Analysis of sparse wide-angle reflection data in the τ(p) domain

Abstract

Uncertainties in ray parameter p for events recorded with small-aperture arrays can be quite large and lead to sizable second-order errors in intercept time τ when source gathers are slant stacked into the τ- p domain. The problem becomes significant for data sets characterized by large gaps in offset and ray parameter coverage. To estimate the uncertainty in ray parameter p arising from small recording apertures, we use a statistical test to evaluate the cross-correlation of a predicted array response with the observed response extracted from the slant stack. The cross-correlation is carried out using the projections of the array responses along the ray parameter axis, following the linear trajectory defined by the average offset over which the slant stack was computed. The method adjusts automatically to the bandwidth of the arrival and flags arrivals for which the ray parameter is poorly resolved because of either high noise levels or overlapping with other arrivals in the stack. We then use the uncertainties in ray parameter to estimate second-order errors in τ. The latter are inversely proportional to curvature in the travel time ( T ) - offset ( X ) domain, or equivalently, are directly proportional to curvature in the τ- p domain. For reflections, therefore, the errors increase with angle of incidence. Errors for S -wave arrivals are smaller than errors for P -wave arrivals for a given ray path because of the stronger curvature of the S -wave travel-time branches. Second-order errors in τ for quarry blast data recorded with small-aperture arrays in eastern Pennsylvania range from 0.05 to 0.4 sec, and in general are greater than errors due to uncertainties in origin times and finite bandwidth. The uncertainties in τ associated with limited-aperture recordings therefore can have a significant effect on the tradeoff between variance and resolution of model parameters when the data are inverted for velocity structure.