Small-scale heterogeneity alters the arrival time of waves in a way that cannot be explained by ray theory. It is because ray theory is a high-frequency approximation that does not take the finite-frequency of wavefields into account. A theory based on the first Rytov approximation is developed wherein the effect of finite-frequency waves is included. It is found that the developed scattering theory predicts well the arrival time of waves propagating in media with small-scale inhomogeneity that has a characteristic length smaller in size than the width of Fresnel zones. In the regime for which scattering theory is relevant, it is found that caustics are easily generated in wavefields, but this does not influence the good prediction of finite-frequency arrival times of waves by scattering theory. The regime of scattering theory is relevant when the characteristic length of inhomogeneity is smaller than the width of Fresnel zones.

By using ray perturbation theory, a condition for the development of triplications is defined for plane wave sources and for point sources in 2-D and 3-D media. This theory is applied to two cases of slowness media: 1-D slowness perturbation models and 2-D Gaussian random media. The focus position in 1-D slowness models is proportional to the inverse of the square root of the relative slowness fluctuations. For Gaussian random media, the distance at which caustics generate is dependent on the relative slowness perturbation to the power of minus two thirds.

The developed scattering theory is tested in a 2-D numerical finite-differences experiment using models with small-scale heterogeneity and in a 3-D physical laboratory experiment where ultrasonic waves propagate through samples of granite with small grain-sizes of the minerals. For both kind of modelling experiments, the length-scale of inhomogeneity of the slowness models is smaller than the width of the Fresnel zones. The results of the 2-D and 3- D modelling experiment confirm that the developed scattering theory is better than ray theory in predicting the arrival time of waves propagating in complex media for which the conditions for ray theory are not valid. In general, ray theory overestimates the observed timeshifts of waves or even worse the ray theoretical timeshifts are anti-correlated with the observed ones for waves propagating in media where the effect of scattering is significant. By comparing the characteristic value of the Fresnel zone with the characteristic length of heterogeneity in different experiments from exploration seismics and seismology, it is shown that the resolution in present-day seismological tomography is that the limits of the validity of ray theory. With an emphasis on surface wave tomography, I show that it is important to implement the effect of finite-frequency waves in tomographic imaging techniques in order to retrieve small-scale structured Earth models with the correct theory.

The theory for the scattering of waves is applied in global surface wave tomography for Love waves at 40 s and 150s. The estimated tomographic surface wave models derived from ray theory and scattering theory are similar, because a restrictive regularisation condition is incorperated in the inversion so that structures with length-scales smaller than the Fresnel zones are mostly suppressed.

I treat a special case of inverse problem theory, namely the spectral leakage problem. The term spectral leakage indicates that observed data affected by structures with a length-scale that is not account for in a given inverse can leak into the long-wavelength structures that are part of the estimated model. In the case of global surface wave tomography, surface wave scattering theory is used in an inversion including the spectral leakage correction of phase velocity measurements for Love waves at periods of 40 s and 150 s. The global phase velocity models from the spectral leakage inversion are obtained without applying any damping, but they show, however, a good correlation with tectonic features such as plate boundaries, ridges and trenches.