Application of Hyperbolic S-transform in Environmental Gravity Investigation

In this paper, a new method is developed based on downward continuation with a varying shape window of hyperbolic S-transforms to estimate the depth of subsurface structures from gravity data. To increase the sensitivity of the algorithm to better define the top surface of a buried structure ( i.e. , enhanced resolution of short wavelength features), the Fourier transform operator in the downward continuation equation is replaced by a hyperbolic S-transform window with critical shape parameters. The hyperbolic S-transform allows us to focus on the high wavenumber features in the gravity data. Using the downward continuation equation, the greater the continuation level, the more unstable the values of the continued gravity data will be because of the exponentially increasing effects of continuation level within the formula. One of the objectives of our method is to employ a new formula for depth estimation based on a variant version of downward continuation, where the exponential operator is replaced by a logarithmic operator. Because the logarithm has a slow increase initially (gravity anomalies are usually located at the early portion of the position–wavenumber space), a result is the gradual increase in depth values, which avoids large and unrealistic variations in the calculated depths. Unlike the Fourier transform, the S-transform provides depth values dependent on both position and wavenumber. This allows us to average different calculated depths at each station to obtain a value for the depth in each position. The estimated depth values for gravity sources are referenced to the x-axis, because the S-transform provides the localized Fourier spectrum of data. The depth values must be averaged to decrease spatial shifting of locally obtained depths. Moreover, the averaging neutralizes the repetitive depth information that occurs through S-transforms of the original gravity data, rather than the S-transform of the analytic signal of gravity data. Analysis of synthetic and field gravity data illustrates the usefulness of the hyperbolic S-transform for estimating depth values of geologic structures.