VTI migration velocity analysis using RTM

VTI reverse-time migration imaging condition

Traditionally, the subsurface image is often considered as the first gradient of an FWI objective function with respect to velocity. In this paper, we are going to derive the VTI reverse-time migration imaging condition according to the same criteria.

We define FWI objective function as

$$J_{\mbox w} = \frac{1}{2} \langle d - d_{\mbox {est}}, d - d_{\mbox {est}} \rangle,$$ (6)

where $d_{\mbox {est}}$ is the data estimated from the current model, which is sampled from wavefield $\bf p$ , and $d$ is the recorded data.

For the first iteration, $d_{\mbox {est}}=0$ . Therefore the first gradient in velocity is:

$$\nabla_c J_{\mbox w} = \left(\frac{\partial {\bf p}}{\partial c}\right)^* d$$

$$= (-{\bf L}^{-1} \frac{\partial {\bf L}}{\partial c} {\bf L}^{-1} f )^* d.$$ (7)

Now we introduce the receiver vector field ${\bf q} = (u_x, ~u_y, ~u_z, ~q_V, ~q_H)^T$ , which is the solution of the following equation:

$${\bf L^*}(c) {\bf q} = {\bf f~'}.$$ (8)

The equivalent source term in equation 8 is defined as ${\bf f~'} = (0, ~0, ~0, ~f'_V, ~f'_H)^T$ , where $f'_V =f'_H=d$ . From equation 5, we have

$$\frac{\partial {\bf L}} {\partial c} = \left \vert \begin{array}{... ...t & 0 \\ 0 & 0 & 0 & 0 & -\frac{2}{c^3} \partial_t \\ \end{array} \right \vert,$$ (9)

If we plug equation 8 and 9 into equation 7 and ignore the velocity dependence, we arrive at the imaging condition as follows:

$$I = {\bf p}^* {\bf M}^* {\bf q},$$

$$= ({\bf Mp})^* {\bf q},$$ (10)

where

$${\bf M} = \left \vert \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0... ...ial_t & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2}\partial_t \\ \end{array} \right \vert.$$ (11)

The explicit form of this imaging condition for acoustic RTM is:

$$I = \int_{0} ^{t_{max}} \frac{1}{2} \left ( (\partial t p_H) q_H + (\partial t p_V) q_V\right ) d t.$$ (12)

The scaling factor $\frac{1}{2}$ is chosen to make sure that when $p_H=p_V$ , equation 12 reduces to the isotropic cross-correlation imaging condition (Claerbout, 1987). For the purpose of velocity analysis, we often work with extended images and generalized imaging conditions. Similarly, we define our subsurface-offset-domain common-image gathers (SODCIGs) ${\bf I}$ as a column vector:

$${\bf I} = [I_{-{\bf h_{\rm max}}},I_{-{\bf h_{\rm max}}+\Delta {\... ...ts,I_{\bf0},\cdots,I_{{\bf h_{\rm max}}-\Delta {\bf h}},I_{\bf h_{\rm max}}]^*,$$ (13)

where ${\bf h}$ is the half-subsurface offset, which ranges from ${\bf -h_{\rm max}}$ to ${\bf h_{\rm max}}$ with an increment of $\Delta {\bf h}$ . For each element $I_{\bf h}$ , the extended imaging condition is as follows (Sava and Formel, 2006) :

$$I_{\bf h} = ({\bf S}_{+{\bf h}} {\bf p})^* {\bf M}^* ({\bf S}_{-{\bf h}} {\bf q}),$$ (14)

where ${\bf S}_{+{\bf h}}$ is a shifting operator which shifts the wavefield by an amount of $+{\bf h}$ in the ${\bf x}$ direction. Notice that $({\bf S}_{+{\bf h}})^* = {\bf S}_{-{\bf h}}$ .

VTI migration velocity analysis using RTM