Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation

Let Φ(x,y,p,t) be a meteorological field of interest, say, height, temperature, a component of the wind field, etc. We suppose that data concerning the field of the form ΦI = LiΦ + εi are where each Li is an arbitrary continuous linear functional and εi is a measurement error. The data Φi may be the result of theory, direct measurements, remote soundings or a combination of these. We develop a new mathematical formalism exploiting the method of Generalized Cross Validation (GCV), and some recently developed optimization results, for analyzing this data. The analyzed field ΦN,m,λ is the solution to the minimization problem: Find Φ in a suitable space of functions to minimize where Functions of d=1, 2 or 3 of the four variables x, y, p, t are also considered. The approach can he used to analyze temperature fields from radiosonde-measured temperatures and satellite radiance measurements simultaneously, to incorporate the geostrophic wind approximation and other information. In a test of the method (for d = 2) simulated 500 mb height data were obtained at discrete points corresponding to the U.S. radiosonde network, by using an analytic representation of a 500 mb wave and superimposing realistic random errors. The analytic representation was recovered on a fine grid with what appear to be impressive results. An explicit representation for the minimizer of Eq. (1) is found, and used as the basis for a direct (as opposed to iterative) numerical algorithm, which is accurate and efficient for N2 somewhat less than the high-speed storage capacity of the computer. The results extend those of Sasaki and others in several directions. In particular, no starting guesses and no preliminary interpolation of the data is required, and it is not necessary to solve a boundary-value problem or even assume boundary conditions to obtain a solution. Different types of data can be combined in a natural way. Prior climatologically estimated covariances are not used. This method may be thought of as a very general form of low-pass filter. The parameter λ controls the half-power point of the implied data filter, while m controls the rate of “roll off” of the power spectrum of the analyzed field. From another point of view. λ and m play the roles of the most important free parameters in an (implicit) prior covariance. The correct choice of the parameter λ and to some extent m is important. These parameters are estimated from the data being analyzed by the GCV method. This method estimates λ and m for which the implied data filter has maximum internal predictive capability. This capability is assessed by the GCV method by implicitly leaving out one data point at a time and determining how well the missing datum can be predicted from the remaining data. The numerical algorithm given provides for the efficient calculation of the optimum λ and m.