Integer aperture ambiguity resolution based on difference test

Abstract

Carrier-phase integer ambiguity resolution (IAR) is the key to highly precise, fast positioning and attitude determination with Global Navigation Satellite System (GNSS). It can be seen as the process of estimating the unknown cycle ambiguities of the carrier-phase observations as integers. Once the ambiguities are fixed, carrier phase data will act as the very precise range data. Integer aperture (IA) ambiguity resolution is the combination of acceptance testing and integer ambiguity resolution, which can realize better quality control of IAR. Difference test (DT) is one of the most popular acceptance tests. This contribution will give a detailed analysis about the following properties of IA ambiguity resolution based on DT:

• 1. The sharpest and loose upper bounds of DT are derived from the perspective of geometry. These bounds are very simple and easy to be computed, which give the range for the critical values of DT.

• 2. The definition of DT integer aperture bootstrapping (IAB) estimator is firstly given. The relationships between DT-IAB and DT-IA are deeply investigated, which also firstly give a new perspective to review the IAB and IA least square (IALS) estimators.

• 3. Based on the properties of the second best integer candidates in integer least square and integer bootstrapping estimators, the definition of DT-IA is given from another perspective, which is mathematically equivalent to its original definition.

• 4. The analytical expressions of the success rate lower bound and upper bound of DT-IA estimator are firstly derived. Then, the quality measure for DT-IA estimator can be completely calculated as integer estimator without measurements. Both sharp and loose bounds of DT-IA success rate are given so that the success rates are easily evaluated, which also can provide reasonable approximation for DT-IA estimator.

All these conclusions are verified based on the single and combination GNSS simulation experiments. The experiment results indicate the correctness of these conclusions. These properties demonstrate the special properties of DT-IA estimator, and also provide the research frame to investigate other IA estimators. They are helpful to realize better use of IA estimators in quality control and precise positioning in future.

Appendix

Proof of Corollary 1

(DT-IAB success rate)

To deduce the success rate of DT-IAB, the success rate of IAB is given below (Teuniseen 2005)

$$\begin{aligned} P(\check{a}_{\text {IAB}}=a)= \prod _{i=1}^n \left( 2 \Phi \left( \frac{\gamma }{2\sigma _{a_{i|I}}}\right) -1 \right) \end{aligned}$$

(71)

with $\gamma$ the scaling ratio of IAB estimator.

We will derive similar formula for DT-IAB from 1-D to $n$-D.

In the scalar case, the probability of DT-IAB is given as:

$$\begin{aligned} P(\check{a}_{\text {DTIAB}}=a)=P(| \hat{a}-a| \le |x|) \end{aligned}$$

(72)

with $x$ the intersecting point between 1-dimensional pull-in interval (region) and coordinate axis.

Then, its success rate of ambiguity resolution becomes

$$\begin{aligned} P(\check{a}_{\text {DTIAB}}=a)= 2 \Phi \left( \frac{|x|}{ \sigma }\right) -1 \end{aligned}$$

(73)

with $\Phi (|x|)= \int _{-\infty }^{|x|} \frac{1}{\sqrt{2 \pi }} \text {exp} \left( - \frac{z^2}{2}\right) dz$. Actually, in the scalar case, the DT-IAB is the same as the scalar case of integer rounding. All IA estimators have the same properties in the scalar case when fixed failure rates are the same.

In the $n$-D (vectorial) case, the bootstrapping probability of correct IA estimation is given as:

$$\begin{aligned} P(\check{a}_{\text {DTIAB}}=a)= P\left( \underset{i=1}{\overset{n}{\bigcap }} \left( | \hat{a}_{i|I}-a_i | \le |x_i| \right) \right) \end{aligned}$$

(74)

with $|x_i|$ the $i$th intersection point between IA pull-in region and coordinate axis.

Based on the chain rule of conditional probabilities

$$\begin{aligned}&P(\check{a}_{\text {DTIAB}}=a) \nonumber \\&=\prod ^n_{i=1} P([\hat{a}_{i|I}]=a_i|[\hat{a}_{1}]=a_1,\cdots ,[\hat{a}_{i-1|I-1}]=a_{i-1}) \end{aligned}$$

(75)

Hence, according to the normal distribution of float ambiguity, the success rate of DT-IAB estimator reads as:

$$\begin{aligned} P(\check{a}_{\text {DTIAB}}=a)=\prod ^n_{i=1} \left( 2\Phi \left( \frac{|x_i|}{\sigma _{\hat{a}_{i|I}}}\right) -1\right) \end{aligned}$$

(76)

End of proof. $\square$

Proof of Corollary 3

(Lower bound of DT-IA estimator)

To prove this corollary, we firstly summarize the relationship between DT-IA and ILS estimator in (51)

$$\begin{aligned} \Omega _{z,\text {DTIA}}&= \bigcup _{c \in \mathbb {Z}^n /\{ 0 \}} T(c) S_{z,\text {ILS}} (c+z) \end{aligned}$$

(77)

$T(c)$ is the scaling ratio of $S_{z,\text {ILS}} (c+z)$ and computed by (52).

