Hamiltonian Description of Seismic Wave Propagation and Symplectic Method

Abstract

AbstractThe process of seismic wave propagation virtually is a process of energy dissipation. While in practice, it is often described by elastic or scalar wave equation with the assumption of no dissipation. Under the mathematic frame of Hamiltonian dynamic system, the propagation of seismic wave is the evolution of the infinite dimensional Hamiltonian system. If without dissipation, the propagation essentially is a symplectic transformation with one parameter, and consequently, the numerical calculation methods of the propagation ought to be symplectic too, which is known as symplectic method. For simplicity, only the symplectic method based on scalar wave equation is given in this paper. Let wave field and its derivative construct a phase space, the scalar wave equation as an evolution equation of a linear Hamiltonian system has symplectic property. Consequently, after discretizing 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 approximation of symplectic method. A second order explicit symplectic scheme 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, otherwise accuracy of FD method may be questioned. The theory and methods in this paper provide a new approach for the theoretic and applied study of wave propagation.