Estimation of apparent slowness vector for a plane wave using data from a three- component seismic array: A statistical problem involving nuisance parameters
This paper is concerned with computational algorithms for estimating the propagation parameters of a plane seismic wave using data from a small-aperture three-component seismic array. This problem is treated as a statistical one involving nuisance parameters arising from a significantly correlated (and frequently coherent) noise that affects the seismometers and an absence of complete prior information on the waveform to be analyzed. The wave is represented by two models: one assumes a stationary random process with a spectral density that is a function of a finite-dimensional nuisance parameter, while the other is assumed to be a time series that is completely unknown to the observer and which has a time averaged autocorrelation function of time (which is unknown as well). For each of these models we construct consistent and asymptotically normal algorithms for estimating the propagation parameters of a wave, the algorithms obeying the asymptotic requirements. The estimators are compared for large and small signal-to-noise ratios. Theoretical analysis and practical tests show that one can estimate the apparent slowness vector with much higher accuracy by using the designed algorithms than with the help of spatial spectrum analysis algorithms conventionally used for array data processing. These statistically optimal algorithms are most efficient when the array noise contains a strong coherent component.