Hamiltonian description and symplectic method 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 property. 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.