Introduction

GPS has been proven to support a wide variety of applications. However, major problems have been encountered in some applications. This is due to the signal multipath, which is the result of signal reflections from objects in the environment (Braash 1996).

The multipath distorts the signal modulation and the carrier phase, thus degrading the accuracy and precision of absolute and relative positioning. Furthermore, this effect can impede the ambiguity solution or even lead to incorrect solutions. The multipath signals are always delayed compared to the line-of-sight signal because of the longer travel paths caused by the reflection. Multipath effects from long delays have characterized high frequencies; otherwise they are of low frequencies. The largest errors are caused by the low frequency multipath (Ray et al. 1999; Souza 2004).

Multipath has been treated in many different ways. The most practical one is to avoid it by means of appropriate site selection. Others refer to improvements of antenna and receiver hardware design. Several post-reception methods to mitigate multipath have also been proposed. Walker and Kubik (1996) developed a parabolic equation technique to model the GPS signal propagation. However, this technique requires a priori environmental knowledge (such as Digital Terrain Model). Axelrad et al. (1996) proposed a carrier phase multipath mitigation technique that relies on the analysis of the signal-to-noise ratio (SNR) values. But this method requires knowledge of the antenna gain pattern. Semi-parametric models and the penalized least square method were presented to model multipath (Jia et al. 2000). One of the limitations of their technique is that it works well only if the antenna environment remains unchanged, in addition it is not efficient computationally.

Multipath spectral analysis using wavelets appears to be another good approach. Geodetic applications of wavelets theory, such as GPS cycle slip correction, geoid computation and solution of large systems of linear equations are discussed by Collin and Warnant (1995), Keller (1998, 2001), and others. Regarding multipath, Xia (2001) used the wavelet transform to separate the multipath from the carrier phase double differences (DD) observation. Satirapod et al. (2003) have used wavelets to mitigate the multipath of permanent GPS stations. In this paper, the wavelet transform is used to decompose the pseudorange and carrier phase DD signals to separate the high frequencies, which are due to multipath from long delays, and the low frequencies effects, associated with multipath from short delays. After decomposition the wavelet shrinkage is performed by thresholding. The signal is then reconstructed without the high frequency effects. The high frequency multipath errors are estimated using the DD wavelet coefficients. These errors could be analyzed to confirm that low elevation satellites cause the larger multipath errors. The daily multipath repeatability was also verified. Furthermore, the method is very efficient computationally.

Wavelet transform

Wavelets are building block functions and localized in time or space. They are obtained from a single function ψ∈L 2(R), called the mother wavelet, by translations and dilations (Daubechies 1992):

$$ \psi _{{a,b}} (x) = \frac{1} {{{\sqrt {|a|} }}}\psi {\left( {\frac{{x - b}} {a}} \right)};\quad a,b \in R,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a \ne 0, $$
(1)

where a represents the dilation parameter and b the translation parameter. The wavelet transform of a signal f is (Daubechies 1992):

$$ T^{{{\text{wav}}}} (a,b) = {\left\langle {f,\psi _{{^{{a,b}} }} } \right\rangle } = {\int {f(x)\frac{1} {{{\sqrt {|a|} }}}\overline{{\psi {\left( {\frac{{x - b}} {a}} \right)}}} } }. $$
(2)

The signal can be reconstructed from

$$ f = \frac{1} {{C_{\psi } }}{\int\limits_{ - \infty }^\infty {{\int\limits_{ - \infty }^\infty {{\left\langle {f,\psi _{{^{{a,b}} }} } \right\rangle }\psi _{{^{{a,b}} }} \frac{{dadb}} {{a^{2} }}} }} }, $$
(3)

where the constant C ψ is the admissibility condition and is given by:

$$ 0 < C_{\psi } = {\int\limits_{ - \infty }^\infty {\frac{{{\left| {\mathcal{F}{\left\{ \psi \right\}}{\left( \xi \right)}} \right|}^{2} }} {{{\left| \xi \right|}}}d\xi < \infty } }, $$
(4)

where \(\mathcal{F}\left\{ \psi \right\}\) is the Fourier-transformation and ξ is the frequency.

A form that facilitates the discretization and speeds up the algorithm becomes possible with the appropriate choice of mother wavelet. If some scale relationships are part of the mother wavelet properties, then the discretization becomes easier to implement and has lower computational complexity.

