Frequency‐domain acoustic‐wave modeling and inversion of crosshole data: Part I—2.5-D modeling method

Abstract

Crosshole experiments usually have sources and receivers confined to a plane, and it is assumed that there is negligible variation in the properties of the medium normal to this plane. Therefore, the problem appears two‐dimensional, except for the sources which are 3-D points rather than lines. This configuration is denoted as two‐and‐one‐half‐dimensional. We present a frequency‐domain approach to modeling acoustic wave propagation in such situations which allows correct treatment of point sources but takes advantage of the assumed 2-D nature of the medium to avoid full 3-D simulations. The approach uses a Fourier transform with respect to the out‐of‐plane coordinate to reduce the problem of modeling in 3-D to repeatedly solving a 2-D equation, which we accomplish using finite differences. The discrete inverse Fourier transform from the out‐of‐plane wavenumber implies the existence of an infinite number of spurious “ghost sources” spaced periodically in the out‐of‐plane direction. These sources generate significant artifacts on the time‐domain traces, even when their spatial period is much greater than the (in‐plane) dimensions of the survey area, because of time‐wrapping in the transform from the frequency domain. We describe two methods for reducing these artefacts, the more effective of which entails exponential damping by adding a positive imaginary part to the frequency, compensated by ramping of the wrapped time domain records. We check the modeling scheme by analysis of direct and scattered arrivals from simple models. The observed seismograms agree well with those calculated using Born theory, and so confirm the potential of this modeling method for use in inversion.