The optimal narrow lane combinations for multi-frequency GNSS orientating

Abstract

Most of global navigation satellite systems (GNSS) are transmitting three or even more frequency signals. Narrow lane combinations of these multi-frequency observations are characterized by short wave length, high precision and have been widely used in positioning and orientating. In this work, the optimal narrow lane combinations were theoretically investigated to improve the solution precision. To this end, the basic multi-frequency linear combination model was introduced and two key factors (i.e. noise factor and lane number) were defined. On this basis, two parameters SPI (Solution Precision Indicator) and SPI ${}_{\text{rel}}$ (Relative Solution Precision Indicator) were derived to evaluate solution precision. Then, the optimal dual-frequency and triple frequency conditions were derived theoretically, which was verified by BDS (BeiDou Navigation System) experiment. The obtained results show that the linear combinations satisfying optimal condition can lead to a higher positioning and orientating precision and the precision increased with more frequencies.

Keywords

GNSS optimal narrow lane combinations GNSS orientating

1. Introduction

Most of global navigation satellite systems (GNSS) are transmitting three or even more frequency signals. BeiDou Navigation Satellite System (BDS) developed by China started its regional services in 2012 and provides signals centered at B1 (1561.09 MHz), B2 (1207.14 MHz) and B3 (1268.52 MHz). Besides the previous L1 (1575.42 MHz) and L2 (1227.6 MHz) frequency, GPS now provides a third frequency L5 (1176.45 MHz). And Galileo system can provide five frequency signals centered at E1 (1575.42 MHz), E2 (1561.1 MHz), E5A (1176.45 MHz), E5B (1207.14 Mhz) and E6 (1278.75 MHz) respectively.

Linear combinations of these multi-frequency observations have been used for various purposes. According to Cocard et al. [1], three groups of combinations can be distinguished: wide lane region, intermediate lane region and narrow lane region. Wide lane and intermediate lane combinations are usually used in ambiguity resolution. Narrow lane combinations are always used to increase positioning or orientating precision after ambiguity parameters have been fixed. Lots of investigations have been conducted on linear combinations of multi-frequency observation in the near decade. Most of them focused on combinations in wide lane region. For example, Neumann et al. [2] showed that dual-frequency narrow lane combinations are capable of mitigating multipath and noise. Cocard and Geiger [3] and Collins [4] described an approach which could be used to search optimal dual-frequency wide lane carrier phase combinations. Han and Rizos [5] developed long wave lengths but low noise linear combinations which were suitable for ambiguity resolution. Teunissen and Odijk [6] and Odijk [7] mainly focused on the ionosphere-free combinations that three frequency can provide and have shown that three frequency observations can lead to much more possible combinations. Feng [8] developed three frequency optimal combinations under three criteria: long wavelength, reduced ionosphere delay, and low combination noise. The combinations were used in geometry-based ambiguity resolution. In order to satisfy the demands of bootstrapping method, Cocard [1] derived rigorous mathematical equations for gaining optimal combinations of three frequency observations. The triple frequency optimal combinations of GLONASS was investigated by Xu et al. [9]. And they were investigated based on principles such as long wavelength, less ionosphere delay and low observation noise. Zhang and He [10] 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. [11] conducted an analytical study on three carrier phase combinations, and presented a general analytical method for solving the optimal integer linear combinations which were used in cycle slip detection and ambiguity resolution.

Through the introduction above, we see that most of the investigations concentrate on the optimal combinations that have long wave length, less ionosphere delay, less troposphere delay and low noise. These combinations are mainly developed for resolving ambiguities and are studied based on their characteristics not on the solution precision. Investigations of optimal narrow lane combinations suitable for positioning and orientating are few and not systematic. In this contribution, we presented a systematic investigation of the optimal narrow lane combinations after ambiguity parameters have been fixed. The basic multi-frequency linear combination model was first introduced and the definitions of noise factor and lane number were defined. Then SPI and SPI ${}_{\text{rel}}$ were derived to evaluate solution precision. On basis of SPI and SPI ${}_{\text{rel}}$ , the dual-frequency and triple frequency conditions of optimal narrow lane combinations were deduced. The theoretical derivations were verified with real BDS observations. The processed data lasted for 24 hours, and precision of various combinations were investigated.

2. The optimal narrow lane combinations

2.1 The basic linear combination model

The following observation equation of carrier phase will be used

$\displaystyle\phi_{i}=\frac{\rho}{\lambda_{i}}+N_{i}+\frac{I_{i}}{\lambda_{i}}% +\frac{\delta_{\textit{trop}}}{\lambda_{i}}+\frac{\delta}{\lambda_{i}}+% \varepsilon_{i}$ (1)

Where $\phi_{i}$ is the phase measurement in units of cycles; $\rho$ is the geometrical distance between receiver and satellite in meters; $\lambda_{i}$ is the phase wavelength in meters; $\delta_{\textit{trop}}$ is the troposphere error in meters; $\delta$ contains other systematic errors; $\varepsilon_{i}$ is the stochastic error in units of cycles; $I_{i}$ is the ionosphere delay; $i$ is the index of frequency. The equation can also be considered in a differential mode. In this case, all terms in the equation are double differenced but ambiguity parameters remain as integers.

