Instantaneous and controllable integer ambiguity resolution: review and an alternative approach

Abstract

In the high-precision application of Global Navigation Satellite System (GNSS), integer ambiguity resolution is the key step to realize precise positioning and attitude determination. As the necessary part of quality control, integer aperture (IA) ambiguity resolution provides the theoretical and practical foundation for ambiguity validation. It is mainly realized by acceptance testing. Due to the constraint of correlation between ambiguities, it is impossible to realize the controlling of failure rate according to analytical formula. Hence, the fixed failure rate approach is implemented by Monte Carlo sampling. However, due to the characteristics of Monte Carlo sampling and look-up table, we have to face the problem of a large amount of time consumption if sufficient GNSS scenarios are included in the creation of look-up table. This restricts the fixed failure rate approach to be a post process approach if a look-up table is not available. Furthermore, if not enough GNSS scenarios are considered, the table may only be valid for a specific scenario or application. Besides this, the method of creating look-up table or look-up function still needs to be designed for each specific acceptance test. To overcome these problems in determination of critical values, this contribution will propose an instantaneous and CONtrollable (iCON) IA ambiguity resolution approach for the first time. The iCON approach has the following advantages: (a) critical value of acceptance test is independently determined based on the required failure rate and GNSS model without resorting to external information such as look-up table; (b) it can be realized instantaneously for most of IA estimators which have analytical probability formulas. The stronger GNSS model, the less time consumption; (c) it provides a new viewpoint to improve the research about IA estimation. To verify these conclusions, multi-frequency and multi-GNSS simulation experiments are implemented. Those results show that IA estimators based on iCON approach can realize controllable ambiguity resolution. Besides this, compared with ratio test IA based on look-up table, difference test IA and IA least square based on the iCON approach most of times have higher success rates and better controllability to failure rates.

Appendix

1.1 Proof of (29), (18) and (20)

In Teuniseen (2001), the general formula to calculate the PMF of IB estimator is given. Here, we give the specific and simple results in decorrelated space.

The original integer bootstrapped PMF is given below

$$\begin{aligned}&P_{\text {IB}}({\check{a}} = {\check{a}}_1) \nonumber \\&\quad = \prod _{i=1}^n \left[ \Phi \left( \frac{ 1 - 2 l^\mathrm{T}_i({\check{a}}-{\check{a}}_1) }{2 \sigma _{\hat{a}_{i|I}}} \right) + \Phi \left( \frac{ 1 + 2 l^\mathrm{T}_i(\check{a}-{\check{a}}_1) }{2 \sigma _{\hat{a}_{i|I}}} \right) -1 \right] \nonumber \\ \end{aligned}$$

(45)

with ${\check{a}}_1 \in \mathbb {Z}^n$ the best integer candidate and $\check{a} \in N(a,Q_{ \hat{a} \hat{a}})$ the fixed solution.

As we know, $Q_{\hat{a} \hat{a}} = LDL^\mathrm{T}$. After decorrelation,

$$\begin{aligned} Q_{ \hat{z} \hat{z}} = Z^\mathrm{T} Q_{\hat{a} \hat{a}} Z = Z^\mathrm{T} L D L^\mathrm{T} Z = D \end{aligned}$$

with Z the decorrelated transformation matrix. We can say that $Q_{\hat{z} \hat{z}} = I^\mathrm{T} D I$ with I the identity matrix.

In decorrelated space, (45) can be written as

$$\begin{aligned} \begin{array}{ll} P_{\text {IB}}(\check{z} = \check{z}_1) = \prod _{i=1}^n \left[ \Phi \left( \frac{ 1 - 2 z_i }{2 \sigma _{\hat{z}_{i|I}}} \right) + \Phi \left( \frac{ 1 + 2 z_i }{2 \sigma _{\hat{z}_{i|I}}} \right) -1 \right] \end{array}\nonumber \\ \end{aligned}$$

(46)

with $z_i = \check{z} - \check{z}_1$. As we know that

$$\begin{aligned} \Phi \left( \frac{ 1 - 2 z_i }{2 \sigma _{\hat{z}_{i|I}}} \right) = 1 - \Phi \left( \frac{ z_i -0.5 }{\sigma _{\hat{z}_{i|I}}} \right) \end{aligned}$$

(47)

Apply (47) into (45). For the k-th integer candidate $z(k) = \check{z}_k - \check{z}_1$ and $z(k) = [z_1(k),\dots ,z_n(k) ]^\mathrm{T}$, we have

$$\begin{aligned} P_{\text {IB}}(\check{z} = \check{z}_k) = \prod _{i=1}^n \left[ \Phi \left( \frac{ z_i(k) +0.5 }{ \sigma _{\hat{z}_{i|I}}} \right) - \Phi \left( \frac{ z_i(k)-0.5}{\sigma _{\hat{z}_{i|I}}} \right) \right] \nonumber \\ \end{aligned}$$

(48)

Obviously, for the best integer candidate, we have the same lower bound as (27). In decorrelated space, $P_{\text {ILS}}(\check{z} = \check{z}_k)$ can be approximated by (46). Hence, (29) is proved when $z(k) \ne 0$.

