Recovery of the acoustic impedance from reflection seismograms

Abstract

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.