Local inversion of the sonar transform regularized by the approximate inverse

A new reconstruction method is given for the spherical mean transform with centers on a plane in which is also called the sonar transform. Standard inversion formulas require data over all spheres, but typically, the data are limited in the sense that the centers and radii are in a compact set. Our reconstruction operator is local because, to reconstruct at , one needs only spheres that pass near , and the operator reconstructs singularities, such as object boundaries. The microlocal properties of the reconstruction operator, including its symbol as a pseudodifferential operator, are given. The method is implemented using the approximate inverse, and reconstructions are given. They are evaluated in light of the microlocal properties of the reconstruction operator.