Understanding the role of geometric spreading and estimating its effects on seismic wave propagation play an important role in several techniques used in seismic exploration. The spreading can be estimated through dynamic ray tracing or determined from reflection traveltime derivatives. In the latter case, derivatives of nonhyperbolic moveout approximations are often used. We offer an alternative approach based on the weak-anisotropy approximation. The resulting formula is applicable to P-waves reflected from the bottom of a stack of horizontal layers, in which each layer can be of arbitrary anisotropy. At an arbitrary surface point, the formula depends, in each layer, on the thickness of the layer, on the P-wave reference velocity used for the construction of reference rays, and on nine P-wave weak-anisotropy (WA) parameters specifying the layer anisotropy. Along an arbitrary surface profile, the number of WA parameters reduces to five parameters related to the profile. WA parameters represent an alternative to the elastic moduli and, as such, can be used for the description of any anisotropy. The relative error of the approximate formula for a multilayered structure consisting of layers of anisotropy between 8% and 20% is, at most, 10%. For models including layers of anisotropy stronger than 20%, the relative errors may reach, locally, even 30%. For any offset, relative errors remain under a finite limit, which varies with the anisotropy strength.

中文翻译：

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任意对称的水平层状各向异性介质中 P 波几何传播的弱各向异性近似：倾斜横向各向同性规范

了解几何传播的作用并估计其对地震波传播的影响在地震勘探中使用的几种技术中起着重要作用。传播可以通过动态光线追踪估计或从反射走时导数确定。在后一种情况下，经常使用非双曲线时差近似的导数。我们提供了一种基于弱各向异性近似的替代方法。得到的公式适用于从一堆水平层的底部反射的 P 波，其中每一层可以具有任意的各向异性。在任意表面点，公式取决于每一层中的层厚度、用于构建参考射线的 P 波参考速度，以及指定层各向异性的九个 P 波弱各向异性 (WA) 参数。沿着任意表面轮廓，WA 参数的数量减少到与轮廓相关的五个参数。WA 参数代表弹性模量的替代方案，因此可用于描述任何各向异性。由各向异性层数在 8% 到 20% 之间的多层结构的近似公式的相对误差最多为 10%。对于包含大于 20% 的各向异性层的模型，相对误差可能会在局部达到 30%。对于任何偏移，相对误差都保持在有限的范围内，该范围随各向异性强度而变化。WA 参数代表弹性模量的替代方案，因此可用于描述任何各向异性。由各向异性层数在 8% 到 20% 之间的多层结构的近似公式的相对误差最多为 10%。对于包含大于 20% 的各向异性层的模型，相对误差可能会在局部达到 30%。对于任何偏移，相对误差都保持在有限的范围内，该范围随各向异性强度而变化。WA 参数代表弹性模量的替代方案，因此可用于描述任何各向异性。由各向异性层数在 8% 到 20% 之间的多层结构的近似公式的相对误差最多为 10%。对于包含大于 20% 的各向异性层的模型，相对误差可能会在局部达到 30%。对于任何偏移，相对误差都保持在有限的范围内，该范围随各向异性强度而变化。