Applications of the Barnes Objective Analysis Scheme. Part I: Effects of Undersampling, Wave Position, and Station Randomness

Abstract

An empirical approach is employed to investigate the accuracy of the Barnes successive corrections objective analysis scheme for discrete samples obtained from a simple sinusoidal function in a two-dimensional domain. Mean absolute error at grid points (MAEG) is the accuracy statistic, and uniform sampling arrays define achievable accuracy. Analysis accuracy as a function of number of samples per wavelength is considered for three arrays of different uniform density. Weight function scale-length parameter (α, the denominator in the exponential Gaussian weight function) is varied over a range of values, and an error-minimizing α is found to exist in each case. The value of this α depends on the number of correction passes being performed, on the density of observations, and for marginally sampled waves, on the phase relationship of the underlying function to the observations. Using a pseudorandomizing process to displace station locations away from uniformity by increasing amounts, it is found that sampling arrays take on a quasi-random character when stations are displaced by 70%-100% of their spacing in an equivalent uniform array. As other authors have found, irregularity of station locations and station clustering are found to have detrimental effects on analysis accuracy. However, a notable economy is discovered: the accuracy achieved from sparse, but uniformly arrayed, observations is greater than that obtained from more than three times as many observations, quasi-randomly arrayed. On average, the MAEG for 77 quasi-randomly arrayed stations is 140% larger than when sampling is by 23 uniformly spaced stations. Some irregular dense arrays produce MAEG that are nearly 40% of the sinusoid's amplitude. Considering these results, a general strategy is suggested for selecting appropriate weight function parameters that can minimize analysis errors.