Interpolation errors and oceanographic sampling

Abstract

Formulae are derived which can be used to give an interpolated value for an oceanographic variable and to estimate the accuracy of the interpolation. The interpolated value is obtained by taking the average of the values obtained from two Lagrange interpolation polynomials: one utilizing two points above and one below the depth of interpolation; and the other, one above and two points below this depth. The error of an interpolated value is composed of two parts, one part which is due to measurement error of the observations, and another part which is the error of the interpolation itself. The difference in the values given by the two interpolation polynomials provides a measure of the error of interpolation. Expressions for the errors resulting from measurement errors, given as the ratio of the standard deviation of the interpolated values to the standard deviation of the individual measurements, depend only on the sampling and interpolation depths. Utilization of these formulae permits the determination of optimum sampling programmes which, with a minimum number of observations, yield the distribution of a variable to a prescribed accuracy.