Transmission of elastic waves through solids containing small-scale heterogeneities

An explicit expression. based on the second-order Neumann expansion. is derived for computing the gradual evolution of a monochromatic compressional seismic wave as it propagates through a medium, the density of which varies in two dimensions. In particular, the case is considered where these variations take place on a scale that is small with respect to the seismic wavelength. For monochromatic plans waves propagating in a particular direction, a medium containing small-scale inhomogeneities can be replaced by a smoother 'apparent' medium. The apparent medium accounts for both first- and second-older scattering effects, is frequency-dependent and also depends on the direction of propagation. Frequency dependent wave speeds, which can be measured very accurately, can therefore be an indication of the presence of heterogeneities of a much smaller scale than the wavelength of the incident wave. This is of particular interest for seismic imaging.