Applications of the Barnes Objective Analysis Scheme. Part II: Improving Derivative Estimates

Abstract

Two recently published objective analysis schemes improve derivative estimates obtained from observed field variables. Achtemeier proposed a hybrid successive-correction scheme that uses a very low pass Gaussian weight function on the first pass through the observations, followed by two correction passes with a higher-pass, fixed-weight function. His experiments demonstrated better separation of short, but resolvable, wavelengths from noise components than does the widely used two-pass Barnes scheme. Both schemes produce derivative estimates using conventional second-order finite-difference formulas. The generality of Achtemeier's improvement over two-pass schemes is demonstrated in an experiment similar to that designed by Achtemeier. It is found that the amount of improvement diminishes the closer the two-pass convergence parameter is to 1. Considering other three-pass scheme designs, it is found that Achtemeier's is not necessarily the “best” design. A three-pass “fixed” scheme that uses but one weight function yields greater improvement, and implies that an even larger number of passes might separate signal from “noise” even more efficiently. Caracena recast successive corrections objective analysis schemes as a matrix problem in which observations are operated upon in a single pass by an effective weight function that represents the combined effect of multiple passes through the data. This so-called analytic approximation scheme produces not only the distribution of the field variable, but the distribution of its first- and higher-order derivatives as well, without resorting to finite-difference approximations. One version of the scheme produces an exact analytic approximation that is equivalent to an infinite number of successive correction passes. A demonstration of the exact scheme shows that it reproduces with very high accuracy all wavelengths of an analytic function down to the station Nyquist interval, and its first three derivatives as well. For practical applications, it is seldom wise to retain waves near the Nyquist interval because of analysis errors due to boundary effects, observational errors, and irregular spacing of observations. Another version of Caracena's scheme, equivalent to a four-pass traditional scheme, is applied to a relatively dense uniform array of observations containing Nyquist-scate “errors.” Under these conditions, this version has the ability to completely suppress the Nyquist wavelength, while passing essentially 100% of resolvable waves and their derivatives up to third order. In this regard, it is comparable to, but more accurate than, the best result obtainable using a four-pass Barnes scheme. The general applicability of Caracena's approach is limited only by computation time (it took twice as long in these tests), and by possible peculiarities in real station distributions that could produce ill-defined matrices.