Geophysical explorations and geophysical signal processing

Geophysical signal processing techniques attempt to extract hidden information from a data set. In case of potential field measurements the goal is to learn about the distribution of the material properties in the subsurface of the earth. Many of these techniques include the Fourier transformation. However, although the Fourier transformation gives the frequency content of a signal, loses completely any information in time domain. This insufficiency urged the development of the wavelet transform having simultaneous time and frequency localization properties. In this doctoral thesis we discus the application of the wavelet transformation in potential field data processing. A brief description of the contents of the thesis is outlined below. In chapter one, the Fourier transforms are presented briefly together with some applications in potential field data transformations. In the second chapter the concept of time-frequency and time-scale analysis is introduced. In this chapter the short-time Fourier transform is described and the story of wavelets is introduced. Also an introduction to multiresolution analysis is presented. The third chapter contains two subjects: The application of the discrete wavelet transform in the regional-residual magnetic field separation problem and data de-noising by thresholding the wavelet coefficients.Thus at first, a new algorithm is proposed for regional-residual magnetic data separation in discrete wavelet domain. The separation is based on the wavelet vanishing moments and an internal model for the regional field. The separation is controlled by the wavelet transformation of the model. The results in trend removal are perfect. The method has benefits over the classical separation with the use of the polynomial regression method, and is capable to detect a spatially varying trend. The same method was tested in the more general case of regional-residual separation and also gives good results. Secondly the de-noising techniques in wavelet domain are presented, and an empirical study of the denoising of the magnetic data is resumed. Based on this study a de-noising schema is proposed. It is a combination of the cyclo-spinning translation invariant de-noising, proposed by the Donoho and Jonhstone, with a variable threshold selection. The selection of the threshold is based on the statistical behavior of the wavelet coefficients of the noise. In chapter four edge detection techniques are discussed. Also, a wavelet edge detection method is tested for the detection of the boundaries of the causative bodies. The method is based on the continuous wavelet transformation (CWT) of the data. The gradient of the data is computed by the use of the DOG (first derivative of a Gaussian) wavelet and the gradient maxima are detected in a Canny like method. The method is analogous to the Canny’s edge detection method, but has the benefit of less sensitivity to the noise. The efficiency of the method is tested with synthetic and real data. The results are very good but in some cases the method fails in the detection of prismatic bodies corners. For this reason the use of steerable filters in edge detection is examined. The method seems to give better results in the detection of the corners. In the appendix A the non-stationary filtering is reviewed and the implementation in wavelet domain is tested. Also the upward and downward continuation of potential fields in wavelet domain is implemented and the results are compared with those given by the chessboard method. An improvement of the smoothing filter for the wavelet domain downward continuation is proposed. Next the results of the thesis are summarized and the prospects of the possible future research are given