The advantages of the original combination over narrow lane combinations in multi-frequency GNSS orientating

Abstract

The narrow lane carrier phase combinations, especially in the case of short baseline, are commonly used to obtain high orientating accuracy using the multi-frequency GNSS (Global Navigation Satellite System) due to their ability to mitigate measurement noise and multipath. However, after reviewing the errors in solutions of NL (narrow lane) combinations and the original combination (combine carrier phase data independently), it is found that the original combination always has a better performance than all NL combinations with positive coefficients regardless of which navigation system is used and how many frequencies are used. To achieve this, the observation equations of linear combination and original combination are first considered with emphasis on the relevant errors and the measurement noise mitigating ability. Then this paper derives the solutions of those combinations and focuses on their performance comparison in aspects of solution accuracy and systematic errors. Finally, the theoretical developments are verified with BDS data and the advantages of the original combination are summarized and discussed.

Keywords

GNSS orientating original combination narrow lane combinations multi-frequency GNSS solution precision

1. Introduction

Most of global navigation satellite systems (GNSS) are transmitting three or even more frequency signals. Beidou navigation system (BDS) developed by China started its regional services in 2012 and provides signals centered at B1, B2 and B3. In addition to the previous L1 and L2 frequency, GPS now provides a third frequency L5. Galileo system can provide five frequency signals respectively centered at E1, E5, E5A, E5B and E6. Linear combinations of these multi-frequency observations have different characteristics. Wide lane combinations are often used to solve ambiguities because of their long wavelength. However, narrow lane combinations are always used to calculate high accuracy position for they are capable of mitigating noise.

Lots of investigations have been conducted on linear combinations of multi-frequency observation in the last decades. Neumann et al. [2] showed that dual-frequency narrow lane combination was capable of mitigating multipath and noise. Dual-frequency linear combinations was systematically investigated by Cocard and Geiger [3] and Collins [4]. Han and Rizos [5] developed long wavelengths but low noise linear combinations which were suitable for ambiguity resolution. Odijk [6] mainly focused on the ionosphere combinations that three frequency can provide. Teunissen and Odijk [7] showed that three frequency observations can lead to much more possible combinations. Petovello [8] mathematically compared the solution precision of narrow lane combination L1 $+$ L2 and the original combination (using the L1 and L2 carrier phase observations independently) and proved that the independent combination can lead to higher solution precision than narrow lane combination L1 $+$ L2. Feng [9] developed three frequency optimal combinations under three criterial including long wavelength, reduced ionospheric errors, and low combination noise and these combinations were used in geometry-based ambiguity resolution. In order to satisfy the demands of bootstrapping method, Cocard and Geiger [3] derived rigorous mathematical equations for gaining optimal combinations of three frequency observations. The three frequency optimal combinations of GLONASS was investigated by Xu et al. [10]. Zhang and He [11] investigated the models and characteristics of BDS triple-frequency carrier phase linear combinations, and developed three categories of combinations: ionosphere free, troposphere free and minimum noise. Li et al. [12] conducted an analytical study on three carrier phase combinations, and presented a general analytical method for solving the optimal integer linear combinations for all GNSS.

Through the introduction above, it is easy to find out that most of the investigations mainly concentrated on wide lane combinations with long wavelength, low ionosphere delay and low tropospheric error. However, the investigations on narrow lane combinations and the original combination are not systematic. Though Petovello [8] mathematically compared the performance of narrow lane combination L1 $+$ L2 and the original combination, this investigation just focused on dual frequency case with GPS and only one narrow lane combination is considered. This paper systematically investigates the relationship between the original combination and narrow lane combinations, mainly focusing on the performance comparison of them in different cases with different navigation system. It will be shown that the original combination always has better performance than those narrow lane combinations with positive coefficients. The conclusion is drawn through analyzing the effects of both stochastic and systematic errors in the aspect of solutions.

