A method for improved analyses of scalars and their derivatives

Analytic observations are used to compare the traditional and triangle methods for the objective analysis of scalar variables. The traditional method for objective analysis assigns gridpoint values based on the distance from the grid points to each member of the set of observations. Subsequently, spatial derivatives are derived by applying a finite-differencing scheme to the field of gridded observations. The triangle method for objective analysis calculates the spatial first-order derivatives directly from each set of nonoverlapping triangles that are formed by the observations, and the derivatives are assigned to the triangle centroids. By calculating the first-order derivatives directly from the observations, the triangle method bypasses the need for finite differencing. Triangle centroid estimates of the scalar field itself are simply arithmetic averages of the three observations comprising each triangle. The centroid estimates of the scalar variable and its spatial derivatives are then treated as observations and mapped to a uniform grid via the traditional method. Results indicate that the traditional method for the analysis of a scalar variable is superior to the triangle method for scalar analysis because the simple averaging involved in creating the triangle centroid estimates of the scalar exposes the triangle analysis to the potential for significant damping of the input field. Indeed, although the patterns of the scalar analyses from the two methods are comparable, the analysis from the triangle method does not reproduce the amplitude of the scalar field as well as the analysis from the traditional method. Gradient and Laplacian fields computed from the triangle method, however, are generally superior to those derived by the traditional method, which tends to force all of the gradient information into the gaps between observing stations. To overcome the deficiency of the triangle method's ability to produce an acceptable scalar analysis and the deficiency of the traditional method's ability to produce an acceptable derivative analysis, a variational objective analysis scheme is developed that combines the best aspect of the triangle method with that of the traditional method. Analyses of the scalar and its spatial derivatives from the variational analysis scheme are generally superior to analyses from both the traditional and triangle methods.