The linear combination of carrier phase is given by

$\displaystyle\Phi_{C}=\sum\limits_{i}^{n}a_{i}\phi_{i}\quad i=1,2,3\ldots n$ (2)

Where $n$ is the number of frequencies used, $a_{i}$ are integer combination coefficients.

After combination, the Eq. (2) has the same form as Eq. (1). Taking $c=\lambda f$ , $f_{i}=k_{i}f_{0}$ . $f_{0}$ is the base frequency, $k_{i}$ are integers. The base frequency of GPS is 10.23 MHZ, $f_{L1}>f_{L2}>f_{L5}$ , then $k_{i}$ of GPS are corresponding to $k_{1}=$ 154, $k_{2}=$ 120, $k_{3}=$ 115. The base frequency of BDS is 2.046 MHZ, $f_{B1}>f_{B3}>f_{B2}$ , so $k_{i}$ of BDS are corresponding to $k_{1}=$ 763, $k_{2}=$ 590, $k_{3}=$ 620.

After combination the wavelength can be written as

$\displaystyle\lambda_{c}=\frac{c}{\sum\limits_{i}^{n}a_{i}f_{i}}=\frac{c}{f_{0% }g}\quad i=1,2,3\ldots n$ (3)

Where $g=\sum\limits_{i}^{n}a_{i}k_{i}$ is defined as lane number, $c$ is the speed of light.

The combined frequency is

$\displaystyle f_{c}=\sum\limits_{i}^{n}a_{i}f_{i}=f_{0}g\quad i=1,2,3\ldots n$ (4)

The combined ambiguity parameter is

$\displaystyle N_{c}=\sum\limits_{i}^{n}a_{i}N_{i}\quad i=1,2,3\ldots n$ (5)

The combined noise is

$\displaystyle\varepsilon_{c}=\sum\limits_{i}^{n}a_{i}\varepsilon_{i}\quad i=1,% 2,3\ldots n$ (6)

In this paper, the noise on all frequencies expressed in cycles is assumed the same and all frequencies are statistically independent. Then variance propagation of combined observations leads to

$\displaystyle D\Phi_{C}=D\left(\sum\limits_{i}^{n}a_{i}\phi_{i}\right)=\sigma_% {0}l$ (7)

Where $\sigma_{0}$ is the standard deviation of carrier phase, $l=\sum\limits_{i}^{n}a_{i}^{2}$ is defined as noise factor.

2.2 Optimal narrow lane combinations

2.2.1 Solution precision indicators and optimal combinations

The double differential mode of Eq. (1) will be used

$\displaystyle\Delta\nabla\phi_{i}=\frac{\Delta\nabla\rho}{\lambda_{i}}+\Delta% \nabla N_{i}+\frac{\Delta\nabla I_{i}}{\lambda_{i}}+\frac{\Delta\nabla\delta_{% \textit{trop}}}{\lambda_{i}}+\frac{\Delta\nabla\delta}{\lambda_{i}}+\Delta% \nabla\varepsilon_{i}$ (8)

Where $\Delta\nabla$ represents the double-differenced (DD) operation. Then the variance-covariance matrix of all the double differenced carrier phase from observed satellites can be written as

$\displaystyle{\rm{\bf Q}}=\text{cov}(\nabla\Delta{\rm{\bf\Phi}}_{i})=\sigma_{0% }^{2}{\rm{\bf C}}$ (9)

Where ${\rm{\bf C}}$ is a constant matrix, $\nabla\Delta{\rm{\bf\Phi}}_{i}=\left[\nabla\Delta\phi_{i}^{s1}\quad\nabla% \Delta\phi_{i}^{s2}\quad{\ldots}\right]^{\text{T}}$ .

The linearized Eq. (8) is expressed as

$\displaystyle{\rm{\bf L}}_{i}={\rm{\bf A}}_{i}{\rm{\bf X}}+{\rm{\bf BN}}_{i}+{% \rm{\bf\varepsilon}}_{i}$ (10)

Where ${\rm{\bf A}}_{i}=\frac{{\rm{\bf A}}_{0}}{\lambda_{i}}$ , ${\rm{\bf A}}_{0}$ is a matrix depending on satellite distribution and reference satellite, ${\rm{\bf L}}=\left[{L_{1}}\quad{L_{2}}\quad{\ldots}\right]$ is double differenced carrier phase measurements vector and consists of the remaining errors, such as atmospheric delay and multipath, ${\rm{\bf N}}_{i}$ is the vector of double differenced ambiguities, ${\rm{\bf X}}$ is the baseline vector, ${\rm{\bf A}}_{i}$ and ${\rm{\bf B}}$ are design matrixes. When ambiguities are fixed, Eq. (10) can be rewritten as

$\displaystyle{\rm{\bf\bar{L}}}_{i}={\rm{\bf A}}_{i}{\rm{\bf X}}+{\rm{\bf% \varepsilon}}_{i}$ (11)

