An alternative approach to calculate the posterior probability of GNSS integer ambiguity resolution

Abstract

When precise positioning is carried out via GNSS carrier phases, it is important to make use of the property that every ambiguity should be an integer. With the known float solution, any integer vector, which has the same degree of freedom as the ambiguity vector, is the ambiguity vector in probability. For both integer aperture estimation and integer equivariant estimation, it is of great significance to know the posterior probabilities. However, to calculate the posterior probability, we have to face the thorny problem that the equation involves an infinite number of integer vectors. In this paper, using the float solution of ambiguity and its variance matrix, a new approach to rapidly and accurately calculate the posterior probability is proposed. The proposed approach consists of four steps. First, the ambiguity vector is transformed via decorrelation. Second, the range of the adopted integer of every component is directly obtained via formulas, and a finite number of integer vectors are obtained via combination. Third, using the integer vectors, the principal value of posterior probability and the correction factor are worked out. Finally, the posterior probability of every integer vector and its error upper bound can be obtained. In the paper, the detailed process to calculate the posterior probability and the derivations of the formulas are presented. The theory and numerical examples indicate that the proposed approach has the advantages of small amount of computations, high calculation accuracy and strong adaptability.

Appendix: Proof of Eq. (42)

For $\forall S_{z_{i,Z} } \in S_{\left( {m+1} \right) \sim \infty }$, the approximate probability that $S_{z_{i,Z} }$ will occur can be expressed as

$$\begin{aligned} \bar{{P}}\left( {S_{z_{i,Z} } } \right)= & {} V_S \cdot R\cdot \exp \left[ {-\frac{1}{2}\left\| {z_{i,Z} -\hat{{a}}_Z } \right\| _{D_{\hat{{a}}\hat{{a}},Z}}^2 } \right] \nonumber \\= & {} V_S \cdot R\cdot \exp \left\{ {-\frac{1}{2}\left\| {z_{i,Z} -\left[ {\hat{{a}}_Z } \right] +\delta } \right\| _{D_{\hat{{a}}\hat{{a}},Z} }^2 } \right\} \end{aligned}$$

(64)

where $\delta =\left[ {\hat{{a}}_Z } \right] -\hat{{a}}_Z$.

Because $S_{1\sim m}$ is a centrosymmetric space to $\left[ {\hat{{a}}_Z } \right]$, there is a subspace $S_{z_{q,Z} } \in S_{\left( {m+1} \right) \sim \infty }$ that is symmetrical to $\left[ {\hat{{a}}_Z } \right]$ with $S_{z_{i,Z} }$. And the relation between $z_{i,Z}$ and $z_{q,Z}$, which are the entre points of the two subspaces, can be expressed as

$$\begin{aligned} z_{q,Z} =2\left[ {\hat{{a}}_Z } \right] -z_{i,Z} \end{aligned}$$

(65)

Referring to Eq. (64), the approximate probability that $S_{z_{q,Z} }$ will occur can be expressed as

$$\begin{aligned} \bar{{P}}\left( {S_{z_{q,Z} } } \right)= & {} V_S \cdot R\cdot \exp \left[ {-\frac{1}{2}\left\| {z_{q,Z} -\hat{{a}}_Z } \right\| _{D_{\hat{{a}}\hat{{a}},Z}}^2 } \right] \nonumber \\= & {} V_S \cdot R\cdot \exp \left\{ {-\frac{1}{2}\left\| {2\left[ {\hat{{a}}_Z } \right] -z_{i,Z} -\hat{{a}}_Z } \right\| _{D_{\hat{{a}}\hat{{a}},Z} }^2 } \right\} \nonumber \\= & {} V_S \cdot R\cdot \exp \left\{ {-\frac{1}{2}\left\| {z_{i,Z} -\left[ {\hat{{a}}_Z } \right] -\delta } \right\| _{D_{\hat{{a}}\hat{{a}},Z}}^2 } \right\} \nonumber \\ \end{aligned}$$

(66)

The sum of Eqs. (64) and (66) can be expressed as

$$\begin{aligned}&\bar{{P}}\left( {S_{z_{i,Z} } } \right) +\bar{{P}}\left( {S_{z_{q,Z} } } \right) \nonumber \\&\quad =V_S \cdot R\cdot \exp \left\{ {-\frac{1}{2}\left\| {z_{i,Z} +\delta -\left[ {\hat{{a}}_Z } \right] } \right\| _{D_{\hat{{a}}\hat{{a}},Z}}^2 } \right\} \\&\qquad +V_S \cdot R\cdot \exp \left\{ {-\frac{1}{2}\left\| {z_{i,Z} -\delta -\left[ {\hat{{a}}_Z } \right] } \right\| _{D_{\hat{{a}}\hat{{a}},Z} }^2 } \right\} \nonumber \end{aligned}$$

(67)

The probability that $S_{z_{i,Z} }$ occur can be expressed as

$$\begin{aligned} P\left( {S_{z_{i,Z} } } \right) =\int _{S_{z_{i,Z} } } {R\cdot \exp \left\{ {-\frac{1}{2}\left\| {Y-\left[ {\hat{{a}}_Z } \right] } \right\| _{D_{\hat{{a}}\hat{{a}},Z}}^2 } \right\} \mathrm{d}Y} \end{aligned}$$

(68)

Because $-\frac{1}{2}\le \delta \left( j \right) \le \frac{1}{2}$, both $z_{i,Z} +\delta$ and $z_{i,Z} -\delta$ are within $S_{z_{i,Z} }$ and symmetric to the entre point $z_{i,Z}$ of $S_{z_{i,Z} }$. When $S_{1\sim m}$ is larger enough, Eq. (67) can be approximately expressed as

$$\begin{aligned} \bar{{P}}\left( {S_{z_{i,Z} } } \right) +\bar{{P}}\left( {S_{z_{q,Z} } } \right) =2P\left( {S_{z_{i,Z} } } \right) \end{aligned}$$

(69)

Further, we have

$$\begin{aligned} \sum _{i=m+1}^\infty {\bar{{P}}\left( {S_{z_{i,Z} } } \right) } =\sum _{i=m+1}^\infty {P\left( {S_{z_{i,Z} } } \right) } =P\left( {S_{\left( {m+1} \right) \sim \infty } } \right) \end{aligned}$$

(70)

End of proof. $\square$