Uniqueness of auto-correlation curves

The auto-correlation inversion has basically the same limits as the dispersion curve inversion, as auto-correlation curves are calculated from dispersion curves: non-uniqueness, loss of resolution with depth and equivalence for profiles with low velocity zones. As we plan to invert auto-correlation curves to obtain $V_s$ profiles, we first address the question of the relationship between auto-correlation and dispersion curves. Obviously, equation (3.49) does not insure a one-to-one relation between the two types of curves, as the arguments for $J_0(x)$ that satisfy equation (3.49) can be numerous for small values of $\overline{\rho(r,\omega)}$ . However, equation (3.49) does not imply any coupling of $c(\omega)$ with the auto-correlation at other frequencies than $\omega$ , meaning that the inversion can be made independently frequency by frequency. Consequently, transforming auto-correlation curves at frequency $\omega$ into their equivalent common dispersion curve is just a matter of solving a system of equations of the same form as (3.49) (one equation by considered ring) and solutions $c(\omega)$ are discrete numbers. If all the auto-correlation curves for the different rings are consistent with each other, there is a minimum of one solution that satisfies all apertures. From the discrete nature of the solutions and the number of rings likely to be considered, there is little chance of having two distinct solutions for $c(\omega)$ that perfectly match all equations.

Figure 5.11: Reference model for auto-correlation inversion. (a) $V_p$ profiles: input average model (plain line), input standard deviations (dotted lines) and generated random models ranked by their auto-correlation misfit (common grey scale). (b) $V_s$ Profiles: same legend as for $V_p$ . (c) Dispersion curves of random models for the fundamental mode of Rayleigh. (d) to (h) Auto-correlation ratios for chosen rings plotted against frequency, average and standard deviation for all samples (dots).