2. The models and their solutions

2.1 Carrier phase observation

In GNSS orientating, the double differential carrier phase model is commonly used in order to remove receiver and satellite clock errors and mitigate other spatially correlated errors. The model is written as

$\displaystyle\Delta\nabla\phi_{i}{\rm=}\frac{1}{\lambda_{i}}\left({\Delta% \nabla\rho+\Delta\nabla\delta_{\textit{trop}}+\Delta\nabla\delta_{\rho}}\right% )+\Delta\nabla N_{i}+\Delta\nabla I_{i}+\Delta\nabla\varepsilon_{i}$ (1)

Where, $\Delta\nabla$ represents the double-differenced (DD) operation; $i$ is the index of frequency; $\phi_{i}$ denotes the carrier phase observation in units of cycles; $\lambda_{i}$ is the phase wavelength in meters; $\rho$ is the geometrical distance between receiver and satellite in meters; $\delta_{\textit{trop}}$ is the troposphere error in meters; $\delta_{\rho}$ denotes other geometry errors in meters; $N_{i}$ is the ambiguity parameter; $\varepsilon_{i}$ is the noise in units of cycles including multipath and measurement noise effects; $I_{i}$ is the ionosphere delay in units of cycles and the first order DD ionosphere delay in cycles can be given as

$\displaystyle\Delta\nabla I_{i}=\frac{\Delta\nabla(40.3\times{\rm TEC)}}{cf_{i}}$ (2)

Where TEC is the total electron content along the propagation path in units of ${\rm electron/m}^{2},c$ is the speed of light, $f_{i}$ is carrier phase frequency in Hz.

2.1.1 Linear carrier phase combination

The linear combination of DD carrier phase observations is given by

$\displaystyle\Delta\nabla{\rm MW}_{\textit{linear}}=\sum\limits_{i{\rm=}1}^{n}% {a_{i}\Delta\nabla\phi_{i}}=\frac{1}{\lambda_{\textit{linear}}}\left({\Delta% \nabla\rho+\Delta\nabla\delta_{\textit{trop}}+\Delta\nabla\delta_{\rho}}\right% )+\Delta\nabla N_{\textit{linear}}+\Delta\nabla I_{\textit{linear}}+\Delta% \nabla\varepsilon_{\textit{linear}}$ (3)

where, $a_{i}$ are integer combination coefficients, $\delta_{\textit{trop}}$ and $\delta_{\rho}$ are frequency independent, $\phi_{\textit{linear}}^{s},\lambda_{\textit{linear}}$ , $N_{\textit{linear}},I_{\textit{linear}}$ and $\varepsilon_{\textit{linear}}$ are respectively combined carrier phase observation, wavelength, ambiguity, ionosphere delay and noise. Supposing original measurements are independent and the variances of them are equal, then those parameters can be written as

$\displaystyle\lambda_{\textit{linear}}=\frac{c}{\sum\limits_{i}^{n}{a_{i}f_{i}% }}=\frac{c}{f_{0}g}\quad i=1,2,3\ldots n$ $\displaystyle I_{\textit{linear}}=\sum\limits_{i}^{n}{a_{i}\Delta\nabla I_{i}=% \frac{40.3\times\Delta\nabla{\rm TEC}}{cf_{0}}}\sum\limits_{i}^{n}{\frac{a_{i}% }{k_{i}}}$ $\displaystyle N_{\textit{linear}}=\sum\limits_{i}^{n}{a_{i}\Delta\nabla N_{i}}% \quad i=1,2,3\ldots n$ $\displaystyle{\rm D}(\varepsilon_{\textit{linear}})={\rm D}\left(\sum\limits_{% i}^{n}{a_{i}\Delta\nabla\varepsilon_{i}}\right)=4l\sigma^{2}$ (4)

where $g=\sum_{i}^{n}{a_{i}k_{i}}$ is defined as lane number, $l=\sum_{i}^{n}{a_{i}^{2}}$ is defined as noise factor, $\sigma^{2}$ is the variance of original carrier phase observation, $f_{0}$ is the base frequency, $k_{i}$ are constant integers satisfying $k_{i{\rm+}1}>k_{i}$ . For GPS, the values of $k_{i}$ are 154, 120 and 115 which are respectively corresponding to frequency L1, L2 and L5. But for BDS, the values of $k_{i}$ are 763, 620, 590 which respectively corresponding to frequency B1, B3 and B2.

In general, three groups of linear combinations can be distinguished, including wide lane region, intermediate region and narrow lane region. The motivation for selecting linear combinations is that combinations from different groups have different characteristics. Combinations from the first two groups are often used to solve ambiguities because of their long wavelength. However, combinations from narrow lane region are always used to calculate high accuracy position for their low noise.