If we apply the decorrelation step to (77), DT-IA can be approximated by DT-IAB. Then, the second best integer candidates $c$ can be seen the same as $c_i$,and their corresponding pull-in region

$$\begin{aligned} S_{z,\text {ILS}} \approx S_{z,\text {IB}} \end{aligned}$$

(78)

Then, the decomposition of $\Omega _{z,\text {DTIA}}$ in (77) can be approximated by $\Omega _{z,\text {DTIAB}}$ in (44). That is

$$\begin{aligned} \Omega _{z,\text {DTIA}} \approx \Omega _{z,\text {DTIAB}} \end{aligned}$$

(79)

after decorrelation.

According to the definition in Teunissen (1999a) and the linear scaling in (77), $x$ in ILS pull-in region also has an elliptically contoured distribution $f_z (x)$ and

$$\begin{aligned} f_z (x) = (2 \pi )^{-\frac{n}{2}} \sqrt{\text {det}(Q^{-1}_{\hat{z} \hat{z}})} \text {exp} \left( -\frac{\Vert x - z \Vert ^2_{Q_{\hat{z} \hat{z}}}}{2}\right) \end{aligned}$$

The ILS pull-in regions for all members of elliptically contoured distributions are formulated as:

$$\begin{aligned} S_{z,\text {ILS}} = \{ x \in \mathbb {R}^n,u \in \mathbb {Z}^n | f_z (x) \ge f_u(x) \} \end{aligned}$$

(80)

Since $\Omega _{z,\text {DTIA}} \subset S_{z,\text {ILS}}$, the pull-in region of $\Omega _{z,\text {DTIA}}$ also can be given as:

$$\begin{aligned} \Omega _{z,\text {DTIA}}&= \bigg \{ y \in \mathbb {R}^n,u \in \mathbb {Z}^n | f_z (y) \ge f_u(y), \nonumber \\&\quad \frac{u^T Q^{-1}_{\hat{z} \hat{z}}(y-z)}{ \Vert u\Vert _{Q_{\hat{z} \hat{z}}}}\le \frac{\Vert u\Vert ^2_{Q_{\hat{z} \hat{z}}}- \mu _{\text {DT}} }{2 \Vert u\Vert _{Q_{\hat{z} \hat{z}}} } \bigg \} \end{aligned}$$

(81)

It follows that

$$\begin{aligned} f_z(y) \ge \sum _{u \in \mathbb {Z}^n} w_u(y) f_u(y), \forall y \in \Omega _{z,\text {DTIA}} \end{aligned}$$

(82)

with the indicator function

$$\begin{aligned} w_u(y) = \left\{ \begin{array}{ll} 1 &{} y \in \Omega _{u,\text {DTIAB}} \\ 0 &{} \text {otherwise} \end{array} \right. \end{aligned}$$

When taking the integral of Eq. (82) over $\Omega _{z,\text {DTIA}}$, we know

$$\begin{aligned} \int _{\Omega _{z,\text {DTIA}}} f_z(y)dy \ge \sum _{u \in \mathbb {Z}^n} \int _{\Omega _{z,\text {DTIA}} \bigcap \Omega _{u,\text {DTIAB}}} f_u(y)dy \end{aligned}$$

(83)

Change the variable in the left side of Eq. (83) based on the replacements: $f_u(y)\rightarrow f_u(x+u-z)= f_z(x)$, $\Omega _{z,\text {DTIA}}\rightarrow \Omega _{2z-u, \text {DTIA}}$ and $\Omega _{u,\text {DTIAB}} \rightarrow \Omega _{z,\text {DTIAB}}$. Thus

$$\begin{aligned} \int _{\Omega _{z,\text {DTIA}}} f_z(y)dy&\ge \sum _{u \in \mathbb {Z}^n} \int _{\Omega _{2z-u,\text {DTIA}} \cap \Omega _{z,\text {DTIAB}}} f_z(x)dx \nonumber \\&= \int _{\Omega _{z,\text {DTIAB}}} f_z(x)dx \end{aligned}$$

