Born approximation paradox of linear finite-frequency theory
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摘要: 对有限频率层析成像线性理论的波恩近似问题进行梳理， 用数值方法统计分析其适用范围， 结果表明波恩近似要求最大速度扰动不超过1%； 然后对相关走时一阶近似进行统计分析， 结果表明它也只适用于最大速度扰动在1%以内的情形. 然而， 结合波恩近似和相关走时一阶近似而得到的有限频率线性理论， 其适用的速度扰动范围最大可达10%. 这个表面上的逻辑悖论， 称为“波恩近似佯谬”. 此佯谬是由于不恰当地使用波恩近似造成的. 本文摒弃波恩近似， 使用泛函的Fréchet微分和隐函数定理推导得到有限频率线性理论， 圆满解释了波恩近似佯谬. 由于有限频率非线性理论早已摒弃了波恩近似， 因此波恩近似概念在有限频率层析成像理论中完全没有必要.Abstract: After reviewing the Born approximation problem of linear finite-frequency tomography theory, its scope of application is statistically analyzed by numerical method. The result indicates that the maximum velocity perturbation should not exceed 1% for Born approximation. Then the statistical analyses on the first-order approximation of cross-correlation travel-time also show that it only meets the case of the maximum velocity perturbation less than 1%. However, the maximum velocity perturbation can be 10% for linear finite-frequency theory, which combines Born approximation with the first-order approximation of cross-correlation travel-time. This apparent logic paradox is called “Born approximation paradox”, which is caused by misusage of Born approximation. Thus, Born approximation is discarded in this study; Fréchet derivative and implicit functional theorem are used to deduce linear finite-frequency theory. As a result, Born approximation paradox is explained thoroughly. Since Born approximation has been discarded early in nonlinear finite-frequency theory, this concept is unnecessary in finite-frequency tomography theory.
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图 2 随机扰动情形下的波恩近似相关走时误差
初始模型为均匀介质，目标模型为高斯型三维随机介质． 图(a)—(d)的最大速度扰动分别为1%，2%，5%和10%
Figure 2. The errors of cross-correlation traveltime of Born approximation with random perturbations The starting model is homogeneous medium.
The object model is 3-D Gaussian random medium. The maximum velocity perturbations of Figs.(a)—(d) are 1%，2%，5% and 10%，respectively
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图 3 随机扰动情形下相关走时泛函的一阶近似误差
模型和震源参数同图2． 图(a)—(d)的最大速度扰动为分别为1%，2%，5%和10%
Figure 3. The first-order approximation errors of the cross-correlation traveltime functional with random perturbations
The parameters of the model and seismic sources are the same as those in Fig.2. The maximum velocity perturbations in Figs.(a)—(d) are separately 1%，2%，5% and 10%
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图 4 随机扰动情形下有限频率方法的相关走时误差
模型和震源参数同图2． 图(a)—(d)的最大速度扰动为分别为1%，2%，5%和10%
Figure 4. The errors of the cross-correlation traveltime of finite-frequency theory with random perturbations
The parameters of the model and seismic sources are the same as those in Fig.2. The maximum velocity perturbations in Figs.(a)—(d) are separately 1%，2%，5% and 10%
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