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Ionospheric tomography systems provide data that can be used to reconstruct images of ionospheric electron density. Since ionospheric tomography systems have fundamentally poor vertical resolution, a priori information on the vertical distribution of ionospheric electron density must be used in the reconstruction algorithm. This can be accomplished using an existing technique called orthogonal decomposition. However, the poor vertical resolution of the imaging system makes the reconstruction problem numerically ill conditioned. This paper presents a new ionospheric reconstruction algorithm called the residual correction method (RCM). The RCM is a fast, efficient, and numerically stable ionospheric tomography algorithm. This paper will present results demonstrating the performance of the RCM algorithm using a realistic example.