(84)

since after decorrelation

$$\begin{aligned}&\left( \bigcup _{u \in \mathbb {Z}^n}\Omega _{2z-u,\text {DTIA}}\right) \bigcap \Omega _{z,\text {DTIAB}} \nonumber \\&\quad \approx \left( \bigcup _{u \in \mathbb {Z}^n}\Omega _{2z-u,\text {DTIAB}}\right) \bigcap \Omega _{z,\text {DTIAB}} \nonumber \\&\quad =\Omega _{z,\text {DTIAB}} \end{aligned}$$

(85)

Hence, we have

$$\begin{aligned} P(\check{z}_{\text {DTIA}}= z) \ge P(\check{z}_{\text {DTIAB}}= z) \end{aligned}$$

(86)

Previously we know that

$$\begin{aligned} P(\check{z}_{\text {DTIAB}}=z) \ge P(\check{a}_{\text {DTIAB}}=a) \end{aligned}$$

(87)

Eventually, we have

$$\begin{aligned} P(\check{z}_{\text {DTIA}}=z) \ge P(\check{a}_{\text {DTIAB}}=a) \end{aligned}$$

(88)

End of proof. $\square$

Proof of Corollary 4

(Upper bound of DT-IA estimator)

Let $\rho = \sqrt{ c_n / \text {ADOP}^2}$ denote the radius of $n-$D Euclidean ball. The volume of this ball is

$$\begin{aligned} V_n= \frac{\rho ^n }{c^{n / 2}_{n}} = \frac{{ \sqrt{{c_n} / \text {ADOP}^2}}^n}{c^{n / 2}_n} \end{aligned}$$

with

$$\begin{aligned} c_n = \frac{ (\frac{n}{2} \Gamma (\frac{n}{2}))^{\frac{n}{2}} }{\pi } \end{aligned}$$

The ILS upper bound based on ADOP can be interpreted as:

$$\begin{aligned} P_{s, \text {ILS}} \le P( \chi ^2(n,0)\le \rho ^2 ) \end{aligned}$$

(89)

When the ILS pull-in regions are scaled down to the IA pull-in regions, the radius of Euclidean ball is also scaled down to certain value. Since the scalings are different in different directions, the Euclidean ball actually is transformed into hyper-ellipsoid. To be convenient, we choose to compute the upper bound of DT-IA after decorrelation step. At that time, we can use DT-IAB to approximate DT-IA. Then, the scaling directions will be limited to those of axes. Based on the previous derivations about the scaling ratios of DT-IAB estimators, these scaled radii or axes can be given as:

$$\begin{aligned} \overset{\sim }{\rho }_i =2 |x_i| \rho \quad i=1,2,...,n \end{aligned}$$

(90)

This means that the Euclidean ball becomes hyper-ellipsoidal after scaling. The mean of $\overset{\sim }{\rho }$ in all directions is denoted as:

$$\begin{aligned} \bar{\rho }&= \frac{\sum _{i=1}^n \overset{\sim }{\rho _i}}{n} = 2 \frac{\sum _{i=1}^n |x_i| \rho }{n} = 2 \bar{|x|} \rho \end{aligned}$$

(91)

with $\bar{|x|}= \frac{\sum _{i=1}^n |x_i| }{n}$. There exists the following relation for the volume of Euclidian ball

$$\begin{aligned} \prod _{i=1}^n \frac{\overset{\sim }{\rho _i}}{c^{1/2}_n }&\le \left( \frac{\sum _{i=1}^n \overset{\sim }{\rho }_i}{n} \right) ^n / c^{n/2}_n \nonumber \\&\le \frac{\bar{\rho }^n}{c^{n/2}_n } \end{aligned}$$

(92)

The equality can be built when all scaling ratios of radius are equal. The upper bound based on ADOP of DT-IA is given as:

$$\begin{aligned} P(\chi ^2(n,0)\le \bar{\rho }^2 )&= P(\chi ^2(n,0)\le 4 \bar{x}^2 \rho ^2 ) \nonumber \\&= P\left( \chi ^2(n,0)\le \frac{ 4 \bar{x}^2 c_n}{\text {ADOP}^2} \right) \end{aligned}$$

(93)

Then, we have

$$\begin{aligned} P(\check{z}_{\text {DTIA}}=z) \le P\left( \chi ^2(n,0)\le \frac{ 4 \bar{x}^2 c_n}{\text {ADOP}^2} \right) \end{aligned}$$

(94)

End of proof. $\square$