| A large number of seismic explorations for petroleum, natural gas, mineral, etc resources are carried out in mountainous regions in current world, especially in western China. Comparing with the seismic explorations in plain, there are some special problems in mountainous regions should be solved, such as:①We have not done some deep enough research on the propagation law of seismic wave in mountainous regions, and many theoretical and practical problems have not been solved for a long time;③There are many difficulties for collecting seismic data, and the quality of the seismic data which is collected in mountainous regions is relatively poor;③The traditional theory of seismic wave and data processing techniques are both based on the assumption that the earth's surface is plane. The static correction is the main data processing technique for treating the undulating earth's surface problems, but it has been shown that most of the effects caused by surface topography cannot be removed by static correction. Specifically, the imaging quality is sometimes still poor, although a careful static correction has been performed.A basic way for solving the above problems is to increase the study on the seismic wave field's feature in mountainous regions by the following two main ways: numerical simulation of wave equation and ray tracing. In this paper, we study the methods for calculating the seismic traveltime and raypath under undulating earth's surface condition. The main contents of our thesis as following:①To select the methods which are suitable for solving the irregular surface problem, we review the conventional methods including two-point ray tracing method, shortest path ray tracing method, finite difference method, linear interpolation method, wavefront construction method and the other methods that are used for calculating seismic traveltime and raypath. We also make a comprehensive comparison among these methods. At last, we choose the fast marching method and linear interpolation method as the basis algorithm for our following research works. ②For solving the eikonal equation in 2D undulating earth's surface condition, we present two schemes:one is an upwind finite difference scheme with ladder-like grids, and another is an finite difference scheme with non-uniform grid spacing. Specifically, the first scheme use the upwind finite difference scheme with ladder-like grids as the local traveltime formulas, and the second scheme deduce the local traveltime formulas by introducing the finite difference scheme with non-uniform grid spacing into upwind finite difference scheme and by integrated applying the Huygens principle and the Fermat principle. Specifically, the two schemes both use the narrow band technique under undulating earth's surface condition as the wavefront expansion scheme. We analyze the stability and accuracy of the two new methods, and make a conclusion that the two methods both have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.③To calculate the seismic traveltime in 2D undulating earth's surface condition, we present a new linear interpolation scheme by improving the conventional linear interpolation method with the hybrid grids that are combined by the triangular and the rectangular grids. This new scheme also uses the narrow band technique under undulating earth's surface condition as the wavefront expansion scheme. We also analyze the stability and accuracy of the new scheme, and make a conclusion that the the new scheme has an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.④We do a tentative research on the curvilinear grid methods for calculating the seismic traveltime under 2D undulating earth's surface condition. We first deduce some theoretical formulas by coordinate transformation method and orthogonal curvilinear grid method and secondly propose the corresponding numerical implementation schemes for these two curvilinear grid methods. Finally, we discuss the technical difficulties of the curvilinear grid methods in their numerical implementation process.⑤We make a comprehensive comparison among the methods (These methods include an upwind finite difference scheme with ladder-like grids, an finite difference scheme with non-uniform grid spacing, a linear interpolation scheme with hybrid grids, and two curvilinear grid methods) are proposed in our thesis for calculating the seismic traveltime under 2D undulating earth's surface condition, and give some numerical examples:Firstly, we make the accuracy and efficiency comparison between the conventional fast marching method and linear interpolation method. Secondly, we make a comparison among the three methods which are all in the Cartesian coordinates. At last, we make a comparison among the methods which are all in the Cartesian coordinates and the curvilinear grid methods.⑥To calculate seismic raypath in 2D undulating earth's surface condition, we study the methods in various grids base on the linear interpolation scheme. The raypath is calculated by linear interpolation scheme basing on the distribution information of the traveltime. Algorithm analysis and some numerical examples show that the method is proposed here have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media.⑦For calculating the seismic traveltime in 3D undulating earth's surface condition, we present a 3D finite difference scheme with non-uniform grid spacing. This scheme uses non-uniform grids for building up the 3D topography model, and deducing the local traveltime formulas by introducing the finite difference scheme with non-uniform grid spacing into upwind finite difference scheme. We also analyze the stability and accuracy of the new scheme, and make a conclusion that the new scheme has an unconditional stability and good accuracy, and good adaptability to any 3D rugged terrain and any 3D complex media.⑧To obtain some propagation laws of seismic wave under the undulating earth's surface condition, we study the methods for calculating traveltime and raypath of some types of seismic wave in complex media under undulating earth's surface condition. Here, the seismic wave types include first-arrival, refraction, transmission, diffraction, reflection, multiple reflection, etc.Based on the above research, we obtain the following results:①We propose three methods including an upwind finite difference scheme with ladder-like grid, an finite difference scheme with non-uniform grid spacing, and a linear interpolation scheme with hybrid grid for calculating the seismic traveltime under 2D undulating earth's surface condition in the Cartesian coordinates. These methods all have an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media;②We propose two curvilinear grid methods including coordinate transformation and orthogonal curvilinear grids for calculating the seismic traveltime under 2D undulating earth's surface condition and discuss the technical difficulties of the curvilinear grid method in their numerical implementation process;③We present a method for calculating the raypath in various grid types in undulating earth's surface condition basing on the linear interpolation method which has an unconditional stability, good accuracy, and good adaptability to any 2D rugged terrain and any 2D complex media;④We propose a 3D finite difference scheme with non-uniform grid spacing for calculating the seismic traveltime under 3D undulating earth's surface condition, and this method has an unconditional stability, good accuracy, and good adaptability to any 3D rugged terrain and any 3D complex media;⑤ccording to the calculation result from the methods that are used for calculating the traveltime and raypath of some types of seismic wave in complex media under undulating earth's surface condition, we obtain some laws of seismic wave propagation laws under undulating earth's surface condition. The studying works and some conclusions in this thesis are of benefit to deeply research some propagation laws of seismic wave under undulating earth's surface condition, and to prove some reference information for the design of seismic data acquisition system under undulating earth's surface condition. The methods are proposed in our thesis can provide a set of tools for improving the effect of seismic data processing and for presenting some new seismic data processing techniques under undulating earth's surface condition.
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