In this article, stable Padé approximations to the function 1+z are derived by choosing a branch cut in the negative half-plane. The Padé coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z<−1, I(z)=0 [I(z) denotes the imaginary part of z]. The characteristics of the corresponding square-root approximation are illustrated for various segments of the complex plane. In particular, for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case. An elastic parabolic equation algorithm is used to illustrate the application of the new Padé approximations to a realistic ocean environment, including elasticity in the ocean bottom.

An algorithm for estimating the acoustic reflection coefficient profile from ultrasonic traces obtained during inspection of layered materials is described. Given the measured trace and the incident wavelet, the inversion proceeds by means of a layer stripping approach combined with high-resolution deconvolution. The inversion algorithm is stable to noise and is suitable for use with bandlimited data. It is particularly suitable for use with materials that exhibit a few large discontinuities in impedance and in which multiple reflections in the data are evident. The performance of the algorithm is illustrated in tests with synthetic and real data. An implementation of the algorithm on a TMS 320C30 signal processing board allowed the inversion of an entire set of 256 traces, each of 256 elements, in 15 s.

Abstmct-In this paper a new method for obtaining a quantitative estimate of an acoustic field consisting of a set of discrete sources and background noise is described. The method is based on the L1 (least absolute values) norm solution to an underdetermined system of linear equations defining the Fourier transform of the signal series. Implementations of the method with either equality or inequality constraints are presented and discussed. The much faster and more compact equality constraint version with a provision for modeling the noise field is recommended in practice. Experience with real data has shown the necessity of correcting for an observed Gaussian decay on the covariances. A simple means of estimating this effect and taking it into account in the signal estimation procedure is discussed, and the implications of this effect in high-resolution beamforming are considered. The effectiveness and versatility of the L1 method indicate that it has a useful role in high-resolution signal estimation.

When analyzing linear systems of equations, the most important indicator of potential instability is the condition number of the matrix. For a convolution matrix W formed from a series w (where Wij wi-, + ,, 1 5 ij + 1 5 k, W,j = 0 otherwise), this condition number defines the stabirity of the deconvolution process. For the larger convolution matrices commonly encountered in practice, direct computation of the condition number (e.g., by singular value decomposition) would be extremely time consuming. However, for convolution matrices, an upper bound for the condition number is defined by the ratio of the maximum to the minimum values of the amplitude spectrum of w. This bound is infinite for any series w with a zero value in its amplitude spectrum; although for certain such series, the actual condition number for W may in fact be relatively small. In this paper we give a new simple derivation of the upper bound and present a means of defining the rate of growth of the condition number of W for a bandlimited series by means of the higher order derivatives of the amplitude spectrum of w at its zeros. The rate of growth is shown to be proportional to m p , where m is the column dimension of Wand p is the order of the zero of the amplitude spectrum.

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.