First-arrival traveltime tomography is an essential method for obtaining near-surface velocity models. The adjoint-state first-arrival traveltime tomography is appealing due to its straightforward implementation, low computational cost, and low memory consumption. Because solving the point-source isotropic eikonal equation by either ray tracers or eikonal solvers intrinsically corresponds to emanating discrete rays from the source point, the resulting traveltime gradient is singular at the source point, and we denote such a singular pattern the imprint of ray-illumination. Because the adjoint-state equation propagates traveltime residuals back to the source point according to the negative traveltime gradient, the resulting adjoint state will inherit such an imprint of ray-illumination, leading to singular gradient-descent directions when updating the velocity model in the adjoint-state traveltime tomography. To mitigate this imprint, we solve the adjoint-state equation twice but with different boundary conditions: one being taken to be regular data residuals and the other taken to be ones uniformly, so that we are able to use the latter adjoint state to normalize the regular adjoint state and we further use the normalized quantity to serve as the gradient direction to update the velocity model; we call this process ray-illumination compensation. To overcome the issue of limited aperture, we have developed a spatially varying regularization method to stabilize the new gradient direction. A synthetic example demonstrates that our method is able to mitigate the imprint of ray-illumination, remove the footprint effect near source points, and provide uniform velocity updates along raypaths. A complex example extracted from the Marmousi2 model and a migration example illustrate that the new method accurately recovers the velocity model and that an offset-dependent inversion strategy can further improve the quality of recovered velocity models.

中文翻译：

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伴随态初到走时层析成像的射线照明补偿

初至走时层析成像是获得近地表速度模型的重要方法。伴随态初到走时断层扫描由于其简单的实现、低计算成本和低内存消耗而具有吸引力。因为通过射线追踪器或 eikonal 求解器求解点源各向同性 eikonal 方程本质上对应于从源点发出离散射线，因此产生的走时梯度在源点处是奇异的，我们将这种奇异模式表示为射线印记 -照明。因为伴随态方程根据负的走时梯度将走时残差传播回源点，因此产生的伴随态将继承这样的射线照射印记，在伴随态走时断层扫描中更新速度模型时导致奇异的梯度下降方向。为了减轻这种影响，我们两次求解伴随状态方程，但使用不同的边界条件：一个被视为常规数据残差，另一个被视为统一数据残差，以便我们能够使用后一个伴随状态来归一化规则伴随状态，我们进一步使用归一化量作为梯度方向来更新速度模型；我们称这个过程为光线照明补偿。为了克服有限孔径的问题，我们开发了一种空间变化的正则化方法来稳定新的梯度方向。一个综合示例表明，我们的方法能够减轻光线照射的印记，消除源点附近的足迹效应，并沿射线路径提供均匀的速度更新。从 Marmousi2 模型中提取的复杂示例和偏移示例说明新方法准确地恢复了速度模型，并且依赖偏移量的反演策略可以进一步提高恢复速度模型的质量。