Gravity sources identification using continuous wavelet transform

Introduction: Wavelet transform is one of the useful and suitable tools for time series and signal analysis. Nowadays wavelet transform is frequently used in geophysical data processing and interpretation, especially seismic data. However, the use of this method isn’t widespread in gravity and geomagnetic. Fedi and Quarta (1998), Martelet et al. (2001) and de Oliveira Lyrio (2004) used the wavelet transform for processing and interpretation of the potential field data. In this paper, a new method based on continuous wavelet transform for determination of depth and location of gravity anomalies is introduced.
Continuous Wavelet Transform and Gravity Source Identification: All of the time-frequency or time-scale transforms intend to show how the energy of a signal is distributed in time-frequency or time-scale plan. The Continuous Wavelet Transform (CWT) maps the time (space) domain signal into the time (space)-scale plan. The CWT of a signal is defined as the convolution of signal with a translated and scaled wavelet (Equation (1)).
(1)
where, denotes the complex conjugate, is scale, is space and is the mother wavelet. Shifted and scaled version of the mother wavelet can be computed as equation (2):
(2)
Any wavelet which is selected as the mother wavelet must meet the zero mean value condition. Mother wavelet selection can affect on the results of wavelet analysis. If the properties of the selected mother wavelet are the same as the signal, then the space-scale representation of the signal can give more useful information about the energy distribution of the signal in space-scale plan.
A buried cylinder can be seen as a rectangle in 2D view. In addition, any body in 2D can be shown by arranged rectangles. Therefore, we use the gravitational anomaly of a buried cylinder and its first and second horizontal derivatives and their vertical derivative as mother wavelets.
The gravitational anomaly of the buried cylinder can be obtained by equation (3):
(3)
where, is the gravitational constant, m is mass of the buried cylinder located at the position and depth . This wavelet does not meet the zero mean value condition and cannot be used as the mother wavelet. But its derivatives are suitable for the mother wavelet. Equations (4) to (7) are the derivatives of the gravitational the anomaly of the buried cylinder.
(4)
(5)
(6)
(7)
When we used these equations as the mother wavelet, and are not needed and is set to one.
Discussion: We tested the efficiency of the CWT method for gravity source identification on various synthetic models such as a simple cube, various type of faults, simple cubes in different depths and real data. The CWT coefficients are computed using the gravitational anomaly and its first and second horizontal derivatives. The obtained results show that the CWT coefficients obtained using first horizontal derivative of data and equations (4) and (6) can estimate precisely the depth and location of the source of gravitational anomaly.