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

Combining Eqs (2) and (11), the linear combination model of double differenced carrier phase observation can be gained

$\displaystyle\Delta\nabla{\rm{\bf\Phi}}_{C}={\rm{\bf a}}^{T}{\rm{\bf\bar{L}}}_% {i}$ (12)

With ${\rm{\bf a}}^{T}=\left[{a_{1}}\quad{a_{2}}\quad{\ldots}\right]$ . The variance-covariance matrix ${\rm{\bf Q}}_{C}$ of Eq. (12) is

$\displaystyle{\rm{\bf Q}}_{C}=\text{cov}\left({\Delta\nabla\Phi_{C}}\right)=% \sum\limits_{i}\text{cov}(a_{i}\bar{L}_{i})=\left(\sum\limits_{i}a_{i}^{2}% \right)\sigma_{0}^{2}{\rm{\bf C}}=l\sigma_{0}^{2}{\rm{\bf C}}$ (13)

Then the least square solution of Eq. (12) is

$\displaystyle{\rm{\bf X}}_{C}={\rm{\bf Q}}_{XC}{\rm{\bf M}}_{XC}$ $\displaystyle{\rm{\bf Q}}_{XC}=({\rm{\bf A}}^{T}{\rm{\bf P}}_{C}{\rm{\bf A}})^% {-1}$ (14) $\displaystyle{\rm{\bf M}}_{XC}={\rm{\bf A}}^{T}{\rm{\bf P}}_{C}{\rm{\bf L}}$

Where ${\rm{\bf X}}_{C}$ is the baseline vector, ${\rm{\bf Q}}_{XC}$ is variance-covariance matrix of ${\rm{\bf X}}_{C}$ , ${\rm{\bf P}}_{C}={\rm{\bf Q}}_{C}^{-1}$ . Substituting Eq. (13) into Eq. (2.2.1), ${\rm{\bf Q}}_{XC}$ is rewritten as

$\displaystyle{\rm{\bf Q}}_{XC}=({\rm{\bf A}}^{T}{\rm{\bf P}}_{C}{\rm{\bf A}})^% {-1}=l\lambda_{c}^{2}\sigma_{0}^{2}({\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}{\rm{% \bf A}}_{0})^{-1}=\frac{c^{2}\sigma_{0}^{2}}{f_{0}^{2}}\frac{l}{g^{2}}({\rm{% \bf A}}_{0}^{T}{\rm{\bf C}}^{-1}{\rm{\bf A}}_{0})^{-1}$ (15)

It is obvious that $l/{g^{2}}$ is defined by combination, ${(c^{2}\sigma^{2})}/{f_{0}^{2}}$ is a constant factor, ${\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}{\rm{\bf A}}_{0}$ is a nearly constant matrix in a short period of observation time.

Transforming ${\rm{\bf X}}_{C}$ into local coordinate yields

$\displaystyle{\rm{\bf X}}_{\textit{CLocal}}={\rm{\bf RX}}_{C}$ (16) $\displaystyle{\rm{\bf Q}}_{\textit{XCLocal}}={\rm{\bf RQ}}_{XC}{\rm{\bf R}}^{T}$

Where ${\rm{\bf R}}$ is the transformation matrix.

Defining ${\rm{\bf X}}_{\textit{CLocal}}=[{\Delta x}\quad{\Delta y}\quad{\Delta z}]^{T}$ , the azimuth can be written as

$\displaystyle\textit{azi}=f(\Delta x,\Delta y)=\arctan\left(\frac{\Delta x}{% \Delta y}\right)$ $\displaystyle\sigma_{\textit{azi}}^{2}=\frac{\partial f}{\partial\Delta x}% \sigma_{\Delta x}^{2}+\frac{\partial f}{\partial\Delta y}\sigma_{\Delta y}^{2}% +2\frac{\partial f}{\partial\Delta x}\frac{\partial f}{\partial\Delta y}\rho_{% \Delta x\Delta y}\sigma_{\Delta x}\sigma_{\Delta y}$ (17)

Where $\rho_{\Delta x\Delta y}$ is the correlation coefficient.

Substituting Eq. (15) into Eq. (2.2.1) the variance of the azimuth can be written as

$\displaystyle\sigma_{\textit{azi}}^{2}=\frac{l}{g^{2}}\left({\frac{\partial f}% {\partial\Delta x}c_{\Delta x}^{2}+\frac{\partial f}{\partial\Delta y}c_{% \Delta y}^{2}+2\frac{\partial f}{\partial\Delta x}\frac{\partial f}{\partial% \Delta y}\rho_{\Delta x\Delta y}c_{\Delta x}c_{\Delta y}}\right)$ (18)

Where $c_{\Delta x},c_{\Delta y}$ are constants.

So $\frac{l}{g^{2}}$ is the key factor capable of indicating solution precision. Defining SPI as the indicator of solution precision

$\displaystyle\text{SPI}=\frac{g^{2}}{l}=\frac{\left({\sum\limits_{i}{k_{i}a_{i% }}}\right)^{2}}{\sum\limits_{i}{a_{i}^{2}}}$ (19)

SPI is inversely proportional to the diagonal elements of ${\rm{\bf Q}}_{XC}$ . The larger the SPI value is, the higher precision of corresponding solution will be. And the solution will theoretically reaches highest precision if SPI reaches its maximum.

