Cedric A. Zala

Rational square-root approximations for parabolic equation algorithms

In this article, stable Pade approximations to the function 1+z are derived by choosing a branch cut in the negative half-plane. The Pade 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 Pade approximations to a realistic ocean environment, including elasticity in the ocean bottom.

High-Resolution Signal and Noise Field Estimation Using the L 1 (Least Absolute Values) Norm

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 L 1 (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 L 1 method indicate that it has a useful role in high-resolution signal estimation.

Comparison of the l 1 and l 2 norms applied to one-at-a-time spike extraction from seismic traces

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 l1 (least‐absolute‐values) and l2 (least‐squares) norms, while the third is a faster and more compact version of the l1, algorithm. The l1 procedures are shown to exhibit different characteristics which are often desirable, and the results are generally superior to those of the l2 procedure for one‐at‐a‐time spike extraction. The use of the fast l1 algorithm is advocated in practice for efficient and effective deconvolution.

Warping aerial photographs to orthomaps using thin plate splines

Using an image rectification application arising in the field of forest management, we demonstrate in this paper the practical feasibility of applying thin plate spline (TPS) techniques to real image warping problems. The use of TPS‐based warping in large problems can be limited by two factors: numerical instability in the calculation of TPS coefficients, and the intensive computation involved in evaluating TPS functions. Both drawbacks can be overcome by taking advantage of some recent advances in the numerical analysis of radial basis functions. Here we relate our experience in applying some of this work to realize successful TPS warping of large forestry images, and some graphical examples are given. Methods for automated control point selection and editing are also presented, and a cross‐correlation technique for evaluating the effectiveness of the warps is described. This experience has guided our development of an effective and efficient software package for control point selection and TPS warping of digital images.

On the rate of growth of condition numbers for convolution matrices

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 W_{ij} = w_{i-j+1}, 1 \leq i-j + 1 \leq k, W_{ij} = 0 otherwise), this condition number defines the stability 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 band-limited 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 mp, where m is the column dimension of W and p is the order of the zero of the amplitude spectrum.

Matched‐field processing in a range‐dependent environment

Matched‐field processing (MFP) is being considered increasingly for three‐dimensional (3‐D) localization of an acoustic source in a noisy ocean environment. MFP consists of comparing the measured acoustic field to the full field computed using a geophysical and a propagation model. Most MFP implementations have involved only range‐independent propagation models, and many have been restricted to vertical arrays. However, many realistic environments cannot be adequately described by range‐independent models, and the problem of localization using more general arrays is of increasing interest. In this paper, a technique is described for range‐dependent MFP with arbitrary arrays, where the field is computed using a parabolic equation (PE) approximation. Using PE, two‐dimensional (2‐D) field values are computed for each sensor in the array for a set of possible source ranges, depths, and (N) bearings to form an N×2‐D field model. Discrete estimates of the position of the source are obtained by applying MFP, wit...

Estimation of Geoacoustic Parameters from Narrowband Data Using a Search-Optimization Technique

Geoacoustic parameters were estimated for vertical array data from the matched-field inversion benchmark data sets. Separate inversions were performed for narrowband data at 25 Hz, 50 Hz and 75 Hz, using a matching function consisting of the incoherent sum of the Bartlett outputs for the five vertical arrays at ranges of 1, 2, 3, 4 and 5 km. Parameter estimation was performed using a parabolic equation sound propagation algorithm to generate the replica fields, and a search-optimization technique to obtain estimates of the optimized parameter values. This technique involved an initial search stage in which the parameter space was sampled, and a second optimization stage in which each of a specified number of the best matches found in the search stage was used as the starting point for optimization. This approach provided multiple independent estimates of the geoacoustic parameters, and allowed assessment of the non-uniqueness of the problem and the sensitivity of the matching function to the individual parameters. A method was developed to combine the results for several frequencies to estimate the parameters. It used a weighted average with weights computed on the basis of the relative sensitivities at those frequencies; these sensitivities were estimated by the root-mean-square (RMS) gradient observed during the optimizations. Strong interdependencies among the parameters were found in the analysis, particularly between the sediment thickness and the sound speed at the bottom of the sediment. For the single-frequency matching function used here, it was observed that the inversion problems were ill-posed in that sets of parameter values from a wide region of the parameter space gave essentially perfect matches. The consistency of the parameter estimates was greatly improved by including a regularization term in the matching function. Regularized search-optimization provided an efficient method for estimating an effective geoacoustic model for acoustic field prediction.

Prewhitening for improved detection by matched‐field processing in ice‐ridging correlated noise

Correlated ambient noise at an array can severely impair the ability of matched‐field processing and conventional processing to detect and localize an acoustic source. When components of the correlated noise field are stationary, however, their effects can be reduced through the technique of prewhitening the data covariance matrix. Presented here are the results of a series of simulations in which prewhitening was applied to synthetic data generated for a correlated noise field. This study is relevant to prewhitening in noise fields that arise from distant localized biological sources, distant shipping, or ice ridging at low frequencies where the noise source remains spatially stable for hours. The present simulations modeled noise from ice ridging in the Arctic. In these simulations, a slanted linear array spanning the water column of a range‐independent environment was used to enable full 3‐D localization. The ability of prewhitening to enhance signal detection for low signal‐to‐noise cases was demonstr...

Comparison of Algorithms for High Resolution Deconvolution of Array Beamforming Output

An estimate of a signal may be obtained by applying beamforming or spectral estimation techniques to the output from an array of sensors in the field. With conventional beamforming techniques the resolution of discrete signals (spikes) is limited by the finite aperture of the array. This limitation has motivated the development of high resolution beamforming methods such as maximum likelihood, maximum entropy and eigenvector-based procedures (Johnson, 1982). We have recently examined an alternative approach to high resolution signal estimation which involves deconvolving the beam function from the output of a conventional beamformer. Based on a comparison of three deconvolution algorithms it is concluded that the most effective procedure is an inverse Fourier transform method based on the L1 (least absolute values) norm.

Efficient estimation of the probability that a source track is examined in a matched-field processing tracking algorithm

Tracking techniques may be used to reduce the ambiguity of an acoustic source’s position in matched-field processing, particularly for low signal-to-noise ratios. An efficient tracking algorithm was recently described, in which the performance would be comparable to that of an exhaustive tracker; that is, the probability that the source track is examined is close to unity for cases of interest. The current paper describes an efficient technique to estimate the probability that the source track is examined. The procedure involves the use of noise-only data to define statistical thresholds for the strongest peaks in the ambiguity surface. Once these thresholds are defined, the required probabilities may be estimated by evaluating the ambiguity function for signal-plus-noise data within a small region surrounding the known source location, and identifying any peaks exceeding the thresholds. Results of simulations obtained for a slanted array indicate that this approach provides an effective way to estimate t...