Time-to-depth conversion and seismic velocity estimation using time-migration velocity

Time Migration Velocity

Kirchhoff prestack time migration is commonly based on the following travel time approximation (Yilmaz, 2001). Let $\mathbf{s}$ be a source, $\mathbf{r}$ be a receiver, and $\mathbf{x}$ be the reflection subsurface point. Then the total travel time from $\mathbf{s}$ to $\mathbf{x}$ and from $\mathbf{x}$ to $\mathbf{r}$ is approximated as

$$T(\mathbf{s},\mathbf{x})+T(\mathbf{x},\mathbf{r}) \approx\hat{T}(\mathbf{x}_0,t_0,\mathbf{s},\mathbf{r})$$ (1)

where $\mathbf{x}_0$ and $t_0$ are effective parameters of the subsurface point $\mathbf{x}$ . The approximation $\hat{T}$ usually takes the form of the double-square-root equation

$$\hat{T}(\mathbf{x}_0,t_0,\mathbf{s},\mathbf{r})= \sqrt{t_0^2+\fra... ...qrt{t_0^2+\frac{\vert\mathbf{x}_0-\mathbf{r}\vert^2}{v_m^2(\mathbf{x}_0,t_0)}},$$ (2)

where $\mathbf{x}_0$ and $t_0$ are the escape location and the travel time of the image ray (Hubral, 1977) from the subsurface point $\mathbf{x}$ . Regarding this approximation, let us list four cases depending on the seismic velocity $v$ and the dimension of the problem:

2-D and 3-D, velocity $v$ is constant.

Equation 2 is exact, and $v_m=v$ .

2-D and 3-D, velocity $v$ depends only on the depth $z$ .

Equation 2 is a consequence of the truncated Taylor expansion for the travel time around the surface point $\mathbf{x}_0$ . Velocity $v_m$ depends only on $t_0$ and is the root-mean-square velocity:

$$v_m(t_0)=\sqrt{\frac{1}{t_0}\int_0^{t_0}v^2(z(t))dt}.$$ (3)

In this case, the Dix inversion formula (Dix, 1955) is exact. We formally define the Dix velocity $v_{Dix}(t)$ by inverting equation 3, as follows:

$$v_{Dix}(t)=\sqrt{\frac{d}{d\,t_0}\left(t_0 v_m^2(t_0)\right)}\;.$$ (4)

2-D, velocity is arbitrary.

Equation 2 is a consequence of the truncated Taylor expansion for the travel time around the surface point $x_0$ . Velocity $v_m(x_0,t_0)$ is a certain kind of mean velocity, and we establish its exact meaning in the next section.

3-D, velocity is arbitrary.

Equation 2 is heuristic and is not a consequence of the truncated Taylor expansion. In order to write an analog of travel time approximation 2 for 3-D, we use the relation (Hubral and Krey, 1980)

$$\tensor{\Gamma}=[v(\mathbf{x}_0)\tensor{R}(\mathbf{x}_0,t_0)]^{-1},$$ (5)

where $\tensor{\Gamma}$ is the matrix of the second derivatives of the travel times from a subsurface point $\mathbf{x}$ to the surface, $\tensor{R}$ is the matrix of radii of curvature of the emerging wave front from the point source $\mathbf{x}$ , and $v(\mathbf{x}_0)$ is the velocity at the surface point $\mathbf{x}_0$ . For convenience, we prefer to deal with matrix $\tensor{K}\equiv\tensor{\Gamma}^{-1}$ , which, according to equation 5 is

$$\tensor{K}(\mathbf{x}_0,t_0)\equiv v(\mathbf{x}_0)\tensor{R}(\mathbf{x}_0,t_0).$$ (6)

The travel time approximation for 3-D implied by the Taylor expansion is

$$\hat{T}(\mathbf{x}_0,t_0,\mathbf{s},\mathbf{r})$$

$$= \sqrt{t_0^2+t_0(\mathbf{x}_0-\mathbf{s})^T [\tensor{K}(\mathbf{x}_0,t_0)]^{-1}(\mathbf{x}_0-\mathbf{s})}$$ (7)

$$+ \sqrt{t_0^2+t_0(\mathbf{x}_0-\mathbf{r})^T [\tensor{K}(\mathbf{x}_0,t_0)]^{-1}(\mathbf{x}_0-\mathbf{r})}.$$

The entries of the matrix $\frac{1}{t_0}\tensor{K}(\mathbf{x}_0,t_0)$ have dimension of squared velocity and can be chosen optimally in the process of time migration. It is possible to show, however, that one needs only the values of

$$\det\left(\frac{\partial }{\partial t_0}\tensor{K}(\mathbf{x}_0,t_0)\right)$$ (8)

to perform the inversion. This means that the conventional 3-D prestack time migration with traveltime approximation 2 provides sufficient input for our inversion procedure in 3-D. The determinant in equation 8 is well approximated by the square of the Dix velocity obtained from the 3-D prestack time migration using the approximation given by equation 2.

One can employ more complex and accurate approximations than the double-square-root equations 2 and 7, i.e. the shifted hyperbola approximation (Siliqi and Bousquié, 2000). However, other known approximations also involve parameters equivalent to $v_m$ or $\tensor{K}$ .

Time-to-depth conversion and seismic velocity estimation using time-migration velocity