有限频率线性理论的波恩近似佯谬

Born approximation paradox of linear finite-frequency theory

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.

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 ap proximation 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.