*O. Talagrand*

- Published in print:
- 2014
- Published Online:
- March 2015
- ISBN:
- 9780198723844
- eISBN:
- 9780191791185
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198723844.003.0001
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

In this chapter, four-dimensional variational assimilation (4D-VAR) is described in the context of statistical linear estimation, in which it defines the best linear unbiased estimate (BLUE) of the ...
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In this chapter, four-dimensional variational assimilation (4D-VAR) is described in the context of statistical linear estimation, in which it defines the best linear unbiased estimate (BLUE) of the state of the observed system from the available data. It consists in minimizing a scalar objective function that measures the quadratic difference between the estimated state and the data, weighted by the inverse covariance matrix of the data errors. 4D-VAR can be extended heuristically to the case of nonlinear models or observation operators. It is made possible in practice through the use of the adjoint equations, which allow explicit computation of the gradient of the objective function at a non-prohibitive cost. 4D-VAR is used operationally in a number of major meteorological centres, where it has brought significant improvement in the quality of the forecasts. 4D-VAR, together with the ensemble Kalman filter, is one of the two most powerful assimilation methods currently available.Less

In this chapter, four-dimensional variational assimilation (4D-VAR) is described in the context of statistical linear estimation, in which it defines the best linear unbiased estimate (BLUE) of the state of the observed system from the available data. It consists in minimizing a scalar objective function that measures the quadratic difference between the estimated state and the data, weighted by the inverse covariance matrix of the data errors. 4D-VAR can be extended heuristically to the case of nonlinear models or observation operators. It is made possible in practice through the use of the adjoint equations, which allow explicit computation of the gradient of the objective function at a non-prohibitive cost. 4D-VAR is used operationally in a number of major meteorological centres, where it has brought significant improvement in the quality of the forecasts. 4D-VAR, together with the ensemble Kalman filter, is one of the two most powerful assimilation methods currently available.

*C. Snyder*

- Published in print:
- 2014
- Published Online:
- March 2015
- ISBN:
- 9780198723844
- eISBN:
- 9780191791185
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198723844.003.0003
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

This chapter introduces The Kalman filter, which implements Bayesian data assimilation for linear, Gaussian systems. Its update equations can also be derived as the best linear unbiased estimator ...
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This chapter introduces The Kalman filter, which implements Bayesian data assimilation for linear, Gaussian systems. Its update equations can also be derived as the best linear unbiased estimator (BLUE) and its covariance. Some of the Kalman filter’s detailed properties are reviewed here: linear transformations of the state and observations, extending the state vector to include observed variables, and temporal correlation in the model or observation errors. The Kalman filter can be applied to nonlinear and non-Gaussian systems via either the extended Kalman filter or the BLUE, although both approaches are clearly sub-optimal. The ensemble Kalman filter (EnKF) employs sample covariances from an ensemble of forecasts at each update time and allows practical implementation of an approximate Kalman filter. The EnKF is consistent with a Monte- Carlo implementation of the BLUE. Many of the EnKF’s properties, including basic effects of sampling error, can be understood in the context of Kalman-filter theory.Less

This chapter introduces The Kalman filter, which implements Bayesian data assimilation for linear, Gaussian systems. Its update equations can also be derived as the best linear unbiased estimator (BLUE) and its covariance. Some of the Kalman filter’s detailed properties are reviewed here: linear transformations of the state and observations, extending the state vector to include observed variables, and temporal correlation in the model or observation errors. The Kalman filter can be applied to nonlinear and non-Gaussian systems via either the extended Kalman filter or the BLUE, although both approaches are clearly sub-optimal. The ensemble Kalman filter (EnKF) employs sample covariances from an ensemble of forecasts at each update time and allows practical implementation of an approximate Kalman filter. The EnKF is consistent with a Monte- Carlo implementation of the BLUE. Many of the EnKF’s properties, including basic effects of sampling error, can be understood in the context of Kalman-filter theory.

*M. Bocquet, L. Wu, F. Chevallier, and M. R. Kookhan*

- Published in print:
- 2014
- Published Online:
- March 2015
- ISBN:
- 9780198723844
- eISBN:
- 9780191791185
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198723844.003.0018
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

This chapter discusses several approaches to developing original theoretical approaches to assimilate information at a given scale and consistently propagate it to other scales. This is motivated by ...
More

This chapter discusses several approaches to developing original theoretical approaches to assimilate information at a given scale and consistently propagate it to other scales. This is motivated by the fact that models and observations both provide information at various spatial and temporal scales, but their respective scales do not necessarily match. This chapter discusses several approaches to this problem, such as the construction of a multiscale best linear unbiased estimator (BLUE). This approach is applied to the design of efficient adaptive grids in control space with minimal aggregation errors. Examples are chosen in the field of atmospheric constituents such as the inverse modelling of mesoscale CO2 fluxes and the observability of a nuclear test from the United Nations International Monitoring System.Less

This chapter discusses several approaches to developing original theoretical approaches to assimilate information at a given scale and consistently propagate it to other scales. This is motivated by the fact that models and observations both provide information at various spatial and temporal scales, but their respective scales do not necessarily match. This chapter discusses several approaches to this problem, such as the construction of a multiscale best linear unbiased estimator (BLUE). This approach is applied to the design of efficient adaptive grids in control space with minimal aggregation errors. Examples are chosen in the field of atmospheric constituents such as the inverse modelling of mesoscale CO2 fluxes and the observability of a nuclear test from the United Nations International Monitoring System.