Waveforms from global and regional networks of digital, broadband seismometers contain information about earth structure over a large range of scales. However, recovering this information demands fully non-linear inversion, which in turn requires fast forward methods. In particular, it is important to model wavefields in structures where the scale lengths of heterogeneity overlap the seismic wavelengths. Many advances in understanding of earth structure over the past few decades have been made with analytic and ray based methods which are fast and accurate, but which require highly symmetric or very smooth structures. Other advances have been made with computationally demanding numerical methods. But neither of these approaches is practical for non-linear inversion in problems where structures have scale lengths that are similar to the wavelengths.

The wavelength-smoothing method (Lomax, 1994) combines an averaging of medium properties over a wavelength with numerical propagation along ray-like wavepaths to efficiently generate broadband waveforms in complicated structures. Though the method had no theoretical justification, it was shown that it correctly reproduce several finite frequency wave phenomena and gave good agreement with more accurate synthetic calculations. However, there were some shortcomings in the results.

Here, we present a local averaging theorem which defines an effective velocity for an incremental movement of a plane wave. This theorem forms a theoretical justification of the wavelength-smoothing method and suggests improvements to the algorithm. We show how the updated method performs, and discuss its behavior with regards to the shortcomings of the earlier results.