Born approximation paradox of linear finite-frequency theory

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.