The modeling of waveforms in a volcanic geometry must take into account topography, very low velocity surface layers, and the possibility of velocity or attenuation anomalies and complexity in the resolved structure. These features may produce shadow zones, diffractions and complicated changes in the amplitude and phase of the waveforms. Non-linear inversion of such waveforms requires fast, finite frequency forward methods. For a volcanic geometry similar to that of Vesuvius, we compare and contrast the performance of discrete wavenumber - boundary integral, finite-differences, and the new path-summation methods with regards to efficiency, accuracy, and the suitability of the results for waveform inversion.

Path-summation is a new, rapid method to calculate approximate waveforms in smooth, two- and three-dimensional media. A solution for the scalar wave equation is obtained by a Monte Carlo summation of elementary signals over a representative sample of all possible paths between the source and receiver. For each path, this elementary signal (a time-derivative of the source function) is summed into the waveform at the travel-time along the path. The constructive and destructive interference of these signals converges to the approximate waveform response, including non-ray phenomena such as diffraction.