In order to make comparison more intuitively, define SPI ${}_{\text{rel}}$ as

$\displaystyle\text{SPI}_{\text{rel}}=\frac{g^{2}}{\text{SPI}_{\max}l}=\frac{% \left({\sum\limits_{i}{k_{i}a_{i}}}\right)^{2}}{\text{SPI}_{\max}\left({\sum% \limits_{i}{a_{i}^{2}}}\right)}$ (20)

Where $0<\text{SPI}_{\text{rel}}\leqslant 1$ , $\text{SPI}_{\max}$ is the maximum of SPI.

2.2.2 Optimal condition of dual frequency

When using dual frequency, SPI is

$\displaystyle\text{SPI}=\frac{g^{2}}{l}=\frac{\left({k_{1}a_{1}+k_{2}a_{2}}% \right)^{2}}{a_{1}^{2}+a_{2}^{2}}$ (21)

Solving the partial derivative of Eq. (21) yields

$\displaystyle k_{1}a_{2}^{2}-k_{2}a_{1}a_{2}=0$ (22)

For $a_{1}\neq 0$ , $a_{2}\neq 0$ , Eq. (22) can be written as:

$\displaystyle\frac{a_{1}}{a_{2}}=\frac{k_{1}}{k_{2}}$ (23)

When integer combination coefficients $a_{1}$ , $a_{2}$ satisfy Eq. (23), SPI reaches its maximum, and ${\rm{\bf X}}_{C}$ vreaches the highest precision.

The maximum value of SPI is

$\displaystyle\text{SPI}_{\max}=k_{1}^{2}+k_{2}^{2}$ (24)

According to Eq. (23), $a_{1}$ , $a_{2}$ have the same sign and have a constant ratio which is defined by the frequencies used. Thus the optimal combinations lie near a line that crosses the first and third quadrant.

Figure 1 shows the relationship between a ${}_{1}$ , a ${}_{2}$ and SPI. Though lots of combinations having different sign are in narrow lane region, it is obvious that the SPI values of combinations with the same sign are higher than those not.

Figure 1.

Relationship between a ${}_{1}$ , a ${}_{2}$ and SPI.

2.2.3 Optimal condition of triple frequency

When using three frequencies, SPI is

$\displaystyle\text{SPI}=\frac{g^{2}}{l}=\frac{\left({k_{1}a_{1}+k_{2}a_{2}+k_{% 3}a_{3}}\right)^{2}}{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$ (25)

Solving the partial derivative of Eq. (25) yields

$\displaystyle(k_{1}-k_{2}m)\left[{(k_{1}^{2}+k_{3}^{2})m^{2}+2k_{1}k_{2}m+(k_{% 2}^{2}+k_{3}^{2})}\right]=0$ (26)

Where $m=\frac{a_{1}}{a_{2}}$ , $n=\frac{a_{3}}{a_{2}}$ .

For equation $(k_{1}^{2}+k_{3}^{2})m^{2}+2k_{1}k_{2}m+(k_{2}^{2}+k_{3}^{2})=0$ without solution, then the solution of Eq. (26) is

$\displaystyle m=\frac{a_{1}}{a_{2}}=\frac{k_{1}}{k_{2}}$ (27) $\displaystyle n=\frac{a_{3}}{a_{2}}=\frac{k_{3}}{k_{2}}$ (28)

Then, when $a_{1}$ , $a_{2}$ , $a_{3}$ satisfy Eqs (27) and (28), SPI reaches its maximum

$\displaystyle\text{SPI}_{\max}=k_{1}^{2}+k_{2}^{2}+k_{3}^{2}$ (29) $\displaystyle\text{SPI}_{\text{rel}}=\frac{\left({k_{1}a_{1}+k_{2}a_{2}+k_{3}a% _{3}}\right)^{2}}{(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})(a_{1}^{2}+a_{2}^{2}+a_{3}^{2% })}$ (30)

In this case, $a_{1}$ , $a_{2}$ , $a_{3}$ also have the same sign and any two of them have a fixed ratio value. Table 1 summarizes the optimal condition and an optimal combination in various cases. In addition, the SPI and SPI ${}_{\text{rel}}$ factor are also listed. It is observed that no matter which navigation system is used, the SPI and SPI ${}_{\text{rel}}$ factors of dual-frequency is always lower than those of triple frequency. This implies that triple frequency combinations will theoretically achieve higher solution precision. In the case of dual-frequency, optimal combinations of L1 and L2 from GPS have the highest SPI value than other combinations. When it comes to BDS, the frequencies are B1 and B3.

