The Kaczmarz method was one of the first methods for image reconstruction in computerized X-ray tomography. In the last decades it has been extended to nonlinear imaging techniques, most notably to solving inverse problems based on the wave equation, such as ultrasound tomography in radiology and full waveform inversion in seismic imaging. We derive Kaczmarz type algorithms in analogy to the ART algorithm of X-ray tomography. We give heuristic conditions on the initial approximation for convergence and study in particular the case of missing low frequencies in the source pulse. The behavior of the algorithm in the presence of caustics and trapped rays is studied. Various methods for speeding up the convergence are discussed. We also outline the application to falling weight deflectometer data.
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