In very short baseline situation, the ionosphere delay and geometric errors are often considered negligible which means that the noise is considered as the main factor affecting orientating accuracy. Therefore, to obtain the highest orientating accuracy, the objective is to mitigate the effect of the noise. The noise amplification factor relative to L1 (or B1) can be written as

$\displaystyle l_{\textit{amp}}=\frac{D(\lambda_{\textit{linear}}\varepsilon_{% \textit{linear}})}{D(\lambda_{1}\varepsilon_{1})}=\frac{k_{1}^{2}l}{g^{2}}$ (5)

It is obvious that some NL combinations can make $l_{\textit{amp}}<1$ meaning that those narrow lane combinations can lead to lower noise level. Therefore, they are commonly thought to be ideal to calculate orientation. However, this only considers the errors in measurement domain. In the following sections, it will be shown that if one considers the effect of errors in the position domain, original combination can always achieve better performance than all those narrow lane combinations with $a_{i}>0$ .

2.1.2 Original carrier phase combination

The original carrier phase combination model is given by

$\displaystyle{\rm{\bf\Phi}}_{\textit{origin}}=\left[{\Delta\nabla\phi_{1}^{s_{% 1}}}∼{}∼{}{\Delta\nabla\phi_{1}^{s_{2}}}∼{}∼{}{\ldots}∼{}∼{}{\Delta\nabla\phi_% {2}^{s_{1}}}∼{}∼{}{\Delta\nabla\phi_{2}^{s_{2}}}∼{}∼{}{\ldots}\right]^{{\rm T}}$ (6)

Taking the assumption made earlier, the covariance matrix of ${\rm{\bf\Phi}}_{\textit{origin}}$ can be gained

$\displaystyle{\rm cov}({\rm{\bf\Phi}}_{\textit{origin}})=\sigma^{2}{\rm{\bf C}}$ (7)

where, ${\rm{\bf C}}$ is a constant matrix.

2.2 Solutions of NL combination and the original combination

Define ${\rm{\bf L}}_{i}$ as a measurement vector which contains the DD carrier phase measurements from all observed satellites. ${\rm{\bf L}}_{i}$ can be further expressed as

$\displaystyle{\rm{\bf L}}_{i}={\rm{\bf A}}_{i}{\rm{\bf X}}+{\rm{\bf BN}}_{i}+{% \bm{\varepsilon}}_{i}$ (8)

where, ${\rm{\bf A}}_{i}$ and ${\rm{\bf B}}$ are design matrixes, ${\rm{\bf A}}_{i}={{\rm{\bf A}}_{0}}/{\lambda_{i}},{\rm{\bf A}}_{0}$ is a matrix depending on satellite distribution, ${\rm{\bf X}}$ is the baseline vector, ${\rm{\bf N}}_{i}$ is the vector of DD ambiguities, ${\rm{\bf\varepsilon}}_{i}$ is the noise vector. Since the focus is on obtaining the highest solution accuracy, one assumption made here is that the carrier phase ambiguities have already been resolved correctly and have been incorporated into the measurement vector as follows

$\displaystyle{\rm{\bf\bar{{L}}}}_{i}={\rm{\bf A}}_{i}{\rm{\bf X}}+{\bm{% \varepsilon}}_{i}$ (9)

where ${\rm{\bf\bar{{L}}}}_{i}={\rm{\bf L}}_{i}-{\rm{\bf BN}}_{i}$ .

2.2.1 Solution of NL combination

Based on Eq. (3), the linear combination of measurement vector ${\rm{\bf\bar{{L}}}}_{i}$ from different frequency is given by

$\displaystyle{\rm{\bf L}}_{\textit{narrow}}=\sum\limits_{i}{a_{i}{\rm{\bf\bar{% {L}}}}_{i}}{\rm=}\sum\limits_{i}{a_{i}{\rm{\bf A}}_{i}}{\rm{\bf X}}+\sum% \limits_{i}{a_{i}{\bm{\varepsilon}}_{i}}$ (10)

where $a_{i}>0$ .

