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ABSTRACT
This paper presents new approximate methods to provide error fields for the spatial analysis tool Data Interpolating Variational Analysis (DIVA). The first method shows how to replace the costly analysis of a large number of covariance functions with a single analysis for quick error computations. Then another method is presented where the error is only calculated in a small number of locations, and from there the spatial error field itself is interpolated by the analysis tool. The efficiency of the methods is illustrated on simple schematic test cases and a real application in the Mediterranean Sea. These examples show that with these methods, one has the possibility for quick masking of regions void of sufficient data and the production of ''exact'' error fields at reasonable cost. The error-calculation methods can also be generalized for use with other analysis methods such as three-dimensional variational data assimilation (3DVAR) and are therefore potentially interesting for other implementations.
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1. Introduction
Spatial analysis of observations, also called gridding, is a common task in oceanography and meteorology, and a series of methods and implementations exists and is widely used. Here Nd data points of values di, i 5 1, . . . , Nd at location (xi, yi) are generally distributed unevenly in space. Furthermore, the values of di are affected by observational errors, including representativity errors. From this dataset an analysis on a regular grid is often desired. It has been quickly recognized that it would be natural to define the best analysis as the one that has the lowest expected error. This definition has led to kriging and optimal interpolation (OI) methods (e.g.,Gandin 1965;Delhomme 1978;Bretherton et al. 1976) and to the Kalman-Bucy filter and data assimilation with adjoint models in the context of forecast models (e.g., Lorenc 1986).
These methods assume that statistics on observational errors and the spatial covariance of the field to be analyzed are available to infer the ''best'' analysis field. As these methods aim at minimizing the analysis error, it is not a surprise that they also provide the theoretical a posteriori error field for the analysis. The practical implementation of these methods can lead to very different performances, also when it is necessary to calculate the error...