Linearized Inversion

Equation (1) can be linearized for small perturbations around the homogeneous solution as follows:

$$\begin{eqnarray} t^0_{i} & = & \sum_{j=J_s}^{J_r} t^0_{ij}, \mbox{\hspace{1.cm}... ... \sum_{j=J_s}^{J_r} {\partial t^0_i \over \partial M^z_j} d\!M^z_j\end{eqnarray}$$ (6)

Using this approximation in equation (2) and minimizing it with respect to all $d \! M$s leads to the following set of linear equations:

$$\pmatrix{ \bf y^x \cr \bf y^z } = \pmatrix{\bf A^{xx} & \bf ... ... \bf A^{zx} & \bf A^{zz}} \pmatrix{\bf d\!M^x \cr \bf d\!M^z},$$

where each element ${\bf A^{\eta \zeta}}$ is an $L \times L$ matrix, and ${\bf y^{\eta}}$ and ${\bf d\!M^{\zeta}}$ are column vectors with L elements, L being the number of layers (or cells) of the model. The elements of the matrix and vectors are given by the equations  

$$\begin{array} {rclcrcl} y^x_j & = & {\displaystyle \sum_i{(t... ...box{\hspace{1.5cm}} & A^{zx}_{jk} & = & A^{xz}_{kj}.\end{array}$$ (7)

Michelena (1991) solved the linearized problem using a conjugate gradients routine. He obtained a stable solution by stopping the process after a prespecified number of iterations, thus avoiding the problems arising from the ill-conditioning of matrix ${\bf A}$. Another way to build a more stable algorithm is to introduce an angle-dependent weighting factor into equation (2) and to apply a weak constraint to the inversion:  

$${\cal F} = \sum_{i=1}^N \left( {t_i - t^{\prime}_i \over t_... ...2 + \lambda^2 \sum_{j=1}^L \left( {d\!M_j \over M_j} \right)^2,$$ (8)

where $\alpha_i$ is the angle of the ray as measured from the horizontal.