On sensitivity kernels for 'wave-equation' transmission tomography
On sensitivity kernels for 'wave-equation' transmission tomography
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DOI：
10.1111/j.1365-246x.2004.02509.x
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发表时间：
2005-02-01
影响因子：
2.8
通讯作者：
van der Hilst, RD
中科院分区：
文献类型：
--
作者：
de Hoop, MV;van der Hilst, RD
我们将地震散射理论与分布理论相结合，研究有限频率地震延迟时间敏感核的一些性质。用于计算核的理论取决于测量的方式。例如，对固有走时的敏感性，即几何射线理论中定义的相位到达瞬时开始的时间，在射线上具有非零值，在其他地方具有零值，而波激励的平滑后期部分的测量可能具有更复杂的内核。基于玻恩近似的分析表明，此类内核的行为是由未受扰动的源接收器射线支持的狄拉克三角洲的 n 阶导数的正则化决定的，其中 n 是空间维度。这种正则化大约引起所需的带限制和相关的菲涅耳区平均。如果正则化关于 delta 的奇异支持对称，则其奇数阶导数在此支持上消失，但偶数阶呈现非零值。这解释了为什么体波延迟时间的 3-D 有限频率内核似乎在特定的、相当严格的情况下对射线的灵敏度为零（如所谓的“香蕉甜甜圈内核”）。事实上，在没有焦散的情况下（即，介质必须是简单的或准均匀的）并且仅当确切的源签名（例如，源时间函数）已知并使用时，核的正则化可以在未受扰动的射线上具有零；这两个条件在合成情况下都可以满足，但在实际数据和异构媒体的应用中通常不能满足。一般来说，人们不知道振荡核在哪里具有零值，这对于使用此类核进行透射断层扫描的线性化具有影响。我们通过调用灵敏度内核的多分辨率分析以及相位到达时间的时频分析来简要阐明这些观察结果。
We combine seismological scattering theory with the theory of distributions to study some properties of sensitivity kernels for finite frequency seismic delay times. The theory to be used for calculating the kernels depends on the way the measurements are made. For example, the sensitivity to the traveltime proper, that is the time to instantaneous onset of the phase arrival as defined in geometrical ray theory, has a non-zero value on the ray and zero elsewhere, whereas measurements of the smooth later part of the wave excitation can have more complicated kernels. Analysis based upon the Born approximation reveals that the behaviour of such kernels is determined by the regularization of the nth derivative of the Dirac delta supported on the unperturbed source-receiver ray, where n is the spatial dimension. Such regularization induces approximately the desired band-limitation and the associated Fresnel zone averaging. If the regularization is symmetric about the singular support of the delta its odd order derivatives vanish on this support, but the even orders render non-zero values. This explains why 3-D finite frequency kernels for body wave delay times seem-in specific, rather restrictive circumstances-to have zero sensitivity on the ray (as in the so called 'banana-doughnut kernels'). Indeed, a regularization of the kernel can have a zero on the unperturbed ray, in the absence of caustics (that is, the medium must be simple or quasi-homogeneous) and only if the exact source signature (e. g., source time function) is known and used; both conditions can be met in synthetic cases but not-in general-in applications to real data and heterogeneous media. In general, one will not know where the oscillatory kernels have zero values, which has implications for the use of such kernels for the linearization of transmission tomography. We briefly clarify these observations by invoking a multiresolution analysis of sensitivity kernels, connected with a time-frequency analysis of phase arrival times.