Comparison of the ℓ1 and ℓ2 norms applied to one‐at‐a‐time spike extraction from seismic traces

Abstract

We present an algorithm for deconvolving a seismic trace by extracting spikes one at a time, thereby obtaining a sparsely populated spike train. Three versions of this algorithm are then compared empirically, by applying them to several examples of synthetic and real seismic data. The first two versions correspond to the use of the [Formula: see text] (least‐absolute‐values) and [Formula: see text] (least‐squares) norms, while the third is a faster and more compact version of the [Formula: see text], algorithm. The [Formula: see text] procedures are shown to exhibit different characteristics which are often desirable, and the results are generally superior to those of the [Formula: see text] procedure for one‐at‐a‐time spike extraction. The use of the fast [Formula: see text] algorithm is advocated in practice for efficient and effective deconvolution.

Approximation performance of model functions for blind deconvolution of ultrasonic reflections

“…Deconvolution can be divided into two minimization problems: i) "pure" deconvolution where signal is treated as convolution of test object reflectivity function with probing (reference) signal [10][11][12] and ii) reference signal production [18,19,23]. Deconvolution is the optimization problem that uses sparse deconvolution methods such as matching pursuit, Prony model or orthogonal matching pursuit in order to deconvolve the mixed data [5,6,15,16].…”

“…Furthermore, limited bandwidth is usually desirable, because signal acquisition system requirements can be relaxed [8], attenuation of ultrasound at high frequencies produces structural noise and signal degradation [9]. Deconvolution is used to address this problem [10][11][12]. It is based on assumption that reflections are sparse (based on usual defects structure which present themselves as layers or distributed cavities) and that every reflection is a time and amplitude translated copy of the original probing signal (reference).…”

“…Wiener filtering [13] can be used to address the problem, but it does not offer the decomposition of the signal into ToF and corresponding amplitudes. Iterative deconvolution techniques allow for efficient signal decomposition either processing one reflection at a time or several reflections simultaneously [6,10,12,15]. Yet, production of the reference signal constitutes a major problem in all deconvolution methods [5,6,16,17].…”

Automatic simultaneous measurement of phase velocity and thickness in composite plates using iterative deconvolution

“…Iterative deconvolution [28,[35][36][37][38][39][40] assumes that if received signal is the sum of the step responses at each reflector, reflections can be separated subtracting a reference signal properly placed and weighted. In our case, iterative deconvolution is performed as follows:…”

Temporal Leakage and Its Effects on Resolution in Deconvolution of Ultrasonic Signals

“…Many deconvolution methods have been developed to remove the oscillatory response of the transducer from the received ultrasonic echo thereby improving the temporal resolution. While these methods have worked weIl on simulated signals, the results on real data have generally been much poorer [1][2][3][4][5][6][7]. Ultrasonic pulse shape variations, nonlinear effects or the breakdown of other model assumptions aIl contribute to this lower performance on real signals.…”