  We study inversion of the spherical Radon transform with centers on a sphere
(the data acquisition set). Such inversions are essential in various image
reconstruction problems arising in medical, radar and sonar imaging. In the
case of radially incomplete data, we show that the spherical Radon transform
can be uniquely inverted recovering the image function in spherical shells. Our
result is valid when the support of the image function is inside the data
acquisition sphere, outside that sphere, as well as on both sides of the
sphere. Furthermore, in addition to the uniqueness result our method of proof
provides reconstruction formulas for all those cases. We present a robust
computational algorithm based on our inversion formula and demonstrate its
accuracy and efficiency on several numerical examples.
