INTRODUCTION TO WAVELET SIGNAL ANALYSIS AND WAVELET DENOISING OF MAGNETIC DATA

• G. Tsokas (Geophysical Laboratory, Dept. of Geology, Aristotle University of Thessaloniki, 541 24 Thessaloniki, Greece)
• B. Tsivouraki-Papafotiou (Depatrment of Automation, School of Technical Applications, Tecnological Educ. Inst. of Thessaloniki, Thessaloniki, Greece)

Wavelet Signal Analysis is nowadays a dynamic research tool in a variety of scientific areas included geophysics. Although, the use of the wavelets and wavelet signal analysis is quite simple and fun, their mathematical support is difficult and can seem discouraging to the beginner.

In this paper we try to explain in a simple manner what are wavelets and how are defined. We describe the conventional Fourier transform and the sort time Fourier transform and demonstrate with examples the benefit of time-frequency analysis over the conventional Fourier transform.

Next we introduce the Morlet's idea of the dilated and translated window, and present the integral wavelet transform as an extension of time - frequency analysis. We also present the discrete wavelet transform, the multiresolution analysis algorithm, perfect reconstruction and the filter banks. The section includes examples of wavelet spectrum with emphasis to the relation between wavelet scale and Fourier frequency.

A second approach to the multiresolution analysis theory and dialation equations is also presented. We also give some of the basic wavelet properties, and the Haar basis.

We use wavelets for denoising magnetic data. The fact that orthogonal wavelet transformations map white noise to white noise gives rise to the use of wavelets for denoising noisy data. The idea is to transform the data into an orthogonal wavelet basis, shrinkage the noisy wavelet coefficients via thresholding and then use the modified wavelet coefficients to reconstruct the signal by the inverse wavelet transform. The two most common threshold functions are hard and soft functions. A combination of a function with a threshold selection rule gives a threshold method. We try several denoising schemes, on synthetic 1-D magnetic data, using several wavelets and thresholding methods .Our empirical results, using the standard deviation of the reconstructed from the original signal as a measure of the quality of denoising, show that for magnetic anomaly data the combination of db4 (Daubechies 4) wavelet and hard thresholding with the universal threshold, gives quite good results. Then we compare the results with spatial filtering. Finally we perform 2d hard thresholding on 2d magnetic data and compare with denoising via 2-D spatial filtering.

Basic tool in our work is Matlab and wavelab.