地震學為探測地球深部構造一個重要的工具。現有地球內核構造研究利用不同分支之PKP波相到時差進行分析，通常利用射線理論 (ray theory) 來簡化波傳問題。射線理論假設波為無限高頻，使得觀察到的觀測與理論模型之間的異常，皆來自於震源至測站間射線路徑上異常構造之貢獻，即觀測異常對於模型的敏感度只在射線路徑上；實際上波係以有限頻寬在三維地球構造中傳遞，此時波之敏感區域將沿著其射線展開至一個寬度，也就是菲聶耳帶 (Fresnel zones)，而敏感度算核 (sensitivity kernel) 提供了三維模型擾動對於觀測異常之關係，此時量測異常不再貢獻至射線上，而是其菲聶耳帶中。至今在計算全球三維模型尺度下的敏感度算核，仍然需要相當的計算與儲存資源。本研究使用軸對稱譜元素法 (axisymmetric spectral-element method, AxiSEM) 進行三維正演波傳模擬，因軸對稱之特性，三維波傳球體模型問題可降階至二維半球面，AxiSEM係解此二維半球面上之運動方程式，三維波場可從二維解以其對稱軸旋轉而得到。此降階使得在計算三維波場所需之計算資源，隨頻率變化減少一次方，因此短至一秒週期之三維理論地震圖與敏感度算核能在合理計算資源下得到。在本篇研究中我們利用AxiSEM計算理論波場至主頻1秒，敏感度算核使用Monte-Carlo Kernel (MC Kernel) 程式讀取AxiSEM正逆波場資料進行摺積運算以計算敏感度算核。我們著重分析在內核研究中，通常使用到的不同分支之PKP波走時敏感度算核：PKPab, PKPbc, PKiKP 與PKIKP；以及其差分到時敏感度算核，以檢驗下部地函之構造影響，如PKPbc – PKIKP, PKPab – PKIKP 與PKiKP – PcP。算核結果提供了兩個針對現有資料敏感之改進：一為正確之深部敏感資訊，現有研究通常將觀測異常歸因至至射線轉折點深度，而在有限頻寬的架構下，因香蕉－甜甜圈 (banana-doughnut) 之算核型態，射線路徑上之異常反而對觀測無影響，影響範圍是展開至一個體積中；另一改進為敏感度算核提供了正確之下部地函敏感區域，以PKP波相進行內核構造分析時，需要將地殼與地函之影響盡量去除，敏感度算核能提供修正之基準，或是在逆推時將觀測異常歸因到實際區域擾動。利用AxiSEM得到之高頻走時敏感度算核，可以提升我們對特定波相真正能解析到的目標構造之能力，也開啟了有限頻寬效應對於內核PKP波相敏感度之認識。

Current studies of the Earth’s inner core structure using body waves typically measure the differential travel times between different branches of PKP phases, which are then modeled by ray theory that does not account for the finite widths of the Fresnel zones of seismic waveforms. The sensitivity kernel of the finite-frequency waves connects such volumetric sensitivities of model perturbations to the observed anomalies. However, it is still challenging for computing relatively high-frequency sensitivity kernels in 3-D volume globally. In this study, we adopt the axisymmetric spectral element method (AxiSEM) which collapsed its computational domain from a 3-D sphere into 2-D semi-disk domain and 3-D wavefield are derived analytically. This dimensional reduction drastically reduces the computational demands of order one and hence the sensitivity kernels of the travel times of relatively short-period (up to 1 Hz) seismic waves can be tackled. The sensitivity kernels are computed by the corresponding software: Monte-Carlo Kernel (MC Kernel) which reads the forward and backward waveform from AxiSEM wavefields. We focus on examining the sensitivities of the phases that are typically used in inner core structural studies such as PKPab, PKPbc, PKiKP, and PKIKP. We also evaluate the PKPbc – PKIKP, PKPab – PKIKP, and PKiKP - PcP differential kernels to investigate the possible effect of mantle heterogeneities on inner core models. The resulting delay time kernel suggests the finite-frequency can improve to current explanations of differential time measurements in two aspects: one is the correct depth sensitivity for mapping the observed anomalies, and the other is the mantle heterogeneities effect can be more explicitly evaluated. These finite-frequency sensitivity kernels improve our understanding on how seismic signals sample the structure in the deep Earth and enable us to analyze the finite frequency effect that has so far been ignored in deriving the inner core models.

摘要 i
ABSTRACT ii
誌謝 iii
CONTENTS iv
LIST OF FIGURES vi
LIST OF TABLES viii
Chapter 1 Introduction 1
1.1 Seismic Phases for Studying the Inner Core 2
1.2 Literature Review 3
Inner Core Structures 3
Finite-frequency Sensitivity Kernel of the Delay Time 5
1.3 Motivation, Goal and the Content of This Thesis 7
Chapter 2 Methodology 11
2.1 Linearization of Seismic Inverse Problem 11
2.2 Finite-frequency Delay Time Sensitivity Kernel 13
2.3 Waveform Simulation: Axisymmetric Spectral-Element Method (AxiSEM) 17
2.4 Forward and Backward Green’s Function Databases Computed by AxiSEM 20
2.5 Kernel Calculation Algorithm: Monte-Carlo Kernel 21
Chapter 3 Numerical Examples of Sensitivity Kernels 29
3.1 Parameters Settings 29
3.2 Popular Phases: P, PP, PcP, Pdiff 31
P 32
PP 32
PcP 33
Pdiff 33
3.3 PKP Branches 34
PKIKP 34
PKPbc 34
PKPab 35
PKiKP 35
3.4 Differential Kernels 36
3.5 Time-lapse Videos of Delay Time Sensitivity Kernels 38
Chapter 4 Discussions and Future Works 61
4.1 Numerical Methods for Global Seismic Wave Propagation 61
4.2 Finite-frequency Effect of the PKP Phase for Probing Inner Core Structure 63
4.3 Summary 65
4.4 Future Works 66
REFERENCES 69

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