We study the analysis error using the 3D-Var assimilation scheme. In order to apply the scheme, the background and observation error covariance matrices must be specified. The first is derived from the statistic properties of the true and background signals. They are modeled as scalar isotropic stationary random fields with given second-order autoregressive correlation functions. In order to compute the observation error covariances, we specify an observation operator and generate a set of observations distributed over the sphere. In Ref. 1, the observation points were placed regularly on the circle. In this study, different kinds of more realistic stochastic, non-regular distributions, e.g., random, uniform or pattern-like, are used. The observation values are computed as a spatial integral (i.e., convolution) of the true signal with the instrument resolution function centered at a certain position in space plus an instrumental error. Different resolution functions, e.g., uniform or gaussian, are possible. Having the observation operator and a way in which the observations can be synthesized, the exact observation error-covariance matrix can be computed. It is a function of the observation operator, the instrument resolution function and the spatial distribution of observations. Varying the configuration of these parameters we study their influence on the analysis error. As expected, obtained results are qualitatively similar to those published in Ref. 1. However, the optimal width of the resolution function is generally smaller than the one estimated by Liu and Rabier. This quantitative difference is discussed in detail.
Following the approach of Ref. 1, we perform our study in the spectral domain, where the true, background, and estimated model signals are represented as a truncated series of spherical harmonics. Naturally, the background and model signals has much lower spectral resolution compared to the truth. The spectral representation has the advantage of the diagonal form of the covariance matrices corresponding to the isotropic stationary signals. Moreover, in this space the convolution of the true signal with the instrument resolution function is simply a multiplication of the corresponding spectral coefficients and thus can be computed easily. Also a model representation of the observations can be easily obtained by the transform of the spectral coefficients of the model vector to the values at the observation positions. This means that the observation operator can be represented in a form of the inverse spherical Fourier transformation, for which efficient algorithms exist.
1. Liu Z.-Q., Rabier F, The interaction between model resolution, observation resolution and observation density in data assimilation: A one-dimensional study. Q.J.R. Meteorol. Soc. (2002). 128. pp. 1367-1386.