Vertical Heterogeneity plus Anisotropy

In a model that includes vertical heterogeneity and anisotropy, both factors affect bending of the rays. However, the weak anisotropy approximation allows us to neglect the effect of anisotropy on ray trajectories and to consider its effect on traveltimes only. This assumption is analogous to the linearization concept, conventional for tomographic inversion. Its application to weak anisotropy has been discussed by Grechka and McMechan 1996. According to the linearization assumption, we can retain isotropic formula (24) as describing the ray trajectories and rewrite formula (23) in the form  

$$t(p) = \int_{0}^{z}\,{{dz} \over {V_g(z,\psi(z))\,\cos{\psi(z)}}}\;,$$ (43)

where V_g is the anisotropic group velocity, which varies both with depth and with the ray angle $\psi$ and has the expression (1). Differentiation of the parametric traveltime formulas (43) and (24) and linearization with respect to Thomsen's anisotropic parameters shows that the general form of equations (27) through (30) remains valid if we change the definition of the root-mean-square velocity V_rms and the parameters S₂ and S₃, as follows:

$$\begin{eqnarray} V_{rms}^2 & = & {1 \over t_z}\,\int_{0}^{t_z}\,V_z^{2k}(t)\, \l... ..., \\ S_k & = & {M_k \over {V_{rms}^{2k}}}\;\;(k = 2, 3, \ldots)\;.\end{eqnarray}$$ (44) (45) (46)

It is easy to verify that in the homogeneous case, expressions (44) through (46) transform series (26) with coefficients (27) through (30) to the form equivalent to series (17). Two important conclusions follow from the mathematical form of equations (44) and (45). First, we see that if the mean value of the anisotropic coefficient $\delta$ is less than zero, the presence of anisotropy can reduce the difference between the effective root-mean-square velocity and the effective vertical velocity V_z=z/t_z. In this case, the effects of anisotropy and heterogeneity partially cancel each other, and the moveout curve behaves at small offsets so as if the medium were homogeneous and isotropic. This behavior has been noticed by Larner and Cohen 1993. On the other hand, if the anelliptic parameter $\eta$ is positive and different from zero, it can significantly increase the values of the heterogeneity parameters S_k. In this case, the nonhyperbolicity of reflection moveouts at large offsets is stronger than in the isotropic case.

To exemplify the general theory, let us consider a simple analytic model with constant anisotropy parameters and a vertical velocity linearly increasing with depth according to the equation  

$$V_z(z) = V_z(0)\,(1 + \alpha\,z) = V_z(0)\,e^{\kappa(z)}\;,$$ (47)

where $\kappa$ is the logarithm of the velocity change. In this case, the analytic expression for the RMS velocity V_rms is found according to formula (44) to be  

$$V_{rms}^2 = V_z^2(0)\,(1 + 2\,\delta)\,{{e^{2\kappa}-1}\over {2\,\kappa}}\;,$$ (48)

while the mean vertical velocity is  

$$\widehat{V}_z = {z \over t_z} = V_z(0)\,{{e^{\kappa}-1}\over {\kappa}}\;,$$ (49)

where $\kappa=\kappa(z)$ is evaluated at the reflector depth. Comparing equations (48) and (49), we can see that the squared RMS velocity V_rms² equals the squared mean velocity $\widehat{V}_z^2$ if  

$$1 + 2\,\delta = {{2\,\left(e^\kappa - 1\right)} \over {\kappa\,\left(e^\kappa + 1\right)}}\;.$$ (50)

For small $\kappa$, the estimate of $\delta$ from equation (50) is  

$$\delta \approx - {\kappa^2 \over 24}\;.$$ (51)

For example, if the vertical velocity near the reflector is four times higher than the velocity at the surface, having the anisotropic parameter $\delta \approx -0.067$ is sufficient to cancel out the effect of heterogeneity on the normal moveout velocity. The values of the parameters S₂ and S₃ are found from formula (46) to be

$$\begin{eqnarray} S_2 & = & (1 + 8\,\eta)\,\kappa\,{{e^{2\kappa}+1} \over {e^{2\k... ...appa} + e^{2\kappa}+1} \over {\left(e^{2\kappa} - 1\right)^2}}\;.\end{eqnarray}$$ (52) (53)

Substituting (52) and (53) into formulas (35) and (41) and linearizing both in $\eta$ and in $\kappa$, we find that the error of anisotropic traveltime approximation (16) in the linear velocity model is approximately  

$${{\Delta t^2(h)} \over t^2(0)} = {{\kappa^2\,(1 - 8\,\eta)} \over 12}\, \left({h \over {t_0\,V_n}}\right)^6\;,$$ (54)

while the error of the shifted hyperbola approximation (33) is  

$${{\Delta t^2(h)} \over t^2(0)} = \left({{\kappa^2\,(1 - 8\,... ...over 24} - \eta\right)\, \left({h \over {t_0\,V_n}}\right)^6\;.$$ (55)

Comparing formulas (54) and (55), we conclude that if the medium is isotropic $(\eta=0)$, the shifted hyperbola can be twice as accurate as the anisotropic formula (assuming the optimal choice of parameters). It is, however, less accurate if the coefficient $\eta$ is positive and satisfies the approximate inequality  

$$\eta \geq {\kappa^2 \over {8\,(1 + \kappa^2)}}\;.$$ (56)