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The BeiDou-2 satellite navigation system broadcasts triple frequency data. In this paper, the pseudorange multipath is extracted by using the geometry-free and ionosphere-free combination of one pseudorange and two phase measurements, and the phase multipath is extracted by using triple frequency phase measurements, respectively. By using several days’ static observation data, we exact the noisy pseudorange and phase multipath of three types of satellites, GEO, IGSO and MEO satellites. Because of the low frequency characteristics of the multipath, the low frequency wavelet filter is further used to recover the high-precision low frequency multipath signals that are specified by their amplitudes, periods and phases. The results show that the multipath periods are about 86160s, 86158s and 46391s for GEO, IGSO and MEO satellites, respectively, which coincide with that of the corresponding satellite orbits. Then we use the extracted multipath signals to correct the pseudorange measurements in order to improve the accuracy of point positioning. The positioning accuracy in East-West direction can be significantly improved by using the multipath corrected pseudorange measurements, and in the other two directions the positioning accuracy can also be improved to some extent.

The BeiDou-2 (COMPASS) satellite navigation system will provide regional service in China and its surrounding area in 2012. Since the multipath cannot be eliminated via combined or differential GNSS observations, they are usually suppressed by using choke antenna and other hardware devices, or computed and then corrected to the measurements [1-3]. In recent years, Feng et al. [

Besides receiving signals transmitted by satellites directly, GNSS receiver also receives the indirect signals reflected from the objects nearby receiving antenna at the mean time. The errors produced by overlapping signals are known as multipath.

As shown in

from the surrounding surface features. θ is the angle between reflecting object and indirect signal and D is the distance between reflecting object and receiver antenna. An indirect signal has longer than a direct signal by; thereby these indirect signals will contaminate the direct pseudorange and carrier phase measurements and reduce the positioning accuracy of BeiDou- 2 satellite navigation system [4,5,8].

The observation equations of pseudorange and carrier phase read:

where, P_{i} and L_{i} represent the pseudorange and carrier phase of frequency, respectively, ρ^{s} is the distance between GNSS receiver and satellite s; Δt and Δt^{s} are the receiver and satellite clock errors; δt_{j} and δt_{j}^{s}^{ } denote the frequency-dependent receiver and satellite hardware biases for phase, while dt_{j} and dt_{j}^{s} for code; T and I_{ } represent the troposphere and ionosphere delays; λ_{j} and N_{j}^{s} represent the phase wavelength and ambiguity including the initial phase biases of receiver and satellite, M_{j} and m_{j} represent multipath of pseudorange and phase, respectively, and represent the pseudorange and carrier phase measurement errors respectively. It is emphasized that all terms in (1) are in unit of meter.

To exact the pseudorange multipath, one needs to eliminate all geometric and ionospheric terms. In general, the geometry-free and ionosphere-free combination is formed using one pseudorange and two phases. With loss of generality, let us form the geometry-free and ionospherefree combination by using the ith frequency pseudorange and the jth and kth frequency phases, the combination is as follows:

Here, according to the condition that the combination is both geometry-free and ionosphere-free, the coeffici- ents are solved:

Obviously,. Substituting (1) and (3) into (2) yields

where

It is well-known that the inter-frequency hardware biases are very stable in time for both receiver and satellite. Thereby, can be deemed as constant. Moreover is constant as well if there is no cycle slip happens. Considering the multipath is periodic and can be average out over a period of n epochs. Therefore the constant can be estimated as. Then the estimat-ed pseudorange multipath is

Assuming that the precision of phase is unique for different frequencies, i.e., and also considering the effect of phase multipath (), then the precision the exacted pseudorange multipath is

In this paper, we use the above methods to exact the pseudorange multipath of triple frequency.

The phase multipath cannot be assessed using triple carrier phase measurements [4,7]. We first form two iono- sphere-free combinations from two pairs of carrier frequen-cies, then subtract the results from each other and obtain the geometry-free and ionosphere-free (GF-IF) combination as follows [6,7,10]:

where, DIF denotes the GF-IF combination. M_{12} and M_{13} are multipath combination of B_{1},B_{2} and B_{1},B_{3}. N_{12} and N_{13} are ambiguity combinations of B_{1},B_{2} and B_{1},B_{3}. The combination in (8) are formed by carrier phase measurements, so we should detect cycle slips before subsequent analysis. If cycle slips were found, we mark it and make these data as a segment. In a data segment of no cycle slips, the GF-IF combination mainly consists of receiver measurement errors and multipath on the respective frequencies plus a constant determined by the carrier phase ambiguities. In the case of no cycle slips the ambiguity is a constant and easy to handle [

Experimental data are collected by BeiDou-2 multi-frequency receiver at 3 stations, CCHU, CKUN and CWUQ, the sampling interval is 1 s. In this paper we utilize consecutive day’s measurements from October 11, 2011 to October 13, 2011 and from July 4, 2012 to July 20 to carry out analysis.

Different receiver surroundings lead to the numerical difference of multipath. By processing 24 hours (Oct 11, 2011) measurements of 3 stations, we have extracted the multipath of B_{1}, B_{2} and B_{3} frequency pseudorange and geometry-free and ionosphere-free phase combination for the GEO satellite C01 from each individual station. _{1}, B_{2} and B_{3} frequencies for three stations.

