射線理論(Ray theory)為對震波作高頻近似假設，經常應用於三維速度構造逆推問題，該方法雖簡化問題且易於實行，但往往受限於兩個因素：(1)目標模型的解析力受限於菲涅爾帶(Fresnel zone)寬度；(2)因為高頻近似假設，致使觀測站所收集波形資訊完全無法應用，使大量寶貴資料被捨棄。然而採有限頻寬方法(finite-frequency approach)透過數值模擬與全波場(full-waveform)計算，我們可以計算參考模型下的理論地震圖，藉由比較理論與觀測地震圖二者差異，進而定量化到時異常(traveltime anomaly)與振幅異常(amplitude anomaly)等波形異常資訊，爾後經由波形異常資訊計算構造異質性對於有限頻寬(finite-frequency)震波的影響，即為敏感度算核。
　　本論文應用應變格林張量(strain Green’s tensor, or SGT)與三維交錯網格有限差分法(3-D staggered-grid finite-difference method)波形模擬，藉此探討模型參數擾動對於有限頻寬震波的影響及其敏感度算核。模型擾動可為彈性波速度擾動或非彈性介質之品質因子擾動，代表介質擾動對於波形的影響，並透過震源－觀測站的互換性(reciprocity)使用以測站位置為震源的應變格林張量，計算參考模型下震源－觀測站間的理論地震圖與到時和振幅異常的敏感度算核。首先在均質模型下計算速度擾動的敏感度算核，其震源－觀測站間的射線路徑為一直線，但實際震波為有限頻寬，因此速度擾動造成到時差異的算核為一中空橢球，或稱香蕉甜甜圈(banana-doughnut)，代表觀測站所收到的波形到時隨模型擾動之影響，不單單只是射線路徑(raypath)所穿越之速度構造所貢獻，而是含蓋更廣泛的區域。而品質因子的擾動可藉由該時間視窗的波形的希爾伯特轉換(Hilbert transform)表示，算核形貌如同前述為橢球型，但正負符號相異，即品質因子對於波形異常的貢獻與彈性波範疇下速度擾動大相逕庭。此外，震源機制與濾波器亦為影響算核形貌與分佈的因素，結果顯示能量輻射與散射深受上述物理機制影響。
　　若在三維模型下計算敏感度算核，則明顯可見在速度愈高的區域，其敏感度算核愈寬、影響波形的範圍愈廣，且相較於速度模型的側向變化，深度方向速度梯度較能影響算核形貌。若將速度擾動的算核推廣至非彈性衰減擾動，則同樣可見類似形貌的算核。

　　Ray theory is based on infinitely-high frequency approach and has been widely used in seismic tomography. It is convenient and is therefore commonly used in seismology. However, ray theory tomography is constrained by two important shortcomings: (1) the resolving power of the tomography model is limited by the width of the first Fresnel zone; (2) the full waveform information (e.g. the time history of the seismic response) is discarded because of the high frequency assumption. The recently developed finite-frequency tomography approach provides a way of fully utilizing the spatial and temporal responses of seismic wavefield. The finite-frequency approach adopts numerical or semi-analytical methods to compute the Green’s functions in reference models. Sensitivity kernels can be expressed in terms of the waveform perturbations due to point sources generated by structural heterogeneities, and these waveform perturbations result in traveltime and amplitude anomalies.
　　In this thesis, we apply the strain Green’s tensor (SGT) approach and 4th-order staggered-grid finite-difference method to solve the wave equations and calculate the sensitivity kernels for both elastic and anelastic perturbations of the medium properties. The ellipsoid and hollow appearance (in the shape of banana-doughnut) of the traveltime anomaly kernels for velocity (νp and νs) perturbations will be shown first in the homogeneous elastic medium for each phase. Sensitivities are close to zero on the raypath and strongly influenced by the free surface. This phenomenon is due to finite-frequency effect on the full-waveform that cannot be seen in ray theory. In addition, the anelastic sensitivity kernels of the quality factor (Qp and Qs) perturbations can be expressed in terms of the original waveform’s Hilbert transform. The shapes of kernels are similar to those for the elastic parameters, but the patterns of kernels in terms of sign changes indicate that the contributions to the waveform by the anelastic perturbation are quite different than velocity perturbations of the reference models. We also discuss other effects, such as filtering and focal mechanism, on the sensitivity kernels. The results show that these effects have great influence on the sensitivity kernel.Finally, we demonstrate the same calculation procedure of a scenario earthquake in southern Taiwan in a 3-D velocity model. The result shows that the appearance of sensitivity kernels strongly depends on vertical velocity gradients. Unlike in homogeneous model, the traveltime anomaly kernel is generally non-zero on the raypath, which can be explained by the multi-pathing effect of seismic wave propagation in a heterogeneous structure.

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