Data truncation and the exterior reconstruction problem in reflection-mode tomography

In reflectivity tomography, a reflecting object that is immersed in an acoustically homogeneous background medium is interrogated with a short ultrasonic pulse and the corresponding reflected signal is measured at the source location as a function of time. From the set of reflected-signal data corresponding to different source-receiver locations, an image that represents the object's spatially inhomogeneous reflectivity is reconstructed. In this work, the reconstruction of reflectivity images from temporally truncated data is investigated. An iterative reconstruction algorithm is proposed and investigated for reconstructing images from such truncated data. Using a potato-peeler procedure and concepts from microlocal analysis, we describe necessary and sufficient conditions that permit for a stable reconstruction from temporally truncated data.