Finite-Frequency Kernels Based on Adjoint Methods
Finite-Frequency Kernels Based on Adjoint Methods
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DOI：
10.1785/0120060041
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发表时间：
2006-12
影响因子：
3
通讯作者：
Qinya Liu;J. Tromp
中科院分区：
文献类型：
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作者：
Qinya Liu;J. Tromp
基于拉格朗日乘子法，推导了与层析反演的Frechet导数计算相关的伴随方程。目标函数χ（m）的弗雷歇导数，其中m表示地球模型，可以写成一般形式δ x =<$K_m（x）δ ln m（x）d^3x，其中δ ln m = δm/m表示相对模型扰动，K_m表示相关的三维灵敏度或弗雷歇核。考虑了区域模拟中人为吸收边界以及有限源引起的复杂性。我们构造了三维有限频率“香蕉甜甜圈”核K_m，同时计算所谓的“伴随”波场的时间向前和重建的规则波场的时间向后。伴随波场是通过使用接收器处的时间反演信号作为虚拟的同时源来产生的，而规则波场是通过在时间上向后传播波场的最后一帧来实时重建的，该最后一帧由先前的前向模拟保存。该方法是基于谱元法，只有两个模拟需要产生的3D有限频率灵敏度内核。该方法适用于1D和3D区域模型。不同的三维剪切波和压缩波灵敏度内核不同的区域体波和面波到达的地震图。这些内核说明了观测的结构参数的敏感性，并形成了全三维层析反演的基础。
We derive the adjoint equations associated with the calculation of Frechet derivatives for tomographic inversions based upon a Lagrange multiplier method. The Frechet derivative of an objective function χ(m), where m denotes the Earth model, may be written in the generic form δχ = ∫ K_m(x) δ ln m(x) d^3x, where δ ln m = δm/m denotes the relative model perturbation and K_m the associated 3D sensitivity or Frechet kernel. Complications due to artificial absorbing boundaries for regional simulations as well as finite sources are accommodated. We construct the 3D finite-frequency “banana-doughnut” kernel K_m by simultaneously computing the so-called “adjoint” wave field forward in time and reconstructing the regular wave field backward in time. The adjoint wave field is produced by using time- reversed signals at the receivers as fictitious, simultaneous sources, while the regular wave field is reconstructed on the fly by propagating the last frame of the wave field, saved by a previous forward simulation, backward in time. The approach is based on the spectral-element method, and only two simulations are needed to produce the 3D finite-frequency sensitivity kernels. The method is applied to 1D and 3D regional models. Various 3D shear- and compressional-wave sensitivity kernels are presented for different regional body- and surface-wave arrivals in the seismograms. These kernels illustrate the sensitivity of the observations to the structural parameters and form the basis of fully 3D tomographic inversions.