Sonic well logging methods and apparatus utilizing parametric inversion dispersive wave processing

Methods and related apparatus for generating dispersion curves and conducting shear slowness logging are set forth. The methods include backpropagating detected dispersive waveforms in the Fourier domain while accounting for dispersion in an anisotropic and inhomogeneous formation, and stacking the processed waveforms. The stacking may occur in the frequency or time domains. The semblance of the stacked and windowed waveforms is compared against the detected waveforms. In accounting for dispersion during backpropagation, a plurality of parameters which describe dispersion curves are varied over ranges in order to optimize semblance. At each depth of interest, it is then possible to obtain a dispersion curve of greatest semblance. Where both an x-dipole and y-dipole are used to generate the waves, a dispersion curve of greatest semblance for each may be generated. In addition, from the dispersion curves, formation shear slowness(es) may be plotted as a function of formation depth and orientation.