Separation of signal and noise applied to vertical seismic profiles

Abstract

Inversion of the band‐limited one‐dimensional VSP response is nonunique because impedance functions with very different statistics produce equivalent responses. Least‐squares methods of inversion linearly transform noise and tend to produce impedance functions with a Gaussian distribution of amplitudes. I modify a least‐squares inversion procedure to exclude nonzero impedance derivatives that are significantly influenced by noise. The resulting earth model shows homogeneous intervals unless the data have reliable information to the contrary. The data are modeled with a one‐dimensional wave equation and three invertible functions: acoustic impedance, a source wavelet, and the traces’ amplification. First, a linearized least‐squares inverse perturbs the source function to model the downgoing wave. A relinearized inverse finds perturbations of all three modeling functions to account for first‐order reflections. Further iterations explain higher order reflections. To estimate the reliability of impedance perturbations, each linearized inversion is repeated for pure noise that equals or exceeds the noise in the data. Amplitude histograms are used to estimate probability density functions for the amplitudes of the signal and of the noise in the perturbations. Nonzero impedance derivatives are accepted as reliable if, according to the probability functions, the perturbations contain, with a high probability, only a small amount of noise. For a set of VSP data provided by L’Institut Francais du Petrole, four iterations allowed only a few nonzero impedance derivatives and modeled a recorded VSP as well as did a least‐squares inversion that accepted all proposed perturbations. Estimated probability densities for the remaining signal and noise were used to extract a tube wave that contained little signal.