The covariance matrix of ${\rm{\bf L}}_{\textit{narrow}}$ can be written as

$\displaystyle{\rm{\bf Q}}_{\textit{narrow}}=l\sigma^{2}{\rm{\bf C}}$ (11)

Then the least square solution of Eq. (10) can be derived

$\displaystyle{\rm{\bf X}}_{\textit{narrow}}={\rm{\bf Q}}_{\textit{Xnarrow}}{% \rm{\bf A}}^{T}{\rm{\bf P}}_{\textit{narrow}}{\rm{\bf L}}_{\textit{narrow}}=% \frac{c}{f_{0}}({\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}{\rm{\bf A}}_{0})^{-1}{% \rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}\sum\limits_{i}{\left({\frac{a_{i}}{g}{\rm% {\bf\bar{{L}}}}_{i}}\right)}$ $\displaystyle{\rm{\bf Q}}_{\textit{Xnarrow}}=({\rm{\bf A}}^{T}{\rm{\bf P}}_{% \textit{narrow}}{\rm{\bf A}})^{-1}=l\lambda_{c}^{2}\sigma^{2}({\rm{\bf A}}_{0}% ^{T}{\rm{\bf C}}^{-1}{\rm{\bf A}}_{0})^{-1}=\frac{l}{g^{2}}\frac{c^{2}\sigma^{% 2}}{f_{0}^{2}}({\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}{\rm{\bf A}}_{0})^{-1}$ (12)

where ${\rm{\bf X}}_{\textit{narrow}}$ is the baseline vector, ${\rm{\bf Q}}_{\textit{Xnarrow}}$ is the covariance matrix of ${\rm{\bf X}}_{\textit{narrow}},{\rm{\bf P}}_{\textit{narrow}}={\rm{\bf Q}}_{% \textit{narrow}}^{-1}$ .

2.2.2 Solution of the original combination

The original combination of DD carrier phase vector ${\rm{\bf\bar{{L}}}}_{i}$ is given as

$\displaystyle{\rm{\bf L}}_{\textit{origin}}=\left[{{\rm{\bf\bar{{L}}}}_{1}}∼{}% ∼{}{{\rm{\bf\bar{{L}}}}_{2}}∼{}∼{}{\ldots}\right]^{{\rm T}}$ (13)

The covariance matrix of ${\rm{\bf L}}_{\textit{origin}}$ is

$\displaystyle{\rm{\bf Q}}_{\textit{origin}}=\sigma^{2}\left[{{\begin{array}[]{% *{20}c}{{\rm{\bf C}}}&&\\ &{{\rm{\bf C}}}&\\ &&{\ldots}\\ \end{array}}}\right]$ (14)

Then, the least square solution of Eq. (13) is obtained

$\displaystyle{\rm{\bf X}}_{\textit{origin}}={\rm{\bf Q}}_{{\rm X}\textit{% origin}}\sum\limits_{i}{{\rm{\bf A}}_{i}^{T}{\rm{\bf P}}_{i}}{\rm{\bf\bar{{L}}% }}_{i}=\frac{c}{f_{0}}\left({{\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}{\rm{\bf A}}% _{0}}\right)^{-1}{\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}\sum\limits_{i}{\left({% \frac{k_{i}{\rm{\bf\bar{{L}}}}_{i}}{\sum\limits_{i}{k_{i}^{2}}}}\right)}$ $\displaystyle{\rm{\bf Q}}_{{\rm X}\textit{origin}}=\left({\sum\limits_{i}{{\rm% {\bf A}}_{i}^{T}{\rm{\bf P}}_{i}{\rm{\bf A}}_{i}}}\right)^{-1}=\left({\sum% \limits_{i}{\frac{1}{\lambda_{i}^{2}\sigma^{2}}{\rm{\bf A}}_{0}^{T}{\rm{\bf C}% }^{-1}{\rm{\bf A}}_{0}}}\right)^{-1}=\left({\frac{1}{\sum\limits_{i}{k_{i}^{2}% }}}\right)\frac{c^{2}\sigma^{2}}{f_{0}^{2}}\left({{\rm{\bf A}}_{0}^{T}{\rm{\bf C% }}^{-1}{\rm{\bf A}}_{0}}\right)^{-1}$ (15)

where ${\rm{\bf X}}_{\textit{origin}}$ is the baseline vector, ${\rm{\bf Q}}_{{\rm X}\textit{origin}}$ is the covariance matrix of ${\rm{\bf X}}_{\textit{ origin}},{\rm{\bf P}}_{i}={\rm{\bf Q}}_{i}^{-1}$ .

