地震波传播的哈密顿表述及辛几何算法

地震波传播过程本质上是能量在传播过程中逐步损耗直至殆尽的过程，而在实际应用中，常在无能量损耗假设下，用弹性波动方程或标量波动方程描述它.在哈密顿(Hamilton)体系表述下，地震波传播过程即为一个无限维的哈密顿系统随时间的演化过程.若不计能量损耗，波场演化过程实质上为一个单参数连续的辛变换，因而对应的数值算法应为辛几何算法.本文首先从地震波标量方程出发，给出哈密顿体系下地震波传播的表述，即任意两个时刻的波场是通过辛变换联系起来的.随后，把波场在时间和相空间离散化后，给出了用于波场计算的一些辛格式，如显式辛格式、隐式辛格式和蛙跳辛格式.并进一步讨论了有限差分格式和辛格式的异同.然后，应用显式辛格式和同阶的有限差分方法给出了同一理论速度模型下的波场和Marmousi速度模型下的单炮记录.数值结果表明，辛算法是一类可行的波场模拟的数值算法.在时间步长较小时，有限差分方法是辛算法的一个很好近似.文中的理论和方法，为地震波传播理论及实际应用研究提供了新的途径.

HAMILTONIAN DESCRIPTION AND SYMPLECTICMETHOD OF SEISMIC WAVE PROPAGATION

Seismic wave propagation is a process of energy dissipation. This process is often described by elastic or scalar wave equation with the assumption of no dissipation. In the Hamiltonian fram, seismic wave propagation is evolution of the infinite dimensional Hamiltonian system. If without dissipation, the propagation is essentially a symplectic transformation with one parameter, and, consequently, the numerical calculation methods of the propagation ought to be symplectic, too. For simplicity, only the symplectic method based on scalar wave equation is given in this paper. A phase space is constructed by using wave field and its derivative the scalar wave equation as an evolution equation of a linearly Hamiltonian system has symplectic propertiy. After discreting the wave field in time and phase space, many explicit, implicit and leap frog symplectic schemes are deduced for numerical modeling. The scheme of Finite difference (FD) method and symplectic schemes are compared, and FD method is a good approximate symplectic method. A second order explicit symplectic sheme and FD method are applied in the same conditions to get a wave field in a synthetic model and a single shot record in Marmousi model. The result illustrates that the two method can give the same wave field as long as the time step is enough little. The theory and methods in this paper, gives a new way for the theoretic and applying study of wave propagation.