Integer Aperture Estimation In The Presence Of Biases

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Integer aperture estimation, also called integer ambiguity resolution, is a critical step to realized fast and precise global navigation satellite system (GNSS) positioning. It is a process which projects a float ambiguity to an integer ambiguity and then judges whether to fix that integer. Decades of research offer many projection methods. Efficiency and success rate are two key points in integer aperture estimation.

Efficiency is mainly determined by integer estimation. Standard and systematic methods have been studied and applied into practice, such as the well-known least-square ambiguity decorrelation adjustment (LAMBDA). Based on LAMBDA, many constrained methods for precise GNSS applications have been proposed and contributed a lot in precise positioning. With the increase of GNSS satellites in space, how to tackle high-dimension integer ambiguity resolution is the main research focus.

Success rate is another key point which influences positioning precision. A wrongly-fixed integer ambiguity can severely influence positioning. As shown in Figure 1, the success rate is the integration over the green region based on normal probability density function. The failure rate is the integration over all red regions. Different from integer estimation, integer aperture estimation has an added ambiguity validation step, which further constrains the integration region into aperture regions shown in Figure 2. In Figure 2, only the green region is taken as the correct region, where the correct integer ambiguity can be projected. Red regions are considered failure regions where wrong integer ambiguities are projected. Since the success region is an aperture region in hexagon region, we name this integer estimation the integer aperture estimation.

Figure 1. The success rate of integer estimation. Green region: success regions. Red region: failure regions.


Figure 2. The integer aperture pull-in regions. Green region: success regions. Red region: failure regions. Pink and blue regions: no fixing regions.

There exists a significant problem in research about ambiguity resolution. Between integer aperture estimation and bias estimation, which one is more important and influential in determining precise positioning? In order to answer this question, the connection between bias estimation and integer aperture estimation has to be found. Furthermore, their influence on positioning precision is necessary to be investigated.

Based on previous analysis, two kinds of relations are investigated. First, the connection between bias and integer aperture estimation using theoretical studies that are verified based on numerical experiments. Due to the existence of biases, the success rate of the integer aperture estimator is severely influenced. It is difficult to directly evaluate the success rate of the integer aperture estimator. Only approximation methods can be given.

Two kinds of methods are implemented and verified, the method regardless of bias, and the upper bound based on biased integer aperture estimator. Their results are given in Figure 3. WTIAB, IAB, WTIA and IALS denote different integer aperture estimators. ‘-WB’ and ‘UB’ mean ‘without bias’ and ‘upper bound’, respectively. For instance, ‘WTIAB-WB’ means ‘WTIAB without bias’, and ‘UB-WTIA’ means ‘upper bound of WTIA’. According to Figure 3, we can give the following remarks:

Figure 3. The expectations of approximation errors for WTIA and IALS in different GNSS combinations. G denotes GPS, GE denotes GPS+Galileo, and GEC means GPS+Galileo+BeiDou.

(1) More GNSS combination means stronger GNSS model. The method, regardless of bias, behaves better when the GNSS model is strong, especially with a triple-GNSS combination. The upper bound approximation method behaves better in a single- and two-GNSS combination.

(2) In practice, since the biases are not known in ambiguity resolution, the practical way to improve the evaluation effect is to enhance the strength of GNSS models.

Second, the influences of bias estimation and integer aperture estimation in positioning precision are analyzed based on field test results. Zenith troposphere delay (ZTD) and ionosphere delay were chosen as the analyzed biases. The positioning precision and ambiguity fix rates are compared based on bias estimation and different integer aperture (IA) estimation, just as shown in Figure 4 and Figure 5. In the two figures, different colors denote ambiguity resolution results for different baseline lengths. ILS, IALS, DTIA, WTIA, RTIA denote different IA estimators. Based on Figures 4 and 5, the following remarks can be made:

Figure 4. Positioning precision and ambiguity fix rates of ILS and IA estimation for GPS without ZTD and ionosphere delay estimation. Left: 3D positioning precision (cm); right: ambiguity fix rates


Figure 5. Positioning precision and fix rates of ILS and IA estimation for GPS with ZTD and ionosphere delay estimation. Left: 3D positioning precision (cm); right: ambiguity fix rates

(1) When biases are not estimated in Figure 4, positioning precision is rather poor and there is no obvious difference between different IA estimators. However, if biases are estimated as in Figure 5, positioning precision is greatly improved.

(2) In Figure 5, we can see that ambiguity fix rates basically conform to positioning precision after bias estimation. Higher ambiguity fix rates correspond to better positioning precision, which cannot be seen when biases are not estimated.

Based on previous analyses and experiment results, we can conclude that the influence of biases in success rate approximation of integer aperture estimation can be suppressed by enhancing the GNSS model strength. Bias estimation is more significant than the choice of integer aperture estimator in positioning precision.

These findings are described in the article entitled Integer aperture estimation in the presence of biases, recently published in the journal Advances in Space Research. This work was conducted by Dr. Jingyu Zhang from Beijing Satellite Navigation Center in collaboration with Prof. Meiping Wu and Dr. Yanqing Hou from the National University of Defense Technology.

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