Nonlinear schemes

In a previous report (Cunha, 1990) I used a set of sine- and cosine- like square functions to describe the model:  

$${\bf M} = c + \sum_{j=1}^N a_j \: {\cal S}_j + \sum_{j=1}^N b_j \: {\cal C}_j.$$ (3)

This choice of model decomposition led to an iterative perturbation scheme that estimates different components of the model at each iteration, starting with the homogeneous solution and successively increasing the frequency of the perturbation. The predicted traveltimes are given by

$$\begin{eqnarray} t^{\prime}_{i} & = & \sum_{j=J_s}^{J_r} {o}_j \sqrt{(M^x_j + {s... ..._j \, d\!M^x_2) x_{ij}^2 + (M^z_j + {s}_j \, d\!M^z_2) z_{ij}^2},\end{eqnarray}$$ (4)

where o_j, e_j, and s_j are defined as follows:  

$${o}_j = {1-(- \! 1)^j \over 2}, \mbox{\hspace{2.0cm}} {e}_j... ... \over 2}, \mbox{\hspace{2.0cm}} {s}_j = (- \! 1)^{(j+3) / 2}.$$ (5)

In the above equations j is the layer index, J_s and J_r are the indices of the source and receiver layers, x_ij and z_ij are the horizontal and vertical distances traveled inside the layer j by the ray associated with source-receiver i. Using equation (4) in the objective function (2), the inversion problem can be easily solved with a nonlinear optimization algorithm (like the Downhill Simplex method) since only four parameters are inverted at each step: $d\!M^x_1, d\!M^x_2, d\!M^z_1, d\!M^z_2$.