Table 1

Optimal narrow lane combinations of GPS and BDS

Navigation system	Combination coefficient	Optimal condition	Optimal combination	SPI ${}_{\max}$	SPI ${}_{\text{rel}}$
GPS	a ${}_{1}\neq$ 0, a ${}_{2}\neq$ 0, a ${}_{3}=$ 0	a ${}_{1}$ /a ${}_{2}=$ 77/60	(77, 60, 0)	38116	0.74
	a ${}_{1}\neq$ 0, a ${}_{2}=$ 0, a ${}_{3}\neq$ 0	a ${}_{1}$ /a ${}_{3}=$ 154/115	(154, 0, 115)	36941	0.72
	a ${}_{1}=$ 0, a ${}_{2}\neq$ 0, a ${}_{3}\neq$ 0	a ${}_{2}$ /a ${}_{3}=$ 24/23	(0, 24, 23)	27625	0.54
	a ${}_{1}\neq$ 0, a ${}_{2}\neq$ 0, a ${}_{3}\neq$ 0	a ${}_{1}$ /a ${}_{2}=$ 154/120	(154, 120, 115)	51341	1
		a ${}_{3}$ /a ${}_{2}=$ 115/120
BDS	a ${}_{1}\neq$ 0, a ${}_{2}\neq$ 0, a ${}_{3}=$ 0	a ${}_{1}$ /a ${}_{2}=$ 763/590	(763, 590, 0)	930269	0.71
	a ${}_{1}\neq$ 0, a ${}_{2}=$ 0, a ${}_{3}\neq$ 0	a ${}_{1}$ /a ${}_{3}=$ 763/620	(763, 0, 620)	966569	0.74
	a ${}_{1}=$ 0, a ${}_{2}\neq$ 0, a ${}_{3}\neq$ 0	a ${}_{2}$ /a ${}_{3}=$ 59/62	(0, 590, 620)	732500	0.56
	a ${}_{1}\neq$ 0, a ${}_{2}\neq$ 0, a ${}_{3}\neq$ 0	a ${}_{1}$ /a ${}_{2}=$ 763/590	(763, 590, 620)	1314669	1
		a ${}_{3}$ /a ${}_{2}=$ 620/590

Table 2 lists several common used combinations and their SPI values. In dual-frequency case, the SPI and SPI ${}_{\text{rel}}$ values of combination (1, 1, 0), (1, 0, 1) and (0, 1, 1) are close to their corresponding optimal combinations in Table 1. Though combination (4, 0, $-$ 3) is a narrow lane combination, it has a significant low SPI ${}_{\text{rel}}$ value compared to others. Having a SPI ${}_{\text{rel}}$ value of 0.98, combination (1, 1, 1) is very close to the triple frequency optimal situation. However, narrow lane combination (1, 6, $-$ 5) just has a SPI ${}_{\text{rel}}$ value nearly 32 times lower than (1, 1, 1). A wide lane combination (1, 0, $-$ 1) is also listed in Table 2, its SPI ${}_{\text{rel}}$ value is only one percent of the optimal situation.

Table 2

The SPI and SPI ${}_{\text{rel}}$ of different combinations

	Combination	g	SPI		SPI ${}_{\text{rel}}$		Combination	g	SPI		SPI ${}_{\text{rel}}$
GPS	(1, 1, 0)	270	37538		0.73	BDS	(1, 1, 0)	1383	956344	.5	0.73
	(1, 0, 1)	269	36180	.5	0.7		(1, 0, 1)	1353	915304	.5	0.7
	(0, 1, 1)	235	27612	.5	0.54		(0, 1, 1)	1210	732050		0.56
	(1, 1, 1)	389	50440	.3	0.98		(1, 1, 1)	1973	1297576		0.99
	(4, 0, $-$ 3)	271	2937	.6	0.06		(4, 0, $-$ 3)	1282	65741		0.05
	(1, 6, $-$ 5)	299	1441	.95	0.03		(1, 6, $-$ 5)	1533	37904	.6	0.03
	(1, 0, $-$ 1)	39	760	.5	0.01		(1, 0, $-$ 1)	173	14964	.5	0.01

2.2.4 The case of single frequency

When using single frequency, the concept of combination will be meaningless, but Eq. (19) is still suitable, the SPI value of single frequency is

$\displaystyle\text{SPI}=k_{i}^{2}$ (31)

Where the value of SPI is a constant defined by its frequency. Table 3 shows the SPI and SPI ${}_{\text{rel}}$ value of single frequency, they are much lower than the optimal case of dual and triple frequency. The highest frequency owns the highest SPI value.

