The problem of reconstructing ionospheric electron density from ground-based receiver to satellite total electron content (TEC) measurements was formulated as an underdetermined discrete linear inverse problem. The fact that electron density, TEC, and the ray path distances which relate them are all positive, coupled with the sparse nature of the equations relating TEC and electron density, has been used to implement a computationally efficient ionospheric tomography reconstruction algorithm. The algorithm uses an iterative cross-entropy optimization technique, where the Kullback-Leibler distance is used to define a functional that is minimized using an alternating projection iterative method. Using PIM (parameterized ionospheric model) generated data as a nonnegative prior estimate of the electron density, both maximum entropy and minimum cross-entropy reconstructions have been produced. The high quality of these reconstructions, coupled with the computational efficiency of this algorithm, indicates the potential utility of this technique for real-time ionospheric tomography.¿ 1997 American Geophysical Union