Application of different inverse methods for combination of vS and vGPR data to estimate porosity and water saturation

Inverse problem is one of the most important problems in geophysics as model parameters can be estimated from the measured data directly using inverse techniques. In this paper, applying different inverse methods on integration of S-wave and GPR velocities are investigated for estimation of porosity and water saturation. A combination of linear and nonlinear inverse problems are solved. Linear least-squares and conjugate gradient are used as linear techniques, whereas grid search and Newton methods are selected as nonlinear ones. It is understood that vS depends on density and Lame Constant (shear modulus) and vGPR on dielectric constant. This combination seems to be logical. Shear modulus is related to porosity using Bruggeman’s rule. Density and dielectric constant is also related to porosity and water saturation. This implies that vS and vGPR are bivariate functions of porosity and water saturation, which are our unknown model parameters. The model parameters are estimated to minimize the cost functional ora system of the equations. In order to convert the nonlinear problem into the linear form, taking logarithm and changing variables were used. The problem was convex, which was inferred from the linear form, so there was just one local minimum as the global minimum of the problem. The grid search method shows that porosity and water saturation cannot be estimated by vGPR or vS uniquely. The results of the four methods were compared with each other and a good agreement was observed.