Table 3

SPI and SPI ${}_{\text{rel}}$ values of single frequency

	Base frequency	Frequency	k	SPI	SPI ${}_{\text{rel}}$
GPS	10.23 MHZ	L1	154	23716	0.46
		L2	120	14400	0.28
		L5	115	13225	0.26
BDS	2.046 MHZ	B1	763	582169	0.44
		B2	590	348100	0.26
		B3	620	384400	0.29

2.3 Systematic errors of optimal combinations

In Eq. (2.2.1) ${\rm{\bf M}}_{XC}$ can be rewritten as

$\displaystyle{\rm{\bf M}}_{XC}={\rm{\bf A}}^{T}{\rm{\bf P}}_{C}{\rm{\bf L}}=% \frac{{\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}}{\sigma^{2}}\sum\limits_{i}{\frac{% a_{i}{\rm{\bf L}}_{i}}{\lambda_{c}l}}=\frac{{\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{% -1}}{\sigma^{2}}\frac{f_{0}}{c}\sum\limits_{i}{\left({q_{Ci}{\rm{\bf L}}_{i}}% \right)}$ (32)

Where $q_{Ci}={ga_{i}}/l$ are defined by combination coefficients, other factors such as ${{\rm{\bf A}}_{0}^{T}{\rm{\bf C}}^{-1}}{\sigma_{0}^{2}}$ is a constant matrix. Thus $q_{Ci}$ denotes the influence of combination factors on ${\rm{\bf M}}_{XC}$ .

In case of dual-frequency, $q_{Ci}$ is

$\displaystyle\left[{{\begin{array}[]{*{20}c}{q_{C1}}\\ {q_{C2}}\\ \end{array}}}\right]=\frac{1}{r_{2}}\left[{{\begin{array}[]{*{20}c}{\left({% \frac{a_{1}}{a_{2}}}\right)^{2}}&{\frac{a_{1}}{a_{2}}}\\ {\frac{a_{1}}{a_{2}}}&1\\ \end{array}}}\right]\left({{\begin{array}[]{*{20}c}{k_{1}}\\ {k_{2}}\\ \end{array}}}\right)$ (33)

Where $r_{2}=\left({\frac{a_{1}}{a_{2}}}\right)^{2}+1$ .

In case of triple frequency, $q_{Ci}$ can be written as

$\displaystyle\left[{{\begin{array}[]{*{20}c}{q_{C1}}\\ {q_{C2}}\\ {q_{C3}}\\ \end{array}}}\right]=\frac{1}{r_{3}}\left[{{\begin{array}[]{*{20}c}{\left({% \frac{a_{1}}{a_{2}}}\right)^{2}}&{\frac{a_{1}}{a_{2}}}&{\frac{a_{1}}{a_{2}}% \frac{a_{3}}{a_{2}}}\\ {\frac{a_{1}}{a_{2}}}&1&{\frac{a_{3}}{a_{2}}}\\ {\frac{a_{1}}{a_{2}}\frac{a_{3}}{a_{2}}}&{\frac{a_{1}}{a_{2}}}&{\left({\frac{a% _{3}}{a_{2}}}\right)^{2}}\\ \end{array}}}\right]\left({{\begin{array}[]{*{20}c}{k_{1}}\\ {k_{2}}\\ {k_{3}}\\ \end{array}}}\right)$ (34)

Where $r_{3}=\left({\frac{a_{1}}{a_{2}}}\right)^{2}+\left({\frac{a_{3}}{a_{2}}}\right% )^{2}+1$ .

According to Eqs (33) and (34), $q_{Ci}$ is a function of $\frac{a_{1}}{a_{2}}$ and $\frac{a_{3}}{a_{2}}$ , in other words, it was defined by the combination coefficients.

The value of $\frac{a_{1}}{a_{2}}$ , $\frac{a_{3}}{a_{2}}$ will corresponding to $\frac{a_{1}}{a_{2}}=\frac{k_{1}}{k_{2}}$ and $\frac{a_{3}}{a_{2}}=\frac{k_{3}}{k_{2}}$ , when combinations used satisfy the optimal condition. Then Eqs (33) and (34) can be given as:

In dual-frequency case

$\displaystyle\left[{{\begin{array}[]{*{20}c}{q_{C1}}\\ {q_{C2}}\\ \end{array}}}\right]=\left({{\begin{array}[]{*{20}c}{k_{1}}\\ {k_{2}}\\ \end{array}}}\right)$ (35)

In triple frequency case

$\displaystyle\left[{{\begin{array}[]{*{20}c}{q_{C1}}\\ {q_{C2}}\\ {q_{C3}}\\ \end{array}}}\right]=\left({{\begin{array}[]{*{20}c}{k_{1}}\\ {k_{2}}\\ {k_{3}}\\ \end{array}}}\right)$ (36)

So when using different optimal narrow lane combinations the $q_{Ci}$ will always be a constant.

In Eq. (32), ${\rm{\bf L}}_{i}$ consists of the remaining systematic errors

$\displaystyle{\rm{\bf L}}_{i}={\rm{\bf L}}_{\textit{obs}}+{\rm{\bf\delta}}_{% \textit{trop}}+{\rm{\bf\delta}}_{\textit{ion}}+{\rm{\bf\delta}}_{\textit{mul}}$ (37)

Where ${\rm{\bf\delta}}_{\textit{trop}},{\rm{\bf\delta}}_{\textit{ion}},{\rm{\bf% \delta}}_{\textit{mul}}$ are respectively troposphere delay vector, ionosphere delay vector and multipath vector. Then the following equation can be obtained:

