探討波傳理論和模型參數化方法對震波走時層析成像的影響

Abstract

透過地震震波層析成像，地球物理學家得以利用不同的走時理論、參數化與正則化方法以求得高解析度的地球內部構造。在近年來的研究中，走時理論上除了利用傳統無限高頻的波線理論之外，也引進了有限頻寬的香蕉甜甜圈理論作為探測地函非均質速度構造的工具。另一方面，在模型參數化方法的改進上，相較於使用一般格點參數化再加上先驗的正則化處理，如尋找儘量平滑的模型，多重尺度參數化的特性是在逆推的過程中儘量由資料本身對模型參數約束的能力來決定模型在空間上解析尺度的變化，因此求得的模型解同時具備各種不同尺度而非一味追求平滑的構造。 本研究利用高斯隨機介質類比真實地球模型，並以均質模型類比無側向速度變化的一維地球速度參考模型；再以平行化虛擬頻譜數值方法（parallel pseudo-spectral method）解波動方程式，計算波場在三維介質中傳遞的過程和波形，然後將在非均質介質中所求得的合成波形與其對應的均質介質合成波形做交叉比對（cross-correlation）測量出真實的走時殘差，再利用奇異值分解（singular value decomposition, SVD）對真實走時殘差建立的數據進行逆推以求得原本的速度構造。 在逆推的方法上，過去由於受限於格拉姆矩陣（Gram matrix）過於巨大，通常無法直接求出反格拉姆矩陣，因此本研究利用 PROPACK 直接對格拉姆矩陣做奇異值分解，可求出模型解析度矩陣（model resolution matrix）與模型共變異數矩陣（model covariance matrix），用以評估在不同走時理論、參數化與正則化之下逆推結果在空間中的解析能力。 分析結果發現：在相同的模型分散程度（model spread）下，使用多重尺度參數化進行逆推所得到的結果其模型共變異（model covariance）比使用一般參數化還要來的低；在正則化方式上，使用阻尼（damping）的方式其模型共變異數比使用閥值（truncation）低，而模型共變異數低的意義為其結果不容易受到觀測資料誤差的影響。最後再以模型離差（model misfit）為基準，在相同的資料擬合程度（variance reduction）下，使用有限頻寬理論進行逆推所得到的結果比使用波線理論更為接近真實的模型。

Whether different forward theories and parameterization methods employed in seismic tomographic imaging lead to the improvement of the resulting Earth structures has been a focus of attention in the seismological community. Recent advance in tomographic theory has gone beyond classical ray theory and incorporated the 3-D sensitivity kernels of frequency-dependent travel-time data into probing the mantle velocity heterogeneity with unprecedented resolution. On the other hand, the idea of multi-scale parameterization has been introduced to deal with naturally uneven data distribution and spatially-varying model resolution for the tomographic inverse problems. The multi-resolution model automatically built through the wavelet decomposition and synthesis results in the non-stationary spatial resolution and data-adaptive resolvable scales. Because the Gram matrix of Frechet derivatives that relates observed data to seismic velocity variations is usually too large to be practically inverted by singular value decomposition (SVD), the iterative LSQR algorithm is instead employed in the inversion which inhibits the direct calculation of resolution matrix to assess the model performance. With the increasing computing power, we are now able to calculate the SVD of the Gram matrix more efficiently using the parallel PROPACK solver. In this study, we compute the ground-truth psudospectral seismograms in random media with certain heterogeneity strengths and scale lengths. The finite-frequency travel-time residuals measured from waveform cross correlation are then used to invert for the implanted random structure based on different forward theory and model parameterization. For each inversion approach, the tradeoff between model covariance and model spread is utilized to determine the optimal solution, showing that the multi-scale model yields a much lower model covariance and remains better spectral resolution for longer-wavelength velocity structures than the simple grid one. The spreadness and geometry of the resulting resolution matrices reveal that both the 3-D finite-frequency kernel and multi-scale parameterization tend to broaden and smooth the structures having less smearing toward the non-crossing ray directions. Moreover, the comparison of the misfits between the resolved and initial random model among all the optimal solutions indicates that the models obtained with finite-frequency theory have better fits to the true model because wavefront healing effect is properly taken into account in modeling cross-correlation travel-time residuals.