Similarly, before we derive formula (18), the PMF of DTIAB is firstly derived. Based on similar derivation steps in Teuniseen (2001), the formula below can be built in decorrelated space

$$\begin{aligned}&P_{\text {DTIAB}}(\check{z} = \check{z}_k) \nonumber \\&\quad = \prod _{i=1}^n \left[ \Phi \left( \frac{ z_i(k) + |x_i| }{ \sigma _{\hat{z}_{i|I}}} \right) - \Phi \left( \frac{ z_i(k)- |x_i|}{\sigma _{\hat{z}_{i|I}}} \right) \right] \end{aligned}$$

(49)

with $|x_i|$ the intersecting points between DTIAB pull-in region and coordinate axes. When $\check{z} = \check{z}_1$, $z(k)=0$. Then

$$\begin{aligned} P_{\mathrm{s},\text {DTIAB}} = \prod _{i=1}^n \left[ \Phi \left( \frac{|x_i|}{\sigma _{z_{i|I}}} \right) - \Phi \left( \frac{-|x_i|}{\sigma _{z_{i|I}}} \right) \right] \end{aligned}$$

(50)

For $z(k) \ne 0$, we will have the PMF of DTIAB for each candidate

$$\begin{aligned} P_{\mathrm{f},\text {DTIAB}}(k) = \prod _{i=1}^n \left[ \Phi \left( \frac{z_i(k)+|x_i|}{\sigma _{z_{i|I}}} \right) - \Phi \left( \frac{z_i(k)-|x_i|}{\sigma _{z_{i|I}}} \right) \right] \end{aligned}$$

In decorrelated space, DTIAB almost has the same pull-in region as DTIA (Zhang et al. 2015). Hence,

$$\begin{aligned} P_{\mathrm{f},\text {DTIA}}(k) \approx P_{\mathrm{f},\text {DTIAB}}(k) \end{aligned}$$

(51)

Then, based on (51)

$$\begin{aligned} P_{\mathrm{f},\text {DTIA}}= & {} \sum \limits _{i=1}^{\infty } P_{\mathrm{f},\text {DTIA}}(k) \nonumber \\\approx & {} \underset{z\in \mathbb {Z}^n \slash \{0\} }{ \sum } \prod _{i=1}^n \left[ \Phi \left( \frac{z_i(k)+|x_i|}{\sigma _{z_{i|I}}} \right) \right. \nonumber \\&\left. - \Phi \left( \frac{z_i(k)-|x_i|}{\sigma _{z_{i|I}}}\right) \right] \end{aligned}$$

(52)

Formula (18) is proved.

Similarly, the PMF of IALS can be derived

$$\begin{aligned} \begin{array}{ll} P_{\mathrm{f},\text {IALS}} &{}= \sum \limits _{i=1}^{\infty } P_{\mathrm{f},\text {IALS}}(k) \\ &{} \approx \sum \limits _{z \in \mathbb {Z}^n \slash \{0\}} \prod _{i=1}^n \left[ \Phi \left( \frac{2 z_i+ \beta }{2 \sigma _{z_{i|I}}} \right) - \Phi \left( \frac{2 z_i- \beta }{2 \sigma _{z_{i|I}}} \right) \right] \end{array} \end{aligned}$$

(53)

Equation (20) is proved. $\square$

1.2 Proof for the properties of probability ratio factor

Property 1 is proved based on the following procedures. As to the PDF of normal distribution, $x \in N(0,\sigma ^2)$,

$$\begin{aligned} f(x)= \frac{1}{ \sqrt{2 \pi } \sigma } \text {exp}\left( -\frac{x^2}{2\sigma ^2 } \right) \end{aligned}$$

Since f(x) is an even function and symmetry around y-axis, we will mainly talk about the property in $(0,+\infty )$ and then the other half can be derived similarly. Its first-order derivative

$$\begin{aligned} \frac{\partial f}{\partial x}= - \frac{x}{ \sqrt{2 \pi } \sigma ^3 } \text {exp}\left( -\frac{x^2}{2\sigma ^2 }\right) <0 \end{aligned}$$

