Computerized ionospheric tomography (CIT) is an imaging technique that produces a two-dimensional image of the electron density in the ionosphere. This technique uses total electron content (TEC) measurements from multiple ground stations as input data to computerized tomography (CT) algorithms. Unfortunately, the TEC data suffer from ambiguities that are introduced during the measurement process. Ionosphere researchers model these ambiguities as an unknown constant which is added to the true TEC data. This is known as the constant of integration problem. Traditional methods for resolving this problem, such as the single- and two-station methods, are based on ionospheric models and may perform poorly when the true ionospheric structure differs greatly from the model. This paper introduces a new method to estimate the unknown constants. The method, called the reconstruction residuals method, does not make use of ionospheric models but is based on the correlation between the unknown constants and the residual errors from a CIT reconstruction. Simulation results using this method are compared with the traditional estimation methods. Results from the new method compare favorably with results from the traditional methods.