Zero- and first-order approximations for least-squares estimation of seismic signal with coherent and random noise

Abstract

We have proposed a new least-squares method for signal estimation with a complicated and therefore more realistic mathematical model of the multichannel seismic record containing random noise and an arbitrary number of coherent noise wavetrains. The signal and all the coherent noise wavetrains are supposed to bear individual trace-independent waveforms being mutually uncorrelated in time stationary stochastic processes. The amplitudes and arrival times of these record components vary from trace to trace in an arbitrary manner. Random noise is assumed to be a stationary stochastic process uncorrelated with the signal and all the coherent noise wavetrains and from trace to trace as well. Its spectral (autocorrelation) function is trace independent to within a scale factor, the variance. Under certain conditions, the method may be reduced to two successive stages, namely preliminary subtraction of estimates of all the coherent noise wavetrains and final estimation of the signal from the residual record. At both stages, optimum weighted stacking is used with reference to the variances of random noise and to the amplitudes and arrival times of the corresponding coherent component. A simplified scheme and an advanced scheme for subtracting coherent noise are proposed, which are called the zero-order and first-order approximations, respectively. The first scheme can be thought of as the generalization of a conventional approach for subtracting coherent noise to the complicated data model adopted in this paper. The second scheme has an obvious advantage over the first scheme, since it allows the distortions that appear when estimating and subsequently subtracting the coherent noise wavetrains to be compensated. A simulation on synthetic data shows the efficiency of the first-order approximation, and it provides a qualitative and quantitative comparison of those results with the results given by the zero-order approximation. Also, testing the zero-order approximation exploiting the singular value decomposition on synthetic and actual data sets demonstrates the advantage of this method over f-k filtering combined with subsequent straight stacking.