967cbad4c129fa79 |
| institution | Directory Open Access Journal |
| issn | 2077-1312 |
| language | English |
| last_indexed | 2024-03-10T03:47:50Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Journal of Marine Science and Engineering |
| spelling | doaj.art-632fcdf4c78f441a967cbad4c129fa792023-11-23T09:04:05ZengMDPI AGJournal of Marine Science and Engineering2077-13122021-12-01912146110.3390/jmse9121461A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean ModellingJose M. Gonzalez-Ondina0Lewis Sampson1Georgy I. Shapiro2University of Plymouth Enterprise Ltd. (UoPEL), Drake Circus, Plymouth PL4 8AA, UKMet Office, FitzRoy Road, Exeter EX1 3PB, UKSchool of Biological and Marine Sciences, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UKData assimilation methods are an invaluable tool for operational ocean models. These methods are often based on a variational approach and require the knowledge of the spatial covariances of the background errors (differences between the numerical model and the true values) and the observation errors (differences between true and measured values). Since the true values are never known in practice, the error covariance matrices containing values of the covariance functions at different locations, are estimated approximately. Several methods have been devised to compute these matrices, one of the most widely used is the one developed by Hollingsworth and Lönnberg (H-L). This method requires to bin (combine) the data points separated by similar distances, compute covariances in each bin and then to find a best fit covariance function. While being a helpful tool, the H-L method has its limitations. We have developed a new mathematical method for computing the background and observation error covariance functions and therefore the error covariance matrices. The method uses functional analysis which allows to overcome some shortcomings of the H-L method, for example, the assumption of statistical isotropy. It also eliminates the intermediate steps used in the H-L method such as binning the innovations (differences between observations and the model), and the computation of innovation covariances for each bin, before the best-fit curve can be found. We show that the new method works in situations where the standard H-L method experiences difficulties, especially when observations are scarce. It gives a better estimate than the H-L in a synthetic idealised case where the true covariance function is known. We also demonstrate that in many cases the new method allows to use the separable convolution mathematical algorithm to increase the computational speed significantly, up to an order of magnitude. The Projection Method (PROM) also allows computing 2D and 3D covariance functions in addition to the standard 1D case.https://www.mdpi.com/2077-1312/9/12/1461data assimilationvariational methodsanalysis of innovationsocean modellingoperational forecast |
| spellingShingle | Jose M. Gonzalez-Ondina Lewis Sampson Georgy I. Shapiro A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling Journal of Marine Science and Engineering data assimilation variational methods analysis of innovations ocean modelling operational forecast |
| title | A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling |
| title_full | A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling |
| title_fullStr | A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling |
| title_full_unstemmed | A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling |
| title_short | A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling |
| title_sort | projection method for the estimation of error covariance matrices for variational data assimilation in ocean modelling |
| topic | data assimilation variational methods analysis of innovations ocean modelling operational forecast |
| url | https://www.mdpi.com/2077-1312/9/12/1461 |
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