Wavelet transform filtering of seismic data by semblance weighting

The presence of noise in a nonstationary signal (i.e., whose frequency content varies with time) complicates the extraction of meaningful information from the signal. The classical Fourier domain of such a signal can only separate the noise from the useful signal energy if they have different frequency content. For nonstationary signals the windowed Fourier transform has been introduced to localize the signal in time. The windowing is accomplished by a weight function that places less emphasis near an interval's endpoints than in the middle. The short-time Fourier transform (STFT) thereby decomposes a signal into a time frequency plane. Nonstationary filtering can be done by prescribing weights that vary with time and frequency, which are applied to the decomposed data. An inverse STFT then reconstructs the filtered signal. Recently, the wavelet transform (WT) has been applied in diverse fields such as mathematics, quantum physics, engineering and geophysics. The WT decomposes a signal in a time-scale frame. The seismic data can be filtered using the WT in a form similar to time-frequency filtering techniques. This paper explores a method of filtering seismic data using the discrete wavelet transform (DWT) with filter weights in the wavelet domain using a time-domain semblance measure. The semblance coefficients, as a measure of multichannel coherence, serve to emphasise the signal in the wavelet coefficients of the decomposed trace. This method has been tested on a Blackfoot final stack where it appears to improve the resolution of the section. INTRODUCTION The time-variant filtering technique was implemented to suppress nonstationary noise bursts in seismic data. To compensate for the limitations of the classical Fourier transform, the STFT was first introduced (Cohen, 1995). The windowed Fourier transform is the most widely used method for studying nonstationary signals. The time-variant spectrum (TVS) is a decomposition of a signal onto a time-frequency matrix. The TVS(τ, f) (Schoepp, 1998) is calculated by taking the STFT: dt e t h t s f TVS ft i ∫ ∞ ∞ − − − = π τ τ 2 ) ( ) ( ) , ( , (1) where s(t) is the signal and h(t-τ) is a time-shifted window with τ the fixed time and t the running time. The window width and the window increment are important parameters in the TVS computation. The width of the time window is determined by a fixed positive constant. In all of the STFT methods, (depending on the windows used) computational complexities arise when either narrowing of the window is required for better localization or widening of the window is required to obtain a better spectral resolution (Chui, 1992). A whitening technique applied to the TVS of Iliescu and Margrave CREWES Research Report — Volume 12 (2000) data (a stacked section from the Blackfoot broad band survey) will be used in this test to equalize the Fourier spectrum and therefore, the resolution. In comparison, the WT distinguishes itself from the STFT in that it has a zoom-in and zoom-out capability. Unlike the STFT in which the length of the window is fixed, the WT localizes signals in a variable window determined by the scale parameter. The WT is a relatively new signal analysis and processing approach. There are few applications of the wavelet transform in geophysical data processing. Some examples are: data compression (Donoho, Ergas and Villasenor, 1995), time frequency analysis, filtering and interpretation using frequency-time plots (Chakraborty and Okaya, 1994), and phase correction using the wavelet transform (Rodriguez and Mansar, 1997). The WT of a signal decaying in time depends on two variables: scale and time. The strength of the WT representation is the separation of the signal in different scale levels. The result of the wavelet decomposition consists of coefficients that are influenced by local events that can be potentially identified, analysed, and filtered. The WT was used to filter the stacked section by applying weights to the wavelet coefficients. The weights are based on semblance in t-x domain. MATHEMATICAL BACKGROUND The WT is, like the Fourier transform, an inner product between the signal and a set of basis functions. The expansion coefficients reflect the similarity between the signal and the elementary basis functions. The elementary functions are also called analysis functions. The result of the inner product represents the expansion coefficient and the set of all expansion coefficients represents the wavelet domain. There are many types of wavelet transforms. The most important are the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT). The CWT The CWT can be thought of as the inner product of the signal with the basis functions ψa,b(t), (Daubechies, 1992) (ψ(t) is called the mother wavelet). dt t t s a CWT t t s b a ab ) ( * ) ( 1 ) ( ), ( ) , ( ψ = = 〉 ψ 〈 ∫ ∞ ∞ − . (2) In this expression, ψa,b*(t) is the complex conjugate of ) ( 1 ) ( , a b t a t b a − = ψ ψ , (3) for ψ(t) real, ψ* = ψ. The scale index, a, can be thought as the reciprocal of the frequency while b indicates time shifting (or translation). The normalizing constant a is chosen so that the total