This paper examines the problem of recovering the acoustic impedance from a band‐limited normal incidence reflection seismogram. The convolutional model for the seismogram is adopted at the outset, and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible. In the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace (that is, the band‐limited version of the reflectivity function) into the standard recursion formula for the acoustic impedance. The formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance. Moreover, the averaging function is identical to that used in deconvolving the initial seismogram. The advantage of these averages is that they are unique; their disadvantage is that low‐frequency information, which is crucial to making a geologic interpretation, is missing. We next present two methods by which the missing low‐frequency information can be recovered. The first method is a linear programming (LP) construction algorithm which attempts to find a reflectivity function made of isolated delta functions. This method is computationally efficient and robust in the presence of noise. Importantly, it also lends itself to the incorporation of impedance constraints if such geologic information is available. A second construction method makes use of the fact that the Fourier transform of a reflectivity function for a layered earth can be modeled as an autoregressive (AR) process. The missing high and low frequencies can thus be predicted from the band‐limited reflectivity function by standard techniques. Stability in the presence of additive noise on the seismogram is achieved by predicting frequencies outside the known frequency band with operators of different orders and extracting a common signal from the results. Our construction algorithms are shown to operate successfully on a variety of synthetic examples. Two sections of field data are inverted, and in both the results from the LP and AR methods are similar and compare favorably to acoustic impedance features observed at nearby wells.

An algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its Fourier components. It is shown that as little as 20–25 percent of the Fourier spectrum is sufficient in practice for a high‐quality reconstruction. The method employs linear programming to minimize the [Formula: see text]‐norm of the output, because minimization of this norm favors solutions with isolated spikes. Given a wavelet, this technique can be used to perform deconvolution of noisy seismograms when the desired output is a sparse spike series. Relative reliability of the data is assessed in the frequency domain, and only the reliable spectral data are included in the calculation of the spike series. Equations for the unknown spike amplitudes are solved to an accuracy compatible with the uncertainties in the reliable data. In examples with 10 percent random noise, the output is superior to that obtained using conventional least‐squares techniques.

The residual wavelet on a processed seismic section is often not zero phase despite all efforts to make it so. In this paper we adopt the convolutional model for the processed seismogram, assume that the residual phase shift can be approximated by a frequency‐independent constant, and use the varimax norm to generate an algorithm to estimate the residual phase directly. Application of our algorithm to reflectivities from well logs suggests that it should work in the majority of cases so long as the reflectivity is non‐Gaussian. An application of our algorithm to stacked data enhances the interpretability of the seismic section and leads to an improved match between the recovered relative acoustic impedance and a measured velocity log.

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JONES, I.F. and LEVY, S. 1987, Signal-to-Noise Ratio Enhancement in Multichannel Seismic Data via the Karhunen-Loeve Transform, Geophysical Prospecting 35, 12-32.The Karhunen-Loeve transform, which optimally extracts coherent information from multichannel input data in a least-squares sense, is used for two specific problems in seismic data processing.The first is the enhancement of stacked seismic sections by a reconstruction procedure which increases the signal-to-noise ratio by removing from the data that information which is incoherent trace-to-trace. The technique is demonstrated on synthetic data examples and works well on real data. The Karhunen-Loeve transform is useful for data compression for the transmission and storage of stacked seismic data.The second problem is the suppression of multiples in CMP or CDP gathers. After moveout correction with the velocity associated with the multiples, the gather is reconstructed using the Karhunen-Loeve procedure, and the information associated with the multiples omitted. Examples of this technique for synthetic and real data are presented.