本文選題：Zoeppritz方程 + 位移函數(shù)��； 參考：《西南石油大學》2016年碩士論文

【摘要】：地震波在遇到分界面時將會產(chǎn)生同類反射波,轉(zhuǎn)換反射波和轉(zhuǎn)換透射波,以及折射和全反射等物理現(xiàn)象,我們稱這種物理現(xiàn)象為地震波在分界面上的廣義散射。這部分的內(nèi)容是地震波場傳播理論的重要組成部分,也廣泛應用于油氣,水合物等地震勘探開發(fā)領域。地震波在自由界面以及彈性界面的散射的研究包含兩個方面：一方面是波動方程和界面邊界條件的邊值定解問題,而另一方面就是該邊值定解問題的求解過程。一言以蔽之,我們求解地震波的分界面上的散射就是確定波動方程滿足的邊值定解問題,然后求出該邊值定解問題的解函數(shù)的過程。然而邊值定解問題又分為位移位和位移這兩種表達形式,故存在兩種不同的研究方式。采用位移位函數(shù)進行研究,我們稱之為”位移位方法”,位移位方法的典型代表即為"Knott"方程。而采用位移函數(shù)進行研究的方法,稱為“位移方法”,位移方法的典型代表即為‘'Zoeppritz"方程。位移方法以及位移位方法在物理含義上保持一致,但在數(shù)學解法上卻存在著些許差異。本文主要論及平面縱波諧波人射到彈性介質(zhì)分界面上產(chǎn)生反射縱波、反射橫波、透射縱波、透射橫波時給位移位振幅方程組。本文采用位移波函數(shù)進行研究,首先講述關于Zoeppritz方程的的一些基本問題,主要包括位移形式的邊值定解問題以及求解該邊值定解問題的解函數(shù)。歸納推導在地震勘探坐標系下P波和SV波分別從四個象限入射的Zoeppritz方程。求解Zoeppritz方程首先設定滿足定解問題波動方程的平面波函數(shù),該波函數(shù)包含有入射波,同類型反射波,同類型透射波以及轉(zhuǎn)換反射波和轉(zhuǎn)換透射波五種具體波,規(guī)定這些波射線上體積元的偏振方向,接著把波函數(shù)代入界面方程,同時我們還定義反射系數(shù)以及透射系數(shù),最后整理出來的解函數(shù)即為相對應的Zoeppritz方程。然后對Zoeppritz方程進行數(shù)值計算,通過數(shù)值計算可以使我們更加清晰更加透徹的認識地震波在不同條件的界面上散射規(guī)律。當入射角到達臨界角附近時,透射波則沿分界面滑行,而當入射角大于臨界角時,透射波此時退化成為不均勻波,不均勻波的波數(shù)以及反射系數(shù)均為復數(shù),廣角反射接收到的就是這段能量。我們通過合成地震記錄發(fā)現(xiàn),在臨界角附近的振幅和相位的急劇變化,由于臨界角的存在使得反射波同相軸扭曲復雜化。
[Abstract]:The seismic waves will produce the same kind of reflection wave, converted reflection wave and converted transmission wave, as well as refraction and total reflection. We call this kind of physical phenomenon the generalized scattering of seismic wave on the boundary surface.This part is an important part of seismic wave field propagation theory and is widely used in seismic exploration and development fields such as oil and gas hydrate and so on.The study of seismic wave scattering at free interface and elastic interface includes two aspects: on the one hand, the boundary value solution problem of wave equation and interface boundary condition, and on the other hand, it is the process of solving the boundary value definite solution problem.In a word, the solution to the scattering on the boundary surface of seismic wave is to determine the boundary value solution problem satisfied by the wave equation, and then to find out the solution function of the boundary value definite solution problem.However, the boundary value determination problem is divided into displacement potential and displacement expression, so there are two different research methods.The displacement potential function is used to study, which is called "displacement potential method", and the typical representation of displacement potential method is "Knott" equation.The method of displacement function is called displacement method, and the typical representative of displacement method is Zoeppritz equation.The displacement method and the displacement potential method are consistent in physical meaning, but there are some differences in the mathematical solution.This paper mainly deals with the equations of displacement potential amplitude when plane longitudinal wave harmonics are emitted onto the elastic medium boundary surface to produce reflected P-wave, reflected S-wave, transmitted P-wave and transmitted S-wave.In this paper, the displacement-wave function is used to discuss some basic problems about the Zoeppritz equation, mainly including the boundary value solution problem in the form of displacement and the solution function for the boundary value definite solution problem.The Zoeppritz equations of P-wave and SV wave incident from four quadrants in seismic exploration coordinate system are deduced.To solve the Zoeppritz equation, a plane wave function satisfying the wave equation of definite solution is first set up. The wave function consists of five concrete waves: incident wave, reflection wave of the same type, transmission wave of the same type, and converted reflection wave and converted transmission wave.The polarization direction of the volume element on these wave rays is defined, and then the wave function is substituted into the interface equation. At the same time, we also define the reflection coefficient and the transmission coefficient. The final solution function is the corresponding Zoeppritz equation.Then the Zoeppritz equation is numerically calculated and the scattering law of seismic waves on the interface of different conditions can be more clearly and thoroughly understood by numerical calculation.When the angle of incidence reaches the critical angle, the transmitted wave glides along the interface, and when the angle of incidence is greater than the critical angle, the transmitted wave degenerates into an inhomogeneous wave, and the wave number and reflection coefficient of the inhomogeneous wave are both complex.The wide angle reflection receives this energy.We find that the amplitude and phase change rapidly near the critical angle by synthetic seismogram, and the coaxial distortion of reflection wave is complicated by the existence of critical angle.
【學位授予單位】：西南石油大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：P631.4

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