Isotropic Case

Nonhyperbolicity of reflection moveout in vertically heterogeneous isotropic media has been extensively studied with the help of the Taylor series expansion in the powers of the offset Al-Chalabi (1973); Bolshykh (1956); Taner and Koehler (1969). The most important property of vertically heterogeneous media is that the ray parameter $p = {{\sin{\psi}(z)} \over {V_z(z)}}$ doesn't change with the depth along each ray (Snell's law). This fact leads to the explicit parametric relationships

$$\begin{eqnarray} t(p) & = & \int_{0}^{z}\,{{dz} \over {V_z(z)\,\cos{\psi(z)}}} =... ...z}\,{{p\,V_z^2(t_z)\,dt_z} \over {\sqrt{1 - p^2\,V_z^2(t_z)}}}\;,\end{eqnarray}$$ (23) (24)

where  

$$t_z = t(0) = \int_{0}^{z}\,{{dz} \over {V_z(z)}}\;.$$ (25)

Straightforward differentiation of parametric formulas (23) and (24) yields the first four coefficients of the Taylor series expansion  

$$t^2(h) = a_0 + a_1\,h^2 + a_2\,h^4 + a_3\,h^6 + \ldots$$ (26)

in the vicinity of the vertical zero-offset ray. Series (26) contains only even powers of the offset h because of the reciprocity principle: the reflection traveltime is an even function of the offset. Taylor coefficients for the isotropic case are defined as follows:

$$\begin{eqnarray} a_0 & = & t_z^2\;, \\ a_1 & = & {1 \over V_{rms}^2}\;, \\ a_2 &... ... a_3 & = & {{2\,S_2^2 - S_2 - S_3} \over {8\,t_z^4\,V_{rms}^6}}\;,\end{eqnarray}$$ (27) (28) (29) (30)

where V_rms² = M₁,

$$\begin{eqnarray} M_k & = & {1 \over t_z}\,\int_{0}^{t_z}\,V_z^{2k}(t)\,dt\;\; (k... ..., \\ S_k & = & {M_k \over {V_{rms}^{2k}}}\;\;(k = 2, 3, \ldots)\;.\end{eqnarray}$$ (31) (32)

Equation (28) shows that, at small offsets, the reflection moveout has a hyperbolic form with the normal moveout velocity V_n equal to the root-mean-square velocity V_rms. At large offsets, however, the hyperbolic approximation is not accurate. Studying the Taylor series expansion (26), Malovichko introduced a remarkable three-parameter approximation for the reflection traveltime in a vertically heterogeneous isotropic medium Malovichko (1978); Sword (1987). Malovichko's formula has the form of a shifted hyperbola Castle (1988); de Bazelaire (1988):  

$$t(h) = \left(1 - {1 \over S}\right)\,t_0 + {1 \over S} \sqrt{t_0^2 + S\,{h^2 \over V_n^2}}\;.$$ (33)

If we set the zero-offset traveltime t₀ equal to the vertical traveltime t_z, the velocity V_n equal to V_rms, and the parameter of heterogeneity S equal to S₂, formula (33) guarantees the correct coefficients a₀, a₁, and a₂ in the Taylor series (26). Note that the parameter S₂ is related to the variance $\sigma^2$ of the squared velocity distribution, as follows:  

$$\sigma^2 = M_4 - V_{rms}^4 = V_{rms}^4\,(S_2 -1)\;.$$ (34)

According to formula (34), this parameter is always greater than 1 (it equals 1 in homogeneous media). In the most common practical cases, the value of S₂ lies between 1 and 2. We can roughly estimate the accuracy of approximation (33) at large offsets by comparing the fourth term of its Taylor series with the fourth term of the exact traveltime expansion (26). According to this estimate, the error of Malovichko's approximation is  

$${{\Delta t^2(h)} \over t^2(0)} = {1 \over 8} (S_3-S_2^2)\, \left({h \over {t_0\,V_n}}\right)^6\;.$$ (35)

As follows from the definition of the parameters S_k (32) and the Schwarz (Cauchy-Bunyakovski) inequality from calculus, expression (35) is greater than zero for any non-uniform velocity distribution V_z(t_z). This means that Malovichko's approximation tends to overestimate traveltimes at large offsets. As the offset approaches infinity, the limit of this approximation is  

$$\lim_{h \rightarrow \infty} t^2(h) = {1 \over S}\,{h^2 \over V_n^2}\;.$$ (36)

