New parabolic equation solutions for high frequency and elastic media problems

Parabolic equation techniques are very efficient and can provide accurate solutions for range-dependent problems, which involve environments that change in the direction of propagation. The classical approach to treat range dependence is to approximate the medium in terms of a series of range-independent regions, in which the factorization of the parabolic equation is exact. This thesis is concerned with two different classes of range-dependent problems. The first is the development of a new solution to the fluid parabolic equation that combines the accuracy of the split-step Pade approach with the efficiency of the split-step Fourier solution. This new method has a greater efficiency for some problems without sacrificing accuracy, because it can tolerate large steps at high frequencies. It handles wide propagation angles and requires only a small number of terms in the rational approximation. The second class of problems concerns solving the parabolic equation in elastic media. Efficiency and accuracy of the elastic parabolic equation can be achieved by the application of single scattering corrections at vertical interfaces between range-independent regions. This approach was previously shown to be very effective for fluid media. The single scattering correction is combined with a new parabolic equation formulation that can account for all types of body, interface, and boundary waves, using improved rational approximations. This correction is applied to the solution of problems in layered media. A generalization of the single scattering is also made for application to problems with a large contrast in material properties across a vertical interface. Since the single scattering method is implemented in terms of an iteration formula, the procedure may not converge for large contrast cases. A technique is developed to overcome this problem and to extend the application of the parabolic equation to a large class of problems. A vertical interface is artificially split into a series of slices and the single scattering correction is applied to each of them, neglecting multiple scattering events. This approach is accurate for cases with sharp scattering features and for problems involving gradual range dependence, such as sloping interfaces.