The Daubechies wavelets are a good option. They are orthonormal bases of compactly supported wavelets (Daubechies 1992). The Daubechies wavelets are computed with finite impulse response filters h n and they need not be truncated in the implementation. So, the Daubechies wavelets allow perfect reconstruction. In this paper, wavelets with 8, 12 and 20 coefficients are analyzed. They are denoted by DAUB8, DAUB12 and DAUB20, respectively.

Daubechies wavelets are very asymmetric. To obtain symmetric wavelets, the filter h must be symmetric with respect to the center of its support. Symmlets wavelets obtained from Daubechies wavelets, however, are “least asymmetric”. More details about these wavelets are presented in Daubechies (1992) and Mallat (1998). In this paper, the Symmlets with 8, 12, and 20 coefficients are tested. They are denoted by SYM8, SYM12, and SYM20, respectively.

Wavelet Shrinkage

The wavelet shrinkage aims at the reduction or even the removal of the noise in a signal by decreasing, or reducing to zero, the magnitude of the wavelet coefficients. The general procedure of wavelet shrinkage can be described in three steps:

  1. 1.

    Take a wavelet transform (WT) of each DD time series to get the wavelet coefficients;

  2. 2.

    Shrink the empirical wavelet coefficients by thresholding using one of the thresholds given by Eqs. 5, 6 and 7;

  3. 3.

    Invert the thresholded wavelet transform coefficients.

There are many ways of applying thresholding. Basically, the process of thresholding wavelet coefficients can be divided into two steps: the choice of the threshold function T and the choice of a threshold parameter.

First, one chooses a threshold function T. Let d i , with i =1,⋯, n, denote the wavelet coefficients. Three standard choices are available: hard (T Hλ ), soft (T Sλ ) and quantitative (T Qλ ) thresholding:

$$ T_\lambda ^{\text{H}} (d_i ) = \left\{ \begin{aligned} 0, & \quad |d_i | < \lambda \\ d_i , & \quad |d_i | \geq \lambda \\ \end{aligned} \right. $$
(5)
$$ T_\lambda ^{\text{S}} (d_i ) = {\text{sign}}(d_i )(|d_i | - \lambda ),\quad |d_i | < \lambda $$
(6)
$$ T_\lambda ^{\text{Q}} (d_i ) = \left\{ \begin{aligned} 0, & \quad |d_i | < p \\ d_i , & \quad |d_i | \geq p \\ \end{aligned} \right. $$
(7)

where λ is the threshold parameter, and p is the value for which a certain percentage of coefficients d i is eliminated.

The second step involves the choice of λ. The choice of T and λ is one of the most important issues in wavelet shrinkage (Chiann and Morettin 1998). Donoho and Johnstone (1994), Donoho (1995) and Donoho et al. (1995) present several proposals about the choice of λ. In particular, the universal threshold given by

$$ \lambda = \hat \sigma \sqrt {2\log n} $$
(8)

is one of the first proposed. It provides a fast and automatic thresholding. The \(\hat \sigma \) in Eq. 8 is the observation noise level and should be estimated from each DD dataset. In this step, however, previous knowledge of \(\hat \sigma \) for an unknown environment is not available. However, it can be estimated from the first decomposition level (the finest scale) because the empirical wavelet coefficients at that level are, with a few exceptions, essentially pure noise. Donoho and Johnstone (1994) proposed the following estimator:

$$ \hat \sigma = {{{\text{Median}}\left\{ {\left| {d_{J - 1,k} } \right|:0 \leq k < n/2} \right\}} \mathord{\left/ {\vphantom {{{\text{Median}}\left\{ {\left| {d_{J - 1,k} } \right|:0 \leq k < n/2} \right\}} {0,6745}}} \right. \kern-\nulldelimiterspace} {0,6745}} $$
(9)

where J−1 is the finest scale.

In general, the wavelet shrinkage methods have a number of theoretical advantages, such as near optimal mean squared error.

Which threshold and mother wavelet to use?

This choice is a very important step to obtain good results. In order to test to compare results from different thresholds and mother wavelets, GPS data were collected in an open field. Reflector objects were collocated near the receiver antenna to cause multipath effects. The proposed method was applied using the Daubechies and Symmlets wavelets and the hard and soft thresholds. The quantitative threshold was not used in this work. This threshold can not be efficiently used in this application, because the chosen percentage cut (p), presented in Eq. 7 may not be the ideal to remove the high frequency multipath. The user normally has no previous knowledge of the data behavior. Furthermore, the same percentage would be applied to all DDs. However, only the DDs affected by multipath and noise should be altered.

