Orthorhombic anisotropy is a modern standard for 3D seismic studies in complex geologic settings. Several seismic data processing methods and wave propagation modeling algorithms in orthorhombic media rely on phase-velocity, group-velocity, and traveltime approximations. The algebraic simplicity of an approximate equation is an important factor in these media because the governing equations are more complicated than transversely isotropic media. To approximate the P-wave kinematics in acoustic orthorhombic media, we have developed a new 3D general functional equation that has a simple rational form. Using the general form, we adopt two versions of rational approximations for the phase velocity, group velocity, and traveltime. The first version uses a simpler functional form and parameter definition within the orthorhombic symmetry planes. The second version is more accurate, using one parameter that is defined out of the symmetry planes. For the phase velocity, we obtain another approximation that is no longer rational but is still algebraically simple, exact for 3D transversely isotropic media, and it is exact within the symmetry planes of orthorhombic media. We find superior accuracy in our approximations compared with previous ones, using numerical studies on multiple moderately anisotropic orthorhombic models. We investigate the effect of the negative anellipticity parameters on the accuracy and find that, in models in which the error of the existing most accurate approximations exceeds 2%, the error of the new approximations remains below 0.2%. The adopted approximations are algebraically simpler and stably more accurate than existing approximations; therefore, they may be considered as attractive alternatives for the existing approximations in many practical applications. We extend the applicability of our approximations by using them to obtain the equations of group direction as a function of phase direction and vice versa, which are useful in wave propagation modeling methods.

中文翻译：

---

P波运动学的有理逼近：第2部分-斜方介质

正交各向异性是在复杂地质环境中进行3D地震研究的现代标准。斜方介质中的几种地震数据处理方法和波传播建模算法都依赖于相速度，群速度和传播时间近似值。近似方程的代数简单性是这些介质中的重要因素，因为控制方程比横向各向同性介质更为复杂。为了逼近正交各向异性介质中的P波运动学，我们开发了一种新的3D通用函数方程，该方程具有简单的有理形式。使用一般形式，我们对相速度，群速度和行进时间采用两个有理近似形式。第一个版本在正交斜对称平面内使用了更简单的功能形式和参数定义。第二个版本使用在对称平面之外定义的一个参数更为精确。对于相速度，我们获得了另一个近似，该近似不再是有理的，而是代数简单的，对于3D横向各向同性介质来说是精确的，并且在正交晶体的对称平面内也是精确的。通过对多个中等各向异性正交晶体模型的数值研究，我们发现与以前的方法相比，我们的近似方法具有更高的精度。我们研究了负椭圆度参数对精度的影响，发现在现有最精确近似值的误差超过2％的模型中，新近似值的误差保持在0.2％以下。与现有的近似相比，采用的近似在代数上更简单且稳定地更准确；因此，对于许多实际应用中的现有近似值，它们可能被视为有吸引力的替代方案。通过使用它们来获得近似的适用性，方法是使用它们来获得作为相位函数的组方向方程，反之亦然，这在波传播建模方法中很有用。