We have developed algorithms for modelling seismic waveforms to constrain regional Earth structure. The seismogram is represented as a sum of locked-mode travelling waves in a layered medium. This representation is convenient as it allows us to model structures with slowly varying heterogeneity and to construct differential seismograms. Describes the techniques we have implemented that enable us to compute synthetic and differential seismograms in an efficient and stable manner. The computational methods are sufficiently rapid that many modes can be included and in some cases the entire seismogram may be modified. These algorithms are applied to model a set of seismograms of southern Mexican earthquakes recorded in northern Mexico. The frequency bandwidth of these data is centred at 0.067 Hz and we demonstrate that even at these relatively high frequencies, many features of the seismogram can be successfully modelled. Our results suggest that the structure within the recording array in northern Mexico is resolvably different from that to the south. We find that the average shear velocity of the lower lithosphere of southern Mexico is very low, approximately 4.3 km s-1. If the low-velocity region is confined to the Trans Mexican Volcanic Belt, the shear velocities between 20-80 km depth are approximately 3.3 km s-1. This may be correlated with partial melt and is consistent with the active volcanism and high heat flow found in the region. -Authors