One-dimensional iterative inversion of vertical seismic profiles

Abstract

In a vertical seismic profile, the earth's response to a surface excitation is measured by geophones located in a borehole. The objective of this research has been to use the recorded up and downgoing waves to estimate depths to reflectors and velocities below the bottom of the well and in the near surface. An inversion scheme was developed to accomplish this. It works by iteratively adjusting a forward model until its output best matches a recorded field trace (in a least squares sense). The forward model is the well known equal-traveltime model. The model consists of reflecting interfaces spaced at uniform traveltime intervals. The output of the model is adjusted by using a steepest descent technique to change the reflection coefficients characterizing the interfaces. By assuming constant density and by knowing the velocity in one layer, the reflection coefficients derived from the inversion scheme may be converted into velocity and depth estimates. This inversion scheme is stable in the presence of noise, yields unique estimates of the velocity and reflector depths, accepts bandlimited source input, and runs on a computer in a reasonable amount of time. It was successfully tested by using data from the upper half of the borehole to estimate velocities and reflector depths in the lower half. The estimates were independently checked using sonic log data. The inversion below TD gave velocity estimates in the range expected for sedimentary materials. The deepest reflector estimate at 12000 ft (3658 m) coincides with the depth of the acoustic basement estimated from surface seismic data. Inversion of the near surface velocities and reflector depths showed the expected rapid increase of velocity with depth.