VTI migration velocity analysis using RTM

Physical interpretation and implementation of the DSO gradient

In this subsection, we interpret each term in the DSO gradient formulation, and provide the readers with some hints for implementation. We find the Lagrangian formulation is easier to interpret, and readers can clearly relate the corresponding terms to the adjoint formulation. We will only discuss the physical meaning and the implementation for the first term in the gradient (Equation 28 and 25). Then similar reasoning can be argued using reciprocity.

First, for each image slice in the subsurface-offset domain $I_{\bf h}$ , we compute a weighted image $\gamma_{\bf h}$ using equation 27. Then we move on to equation 25. We can rearrange the independent and commutable operators as follows:

$${\bf L^*} (c) \bf\lambda = \sum_{\bf h} {\bf S}_{-{\bf h}} \left ({\bf S}_{-{\bf h}} {\bf M}^* {\bf q}\right) \gamma_{\bf h}.$$ (33)

Operator ${\bf M}^*$ corresponds to differentiating $q_V$ and $q_H$ once reversely in time and setting $u_x, u_y$ , and $u_z$ fields to zero. Notice that the directions of propagation and differentiation in time of wavefield ${\bf q}$ are the same. Therefore, we can compute the time derivative during the same process as the propagation. Then we shift the reverse-time derivative ${\bf q}$ by $-{\bf h}$ in ${\bf x}$ , and multiply it with the weighted image $\gamma_{\bf h}$ . This product is shifted again by $-{\bf h}$ . Finally, we sum over the contributions from all subsurface-offset image slices to get an effective source term ${\bf f}_p$ . Next, we solve equation 33 for $\bf\lambda$ backward in time, using ${\bf f}_p$ as the source.

At the same time, in equation 28 $-\frac{\partial {\bf L}}{\partial c}$ is a sparse matrix, with non-zero elements only for $p_V$ and $p_H$ . We can therefore write everything out explicitly:

$$\left ( \nabla_c J \right )_1 = \int_{0} ^{t_{max}} \frac{2}{c^3} \left[ (\partial_t p_H) \lambda_{H} + (\partial_t p_V) \lambda_{V} \right] d t.$$ (34)

The explicit forms for the complete gradients are:

$$\nabla_c J = \int_{0} ^{t_{max}} \frac{2}{c^3} \left[ (\partial_t p_H) \lambda_{H} + (\partial_t p_V) \lambda_{V} \right] d t$$

$$+ \int_{0} ^{t_{max}} \frac{2}{c^3} \left[ (\partial_t q_H) \mu_{H} + (\partial_t q_V) \mu_{V} \right] d t$$ (35)

and

$$\nabla_{\epsilon} J = -\int_{0} ^{t_{max}} [( \partial_x v_x) \lambda_{H} + (\partial_y v_y ) \lambda_{H}] d t$$

$$+\int_{0} ^{t_{max}} [( \partial_x q_{H}) \mu_x + (\partial_y q_{H}) \mu_y ] d t.$$ (36)

VTI migration velocity analysis using RTM