Automatic dispersion extraction using continuous wavelet transform

Abstract

In this paper we present a novel framework for automatic extraction of dispersion characteristics from acoustic array data. Traditionally high resolution narrow-band array processing techniques such as Prony's polynomial method and forward backward matrix pencil method have been applied to this problem. Fundamentally these techniques extract the dispersion components frequency by frequency in the wavenumberfrequency transform domain of the array data. The dispersion curves are subsequently extracted by a supervised post processing and labelling of the extracted wavenumber estimates, making such an approach unsuitable for automated processing. Moreover, this frequency domain processing fails to exploit useful time information. In this paper we present a method that addresses both these issues. It consists in taking the continuous wavelet transform (CWT) of the array data and then applying a wide-band array processing technique based on a modified Radon transform on the resulting coefficients to extract the dispersion curve(s). The time information retained in the CWT domain is useful not only for separating the components present but also for extracting group slowness estimates. The latter help in the automated extraction of smooth dispersion curves. In this paper we will introduce this new method referred to as the Exponential Projected Radon Transform (EPRT) in the CWT domain and limit ourselves to the analysis for the case of one dispersive mode. We will apply the method to synthetic and real data sets and compare the performance with existing methods.

Joint Multi-Mode Dispersion Extraction in Frequency-Wavenumber and Space-Time Domains

“…It was shown in [1] that 2 the CWT of the noisy data in band denoted by at receivers can be written as (16) with the vectorized form of an unknown coefficient matrix whose -th column corresponds to the CWT coefficient vector at the reference receiver , corresponding to the mode with phase slowness and group slowness . The (row) vectors represent the Fourier transform of the data at frequency .…”

“…However, this approach is limited to a particular type of borehole mode namely the lowest order flexural mode of an open borehole or to the case when the effect of the tool is well characterized. Authors in [15] and [16] propose a more general approach for extracting dispersion of a single mode without reference to a physical model or restriction of the type of mode. They exploit the continuous wavelet transform (CWT) of the array data along with a parameterization of the wavenumber dispersion in the domain in terms of a local first order linear approximation at each frequency.…”

“…This approach is valid for non-stationarity assumptions and is shown to be robust to time overlap across modes and heavy noise. However, the approach is not as reliable compared to semblance based processing techniques such as the space-time coherence (STC) processing of non-dispersive waves, [20], or of dispersive modes such as EPRT, [16], which provide better group slowness estimates. The main reason is that unlike STC based techniques such as EPRT, the method in [1] doesn't exploit the local coherence of modes in space-time domain.…”

“…This results in a larger variance in the group slowness estimates from domain processing as shown by a Cramer-Rao Bound (CRB) analysis in the Appendix of this paper. In this paper we overcome this drawback by combining the two techniques of [1] and [16] and propose a joint and space-time domain multimode dispersion extraction method.…”

Broadband Dispersion Extraction Using Simultaneous Sparse Penalization

“…Special purpose dispersion extraction methods such SPI [17] have been proposed but are applicable to one particular kind of mode, e.g., flexural waves. More recently, new broadband methods have been proposed that exploit the time-frequency localization of the modes for dispersion extraction; see [18] and [19]. Similar ideas that exploit the time frequency localization have been considered before, e.g., the energy reassignment method of [20] that has been applied in [21] for analysis of dispersion of Lamb waves and in [22] for improving the group velocity measurements.…”