Partial integer decorrelation: optimum trade-off between variance reduction and bias amplification

TLDR: In this article, a linear combination of E1 and E5 code and carrier phase measurements with a wavelength of 3.285 m and a noise level of a few centimeters is derived.

Abstract: Different techniques have been developed for determining carrier phase ambiguities, ranging from float approximations to the efficient solution of the integer least square problem by the LAMBDA method. The focus so far was on double-differenced measurements. Practical implementations of the LAMBDA method lead to a residual probability of wrong fixing of the order one percent. For safety critical applications, this probability had to be reduced by eight orders of magnitude, which could be achieved by linear multi-frequency code–carrier combinations. Scenarios with single or no differences include biases due to orbit errors, satellite clock offsets, as well as residual code and phase biases. For this case, a linear combination of Galileo E1 and E5 code and carrier phase measurements with a wavelength of 3.285 m and a noise level of a few centimeters is derived. This ionosphere-free combination preserves the orbit and clock errors, and suppresses the E1 code multipath by 12.6 dB. Since integer decorrelation transformations, as used in the LAMBDA method, inflate biases, the number of such transformations must be limited, and applied in a judicious order. With a Galileo type constellation, this leads to a vertical standard deviation of ca. 20 cm, while keeping the probability of wrong fixing extremely low for code biases of 10 cm, and phase biases of 0.1 cycle, combined in a worst case.