| 有限频率线性理论的波恩近似佯谬 |
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| 引用本文： | 江,燕,陈晓非.有限频率线性理论的波恩近似佯谬[J].地震学报,2014,36(3):372-389. |
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| 作者姓名： | 江 燕 陈晓非 |
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| 作者单位： | 1.中国 北京 100081 中国地震局地球物理研究所 |
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| 基金项目： | 国家自然科学基金(41090292);中央级公益性科研院所基本科研业务费专项(DQJB13B15)共同资助 |
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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.
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| 关 键 词： | 有限频率层析成像 波恩近似 相关走时 Fréchet微分 隐函数定理 |
| 收稿时间： | 2013-05-27 |
Born approximation paradox of linear finite-frequency theory |
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| Affiliation: | 1.Institute of Geophysics, China Earthquake Administration, Beijing 100081, China2.School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China |
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| 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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