3. Comparing the solutions

Two approaches are required to compare those solutions; one for stochastic errors and one for systematic errors.

3.1 Comparison of stochastic errors

According to Eqs (2.2.1) and (2.2.2), ${\rm{\bf Q}}_{{\rm X}\textit{narrow}}$ and ${\rm{\bf Q}}_{{\rm X}\textit{origin}}$ only differ by a scale factor. Thus, the ratio of them can be presented as below

$\displaystyle r_{{\rm stoch}}={\rm{\bf Q}}_{{\rm X}\textit{narrow}}{\rm{\bf Q}% }_{{\rm X}\textit{origin}}^{{\rm-1}}=\sum\limits_{i}{k_{i}^{2}}\frac{l}{g^{2}}% =\frac{\sum\limits_{i}{k_{i}^{2}}\sum\limits_{i}{a_{i}^{2}}}{\left({\sum% \limits_{i}{a_{i}k_{i}}}\right)^{2}}\geqslant 1$ (16)

It can be proved that $r_{{\rm stoch}}$ is always equal or greater than 1. Therefore, original combination theoretically has highest solution precisions than the narrow lane combinations regardless of the number of frequencies.

3.1.1 Dual-frequency case

In dual-frequency case, the variation of $r_{stoch}$ can be given as

$\displaystyle r_{{\rm stoch}}=\left({k_{1}^{2}+k_{2}^{2}}\right)\frac{m^{2}{% \rm+}1}{\left({k_{1}m+k_{2}}\right)^{2}}$ (17)

where $m=a_{1}/a_{2}$ . The curve of this equation is shown in Fig. 1. It is easy to obtain that this equation has a minimum point and the minimum is 1. For the commonly used combination (1, 1), the values of $r_{{\rm stoch}}$ (Table 1) are greater but very close to 1.

Table 1

$r_{{\rm stoch}}$ with combination (1,1)

GNSS	Frequency	$r_{{\rm stoch}}$	$1/r_{{\rm stoch}}$
	L1, L2	1.015	0.985
GPS	L1, L5	1.021	0.979
	L2, L5	1.001	0.999
	B1, B2	1.016	0.984
BDS	B1, B3	1.011	0.989
	B2, B3	1.001	0.999

Figure 1.

$r_{{\rm stoch}}$ with respect to $m$ .

3.1.2 Triple-frequency case

In triple-frequency case, $r_{{\rm stoch}}$ can be given as

$\displaystyle r_{{\rm stoch}}=\left({k_{1}^{2}+k_{2}^{2}+k_{3}^{2}}\right)% \frac{m^{2}+n^{2}+1}{\left({k_{1}m+k_{2}+k_{3}n}\right)^{2}}$ (18)

where $m={a_{1}}/{a_{2}}$ and $n={a_{3}}/{a_{2}}$ . The surface of Eq. (18) is shown in Fig. 2. Its minimum point is $\left({{k_{1}}/{k_{2}},{k_{3}}/{k_{2}}}\right)$ and the minimum is 1. The $r_{{\rm stoch}}$ values of combination (1, 1, 1) is shown in Table 2. It is easy to find that they are also greater but very close to 1.

Table 2

$r_{{\rm stoch}}$ with combination (1, 1, 1)

GNSS	Frequency	$r_{{\rm stoch}}$	$1/r_{{\rm stoch}}$
GPS	L1, L2, L5	1.018	0.9823
BDS	B1, B2, B3	1.013	0.9872

Figure 2.

$r_{{\rm stoch}}$ with respect to $m$ and $n$ .

From above analysis, the original solution is more accurate in terms of standard deviations when the problem is considered in the position domain. But for commonly used combination (1, 1) and (1, 1, 1), the difference is very small.

