A sparse spectrum technique for gridding and separating potential field anomalies

The separation of regional and residual potential field anomalies, regarded as a spectral problem, can be greatly facilitated when a spectrum estimate shows a clear break between low- and high-frequency components, a feature that normal fast-Fourier-transform (FFT) methods fail to present. In this work, we model the discrete Fourier transform of a potential field, measured at stations irregularly distributed on a surface, by means of a high-resolution sparse estimate derived originally for seismic signal processing. The coefficients of this estimate, which are distributed according to the Cauchy probability law, produce a model with only few components having a significant value. A steepest-descent algorithm gives a computing alternative to large matrix multiplications and inversions. Advantages of taking this approach are twofold. First, the high-resolution transform can be used as a gridding tool to evaluate the potential field either on a horizontal plane or on the topographic surface. The enhancement of the spectral peaks and the virtual absence of side-lobes prevents oscillations and edge effects in the result. Secondly, the sparse distribution of the spectral elements allows the interpreter to locate clearly the low-frequency components related to the regional field. After a second and faster pass, the values of those coefficients can be redefined in order to obtain a more robust separation, ajusting the residuals by the Cauchy criterion. A theoretical noise-free example to separate the magnetic anomaly of a prism from a polynomial background illustrates well the difference between sparse and FFT spectra, An example with real Bouguer anomalies in the Interserrana basin, Argentina, shows that gridding results, in this case reduced to sea level, compare well with those obtained by other gridding methods, and that the separation procedure is able to outline well defined areas of positive and negative residual anomalies.