Diffraction tomography using multimode surface waves

A new method is described that makes it feasible to include scattered and converted surface waves into waveform inversions for the three-dimensional (3-D) structure of the Earth. The single scattering (Born) approximation forms the basis of the method. In order to minimize the amplitude of the scattered wave field, the background model is first adapted to correct for nonconverted, forward-scattered wave energy. We then perform Born inversion of the difference between the measured and synthetic waveforms, including it suite of Love and Rayleigh modes. The Born approximation yields linear equations of the form A delta gamma = delta u(Born), which allow the determination of the three-dimensional perturbations gamma to the background model from the scattered wave field delta u(Born). This procedure is followed separately for each source-receiver pair to allow for optimized background models for each signal, as well as to minimize the computational burden. We winnow the data vector for each path by performing singular value decomposition using a diagonalization of AA(T). In a realistic example we found that each vertical component seismogram yields 30-40 Linear constraints on the 3-D Earth, significantly more than with conventional pure-path (WKBJ) inversions. In a synthetic test, one seismogram is shown to be able to image a simple model of a point scatterer off the great circle. As a spin-off of the formulation of the multimode inverse scattering problem, we not only obtain a series of eigenvectors that rank the sensitivity of a seismogram to Earth structure in a series of geometrical patterns, we also can compute the surface wave equivalent of a Fresnel zone.