$\displaystyle\sum\limits_{i}{q_{Ci}{\rm{\bf L}}_{i}}=\sum\limits_{i}{q_{Ci}{% \rm{\bf L}}_{\textit{obs}}^{i}}+\sum\limits_{i}{q_{Ci}{\rm{\bf\delta}}_{% \textit{trop}}^{i}}+\sum\limits_{i}{q_{Ci}{\rm{\bf\delta}}_{\textit{ion}}^{i}}% +\sum\limits_{i}{q_{Ci}{\rm{\bf\delta}}_{\textit{mul}}^{i}}$ (38)

Combining Eqs (32), (33) and (35) yields

$\displaystyle\sum\limits_{i}{q_{Ci}{\rm{\bf L}}_{i}}=\sum\limits_{i}{k_{i}{\rm% {\bf L}}_{\textit{obs}}^{i}}+\sum\limits_{i}{k_{i}{\rm{\bf\delta}}_{\textit{% trop}}^{i}}+\sum\limits_{i}{k_{i}{\rm{\bf\delta}}_{\textit{ion}}^{i}}+\sum% \limits_{i}{k_{i}{\rm{\bf\delta}}_{\textit{mul}}^{i}}$ (39)

Therefore, when using different optimal narrow lane combinations, the impact of systematic errors on solution will theoretically remain the same.

3. Numerical experiment

The theoretical derivation of optimal narrow lane combinations is verified with real BDS data. The experiment was carried out in 11 ${}^{\text{th}}$ April, 2016 using a dual-frequency receiver made by HWA Create with a sampling rate of 1 Hz. The receiver consists of two boards respectively connected to two antennas which are about 10 meters away from each other. To ensure the distribution of satellite unchanged, we used the data collected just in an hour, during which the carrier phase measurements on B1 and B3 frequency are recorded. Figure 2 shows the experiment equipment, which is laid on the top of a building with no obstacles.

Figure 2.

Experiment equipment.

Recorded data was processed by GNSS Multiple Frequency ToolBox, a self-developed software. Figure 3 illustrates the elevation variation of all observed satellites. There is no significant change in their elevation in an hour. Then the influence of satellite distributing variation can be disregard and the influence of combination coefficients were mainly focused on.

Figure 3.

Azimuth and elevation variation of observed satellites.

3.1 Hour experiments

Solutions of optimal combination (763,620) are first compared to several common used combinations such as (1,1), (4, $-$ 3) and single frequency. Figure 4 shows the least square results of combination (763,620) and (1,1) for each epoch. Figure 4a shows the relative error of the results in form of position while (b) shows the results in form of azimuth. The red dots and black dots respectively represent results of combination (763,620) and (1,1). It is hard to distinguish the results of those two combinations, which means that they have very close precision. Similarly, Fig. 5 shows the solutions of combination (763,620) and (4, $-$ 3). It is obvious that the solution precision of the former combination is much higher than the latter.

Figure 4.

The results of combination (763,620) and (1,1).

Table 4 summarized the solution precision of all combinations mentioned above. The azimuth standard deviation (STD) of combination (763,620) and (1,1) is corresponding to 16. 84 ${}^{\prime\prime}$ and 17 ${}^{\prime\prime}$ ; (4, $-$ 3) has a STD of 75.33 ${}^{\prime\prime}$ which is much more inaccurate than other combinations. As for single frequency, B3 has an azimuth deviation of 25.74 ${}^{\prime\prime}$ , and it is 23.78 ${}^{\prime\prime}$ for B1 frequency. In general, the combination (763,620) is relatively more optimal than other form of combinations.

Table 4

The orientating results of different combinations

	Combination	g	$\sigma_{\textit{azimuth}}$ ( ${}^{\prime\prime}$ )	$\sigma_{x}$ (mm)	$\sigma_{y}$ (mm)	$\sigma_{z}$ (mm)
BDS	(1,0)	763	23.78	1.99	1.18	3.45
	(0,1)	620	25.74	1.66	1.07	2.83
	(763,620)	966569	16.84	1.29	0.823	2.25
	(1,1)	1353	17.00	1.33	0.8	2.33
	(4, $-$ 3)	1282	75.33	5.89	3.51	10.04

Table 5

Value of a ${}_{1}$ /a ${}_{2}$ and the orientating results

a ${}_{1}$ /a ${}_{2}$	Mean Azimuth ( ${}^{\circ})$	$\sigma_{\textit{azimuth}}$ ( ${}^{\prime\prime}$ )	a ${}_{1}$ /a ${}_{2}$	Mean Azimuth ( ${}^{\circ})$	$\sigma_{\textit{azimuth}}$ ( ${}^{\prime\prime}$ )
1.0	30.4202	16.84	0.1	30.4209	21.95
2.0	30.4205	17.98	0.2	30.4195	20.85
3.0	30.4206	19.12	0.3	30.4197	19.42
4.0	30.4206	19.94	0.4	30.4198	18.43
5.0	30.4207	20.521	0.5	30.4205	17.985
6.0	30.4208	20.95	0.6	30.4200	17.32
7.0	30.4208	21.29	0.7	30.4200	17.04
8.0	30.4208	21.55	0.8	30.4200	16.90
9.0	30.4208	21.77	0.9	30.4201	16.84
10.0	30.4209	21.95	1.0	30.4202	16.84

Figure 5.

