COMPARING THE DIFFERENT SCHEMES

I use the same simple synthetic model as in Cunha (1990) to compare the results obtained with different inversion schemes. The model consists of an isotropic layer in an isotropic background (V^z=V^x in both the layer and the background). The layer is not centered, and its size is not a multiple of the layer size used in the inversion schemes.

 

isoinv
Figure 3 Inversion results for the one-layer isotropic model, using different algorithms: (a) square function decomposition (nonlinear), (b) Walsh function decomposition (nonlinear), (c) linearized inversion with layer-parametrization, and (d) the same as in c, but with the derivative term introduced into the objective function.continuous line represents V_x and dotted line represents V_z.

Figures a and b show that, contrary to what we would expect, the results obtained with the Walsh functions are inferior to the ones obtained with the sine- and cosine-like square functions. Although Walsh functions form a complete set, their orthogonality impedes further steps from correcting errors in previously estimated components of the model in this kind of iterative scheme. Moreover, both nonlinear schemes are inferior to the linearized algorithms shown in Figures c and d. The result in c was obtained with an iterative scheme that uses the least-squares solution of equation (8) by successively increasing (by a power of 2) the number of layers at each step. The difference between c and d is the introduction into the objective function of an extra term that corresponds to the difference between the traveltime-derivatives relative to the receiver positions. As a result the solution for V^z in d becomes more stable since with this term the inversion will try to fit not only the traveltimes, but also the form of the traveltime curves.