The Approximation And Analysis Of Dispersion Function Of Rayleigh Waves With CHEBYSHEV Polynomials

Rayleigh waves method is a new geophysical exploration method. It mainly uses the dispersion characteristics of Rayleigh waves in the multilayered medium, including three problems: data collection, dispersion theory and inversion application. As the mathematics theory is not perfect yet, there is not a separate formula for each dispersion curve, which, to some extent, makes an obstruction against the inversion based on it, thereby greatly limits the application in enginnering. The contents are as follows:This article discusses the approximation of secular function, compares the advantage and shortcoming of different approximation methods on exploration, such as polynomial approximation. Because of the highly oscillatory of the secular function, this paper employs the chebyshev polynomial approximation, and simulates the obvious form of dispersion under the situation of high and low frequency respectively. Through comparison, we find the precision of approximation is high, and the approximation polynomial is stable. Meanwhile, this essay simulates the secular function of complex media, The result shows, the secular function transmits as sinusoidal as the increasing of velocity.The paper analyses the effect of stratum parameter to the coefficient of approximation function. We arrive at the conclusion that the velocity of S-wave and depth are relevant to the secular function. The velocity of P-wave and the density are not relevant to the form of approximation function of secular function.In order to faciliate the application of the approximation of chebyshev method, the paper creates a connect approximation method with 3-order chebyshev polynomial. This method not only can keep the continuity of approximation function but also can keep the smooth. So that we can get a simple formula. Meanwhile, the paper adopts the generalized to 3-D chebyshev approximation method together with the surface reconstitution technology to approximate the dispersion function from local part. The result is stable.The research of this article provides convenience for the geophysical exploration method, and presents a new thought of studying the dispersion of complex media.