A computer-implemented method of generating and storing to an electronic storage medium approximations of a set of wave equations describing the propagation of seismic waves of one or more frequencies through a seismic volume of interest, the method comprising: (a) determining a ray-based initial beam solution that approximates the propagation of seismic waves at a first frequency through the seismic volume of interest as a set of one or more beams; (b) generating a system of equations that define a finite element approximation of a wave equation describing the propagation of the seismic waves of the first frequency through the seismic volume of interest; (c) solving the system of equations generated at step (b) implementing the ray-based initial beam solution determined at step (a) as a starting solution to the system of equations, thereby creating a refined finite element beam solution approximating the propagation of the seismic waves of the first frequency through the seismic volume of interest with an enhanced accuracy over the approximation of the ray-based initial beam solution; (d) storing the refined finite element beam solution created at step (c) to an electronic storage medium; (e) identifying a second frequency; (f) determining an approximate initial beam solution of seismic wave propagation at the second frequency through the seismic volume of interest, wherein the approximate initial beam solution at the second frequency is obtained using ray-based approximations to modify the refined finite-element beam solution for the first frequency achieved at step (c); (g) generating a system of equations that define a finite element approximation of a wave equation describing the propagation of the seismic waves of the second frequency through the seismic volume of interest; (h) solving the system of equations generated at step (g) implementing the approximate initial beam solution determined at step (f) as a starting solution to the system of equations generated at step (g), thereby creating a refined finite element beam solution approximating the propagation of the seismic waves of the second frequency through the seismic volume of interest; and (i) storing the refined finite element beam solution created at step (h) to an electronic storage medium. (a) determining a ray-based initial beam solution that approximates the propagation of seismic waves at a first frequency through the seismic volume of interest as a set of one or more beams; (b) generating a system of equations that define a finite element approximation of a wave equation describing the propagation of the seismic waves of the first frequency through the seismic volume of interest; (c) solving the system of equations generated at step (b) implementing the ray-based initial beam solution determined at step (a) as a starting solution to the system of equations, thereby creating a refined finite element beam solution approximating the propagation of the seismic waves of the first frequency through the seismic volume of interest with an enhanced accuracy over the approximation of the ray-based initial beam solution; (d) storing the refined finite element beam solution created at step (c) to an electronic storage medium; (e) identifying a second frequency; (f) determining an approximate initial beam solution of seismic wave propagation at the second frequency through the seismic volume of interest, wherein the approximate initial beam solution at the second frequency is obtained using ray-based approximations to modify the refined finite-element beam solution for the first frequency achieved at step (c); (g) generating a system of equations that define a finite element approximation of a wave equation describing the propagation of the seismic waves of the second frequency through the seismic volume of interest; (h) solving the system of equations generated at step (g) implementing the approximate initial beam solution determined at step (f) as a starting solution to the system of equations generated at step (g), thereby creating a refined finite element beam solution approximating the propagation of the seismic waves of the second frequency through the seismic volume of interest; and (i) storing the refined finite element beam solution created at step (h) to an electronic storage medium.