Formulation of nonlinear inversion

The tomographic inversion is usually done through the optimization process. Given a slowness model, we can simulate the wave propagation between two wells and compute the first arrival traveltime, denoted by $\{\tau_i(r);\ i=1,2,\cdots,N\}$.We want to find a set of parameters $\{m_j;\ j=1,2,\cdots,M\}$ that minimizes the differences between the traveltimes picked from the recorded data and the traveltimes computed from the model, subjected to some linear constraints. The mathematical description of this problem is as follows:  

$$\min_{\{m_j\}}\left\{\sum^N_{i=1} \int_{r}\left[\tau_i({\bf m})-t_i\right]^2dr+ \Vert{\bf B}{\bf m}+{\bf c}\Vert^2\right\},$$ (2)

where ${\bf m}^T=(m_1\ m_2\ \cdots\ m_M)$ is the unknown slowness parameter vector. Matrix ${\bf B}$ and vector ${\bf c}$ are chosen to constrain the inversion process. Because the traveltimes are generally nonlinear functions of the slowness model parameters, equation (2) defines a nonlinear optimization problem. A nonlinear optimization is usually accomplished iteratively. In each iteration, we linearize the traveltime functions as follows:

$$\tau_i(r) \approx \hat{\tau}_i(r)+\sum^M_{j=1}{\partial \tau_i \over \partial m_j}(m_j-\hat{m}_j),$$ (3)

where $\{\hat{m}_j;\ j=1,2,\cdots,M\}$ are the slowness model parameters obtained in the previous iteration, and $\{\hat{\tau}_i(r);\ i=1,2,\cdots,N\}$ are the traveltimes calculated from the previous model. We define $\delta m_j=m_j-\hat{m}_j$ and $\delta t_i(r) = t_i(r)-\hat{\tau}_i(r)$, and then form two vectors as follows:

$${\bf \Delta m}^T = \pmatrix{\delta m_1 & \delta m_2 & \cdots & \delta m_M},$$

$${\bf \Delta t}^T = \pmatrix{\delta t_1(r) & \delta t_2(r) & \cdots & \delta t_N(_r)}.$$

We also define ${\bf A}$ to be an $N\times M$ matrix with the elements

$$a_{ij}(r)={\partial \tau_i \over \partial m_j}$$

and define

$${\bf \Delta c} = {\bf B}\hat{\bf m}-{\bf c}.$$

Then, equation (2) can be approximated as  

$$\min_{\{m_j\}}\left\{\int_{r}\Vert{\bf A}{\bf \Delta m}-{\bf... ...^2dr+ \Vert{\bf B}{\bf \Delta m}-{\bf \Delta c}\Vert^2\right\}.$$ (4)

Now the problem becomes a linear least squares problem. We can use a gradient method to solve it. If we use the steepest descent method, we need to calculate the gradient vector defined as follows:  

$${\bf g}={\bf g}^{(1)}+{\bf g}^{(2)} =\int_{r}{\bf A}^T {\bf \Delta t}dr+{\bf B}^T{\bf \Delta c}.$$ (5)

This vector has a dimension of M. If we choose the conjugate gradient method, we need to calculate, in addition to the gradient vector, the conjugate gradient vector defined as follows:

$${\bf h}(r)=\pmatrix{{\bf h}^{(1)}(r) \cr {\bf h}^{(2)}(r)} =\pmatrix{{\bf A} \cr {\bf B}} {\bf g},$$ (6)

which has a dimension of N plus the number of conditions.