Integrity monitoring-based ambiguity validation for triple-carrier ambiguity resolution

Abstract

Frequency diversity is expected to benefit the ambiguity resolution (AR) with the modernization of global navigation satellite system. As one classic AR method, triple-carrier AR (TCAR) is quite computation efficient for AR by exploring the geometric consistency between the code and phase observations. However, it is difficult for the classical TCAR method to support the safety-critical applications because of insufficiently controlling false alarm and missed detection errors simultaneously. In order to monitor the effect of an unknown bias on ambiguity validation, an integrity monitoring method is developed to test the correctness of TCAR in ambiguity domain. The proposed integrity monitoring method uses double test statistics to satisfy the required probabilities of false alarm and missed detection. The simulated as well as real observed BDS triple-frequency data are used to evaluate the performance of the proposed method. The static results show that the proposed method not only is capable of constraining the false alarm and missed detection errors simultaneously, but also can control the moving average length over multiple epochs. The kinematic experiment reveals that the proposed method can achieve a comparable positioning accuracy and continuity improvement of 9 %, when compared to the geometry-based LAMBDA method with fixed detection threshold.

Appendix: optimal detection power of the proposed test method

Based on (22), according to Neyman–Pearson lemma, if

$$\frac{{f_{{\hat{a}}} \left( {x|y = 0,z} \right)}}{{f_{{\hat{a}}} \left( {x|y \ne 0,z} \right)}} = \exp \left\{ {\frac{1}{{\sigma_{{\hat{a}}}^{2} }}\left( {xy - \frac{1}{2}y^{2} } \right)} \right\} > k$$

(30)

can be satisfied to reject the bias-free hypothesis, the probability of missed detection can be minimized, which means the detection power is maximized. Rearranging (30), we have

$$x > \frac{1}{2}y + \frac{{\sigma^{2} }}{y}\ln k$$

(31)

The proposed integrity monitoring method is based on the following test,

$$x > \mu \sigma_{{\hat{a}}} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} u = {\kern 1pt} \;{\text{erfinv}}\left( {\frac{{1 - 0.5P_{\text{FA}} }}{\sqrt 2 }} \right)$$

(32)

If (32) is satisfied, the bias-free hypothesis is rejected. If the proposed integrity monitoring method is consistent with the Neyman–Pearson lemma, then

$$\frac{1}{2}y^{2} - u\sigma_{{\hat{a}}} y + \sigma_{{\hat{a}}}^{2} \ln k = 0$$

(33)

should be met. In order to make sure at least the real value of y in (33) exists,

$$u^{2} \sigma_{{\hat{a}}}^{2} - 4 \cdot \frac{1}{2}\sigma_{{\hat{a}}}^{2} \ln k \ge 0$$

(34)

needs to be satisfied. If there is a k value satisfying $k \le \exp (0.5u^{2} )$, the proposed integrity monitoring method can obtain minimum missed detection error.