Double differencing of body‐wave arrival times has proved to be a useful technique for increasing the resolution of earthquake locations and elastic wavespeed images, primarily because (1) differences in arrival times often can be determined with much greater precision than absolute onset times and (2) differencing reduces the effects of unknown, unmodeled, or otherwise unconstrained variables on the arrival times, at least to the extent that those effects are common to the observations in question. A disadvantage of double differencing is that the system of linearized equations that must be iteratively solved generally is much larger than the undifferenced set of equations, in terms of both the number of rows and the number of nonzero elements. In this article, a procedure based on demeaning subsets of the system of equations for hypocenters and wavespeeds that preserves the advantages of double differencing is described; it is significantly more efficient for both wavespeed‐only tomography and joint hypocenter location‐wavespeed tomography. Tests suggest that such demeaning is more efficient than double differencing for hypocenter location as well, despite double‐differencing kernels having fewer nonzeros. When these subsets of the demeaned system are appropriately scaled and simplified estimates of observational uncertainty are used, the least‐squares estimate of the perturbations to hypocenters and wavespeeds from demeaning are identical to those obtained by double differencing. This equivalence breaks down in the case of general, observation‐specific weighting, but tests suggest that the resulting differences in least‐squares estimates are likely to be inconsequential. Hence, demeaning offers clear advantages in efficiency and tractability over double differencing, particularly for wavespeed tomography.

中文翻译：

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贬低的双重差分：在震源位置和波速断层扫描中的应用

体波到达时间的双重差分已被证明是提高地震位置和弹性波速图像分辨率的有用技术，主要是因为（1）到达时间的差异通常可以比绝对开始时间更精确地确定，并且（ 2) 差分减少了未知的、未建模的或其他不受约束的变量对到达时间的影响，至少在这些影响对于所讨论的观察是常见的程度。双差分的一个缺点是，就行数和非零元素的数量而言，必须迭代求解的线性方程组通常比无差方程组大得多。在本文中，描述了一种基于震源和波速方程组的贬低子集的程序，该程序保留了双重差分的优点；对于仅波速断层扫描和联合震源位置波速断层扫描，它的效率要高得多。测试表明，尽管双微分内核具有较少的非零值，但这种贬低也比震源位置的双微分更有效。当降低系统的这些子集被适当地缩放并使用观测不确定性的简化估计时，对震源和来自降低的波速的扰动的最小二乘估计与通过双重差分获得的相同。在一般的、特定于观察的加权的情况下，这种等价性会失效，但测试表明，由此产生的最小二乘估计差异可能无关紧要。因此，与双差分相比，贬低在效率和易处理性方面具有明显的优势，尤其是对于波速层析成像而言。