3.2 Comparison of systematic errors

Frequency independent errors and ionosphere delay in solution are assessed according to Eq. (2.2.1) and Eq. (2.2.2). In those two equations, $\sum_{i}{\left({\frac{k_{i}}{\sum_{j}{k_{j}^{2}}}{\rm{\bf\bar{{L}}}}_{i}}% \right)}$ and $\sum_{i}{\left({\frac{a_{i}}{g}{\rm{\bf\bar{{L}}}}_{i}}\right)}$ are mainly focused because ${\rm{\bf\bar{{L}}}}_{i}$ contains the frequency independent errors and ionosphere delay. According to previous discussion, the systematic errors included in ${\rm{\bf\bar{{L}}}}_{i}$ can be written as

$\displaystyle{\rm{\bf err}}=\frac{1}{\lambda_{i}}(\Delta\nabla{\bm{\delta}}_{% \textit{trop}}+\Delta\nabla{\bm{\delta}}_{\rho})+\Delta\nabla{\rm{\bf I}}_{i}$ (19)

Then, the frequency independent errors for original combination is

$\displaystyle\sum\limits_{i}{\left({\frac{k_{i}}{\sum\limits_{j}{k_{j}^{2}}}% \frac{1}{\lambda_{i}}(\Delta\nabla{\bm{\delta}}_{\textit{trop}}+\Delta\nabla{% \bm{\delta}}_{\rho})}\right)}=\frac{(\Delta\nabla{\bm{\delta}}_{\textit{trop}}% +\Delta\nabla{\bm{\delta}}_{\rho})f_{0}}{c}$ (20)

The frequency independent errors for NL combination is

$\displaystyle\sum\limits_{i}{\left({\frac{a_{i}}{g}\frac{1}{\lambda_{i}}(% \Delta\nabla{\bm{\delta}}_{\textit{trop}}+\Delta\nabla{\bm{\delta}}_{\rho})}% \right)}=\frac{(\Delta\nabla{\bm{\delta}}_{\textit{trop}}+\Delta\nabla{\bm{% \delta}}_{\rho})f_{0}}{c}$ (21)

Equations (19) and (20) indicate that the frequency independent errors in both solutions are equivalent. Therefore, the following part mainly focuses on ionosphere delay. The ionosphere delay can be given as

$\displaystyle{\rm{\bf ion}}_{{\rm origin}}=\left({{\sum\limits_{i}{k_{i}}}% \Bigg{/}{\sum\limits_{i}{k_{i}^{2}}}}\right)\Delta\nabla{\rm{\bf I}}_{i}$ $\displaystyle{\rm{\bf ion}}_{{\rm narrow}}=\left({{\sum\limits_{i}{a_{i}}}% \Bigg{/}g}\right)\Delta\nabla{\rm{\bf I}}_{i}$ (22)

Then, the ratio of them can be obtained

$\displaystyle r_{{\rm ion}}=\frac{{\rm{\bf ion}}_{{\rm narrow}}}{{\rm{\bf ion}% }_{{\rm origin}}}{\rm=}\frac{\sum\limits_{i}{a_{i}}\sum\limits_{i}{k_{i}^{2}}}% {\sum\limits_{i}{a_{i}k_{i}}\sum\limits_{i}{k_{i}}}$ (23)

It is difficult to judge the relationship between Eq. (23) and (1) in this general form. Thus, Eq. (23) is discussed in two specified cases as following.

3.2.1 Dual-frequency case

In this case, $r_{{\rm ion}}$ is given as

$\displaystyle r_{{\rm ion}}=\frac{k_{1}^{2}+k_{2}^{2}}{2∼{}k_{1}^{2}}\left({1+% \frac{k_{1}^{2}-k_{2}^{2}}{k_{1}k_{2}m+k_{2}^{2}}}\right)$ (24)

$r_{{\rm ion}}$ is a monotonic decreasing equation with $m$ as an independent variable, as shown in Fig. 3 (red line).

Let $r_{{\rm ion}}\geqslant 1$ , then we can get

$\displaystyle m\geqslant{k_{1}}/{k_{2}}$ (25)

This implies that when $m>{k_{1}}/{k_{2}}$ the NL combinations have lower ionosphere delay. However, according to previous analysis, this condition will lead to a larger value of $r_{{\rm stoch}}$ , as demonstrated in Fig. 3 (blue line). In other words, NL combinations satisfying $m>{k_{1}}/{k_{2}}$ will have lower solution precision comparing to the original combination. Besides, the larger the value of $m$ , the shorter the combined wavelength. This may makes us unable to resolve the ambiguity parameters. So it is not worth increasing the value of $m$ just in order to obtain slightly lower ionosphere delay.

