This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phase-space, i.e., functions of space @math and frequency @math . The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow to write fast-converging, non-asymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines. The classical results of closedness of such symbol classes under multiplication, inversion and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbol-based numerical methods handle operators and not functions, their complexity depends on the desired resolution @math very weakly, typically only through @math factors. We present three applications to computational problems related to wave propagation: 1) preconditioning the Helmholtz equation, 2) decomposing wavefields into one-way components and 3) depth-stepping in reflection seismology.
 A phase screen method for vector elastic waves is developed. The method allows rapid calculation of the propagation of elastic waves for problems where backscatter is small; only forward propagation is included in the analysis. The method can be used for propagation through either deterministic or stochastic media. The work described here generalizes existing phase screen methods for scalar waves, accounting for the difference in phase velocities of the transverse and longitudinal polarizations. The method is tested on a two-dimensional problem, whose exact solution is also computed, for constant Poisson ratios. Synthetic seismograms for two-dimensional random media using this method are compared with existing finite difference synthetics. An important use of the method is to estimate the rate of interconversion of S and P waves in terms of the energy flux to analyze the effectiveness of potential discrimination techniques. We describe the propagation and scattering of elastic waves in heterogeneous media. Decomposing the elastic wavefield into up- and down-going constituents allows the introduction of the 'one-way' wave equations and propagators. Such propagators account for transverse scattering and mode coupling. The generalized-screen expansion of the symbol of the one-way wave equation in medium contrast and medium smoothness induces an approximation of the propagator with an associated computational complexity of the one of the phase screen approximation. The generalized-screen expansion extends the phase-screen approach. It allows for larger medium fluctuations and wider-angle propagation. We illustrate the accuracy of the generalized screen with numerical examples.