AMINZADEH, F. and MENDEL, J.M. 1982, Non-Normal Incidence State-Space Model and Line Source Reflection Synthetic Seismogram, Geophysical Prospecting 30, 541-568. This paper is directed at modeling layered media. We extend the plane-wave normalincidence state-space model developed by Mendel, Nahi and Chan in 1979, to the non-normal incidence case. To do this we introduce a shifting principle, a zero-offset wavefront, and zero-offset travel times for different layers. We also develop an algorithm for obtaining a synthetic line source reflection seismogram. In this algorithm non-normal incidence planewave seismograms are summed over a range of incident angles. The algorithm is based on a modified version of Sommerfield's (1896) theorem. Simulations of acoustic and elastic media are included which illustrate the applicability of our plane-wave and line source seismograms for both elastic and acoustic cases.
An important part of the processing of vertical seismic profiling (VSP) data is the separation of upgoing and downgoing waves. I introduce a new method for separation based on a time‐domain recursive linear filter. The separation method uses an approximation to an optimal, frequency‐domain, nonlinear filter solution as the starting point. The time‐domain recursive linear (approximate) filter converges to the optimal (exact) solution. Since the computation is in the time domain and since this filter is linear, some of the temporal aliasing and other problems resulting from the forward and inverse Fourier transforms are avoided. Specifically, instability for some frequencies (spectral singularities) is not experienced here. This method uses a priori information of the opposite stepouts of the upgoing and downgoing waves. Equal spacing between borehole measurement points is not required. Further, the computational time may be controlled according to the desired accuracy. An important feature of this method is that it locates the reflecting boundaries of the subsurface. Having located the homogeneous layers, it allows variable‐length windows of traces for separation, which eliminates the undesirable effects of smearing and extending wave fields beyond their origins. Also, knowledge of acoustic impedances for accurate implementation of the optimum filter is no longer required.
We show how to include absorption effects in a normal incidence state‐space model of a layered media system. Two models are discussed: a convolutional state‐space model and a differential‐delay state‐space model. Extensive results are presented for the former model, whereas only formulative results are presented for the latter. For the convolutional state‐space model we develop absorption impulse responses in closed form for Gaussian and exponential absorption operators. Simulation results demonstrate that this model has the desired low‐pass filtering effect expected of a system with absorption losses. The differential‐delay state‐space model does not appear to have been applied before in the context of seismic modeling. We suggest it as a useful model.
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