ABSTRACT Wave equation–based migration velocity analysis techniques aim to construct a kinematically accurate velocity model for imaging or as an initial model for full waveform inversion applications. The most popular wave equation–based migration velocity analysis method is differential semblance optimization, where the velocity model is iteratively updated by minimizing the unfocused energy in an extended image volume. However, differential semblance optimization suffers from artefacts, courtesy of the adjoint operator used in imaging, leading to poor convergence. Recent findings show that true amplitude imaging plays a significant role in enhancing the differential semblance optimization's gradient and reducing the artefacts. Here, we focus on a pseudo‐inverse operator to the horizontally extended Born as a true amplitude imaging operator. For laterally inhomogeneous models, the operator required a derivative with respect to a vertical shift. Extending the image vertically to evaluate such a derivative is costly and impractical. The inverse operator can be simplified in laterally homogeneous models. We derive an extension of the approach to apply the full inverse formula and evaluate the derivative efficiently. We simplified the implementation by applying the derivative to the imaging condition and utilize the relationship between the source and receiver wavefields and the vertical shift. Specifically, we verify the effectiveness of the approach using the Marmousi model and show that the term required for the lateral inhomogeneity treatment has a relatively small impact on the results for many cases. We then apply the operator in differential semblance optimization and invert for an accurate macro‐velocity model, which can serve as an initial velocity model for full waveform inversion.

中文翻译：

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伪逆扩展出生算子处理成像和速度分析应用的横向异质性的有效性

摘要 基于波动方程的偏移速度分析技术旨在构建一个运动学精确的成像速度模型或作为全波形反演应用的初始模型。最流行的基于波动方程的偏移速度分析方法是微分外观优化，其中通过最小化扩展图像体积中的未聚焦能量来迭代更新速度模型。然而，由于成像中使用的伴随算子，差分外观优化受到人工制品的影响，导致收敛性不佳。最近的研究结果表明，真实振幅成像在增强差异外观优化的梯度和减少伪影方面起着重要作用。在这里，我们专注于水平扩展 Born 的伪逆算子作为真正的幅度成像算子。对于横向非均匀模型，操作员需要相对于垂直位移的导数。垂直扩展图像以评估这样的导数是昂贵且不切实际的。可以在横向齐次模型中简化逆算子。我们推导出该方法的扩展以应用完整的逆公式并有效地评估导数。我们通过将导数应用于成像条件并利用源和接收器波场与垂直位移之间的关系来简化实现。具体来说，我们使用 Marmousi 模型验证了该方法的有效性，并表明横向不均匀性处理所需的术语对许多情况下的结果影响相对较小。