*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 ...
More

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.

*A. C. Lorenc*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198723844.003.0002
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

In this chapter, four-dimensional variational data assimilation (4D-VAR) is discussed in the context of numerical weather prediction (NWP). The analysis step in an NWP data assimilation cycle ...
More

In this chapter, four-dimensional variational data assimilation (4D-VAR) is discussed in the context of numerical weather prediction (NWP). The analysis step in an NWP data assimilation cycle combines observations with a background forecast. Plausible models of error distributions involve transforms and statistics to describe the structure of errors at one time, plus a forecast model constraining the time evolution. They allow a Bayesian derivation of equations for the optimal analysis, by minimizing a 4D-Var penalty function using an adjoint model. Difficulties with the deterministic best fit of a nonlinear NWP model are discussed and a statistical approach to 4D-VAR based on the extended Kalman filter is presented. Advanced extensions to 4D-VAR can allow for nonlinearities and non-Gaussian distributions, arising from the physical limits to humidity, and from the possibility of erroneous observations. Ensembles provide useful information about likely background errors, which can be used in hybrid ensemble–variational data assimilation.Less

In this chapter, four-dimensional variational data assimilation (4D-VAR) is discussed in the context of numerical weather prediction (NWP). The analysis step in an NWP data assimilation cycle combines observations with a background forecast. Plausible models of error distributions involve transforms and statistics to describe the structure of errors at one time, plus a forecast model constraining the time evolution. They allow a Bayesian derivation of equations for the optimal analysis, by minimizing a 4D-Var penalty function using an adjoint model. Difficulties with the deterministic best fit of a nonlinear NWP model are discussed and a statistical approach to 4D-VAR based on the extended Kalman filter is presented. Advanced extensions to 4D-VAR can allow for nonlinearities and non-Gaussian distributions, arising from the physical limits to humidity, and from the possibility of erroneous observations. Ensembles provide useful information about likely background errors, which can be used in hybrid ensemble–variational data assimilation.

*F. Rabier and M. Fisher*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198723844.003.0019
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

This chapter discusses some of the implementation details that are necessary to apply data assimilation in the context of numerical weather prediction (NWP). It is divided into three parts. The first ...
More

This chapter discusses some of the implementation details that are necessary to apply data assimilation in the context of numerical weather prediction (NWP). It is divided into three parts. The first part addresses the processing of observations, which includes the transformation of raw data into a form that can be processed by a data assimilation system, quality control, and data thinning. The second part discusses two important aspects of data assimilation for NWP: (i) filtering of the analysis to remove spurious inertia–gravity waves and (ii) methods to handle nonlinearities and non-Gaussian error statistics. The third part discusses the development of parallel algorithms for four-dimensional variational data assimilation (4D-VAR), in order to better exploit the parallel nature of the computers on which it is run and to maintain its status as an important and viable NWP data assimilation algorithm into the foreseeable future.Less

This chapter discusses some of the implementation details that are necessary to apply data assimilation in the context of numerical weather prediction (NWP). It is divided into three parts. The first part addresses the processing of observations, which includes the transformation of raw data into a form that can be processed by a data assimilation system, quality control, and data thinning. The second part discusses two important aspects of data assimilation for NWP: (i) filtering of the analysis to remove spurious inertia–gravity waves and (ii) methods to handle nonlinearities and non-Gaussian error statistics. The third part discusses the development of parallel algorithms for four-dimensional variational data assimilation (4D-VAR), in order to better exploit the parallel nature of the computers on which it is run and to maintain its status as an important and viable NWP data assimilation algorithm into the foreseeable future.