Non‐minimum phase wavelet estimation by non‐linear optimization of all‐pass operators

Abstract

A B S T R A C TConvolution of a minimum-phase wavelet with an all-pass wavelet provides a means of varying the phase of the minimum-phase wavelet without affecting its amplitude spectrum. This observation leads to a parametrization of a mixed-phase wavelet being obtained in terms of a minimum-phase wavelet and an all-pass operator. The WienerLevinson algorithm allows the minimum-phase wavelet to be estimated from the data. It is known that the fourth-order cumulant preserves the phase information of the wavelet, provided that the underlying reflectivity sequence is a non-Gaussian, independent and identically distributed process. This property is used to estimate the all-pass operator from the data that have been whitened by the deconvolution of the estimated minimum-phase wavelet. Wavelet estimation based on a cumulant-matching technique is dependent on the bandwidth-to-central-frequency ratio of the data. For the cumulants to be sensitive to the phase signatures, it is imperative that the ratio of bandwidth to central frequency is at least greater than one, and preferably close to two. Pre-whitening of the data with the estimated minimum-phase wavelet helps to increase the bandwidth, resulting in a more favourable bandwidth-to-central-frequency ratio. The proposed technique makes use of this property to estimate the all-pass wavelet from the prewhitened data. The paper also compares the results obtained from both prewhitened and non-whitened data. The results show that the use of prewhitened data leads to a significant improvement in the estimation of the mixed-phase wavelet when the data are severely band-limited. The proposed algorithm was further tested on real data, followed by a test involving the introduction of a 90 • -phase-rotated wavelet and then recovery of the wavelet. The test was successful.

Bayesian frequency-domain blind deconvolution of ground-penetrating radar data

“…Two-stage wavelet-estimation approaches similar to our technique were previously proposed by Misra and Sacchi (2007), Porsani and Ursin (1998), and Ursin and Porsani (2000). These techniques require the length of the all-pass to be either pre-set or included in the inversion as unknown.…”

“…Wood (1999) proposed a simultaneous deconvolution and wavelet estimation technique that inverts in the frequency domain for the wavelet phase spectrum given the wavelet's amplitude spectrum. The decomposition of a mixed-phase wavelet into a minimum-phase and maximum-phase component (Eisner and Hampson, 1990), or reformulated, the parametrization of a mixed-phase wavelet as the convolution of the wavelet's minimum-phase equivalent with an all-pass filter (Claerbout, 1985) was used by Misra and Sacchi (2007), Porsani and Ursin (1998), and Ursin and Porsani (2000) to estimate mixedphase wavelets. Porsani and Ursin (1998) and Ursin and Porsani (2000) estimated an optimum all-pass filter by solving extended normal equations in an exhaustive search manner.…”

“…Porsani and Ursin (1996) applied this approach also to GPR data. Misra and Sacchi (2007) inverted for the all-pass filter using fourth-order cumulant matching and employed a global non-linear optimization scheme for computing the all-pass filter coefficients.…”

A New Stochastic Process of Prestack Inversion for Rock Property Estimation

“…R. P. Srivastava and M. K. Sen used fractal-based initial models combining very fast simulated annealing (VFSA) to improve impedance inversion of prestack data, which addressed frequencies missing compared to deterministic inversion [12]. Somanath Misra and Mauricio D. Sacchi also applied VFSA to AVO inversion with conditionalized model-space, not only to speed convergence, but also to retrieve blocky solutions [13]. Patrick A. Connolly and Matthew J. Hughes used pseudo-wells and generated synthetic traces to match seismic data via stochastic inversion [14].…”

Reflectivity estimation using minimum-delay seismic trace decoposition

“…The ltered seismic trace is then an estimate of the reectivity series. Misra and Sacchi (2007) and Misra and Chopra (2010) deconvolve the data with a standard spiking deconvolution lter. From the ltered, whitened data they estimate an allpass phase lter which is then applied to the whitened data.…”