For NL combinations satisfying $m<{k_{1}}/{k_{2}}$ , they always have larger ionosphere delay and lower solution precision than the original combination, as shown in Fig. 3. The commonly used narrow lane combination (1, 1) are one of those combinations. The values of $r_{{\rm ion}}$ with this combination is listed in Table 3.

Though there is a point where $r_{{\rm ion}}=1$ , combinations satisfying this condition have very short wavelength.

In general, combining the analysis in Section 3.1 the conclusion that in dual frequency case the original combination is characterized by best solution precision, lower ionosphere delay and longer wavelength comparing to the NL combinations with positive coefficients can be obtained.

Table 3

$r_{{\rm ion}}$ with combination (1, 1)

GNSS	Frequency	$r_{ion}$	$1/{r_{{\rm ion}}}$
	L1, L2	1.0313	0.9697
GPS	L1, L5	1.0429	0.9588
	L2, L5	1.0009	0.9991
	B1, B2	1.0332	0.9678
BDS	B1, B3	1.0216	0.9788
	B2, B3	1.0012	0.9988

3.2.2 Triple-frequency case

$r_{{\rm ion}}$ in this case is given as

$\displaystyle r_{{\rm ion}}=\frac{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}}{3∼{}k_{1}}% \frac{k_{2}(k_{3}m+k_{1}n)+k_{1}k_{3}}{k_{2}k_{3}(k_{1}m+k_{2}+k_{3}n)}$ (26)

Taking the partial derivative with respect to $m$ and $n$ , it can be proved that $r_{{\rm ion}}$ has no extreme point, as shown in Fig. 4 (the graphic below). The intersection curve of Eqs (18) and (26) is derived as

$\displaystyle m^{2}+n^{2}-\frac{1}{2}\left({\frac{k_{1}}{k_{3}}+\frac{k_{3}}{k% _{1}}}\right)mn-\frac{1}{2}\left({\frac{k_{1}}{k_{2}}+\frac{k_{2}}{k_{1}}}% \right)m-\frac{1}{2}\left({\frac{k_{2}}{k_{3}}+\frac{k_{3}}{k_{2}}}\right)n+1=0$ (27)

If the extreme point $\left({{k_{1}}/{k_{2}},{k_{3}}/{k_{2}}}\right)$ of $r_{{\rm stoch}}$ is substitute into Eq. (27), its value is zero which means that this point is on the intersection curve. If this point is substitute into $r_{{\rm ion}}$ , we can get $r_{{\rm ion}}=1$ . Thus, it can be summarized that point $\left({{k_{1}}/{k_{2}},{k_{3}}/{k_{2}}}\right)$ is a cross point where $r_{{\rm stoch}}=r_{{\rm ion}}=1$ as shown in Fig. 4.

Table 4

$r_{{\rm ion}}$ with combination (1, 1, 1)

GNSS	Frequency	$r_{ion}^{tri}$	$1/{r_{ion}^{tri}}$
GPS	L1, L2, L5	1.0348	0.9664
BDS	B1, B2, B3	1.0258	0.9748

Figure 3.

$r_{{\rm ion}}$ with respect to $m$ .

Figure 4.

$r_{{\rm ion}}$ and $r_{{\rm ion}}$ in triple frequency case (GPS).

It should be noticed that there are combinations satisfying $r_{{\rm ion}}<1$ which means that those NL combinations have lower ionosphere delay than the original combination. However, those combinations lead to larger $m, n$ and larger $r_{cov}$ , as shown in Fig. 4. Thus, they have very short wavelength and lower solution precision comparing to the original combination.

Though the cross point $\left({{k_{1}}/{k_{2}},{k_{3}}/{k_{2}}}\right)$ may theoretically lead to an equivalence between NL combinations and the original combination. The short wavelength problem still need to be considered.

For the narrow lane combination (1, 1, 1), the $r_{{\rm ion}}$ values for GPS and BDS are respectively 1.0348 and 1.0258, as listed in Table 4. It has slightly larger ionosphere delay than the original combination.

In general, we can derive the conclusion that in triple frequency case the original combination is also superior to NL combinations in higher solution precision, lower ionosphere delay and longer wavelength.

