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Olivier Pannekoucke

Abstract

This article discusses several models for background error correlation matrices using the wavelet diagonal assumption and the diffusion operator. The most general properties of filtering local correlation functions, with wavelet formulations, are recalled. Two spherical wavelet transforms based on Legendre spectrum and a gridpoint spherical wavelet transform are compared. The latter belongs to the class of second-generation wavelets. In addition, a nonseparable formulation that merges the wavelets and the diffusion operator model is formally proposed. This hybrid formulation is illustrated in a simple two-dimensional framework. These three formulations are tested in a toy experiment on the sphere: a large ensemble of perturbed forecasts is used to simulate a true background error ensemble, which gives a reference. This ensemble is then applied to compute the required parameters for each model. A randomization method is utilized in order to diagnose these different models. In particular, their ability to represent the geographical variations of the local correlation functions is studied by diagnosis of the local length scale. The results from these experiments show that the spectrally based wavelet formulation filters the geographical variations of the local correlation length scale but it is less able to represent the anisotropy. The gridpoint-based wavelet formulation is also able to represent some parts of the geographical variations but it appears that the correlation functions are dependent on the grid. Finally, the formulation based on the diffusion represents quite well the local length scale.

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Olivier Pannekoucke
and
Laurent Vezard

Abstract

In this note, a stochastic integration scheme is proposed as an alternative to a deterministic integration scheme, usually employed for the diffusion operator in data assimilation. The stochastic integration scheme is no more than a simple interpolation of the initial condition in lieu of the deterministic integration. Furthermore, this also presents a potential in high performance computing. For the classic preconditioned minimizing problem, the stochastic integration is employed to implement the square root of the background error covariance matrix, while its adjoint is obtained from the adjoint code of the square root code. In a first part, the stochastic integration method and its weak convergence are detailed. Then the practical use of this approach in data assimilation is described. It is illustrated in a 1D test bed, where it is shown to run smoothly for background error covariance modeling, with nearest-neighbor interpolations, and O(100) particles.

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Sébastien Massart
,
Benjamin Pajot
,
Andrea Piacentini
, and
Olivier Pannekoucke

Abstract

Three-dimensional variational data assimilation (3D-Var) with the first guess at appropriate time (FGAT) appears to be an attractive compromise between accuracy and overall computing time. It is computationally cheaper than four-dimensional (4D)-Var as the increment is not propagated back and forth in time by a model, yet the comparison between the model and the observations is still computed at the right observation time. An interesting feature of the 4D-Var is the iterative process known as the outer loop. This outer-loop approach can also be used in conjunction with 3D-FGAT. But it requires the application of the 3D-FGAT analysis increment at the beginning of the assimilation window.

The pros and cons of using this unusual 3D-FGAT variant are illustrated in this paper on two applications focused on the transport, one of the main phenomena governing the atmospheric evolution. The first one is the one-dimensional advection of a passive tracer. By three representative situations, it shows the benefits of the outer loop, except for practical situations driven by very rapid dynamics such as a zonal wind of 50 m s−1 on the earth’s great circle, when the assimilation window has a size of 3 h. The second application is the 3D-FGAT assimilation of true ozone measurements into a chemical–transport model. It confirms the previous results, showing that the 3D-FGAT analysis with the outer loop produces an overestimation of the ozone increment in regions where the wind speed is high compared to the time length of the assimilation window.

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