The results of combination (763,620) and (4, $-$ 3).

To further prove previous theoretical developments, Table 5 lists 19 categories of combinations which are distinguished by the value of a ${}_{1}$ /a ${}_{2}$ . Among them, 9 categories are selected with the range of a ${}_{1}$ /a ${}_{2}$ from 0.1 to 0.9, and the other10 categories are selected with the range from 1 to 10. From each category, only one combination is used to solve the azimuth. The solved azimuth and its standard deviation are also showed in the table. Figure 6 illustrates the relation between a ${}_{1}$ /a ${}_{2}$ and orientating precision. The highest precision is reached when a ${}_{1}$ /a ${}_{2}$ is around 1, for BDS the value should theoretically be 1.23. And the azimuth variances declines rapidly before a ${}_{1}$ /a ${}_{2}$ reaches the optimal point 1.23 and increases slowly afterwards.

Figure 7 shows the variation of SPI vs. a ${}_{1}$ /a ${}_{2}$ . The variation is similar to azimuth variances but in a reversed version. SPI reaches its maximum when a ${}_{1}$ /a ${}_{2}$ equals to 1.23. To further investigate the effect of SPI on orientating precision, azimuth variances and SPI are drawn in Fig. 8. It is obvious that there is a linear relationship between SPI and orientating precision, and the orientating precision varies inversely with SPI. This implies that SPI can be used as an index for evaluating orientating precision. However, Fig. 8 consists of two independent line instead of a single line. The reasons may be associated with our assumption of frequency independent and same measure noise.

Figure 6.

a ${}_{1}$ /a ${}_{2}$ and azimuth variances.

Figure 7.

a ${}_{1}$ /a ${}_{2}$ and SPI.

Figure 8.

SPI and solution precision.

3.2 24 hour experiments

Figure 9 demonstrates the azimuth STD variation of 8 combinations in 24 hours during which the azimuth was calculated once every two hours, and each time lasts for an hour. The azimuth STD is the statistical result of azimuth values calculated in an hour. In Fig. 9, optimal combination solutions are marked by triangle, single frequency solutions are marked by asterisk and circles marks other ordinary combinations. It can be easily find out that optimal combinations can lead to more stable and more precise solutions than other combinations. The variation of STD may caused by the variation of satellite geometry and the influence of external errors.

Figure 9.

The azimuth STD in a day of different combinations.

4. Discussion

1)
The optimal condition for narrow lane combination is suitable for any global navigation systems. For BDS, GPS and Galileo, the basic carrier phase observation equations are the same, so the derivation of optimal condition will be suitable for any systems.
2)
The optimal condition will lead to a great deal of combinations. When using BDS, combinations such as (763,620), (5,4), (9,8) etc. will lead to similar precision. For double frequency the optimal condition is a ${}_{1}$ /a ${}_{2}=$ k ${}_{1}$ /k ${}_{2}$ ; for three frequency is a ${}_{1}$ /a ${}_{2}=$ k ${}_{1}$ /k ${}_{2}$ , a ${}_{3}$ /a ${}_{2}=$ k ${}_{3}$ /k ${}_{2}$ . The condition just limits the ratio of combination parameters, so the number of combinations that satisfy the condition is countless, but we always tend to use those combinations with low noise.
3)
The optimal condition will be tenable in four frequency case or in the case of even more frequencies, where the condition can be written as: a ${}_{i}$ /a ${}_{1}=$ k ${}_{i}$ /k ${}_{1}$ ( $i=$ 2, 3, 4…). The $i$ represents the $i$ th frequency.
4)
From our previous analysis, SPI and orientating precision have a linear relationship, so SPI can be used as indicators of orientating precision. In addition, the factor is easy to calculate.
5)
The more frequencies are used, the higher orientating precision will be. SPI is closely related to the number of frequencies used. At present, some satellites of BDS are broadcasting a fourth frequency centered at 2492.028 MHz. If use all these four frequency, SPI value will reach 2798193 which is nearly two times higher than using three frequency. This will theoretically lead to a higher orientating precision.

5. Conclusion

In this work, we conducted a systematic investigation on optimal narrow lane combination and presented the optimal condition and two precision indicators. The general multi-frequency linear combination models are used, where the noise factor and lane number are defined. Then SPI and SPI ${}_{\text{rel}}$ were derived to evaluate positioning and orientating precision. We analytically demonstrate that they have a closely linear relationship with solution precision. Based on this relation, the dual-frequency and triple-frequency condition of optimal narrow lane combination were deduced. It can be written as: $a_{i}/a_{1}=k_{i}/k_{1}$ , where $a_{i}$ denotes the combination parameter and $k_{i}$ is a constant defined by frequency. And the condition will also be suitable in four frequency case or in even more frequency case. Those theoretical derivations were verified with real BDS data. The results showed that linear combinations which basically satisfy the optimal condition can lead to a higher precision. And the more frequencies are used, the higher precision will be achieved.

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