Regularized solution to fast GPS ambiguity resolution

In rapid global positioning systems (GPS) positioning one of the key problems is to quickly determine the ambiguities of GPS carrier phase observables. Since carrier phase observations are generally collected only for a few minutes in the mode of rapid GPS positioning, the least squares floating solution of the ambiguities will be highly correlated and the decorrelation approach has often been used in order to reduce the search space of integer ambiguities. In this paper we propose a regularized algorithm as an alternative approach to decorrelation, and compute the regularization parameter by minimizing the trace of mean squared errors. Since regularization has been essential to solve inverse ill-posed problems and shown to be very significant in reducing the condition number of normal matrices, we will explore possible applications of regularization for improving the high correlation of the estimated float ambiguities. Numerical experiments with 50 epochs of single frequency observations show that the condition number after regularization reduces to half of that of the floating solution if the ambiguities could be known to 2-3 cycles. If better knowledge about the ambiguities could be obtained to within 1 cycle, further improvement can be achieved. The results indicate that regularization could be used for fast GPS ambiguity resolution. Our experiments also demonstrate that a scale factor of about 8 is needed to multiply the estimated variance of unit weight for obtaining a reasonable estimator for the accuracy of float ambiguities.