The second-derivative is

$$\begin{aligned} \frac{\partial f^2}{\partial ^2 x}= - \frac{1}{ \sqrt{2 \pi } \sigma ^3 } \text {exp}\left( -\frac{x^2}{2\sigma ^2 }\right) + \frac{ x^2 }{\sqrt{2 \pi } \sigma ^5 } \text {exp}\left( -\frac{x^2}{2\sigma ^2 }\right) \end{aligned}$$

When $x\in U \subset (\sigma ,\infty )$, f(x) is a monotonously decreasing and convex function. Based on the Jensity inequality (Chandler 1987), we have

$$\begin{aligned} \int _U f(x) \mathrm{d}x = f(\xi (k)) \ge f(x_0) \end{aligned}$$

(54)

with $x_0 = z_i(k)-a_i$. Similarly, we also have $f(\xi _x(k)) \ge f(x_0)$. It is obvious that $\xi (k) \le x_0$ and $\xi _x(k) \le x_0$.

If $\xi _x (k)< \xi (k)$, then $f(\xi _x(k))>f(\xi (k))$. When $|x_i| \rightarrow 0$

$$\begin{aligned} \underset{|x_i| \rightarrow 0}{ \text {lim}} f(\xi _x(k)) = f(x_0)>f(\xi (k)) \end{aligned}$$

(55)

This contradicts the conclusion with the inequality (54). Then we know that

$$\begin{aligned} \xi _x (k) \ge \xi (k) \quad \text {and} \quad f (\xi _x (k)) \le f(\xi (k)) \end{aligned}$$

Since f(x) is an even function, when $U \subset (-\infty , -\sigma )$, f(x) is a monotonously increasing and convex function. Based on the proof by contradiction, we can derive that $\xi _x (k) \le \xi (k)$ and

$$\begin{aligned} f (\xi _x (k)) \le f(\xi (k)) \end{aligned}$$

Hence, we can summarize that when

$$\begin{aligned} U \subset (-\infty ,-\sigma ) \cup (\sigma ,\infty ) \end{aligned}$$

we have $f(\xi _x (k)) \le f(\xi (k))$. Furthermore $\xi _x (k) \le \xi (k)$ in $(-\infty ,-\sigma )$, and $\xi _x (k) \ge \xi (k)$ in $(\sigma ,\infty )$.

The proof of property 2 is briefly given below. If $\xi _x = x_0 + \delta _x$, $\xi = x_0 + \delta$ and $x_0 = z_i(k) -a_i$, when $U \subset (\sigma ,\infty )$, we have $\delta _x \ge \delta$ based on property 1. Then

$$\begin{aligned} \frac{f (\xi _x (k))}{f(\xi (k))} = \text {exp}\left( \frac{ 2(\delta -\delta _x)(z_i(k) -a_i)+\delta _x^2-\delta ^2 }{ 2 \sigma ^2} \right) \end{aligned}$$

(56)

When $z_i(k) \rightarrow \infty$, $2(\delta -\delta _x)(z_i(k) -a_i) \rightarrow -\infty$ and , hence

$$\begin{aligned} \underset{z_i(k) \rightarrow \infty }{ \text {lim}} \frac{f (\xi _x (k))}{f(\xi (k))} = 0 \end{aligned}$$

Since $0<|x_i| \le 1$, $2|x_i|\le 1$, then

$$\begin{aligned} \underset{z_i(k) \rightarrow \infty }{ \text {lim}} 2|x_i| \frac{f (\xi _x (k))}{f(\xi (k))} = 0 \end{aligned}$$

(57)

If $z_i(k) \rightarrow \infty$, the integer candidate based on $z_i(k)$ will rank as $k \rightarrow \infty$-th candidate. Eventually, the limiting case of probability ratio factor in Eq. (31) is

$$\begin{aligned} \underset{k \rightarrow \infty }{ \text {lim}} \frac{P_{\mathrm{f},\text {DTIA}}(k,\mu )}{P_{\mathrm{f},\text {ILS}}(k)} \approx \underset{k \rightarrow \infty }{ \text {lim}} \prod _{i=1}^n 2|x_i| \frac{f (\xi _x (k))}{f(\xi (k))} = 0 \end{aligned}$$

Similarly, when $U \subset (-\infty ,\sigma )$, we still can derive that

$$\begin{aligned} \underset{k \rightarrow \infty }{ \text {lim}} \frac{P_{\mathrm{f},\text {DTIA}}(k,\mu )}{P_{\mathrm{f},\text {ILS}}(k)} = 0 \end{aligned}$$

$\square$