A test of ray theory and scattering theory based on a laboratory experiment using ultrasonic waves and numerical simulation by finite-difference method

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The structure of the Earth is represented by a wide spectrum of small- and large-scale structures. However, tomographic imaging techniques based on ray theory are often applied inappropriately in models with a characteristic length of heterogeneity smaller than the wavelength and width of the Fresnel zone. In other words, the conditions for ray theory are not satisfied in such models. It is therefore necessary to apply the diffraction theory of waves in tomographic reconstruction techniques in order to retrieve images of the Earth with a more general theory for wave propagation than ray theory. Physically speaking, scattering theory takes the finite-frequency effect of waves into account. We performed a test of ray theory and scattering theory in an ultrasonic wave experiment and in a numerical finite-difference experiment using random media with correlation lengths smaller than the width of the Fresnel zone. We used a stochastic approach to compute the mean squared value of time-shift variations calculated from ray theory and diffraction theory. The theoretical results were compared with the experimental values obtained in the laboratory experiment using rock samples with different length-scales of heterogeneity and from numerical experiments on wave propagation in quasi-random media. We observed that ray theory systematically overestimates the mean squared value of time-shift variations, while the observed statistical values from the laboratory experiments are well predicted by scattering theory. This means that tomographic imaging techniques based on ray theory suffer from a loss of resolution when the reconstructing models have a characteristic length of heterogeneity smaller than the width of the Fresnel zone.