Formula (36) indicates that the effective horizontal velocity for Malovichko's approximation (the slope of the shifted hyperbola asimptote) is different from the normal moveout velocity. We can interpret this difference as an evidence of the effective depth-variant anisotropy. However, the anisotropic effect implied in formula (33) is different from the effect of a homogeneous transversely isotropic medium described by Thomsen's formula (1). To reveal this difference, let us substitute the effective values $t(h) = {\sqrt{z^2 + h^2} \over {V_g(\psi)}}$, $t_0 = {z \over V_z}$, $h = z\,\tan{\psi}$, and $S = {V_x^2 \over V_n^2}$ into (33). After we eliminate the variables z and h, the resultant expression takes the form  

$${1 \over {V_g(\psi)}} = {1 \over V_z}\,\left\{ \cos{\psi}\,\... ...^2}\,\sin^2{\psi} + {V_n^4 \over V_x^4}\,\cos^2{\psi}}\right\}.$$ (37)

If the anisotropic effect is induced by a vertical heterogeneity, V_x is greater than V_n, while V_n is greater than V_z. Both of these inequalities follow from the definitions of V_rms, t_v, and S₂ and the Schwartz inequality. They reduce to equalities only in the case of a constant velocity. Linearizing expression (37) with respect to Thomsen's anisotropic parameters $\delta$and $\epsilon$, we can transform it to a form analogous to that of equation (9), as follows:  

$$V_g^2(\psi) = V_z^2\,\left(1 + 2\,\delta\,\sin^2{\psi} + 2\,\eta\,(1 - \cos{\psi})^2\right)\;,$$ (38)

Figure 3 illustrates the difference between the weak transversally isotropic model and the effective anisotropy implied by Malovichko's approximation. The difference is noticeable in the shapes of both the effective wavefront (left plot) and the traveltime curve (right plot).

 

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Figure 3 Comparing a weak transversally isotropic model and Malovichko's shifted hyperbola approximation. The left plot shows effective wavefronts; right: reflection moveouts. Solid lines correspond to the anisotropic model; dashed lines: Malovichko's approximation. The values of the effective vertical, horizontal, and moveout velocities are the same in both cases and correspond to Thomsen's parameters $\epsilon = 0.2$, $\delta = 0.1$.

Deriving formula (38), we have assumed the correspondence  

$$S = {V_x^2 \over V_n^2} = {{1 + 2\,\epsilon} \over {1 + 2\,\delta}} \approx 1 + 2\,\eta\;.$$ (39)

We could also take the value of the parameter of heterogeneity S so as to match the coefficient a₂ given by formula (29) with the corresponding term in the Taylor series (17). In this case, the value of S would be Alkhalifah (1996)  

$$S = 1 + 8\,\eta\;.$$ (40)

The difference between equations (39) and (40) is an additional indicator of the fundamental difference between the homogeneous VTI model and the vertically heterogeneous model. The three-parameter anisotropic approximation (16) can match the reflection moveout curve in the isotropic model up to and including the fourth-order term in the Taylor series expansion, if the value of $\eta$ is chosen in accordance with formula (40). We can estimate the error of such an approximation with an equation analogous to (35). It takes the form  

$${{\Delta t^2(h)} \over t^2(0)} = {1 \over 8} (S_3-2 + 3\,S_2-2\,S_2^2)\, \left({h \over {t_0\,V_n}}\right)^6\;.$$ (41)

The difference between the error estimates (35) and (41) is  

$${{\Delta t^2(h)} \over t^2(0)} = {1 \over 8} (2 - S_2)\,(S_2-1)\, \left({h \over {t_0\,V_n}}\right)^6\;.$$ (42)

For the usual values of the parameter of heterogeneity S₂, which range from 1 to 2, expression (42) is greater than zero. This means that anisotropic approximation (16) overestimates the traveltimes in the isotropic heterogeneous model even more than the shifted hyperbola approximation (33) (as shown in the right plot of Figure 3). Which of the two approximations is more suitable if the model includes both vertical heterogeneity and anisotropy? We address this question in the following subsection.