As illustrated in

As illustrated in

BeiDou-2 navigation satellite constellation consists of three different types of satellites: GEO, IGSO and MEO satellites. As we know, the multipath is affected due to the reflection of the environment nearby the receiver. The satellite constellation is periodic such that after a certain period time, the receiver can suffer the basically same observation environment. As a result, the similar multipath can be introduced. To address this issue latter from our exacted pseudorange multipath, we firstly compute the theoretical period of the different types of satellites. The theoretical period is computed based on the mean motion velocity n as, where n is computed as

where is the Earth, a and are the semimajor axis of orbit ellipse and the perturbation of mean velocity, both of them come from broadcast ephemeris [

We have extracted the pseudorange multipath of GEO, IGSO and MEO satellite at CCHU station using the data observed from Oct. 11, 2011 to Oct. 13, 2011 and Jul. 01 to Jul. 03 2012, respectively. The results are shown in

As illustrated in

With an appropriate time shift to align geometric repeatability the constructed multipath correction profile can be applied to correct the measurements for another day [

correlated with satellite orbital period.

Carrying out the fast Fourier transformation (FFT) to the B_{1} frequency multipath time series of C01, C03 and C04 satellites at CCHU station for three consecutive day (Oct. 11, 2011-Oct. 13, 2011), the spectrums of these time series are obtained. The spectrum of C01 is shown in

The IGSO satellites are not tracked over a full day, we cannot exact the multipath series for 24 hours. Therefore, we determine the period of multipath by means of calculating cross correlation of different day’s multipath errors [12,13]. The result of cross correlation is plotted in

The cross correlation peak between the first day and the second day is centered at 242s, the second and the third day is centered at 242s, the first day and the third day is centered at 483s, which is less than the sum of the two adjacent days. With the increasing number of days between the maximum correlation will gradually decrease. So the multipath of IGSO satellites appear a period ahead of time 242s every day, namely the period is about 86158s and it is basically consistent with the orbital period (86164s).

The multipath period of MEO satellites is determined similar to IGSO satellites. BeiDou-2 MEO satellite orbital period is about 12.92 hours [

we can hardly determine the multipath period simply by analyzing the cross correlation of adjacent day’s multipath. Since the MEO satellite will appear at nearly the same time after a week, we compute the cross correlation of MEO multipath using the data of adjacent weeks in order to get a more accurate period. The result of cross correlation is plotted in

From the above statistics in

We will extract the multipath of GF-IF combinations. Since linear combination will amplify measurement errors, the measurement errors of GF-IF combination are as large as multipath. The high-precision phase multipath of GF-IF combinations can be extracted according to their low-frequency characteristics related to the high fre-

quency characteristics of measurement errors by using 7 layer wavelet de-nosing of db8 wavelet.

We can see that the phase multipath are seriously contaminated by measurement errors in

By using the foregoing spectrum analysis methods, we depict GEO satellite (C01) phase multipath spectrum and calculate the period corresponding to the maximum amplitude peak as shown in

By comparing the figures of Figures 7 and 12 and the tables of Tables 1-4, we can conclude that the periods of phase and pseudorange multipath are basically the same. Both statistics show that the repeat time is variable across the constellation, at the few-second level for most satellites so the multipath periods of these three kinds of satellites are basically consistent with the orbital period of each kind satellite.

As illustrated in

of positioning errors at adjacent days are basically the same. This cycle repeatability indicates the existence of multipath. Multipath will be recurring day after day at the same station [

Therefore, at the same stations, under the circumstances of same satellites distribution, the effects of multipath are highly related with the surrounding environments. If we extract multipath and correct to the measurements, the positioning accuracy is looking forwards to be improved.

In order to verify the effect of multipath correction we carried out single point positioning using 4 station measurements including CLIN, CCHU, CSHA and CKUN. Since MEO satellites ephemerides are not available currently, we only use 3 GEO and 3 IGSO satellites constellation to carry out single point positioning. Extracting the pseudo range multipath and carry out positioning before and after multipath correction. The results are listed in

The results of

Based on the combinations of triple frequency measurements, we extract the multipath of pseudorange and phase, respectively. According to the above results and analysis, we can draw the following conclusions:

• The multipath of pseudorange and phase measurements is at the meter level and centimeter level, respectively.

The multipath of GEO satellites presents low-frequency changes. The multipath errors of IGSO, MEO satellites present high-frequency changes relative to GEO satellites.

The periods of multipath of the GEO, IGSO and MEO satellites are about 86160s, 86158s and 46391s, respectively. The periods of multipath are basically consistent with satellite orbit period.

Correcting multipath will result in better positioning accuracy, especially in east-west direction.

This work was mainly sponsored by Natural Science Foundation of China (Projects: 41074018), as well as

partly supported by Kwang-Hua Fund for College of Civil Engineering, Tongji University. Li Bofeng provided useful guidance during data processing and paper writing. Jiao Wenhai checked some of the calculated results. The writers thank Tang Chengpan and the anonymous reviewer for their comments.