Local skewness attribute as a seismic phase detector

INTRODUCTION

Wavelet phase is an important characteristic of seismic signals. Physical causal systems exhibit minimum-phase behavior (Robinson and Treitel, 2000). For purposes of seismic interpretation, it is convenient to deal with zero-phase wavelets with minimum or maximum amplitudes centered at the horizons of interest because it leads to the highest resolution as well as more accurate estimates of both reflection times and spacings (Schoenberger, 1974). Zero-phase correction is therefore a routine procedure applied to seismic images before they are passed to the interpreter (Brown, 1999).

It is important to make a distinction between phases of propagating and locally observed wavelets (Van der Baan et al., 2010b,a). The former is the physical wavelet that propagates through the Earth, thereby sampling the local geology. It is subject to geometric spreading, attenuation, and concomitant dispersion. The latter is the wavelet as observed at a certain point in space and time. Its immediate shape results from its interaction (convolution) with the reflectivity of the Earth and the current shape of the propagating wavelet. For instance, a thin layer with a positive change in seismic impedance has opposite polarities of seismic reflectivities at the top and the bottom interface, which make it act like a derivative filter and generate a wavelet with the locally observed phase subjected to a $90^{\circ }$ rotation (Zeng and Backus, 2005). In the absence of well log information, it is usually difficult to separate unambiguously the locally observed phase from the phase of the propagating wavelet. Nevertheless, measuring the local phase can provide a useful attribute for analyzing seismic data (Van der Baan and Fomel, 2009; Fomel and van der Baan, 2010; Xu et al., 2012; Van der Baan et al., 2010a).

Levy and Oldenburg (1987) proposed a method of phase detection based on maximization of the varimax norm or kurtosis as an objective measure of zero-phaseness. By rotating the phase and measuring the kurtosis of seismic signals, one can detect the phase rotations necessary for zero-phase correction (Van der Baan, 2008). Van der Baan and Fomel (2009) applied local kurtosis, a smoothly nonstationary measure (Fomel et al., 2007), and demonstrated its advantages in measuring phase variations as compared with kurtosis measurements in sliding windows. Local kurtosis is an example of a local attribute (Fomel, 2007a) defined by utilizing regularized least-squares inversion.

In this paper, we revisit the problem of phase estimation and propose a novel attribute, local skewness, as a phase detector. Analogous to local kurtosis, local skewness is defined using local similarity measurements via regularized least squares. This attribute is maximized when the locally observed phase is close to zero. Advantages of the new attribute are a higher dynamical range and a better stability, which make it suitable for picking phase corrections. Using synthetic and field-data examples, we demonstrate properties and applications of the proposed attribute.

Local skewness attribute as a seismic phase detector