Fast and accurate calculation of travel time and ray path of seismic waves are two key problems in the field of seismic imaging and inversion. Solving these two problems plays a key role in the in-depth study of the fine structure of the Earth’s interior, the efficient exploration of the distribution of different types of resources underground, and even the scientific prediction of possible natural disasters. However, the situation becomes more complex when anisotropic media are involved. In such media, the velocity of seismic waves does not remain constant as in isotropic media, but varies with the propagation direction of seismic waves. This characteristic makes it a challenging task to calculate the propagation time of seismic waves and to determine the ray paths in anisotropic media. The conventional isotropic fast marching method （FMM） shows excellent performance advantages in solving the travel time problem in isotropic media. It is able to produce results relatively quickly and accurately due to its efficient algorithm and stable computational process. However, when we try to apply the method to solve the Eikonal equation under anisotropic conditions, we are faced with a critical aspect: the need for an accurate estimation of the phase velocity. This is because in an anisotropic medium, the velocity, anisotropy parameter and stratigraphic dip are not constant, and their dynamic changes make the calculation of the propagation direction complicated. The complex calculation of the propagation direction further increases the difficulty of phase velocity estimation. As a key parameter describing the velocity of seismic waves in a given direction, the accuracy of the phase velocity estimation is like the first domino in the chain, which directly affects the subsequent propagation time and the final accuracy of the ray path calculation. Any small deviation in the phase velocity estimate can be amplified in the subsequent calculations, leading to a large deviation of the final results from the actual situation.

In this paper, the fast marching method （FMM） for anisotropic media, which is a mature and widely used method, is taken as a solid basis for an in-depth study. Through rigorous theoretical derivation and practical verification, a phase velocity estimation method for tilted transversely isotropic （TTI） media and an anisotropic ray tracing method based on the principle of phase velocity prediction （PVP） are proposed. The method cleverly inherits the outstanding features of speed and stability of isotropic FMM, while maintaining the same level of computational accuracy as isotropic FMM, ensuring reliability and accuracy of results. A phase velocity prediction formula for TTI media has been derived. The formula can directly and accurately calculate the corresponding changes in phase velocity based on the changes in key factors such as seismic wave velocity, anisotropy parameter and stratigraphic dip. It greatly simplifies the complicated process of solving the anisotropic Eikonal equation and breaks the bottleneck of the previous methods in terms of computational efficiency and accuracy. It also greatly improves the computational efficiency of ray tracing, making the calculation process more efficient and faster, and at the same time significantly improves the computational accuracy, providing a more reliable and accurate basis for subsequent processing and interpretation of seismic exploration data.

In order to comprehensively and thoroughly verify the computational efficiency and accuracy of the method in practical applications, simulation experiments are carried out on a number of representative typical models. First, a unified theoretical model is used as the first test object. The model is relatively simple in structure and has a clear theoretical analytical solution, which can provide a clear reference standard for the accuracy verification of the method. Through comparative analyses, the results clearly show that the computational results obtained by the method are in high agreement with the theoretical results, which strongly verifies the excellent performance of the method in terms of accuracy. Then, the vertical transverse isotropy （VTI） reference model and Marmousi model are further tested. The results show that this method has small calculation error and good stability, and can be applied to ray tracing of complex structures. In addition, in order to further quantify the performance of the method, a multi-source near-surface numerical simulation test is also carried out. A model of complex near-surface structure is designed, and the wave fields of multiple sources are simulated by finite difference method. The seismic record of the model is simulated and its first arrival time is picked up. Then, ray tracing and travel time calculation are carried out by using this method, and the results are compared with those of finite difference simulation. The calculation errors and time of isotropic FMM and anisotropic PVP-FMM are statistically analyzed. The results show that both of them have the same calculation error, and the average error is very small. The error of anisotropic PAP-FMM is 43% less than that of anisotropic FMM without phase velocity prediction. A computer with 2.4 GHz CPU is used for ray tracing. The total time consumption of 913 source for isotropic FMM and PAP-FMM are 326 seconds and 372 seconds respectively. Because anisotropic FMM increases the calculation links of phase velocity and group velocity, the time consumption is increased by 10% compared with isotropic FMM. PVP-FMM added a phase angle prediction link, which increased the time consumption by 14%. In summary, this method has the same computational efficiency as isotropic FMM, and also has good adaptability and stability for near-surface models.