Evaluation of the Regularization Algorithm to Decorrelation of Covariance Matrix of Float Ambiguity in Fast Resolution of GPS Ambiguity Parameters

Precise positioning in Real Time Kinematic (RTK) applications depends on the accurate resolution of the phase ambiguities. In RTK positioning, ambiguity parameters are highly correlated, especially when the positioning rate is high. Consequently, application of de-correlation techniques for the accurate resolution of ambiguities is inevitable. Phase ambiguity as positioning observations by the Global Positioning System (GPS) is referred to the number of the complete cycles of the signal emitted by a satellite, just before its reception by a receiver. Fast and accurate estimation of the phase ambiguities still is a challenge in real time positioning by GPS. Various methods have been developed for the ambiguity resolution. Initialization time, reliability and accuracy of the resolved ambiguities are the key sectors in each resolution technique. 
Methods of ambiguity resolution usually start with the float solution of the ambiguity parameters and end up with their integer values. The method of least-squares is usually used for computing the float solution for ambiguity parameters. To search and fix the corresponding integer values, conditional least-squares is normally used. Search-based methods, as the most commonly used techniques, are usually executed in three successive steps. At first, standard least-squares is used for estimating a float solution for ambiguity parameters and their associated variance-covariance (V-C) information. In this step, the integer nature of the ambiguity parameters is ignored. Next, the method of Weighted Integer Least-Squares (WILS) is used for resolving the integer values of the ambiguity parameters. Real-valued unknowns are then estimated using the integer estimates of the phase ambiguities. The previous step is the most important part of the problem. De-correlation of the VC matrix of the ambiguities' float solution was firstly suggested by Teunissen in order to increase the reliability and speed up the resolution process.
This paper proposes a new method for de-correlating the V-C matrix of ambiguity parameters. A regularization algorithm has been used to achieve the lowest correlation between floating ambiguities. The regularization parameter has been selected in such a way so that the traces (sum of diagonal elements of V-C matrix) of V-C matrix of floating ambiguities are minimized. In order to investigate the de-correlating and efficiency of the proposed method, two criteria of the condition number and also the trace of V-C matrix of floating ambiguities have been used. After de-correlation of V-C matrix of floating ambiguities and space transformation, the sequential conditional least squares is used to search for the integer ambiguities. This method calculates the phase ambiguity with considering the correlation between them. Also, a-posteriori variances of unit weight for the float and fixed solutions can be used to check the consistency of a resolved ambiguity with the measurements. When some of the ambiguities are not correctly resolved, a-posteriori estimate of the variance of unit weight may not statistically conform to the estimate of this parameter in the float solution. Therefore, the comparison of the a-posteriori estimates of this parameter for the float and fixed solutions provides a measure to analyze the consistency of the resolved ambiguities with measurements. All results from the proposed method of this paper have been compared with the results of the famous Lambda method.