Interpolation, smoothing, and differentiation of gravity anomalies

Abstract

Numerical methods are examined whereby gravity anomalies observed at randomly distributed points may be transformed into a continuous harmonic function on a level surface. The interpolated value at a height z at the origin is found from the observed values gⁱ at distances heights ζ_i and representing areas a_i by a linear summation The weighting functions f_i are derived from the Fourier-Bessel integral expansion of the gravity anomaly field, and are designed to act as low pass filters. Two cases are considered: the first is a sharp cut-off filter, where the frequency v has an upper limit ω, so that the frequency response is 1 for 0 ⩽ v ≪ ω and 0 for v > > and the second is an exponential filter of theoretical frequency response exp(−v ²/ω²) for which Differentiation of these formulae with respect to z yields the higher vertical derivatives directly from the observed anomalies. The methods include allowance for local variations in the density of observation points, and are designed to prepare data for the drawing of contour maps on a uniform basis by mechanical means; of the two methods that of the exponential filter exhibits much greater numerical stability in handling irregularly distributed observations.