Study Of The Mathematical Approximation Problems In Seismic Exploration

Seismic exploration is a geophysical method which is applied to infer the elasticity and density differences underground, through bombing with man-made sources and observing, analyzing the seismic wave responses.In this thesis, normal move out(NMO), Zoeppritz equations and converted wave sorting are implied to discuss the relative error of the current approximated formula and exact solution applied in seismic exploration so that these approximate formula could be utilized reasonably. Three sections are diagramed in this thesis: 1) introduction of some basic knowledge about the seismic exploration; 2) numerical simulation and analyzing, i.e. according to physics-mathematical models, solving the exact solutions, analyzing approximated formula, and discussing the relative errors between approximated formula and exact solutions; 3) summarization. At the basis of the upper study, the following conclusions can be deduced.1) After analyzing the relative errors of NMO. It was founded that the bigger the deflection deep ration is, the bigger the relative error is, and the relative error of the high-order approximation is less than the one of low-order approximation. When the deflection deep ratio ranges from zero to two, the relative error of first-order approximation is less than 25% and the ones of second-order and third-order approximation are both less than 10%. From the point of increasing the level of precision and computation, the second-order approximation is favorable for general computing.2) The exact solution of the Zoeppritz equations is extremely complicated. When the incident angle reaches the critical angle, the reflection coefficient becomes plural, the argument becomes the reflectance phase angle and the energy mutates suddenly. In Ostrander, Goodway model, the changes of the phase angle has some relationship with the changes of incident angle. Along with the increasing of the incidence angle, the magnitude of the phase angle also increases, which is relatively a linear relationship. On the condition of same incident angle and same , the bigger P-wave to S-wave velocity ratio is, the bigger the low-order error range is. On the condition of same ratio of reflected wave velocities and same , the smaller the incident angle is, the larger the low-order range is. On the condition of same incident angle and same ratio of reflected wave velocities, the more is close to 0, the larger the low-order error range.3) Sorting conversion point is not very complicated, and computer could quickly solve up now. However, the process of calculation is relatively complex comparing with the approximated formula. It takes long time when calculating the instant mass data. Among these current approximated formulas, the relative error of first-order approximate is the biggest (less then 14%) and Thomsen approximated formula is the best (less then 1.1%). In addition, the precision of iterative formula is also in a high level, but increasing the number of iteration can also increase the calculation amount and computer running time.