Waveform inversion in the Laplace domain
Waveform inversion in the Laplace domain
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DOI：
10.1111/j.1365-246x.2008.03768.x
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发表时间：
2008-06
影响因子：
2.8
通讯作者：
C. Shin;Y. Cha
中科院分区：
文献类型：
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作者：
C. Shin;Y. Cha
在过去的30年里，自从Tarantola开创性的波形反演理论研究以来，地球物理学家和应用数学家已经利用波形反演来描绘地球的结构。然而，波形反演到真实的数据的成功应用是名义上的。故障主要是由于波形反演的高度非线性和真实的数据包含的低频分量不足造成的。通过利用拉普拉斯域中的波场和波动方程的伴随性质，我们提出了一种鲁棒且对初始模型不敏感的波形反演算法。拉普拉斯域中的波场等效于阻尼波场的零频率分量。因此，电法勘探中泊松方程的反演可以看作是利用无阻尼波场零频率分量的波形反演问题。由于我们在拉普拉斯域的反演算法与电法勘探中泊松方程的直流反演的性质密切相关，因此我们的算法可以生成等效于长波长速度模型（平滑速度模型）的速度模型。通过数值试验，我们发现，由于低频分量在频域波形反演中是必不可少的，所以我们的算法中迫切需要最佳低拉普拉斯阻尼系数的拉普拉斯变换波场。我们注意到，在拉普拉斯域中的对数波场的l2范数的目标函数表现为，如果它没有在低和高拉普拉斯阻尼常数的局部极小点。此外，我们注意到，在拉普拉斯域中的正演模拟可以通过使用比频率域的网格粗几百米的网格来精确地计算。数值试验的盐丘模型与高速对比表明，我们的算法的鲁棒性合成和真实的数据。为了将我们的算法更成功地应用于真实的数据，我们需要提高时间域中波场的拉普拉斯变换的精度。然而，我们在拉普拉斯域的波形反演可以成功地应用到真实的数据，因为我们的算法具有更多的优势，在正演和反演比那些在频率域。
SUMMARY For the last 30 yr, since Tarantola’s pioneering theoretical study of waveform inversion, geophysicists and applied mathematicians have utilized a waveform inversion to delineate the earth’s structures. However, successful applications of waveform inversion to real data are nominal. The failures are mainly caused by the high non-linearity of the waveform inversion and the real data containing insufficient low-frequency components. We propose a waveform inversion algorithm that is robust and is not sensitive to the initial model by exploiting the wavefield in the Laplace domain and the adjoint property of the wave equation. The wavefield in the Laplace domain is equivalent to the zero frequency component of the damped wavefield. Therefore, the inversion of Poisson’s equation in electrical prospecting can be viewed as a waveform inversion problem, exploiting the zero frequency component of an undamped wavefield. Since our inversion algorithm in the Laplace domain is strongly associated with the nature of DC inversion for Poisson’s equation in electrical prospecting, our algorithm can generate a velocity model that is equivalent to a long-wavelength velocity model (a smooth velocity model). Through numerical tests, we found that the Laplace-transformed wavefields for optimally low Laplace damping constants are critically needed in our algorithm as the low-frequency components are necessary in the waveform inversion of the frequency domain. We note that the objective function by l2 norm of the logarithmic wavefield in the Laplace domain behaves as if it has no local minimum points in low and high Laplace damping constants. Moreover, we note that the forward modelling in the Laplace domain could be accurately calculated by using a coarser grid of several hundreds of metres than that of the frequency domain. Numerical tests of the salt dome model with high-velocity contrast demonstrate the robustness of our algorithm to both synthetic and real data. To apply our algorithm to real data more successfully, we need to improve the accuracy of the Laplace transformation for the wavefields in the time domain. However, our waveform inversion in the Laplace domain can be successfully applied to real data since our algorithm has more advantages in forward modelling and inversion than those in the frequency domain.