Solving 1-D inverse problems by Chebyshev polynomial expansion

Abstract

Seismic wave propagation described by differential equations with variable coefficients may be solved by the Chebyshev polynomial expansion method (CPEM). This method approximates a model and forward solutions by orthonormal Chebyshev polynomials. The CPEM provides an alternative to the usual formulation. In CPEM, the model is approximated globally and the forward solutions can explicitly depend on the parameters of the model. The former ensures that a smooth model is produced by inversion. The latter produces partial derivatives of the forward solutions directly with respect to the model parameters, which streamlines the inversion and also gives a quantitative tool for determining the feasibility of inversion in the presence of noise (by singular value decomposition). Two examples of inversion demonstrate the potential of the CPEM. The first is nonlinear inversion for velocity and density using borehole SH‐wave data. The second is linear inversion for interval velocities from rms velocity data. Estimation of interval velocity from stacking velocity is illustrated using field data from the Gulf of Mexico.