Thus, just the hard (HT) and the soft (ST) thresholds, given by Eqs. 5 and 6, were applied to pseudorange and carrier phase DD observations of several data sets with differing session lengths. The Daubechies and Symmlets wavelets with 8, 12 and 20 coefficients were compared for these tests. Table 1 shows the average residuals (V a) and respective standard deviation (SD) for the experiments carried out without (WWT) and with the proposed method and using the HT. The mother wavelets DAUB8, DAUB12, DAUB20, SYM8, SYM12 and SYM20 are compared. The respective results using the ST are shown in Table 2.

Table 1 Average (V a) and standard deviation (SD) residuals from pseudorange DD using HT (m)
Full size table
Table 2 Average (V a) and standard deviation (SD) residuals from pseudorange DD using ST (m)
Full size table

Tables 1 and 2 show that the ST did not perform well in smoothing the DD observations. The average residuals and the standard deviations using the ST (Table 2) were always worse than those using the HT (Table 1), and even of the WWT solution. As this fact appears in all data sets tested, the HT was chosen to be applied to the DD observations in this work. Regarding the choice of the mother wavelet, it is known that the wavelet will be smoother for larger numbers of coefficients. However, DAUB20 and SYM20 did not present good results. One can verify from Table 1 that the best average residuals and standard deviation were obtained from SYM12, and the second and the third best ones from SYM8 and DAUB8, respectively. With SYM12, there was improvement of up to 42 and 30% on the V a and SD, respectively. Similar results were obtained for all other tests carried out, but these are not presented here.

In order to confirm the above conclusions, the quality of the pseudorange and carrier phase DD observations model, before and after the use of the proposed method, was analyzed by applying the Global Overall Model (GOM) test statistic (Teunissen 1998a), which is a recursive test statistic based on a Chi-Squared distribution. Many problems were detected with the model used. Figure 1 shows the values of this test for each mother wavelet used.

Fig. 1
figure 1

GOM test statistic

Full size image

From Fig. 1 one can verify that the GOM statistic provided the best results with DAUB8, SYM8 and SYM12.

Aiming to verify the reliability of the ambiguities solution, the values of the ratio test statistic (Teunissen 1998b), without and with the proposed method are presented in Fig. 2. This test gives an indication of the likelihood of the second most likely ambiguity vector relative to the first one. The fixed baseline solution is more reliable if the value of this ratio is significantly larger than one.

Fig. 2
figure 2

Ratio test statistic

Full size image

It is known that the larger the value of the ratio the larger the probability that the ambiguities vector has been solved correctly. Therefore, from Fig. 2 it can be verified that the ambiguities solution using SYM12 is slightly more reliable, although quite similar to SYM8 and DAUB8.

The impact of using different wavelets on the positioning accuracy was verified by analyzing the discrepancies between the estimated coordinates using each of the mother wavelets presented and the “true coordinates”. These coordinates were estimated from data collected without reflector objects nearby the station. The discrepancies are shown in Fig. 3.

Fig. 3
figure 3

Coordinate discrepancies

Full size image

From Fig. 3 one can verify again that the best results were obtained using SYM12. The worse results were obtained using DAUB20.

Thus, based on the results presented so far, the SYM12 mother wavelet and the HT (threshold) were used in this work.

Experiment and analysis

In order to test the wavelet shrinkage method to reduce the high frequency multipath, an experiment was conducted at Presidente Prudente, Brazil on September 12 and 15, 2003. A Trimble 4600 LS receiver was placed in an open field, at a distance of about 90 m from three buildings (Fig. 4).

Fig. 4
figure 4

Test environment

Full size image

The data was collected at a sample rate of 15 s and a cutoff elevation angle of 5°. The permanent GPS station (UEPP) was used as a reference station, where there is a Trimble 4000 SSI receiver with a choke ring antenna centered in a concrete pillar of 3 m height. This station can be considered a multipath-free site. Since the baseline length is about 800 m, errors resulting from ionosphere, troposphere and orbits are assumed to be insignificant. Therefore, the DD carrier phase and pseudorange estimated residuals may exhibit, mainly, multipath and observation noise.

The pseudorange and carrier phase DDs were processed using GPSeq software (Machado and Monico 2002), which is under development at FCT/UNESP, Presidente Prudente.

To visualize the results, the PRN 10 was chosen because it has at the lowest elevation angle (19 – 32°) during the experiment period, and, therefore, most likely to be affected by multipath. For comparison PRN 05 was used because of its highest elevation angle (61– 69°) and an azimuth less favorable to suffer reflections from the buildings. The base satellite was PRN 17 (elevation angle 60 – 88°). The estimated pseudorange and carrier phase DD residuals of satellite pairs 17-10 and 17-05 are plotted in Figs. 5a, b and 6a, b, respectively, for the 2 days of experiment.