4. Numerical experiment

The theoretical development of previous section is verified with real BDS data collected during an experiment. The experiment is carried out in 22 ${}^{\rm th}$ Feb, 2017 using a dual frequency receiver with a sample rate of 1 s. The receiver consists of two boards respectively connected to two antennas which are about 10 meters away from each other. The receiver is capable of receiving frequency B1 and B3. Figure 5 shows the experiment equipment. The antennas are mounted on the top of a building with no obstacles.

Recorded data was processed by self-developed GNSS Multiple Frequency ToolBox. Ambiguity parameters can be resolved in tens of seconds and the detection for cycle slip is conducted all the time to make sure the correction of ambiguities.

Table 5
$r_{{\rm stoch}}$ and $r_{{\rm ion}}$ with four NL combinations

Ratio	(1, 1)	(763, 620)	(11, 10)	(3, 1)
$r_{{\rm stoch}}$	1.0107	1	1.0031	1.1422
$r_{{\rm ion}}$	1.0216	1	1.0116	0.9212

Figure 5.

Experiment equipment.

4.1 Short term experiment: 1 hour

Four NL combinations are selected to compare with the original combination: (1, 1), (763, 620), (11, 10) and (3, 1). Table 5 lists the values of $r_{{\rm stoch}}$ and $r_{{\rm ion}}$ . Figure 7 demonstrates the solution precision of all 36 groups. It is obvious that original combination has higher solution precision than NL combinations. But the improvement is very small which is consistent with Table 5. Though theoretically combination (763, 620) should have the same solution precision to the original combination, the result shows that the equivalence does not exist. This may be associated with other external factors.

Figure 7 illustrates the solution errors (solutions subtract their mean values) of those combinations. The true values of $x, y$ and $z$ are not known, thus it is unable to calculate the absolute error. But it can be find that the differences between those combinations are very small. Those differences may mainly be caused by external errors like multipath, because for the baseline is ultra short, ionospheric error will be small enough to ignore after the DD operation of carrier phase. Therefore, there is no too much meaning to compare the solution errors.

Figure 6.

Solution precision of NL combinations and the original combination.

Figure 7.

Solutions error of NL combinations and the original combination.

4.2 Long term experiment: 24 hour

To further prove the derivation, a 24–hour experiment was conducted. Figure 8 illustrates the solution standard deviations of three narrow lane combinations and the original combination. As can be seen, though it is very small, we still able to confirm that the original combination still generally has higher solution precision than narrow lane combinations.

Figure 8.

Solution precision of NL combinations and the original combination in 24 h.

5. Summary and discussion

In this paper, we compared the performance of NL combinations and original combination in dual and triple frequency case under different navigation systems. Through the above theoretical development, it follows that the original combination is superior to the commonly used narrow lane combination no matter which navigation system is used and how many frequencies are used. Though the improvement is small, the original combination preserves some advantages over the NL combinations which is summarized as following:

1) 1)

Longer wavelength. The virtual wavelength of narrow lane combinations is always shorter than the original. Since the original combination uses carrier phase measurements independently, its shortest wavelength is longer than that of the narrow lane combination. In other words, it is easier to resolve the ambiguities of original combination, than it is to resolve the ambiguities of narrow lane combinations directly. Besides, using the original combination will be more convenient for there is no need to transit other ambiguities to narrow lane ambiguities.

More robust. To use narrow lane combinations, it is needed to make sure that carrier phase measurements from all the frequencies are usable. Otherwise, we will unable to form the narrow lane equation. However, the original combinations uses the carrier phase independently which means that if there are observations from a satellite or a frequency are not available the original combination equation can still be formed and computed.

Higher solution accuracy. Though the improvement is slight, it still can be noticed that the original combination has the highest solution precision.

The main disadvantage of the original combination is that the measurement number of the original combination is at least two times higher than narrow lane combinations, especially in the multi-frequency and multi-system case. This will lead to a high dimension of coefficient matrix and thus affect the computational efficiency. Nevertheless considering the powerful processing ability of hardware nowadays, performing such a computation is not a big problem.

Footnotes

Acknowledgments

This work is supported by Aeronautic Science Foundation of China (grant nos. 201551U8008).

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