Statistical denoising of signals in the S-transform domain

Abstract

In this paper, the denoising of stochastic noise in the S-transform (ST) and generalized S-transform (GST) domains is discussed. First, the mean power spectrum (MPS) of white noise is derived in the ST and GST domains. The results show that the MPS varies linearly with the frequency in the ST and GST domains (with a Gaussian window). Second, the local power spectrum (LPS) of red noise is studied by employing the Monte Carlo method in the two domains. The results suggest that the LPS of Gaussian red noise can be transformed into a chi-square distribution with two degrees of freedom. On the basis of the difference between the LPS distribution of signals and noise, a denoising method is presented through hypothesis testing. The effectiveness of the method is confirmed by testing synthetic seismic data and a chirp signal.

Introduction

Time–frequency analysis has many applications in geophysical data analysis and other disciplines. S-transform (ST), proposed by Stockwell, is also a time–frequency method. As compared with the wavelet and windowed Fourier transform, the ST provides a frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum (Stockwell et al., 1996). According to the results of Stockwell et al., the expression of the ST is

$$S(\tau ,f)={\mathit{\int }}_{-\infty }^{\infty }h(t)\frac{|f|}{\sqrt{2\pi }}\exp (-i2\pi \mathit{ft})\exp (-(t-\tau ){}^2{f}^2/2)\mathit{dt}$$

In Eq. (1), S(τ, f) denotes the ST of a time series h, which is a continuous function of time t; τ is a parameter that controls the position of the window function on the t-axis; and f is the frequency. The inverse transform of Eq. (1) is

$$\mathit{\int }S(\tau ,f)d\tau =H(f)$$

where H(f) is the Fourier transform of the time series h(t). We can derive h(t) by using the inverse Fourier transform of H(f). In Stockwell's paper, the window function is a Gaussian function. There are several papers that generalized the ST by adopting different windows and their parameters (Gao et al., 2003; McFadden et al., 1999; Pinnegar and Mansinha, 2003a, Pinnegar and Mansinha, 2003b, Pinnegar and Mansinha, 2004). The GST can be denoted as

$$S(\tau ,f)={\mathit{\int }}_{-\infty }^{\infty }h(t)\psi (t-\tau ,f,\mathbf{p})\exp (-i2\pi \mathit{ft})\mathit{dt}$$

where ψ is the window function, and p represents a set of parameters that determine the shape and properties of ψ. However, ψ was not optimized to the analyzed signal. Recently, several algorithms have been presented, which relate the energy concentration with the optimization of the window width of ST (Djurovic et al., 2008; Man et al., 2007). In Eq. (1), the standard deviation of the Gaussian window is a reciprocal to the frequency. The standard deviation of a Gaussian window can be defined as 1/(|f|a(f)) (or 1/|f|^α); then, the GST can be given as

$$S^a(\tau ,f)=\frac{|\mathit{fa}(f)|}{\sqrt{2\pi }}{\mathit{\int }}_{-\infty }^{\infty }h(t)\exp (-(t-\tau ){}^2{a}^2(f)f^2/2)\exp (-i2\pi \mathit{ft})\mathit{dt}$$

where a(f) is an additional parameter that can be used to optimize the window width. The optimal value of a(f) can be found for each frequency based on the variation of the concentration measure. The definition of the frequency-dependent concentration measurement can be found in Djurovic et al. (2008) and Man et al. (2007). Meanwhile, optimal a(f) for the analyzed signal can be determined by maximizing (or minimizing) the concentration measure.

In general, for a nonstationary time series buried in noise, wavelet-based techniques (Donoho and Johnstone, 1994; Moulin, 1994; Torrence and Compo, 1998) become effective methods for suppressing the noise in the wavelet domain. However, in this paper, we show that noise processing in the S domain has a different characteristic than that in the wavelet domain. Although there are already several papers discussing noise processing in the ST domain (Pinnegar and Eaton, 2003; Pinnegar and Mansinha, 2003c), the denoising method proposed here differs from previous studies, that is, it utilizes hypothesis testing. Moreover, as compared with the time–frequency filtering technique in ST (Pinnegar, 2005; Schimmel and Gallart, 2005), our study presents a “blind” method.

This paper is organized as follows. In Section 2, the mean power spectrum (MPS) of white noise is derived in the ST and GST domains, while the red noise distribution in the ST and GST domains is analyzed by means of a Monte Carlo simulation. In Section 3, a denoising method is presented and verified by testing a set of synthetic signals. Conclusions are then presented in Section 4.