Fig. 5a, b
figure 5

Pseudorange DD residuals. a Satellite pair 17-10. b Satellite pair 17-05

Full size image
Fig. 6a, b
figure 6

Carrier phase DD residuals. a Satellite pair 17-10. b Satellite pair 17-05

Full size image

As expected, the DD residuals from the satellite pair 17-05 (Figs. 5b and 6b) were smaller than the satellite pair 17-10 (Figs. 5a and 6a). Daily pseudorange and carrier phase multipath repeatability can also be confirmed from Figs. 5 and 6, respectively. Thus, as expected, it can be seen that multipath errors are similar after a sidereal day.

SYM12 was applied to the DD signals. The noise level is computed in the first level of the decomposition (finest scale) followed by the HT, in order to separate the high frequency multipath components. The next step is to perform the reconstruction. Thus, the estimated pseudorange and carrier phase high frequency multipath from the DD can be seen in Figs. 7a, b and 8a, b, respectively.

Fig. 7a, b
figure 7

Pseudorange multipath. a Satellite pair 17-10. b Satellite pair 17-05

Full size image
Fig. 8a, b
figure 8

Carrier phase multipath. a Satellite pair 17-10. b Satellite pair 17-05

Full size image

As expected, Figs. 7 and 8 show that the multipath errors are larger for the DD relative to satellite pair 17-10 than for 17-05. It is necessary to emphasize that these estimated errors also contain high frequency receiver noise, in addition to multipath.

After the DD observables have been corrected from the high frequency multipath, they are used in the baseline processing software to compute new results. The estimated residuals of this processing are compared with the original ones. A summary of the results is shown in Tables 3 and 4 for pseudorange and carrier phase, respectively.

Table 3 Comparison between pseudorange average (V a) and standard deviation (SD) residuals with and without multipath correction
Full size table
Table 4 Comparison between carrier phase average (V a) and standard deviation (SD) residuals with and without multipath correction
Full size table

From Tables 3 and 4 one can verify the effectiveness of the method for reducing the high frequency multipath in the pseudorange and carrier phase DD observations. In the former one, the average residual reduction was of up to 30% (Table 3), while for the last one it reached 24% (Table 4). The improvements for the carrier phase are smaller, because such errors are also of smaller magnitude.

The following analyses are just based on the results from the September 12, 2003, experiments. However, the results from the September 15 experiment are quite similar, but are not presented here.

To compare the quality of the pseudorange and carrier phase DD observations model, before and after the high frequency multipath correction, the GOM test statistic (Teunissen 1998a) was used (Fig. 9).

Fig. 9
figure 9

GOM test statistic

Full size image

From Fig. 9, it can be verified that GOM statistic improved after the corrections of high frequency multipath, showing that this systematic effect was reduced.

As multipath can lead to incorrect ambiguity resolution and the fixing of ambiguities to wrong values, it can have major consequences on the baseline estimation. Aiming to verify the reliability of the ambiguities solution, the value of the ratio test statistic (Teunissen 1998b), without and with the high frequency multipath corrections is presented in Fig. 10. The ambiguities solution after the multipath correction is more reliable.

Fig. 10
figure 10

Ratio test statistic

Full size image

Regarding processing time, this method proved to be very efficient computationally. This is due to the fact that the wavelets allow the implementation through very fast algorithms. The extra computational time required for applying the method using wavelets is 0.55 s, which is not significantly larger than the 1.50 s required for conventional processing. These values apply to a session of 250 epochs (more than 1 h of data).

Conclusions

Applying the wavelet shrinkage method to reduce the high frequency multipath in GPS relative positioning is practical and comes at low costs of development because it is accomplished during data processing.

It was verified that SYM12 presented the best performance for reconstruction of the GPS DD signal, compared to SYM8, SYM12, DAUB8, DAUB12 and DAUB20. Using SYM12, smaller errors and residuals were obtained.

One can also conclude that the HT is more suitable for GPS applications. In several other applications, the ST presents better results, and is usually the threshold selected by the user community in general.

Furthermore, the high frequency multipath causes the smallest errors. However, just correcting for this effect improves the results for the ambiguities solution, the average residuals (up to 30%), and standard deviations (up to 28%).

Another advantage of this method is its computational efficiency. At a practical level, there was no noticeable difference in processing time with and without the application of the proposed method.