TI MOVEOUT IN TERMS OF LATERAL HETEROGENEITY

In a simple model with one horizontal reflector, the anisotropic effect of the group velocity changing with the angle of propagation is somewhat similar to the effect of lateral heterogeneity. In this section, we address the question of whether nonhyperbolic moveout in isotropic weakly heterogeneous model can mimic that in a homogeneous weakly anisotropic model. The analysis follows the results of Grechka 1996.

The angle dependence of the group velocity in formulas (1) and (9) is characterized by small anisotropic coefficients. Therefore, we can assume that an analogous effect of lateral heterogeneity might be cause by a small velocity perturbation. The appropriate model is a laterally heterogeneous (LH) medium with velocity  

$$V(x) = V_0\,\left[ 1 + c(x) \right]\;,$$ (85)

where $\vert c(x)\vert \ll 1$ is a dimensionless function. The velocity function in formula (85) has the generic perturbation form that allows us to use the tomographic linearization assumption. That is, we neglect the ray bending caused by the small velocity perturbation c and compute the perturbation of traveltimes along straight rays in the constant velocity V₀. Thus, we can rewrite equation (13) for this case as  

$$t(h) = {\sqrt{z^2 + h^2} \over {2\,h}}\,\int_{y-h}^{y+h}{ d\xi \over V_z(\xi) }\;,$$ (86)

where y is the midpoint location, and the integral limits correspond to the source and receiver locations. For simplisity, and without loss of generality, we can set y to zero. Linearizing with respect to the small perturbation c(x), we get  

$$t(h) = { \sqrt{z^2 + h^2} \over V_0 } \left[ 1 - {1 \over {2\,h}} \int_{-h}^{h} c(\xi) d\xi \right]\;.$$ (87)

From the form of equation (87) it is clear that lateral heterogeneity can cause many different types of nonhyperbolic moveout shapes. In particular, comparing equations (87) and (15), we conclude that a pseudo-anisotropic behavior of traveltimes is caused by lateral heterogeneity of the form  

$$c(h) = { d \over {d h}} \left[{ {h^3 (h^2 \epsilon + z^2 \delta )} \over {(h^2 + z^2)^2} } \right]$$ (88)

or, in the linear approximation,  

$$c(h) = \left[\delta\,t_0^2\,V_n^2\,h^2\,(3 t_0^2 V_n^2 - h^2... ...0^2 V_n^2 + h^2) \right] / \left(t_0^2 V_n^2 + h^2 \right)^3\;,$$ (89)

where $\delta$ and $\epsilon$ should be considered now as the parameters of the isotropic lateral heterogeneous velocity field. Equation (89) indicates that the velocity heterogeneity c(x), reproducing moveout (16) in a homogeneous TI medium, is the symmetric function of the offset h. It is not surprising because the velocity function (1), corresponding to transverse isotropy, is symmetric as well.

For more details on the relation between lateral heterogeneity and transevse isotropy in interpreting P-wave reflection moveout, see Grechka (1996).