Introduction

GNSS are an efficient and reliable tool for accurate positioning and requires a rigorous characterization of different error effects. Despite the considerable improvements that have been achieved in modeling satellite-specific and atmospheric effects, a generally valid multipath model is still lacking. Therefore, site-specific multipath effects still represent a major error source in mm- to cm-level positioning and distort the original GNSS signal through interference. Diffuse multipath, where the reflected signal is scattered in multiple directions, exhibits noise-like behavior and is considered to be temporally uncorrelated (Braasch 1996, p. 561). In contrast, specular multipath causes systematic errors of up to a quarter of a cycle (Hofmann-Wellenhof et al. 2008, p. 157). Depending on the antenna-reflector distance, multipath effects are classified; while far-field effects show short-periodic properties (up to half an hour; Seeber 2003, p. 317) and can be averaged out over a long observation period, the near-field effects possess nonzero mean and long-periodic characteristics of up to several hours (Dilßner et al. 2008) and can be determined by absolute antenna calibration (Wübbena et al. 2011).

A straightforward option for multipath mitigation is to prevent and reduce reflections in the vicinity of the receiver antenna, e.g., site selection, microwave-absorbing materials (Ning et al. 2011). Other mitigation options can be classified into (1) antenna design (Bedford et al. 2009), e.g., polarization-related antenna gain pattern, choke rings, ground planes, antenna arrays, (2) receiver architecture, e.g., correlator techniques (Ray 2006; Lau 2012), and (3) data processing. The third category is preferred for handling multipath errors in carrier phase measurements and includes the approaches summarized in Table 1.

Table 1 Multipath mitigation approaches based on data processing
Full size table

Since multipath effects are strongly site-specific, the analytical modeling is difficult. In contrast, empirical methods such as stacking enable a flexible multipath characterization that copes with site-specific properties. Making use of observation residuals, a sophisticated, cell-related stacking approach should be (1) applicable to all satellite-based navigation systems, (2) capable of detecting both far-field and near-field multipath signals, (3) robust against satellite maneuvers, (4) independent of GNSS equipment and data sampling rates, (5) usable in real-time and post-processing, and (6) suitable for easy assessment of site quality.

Taking these aspects into account, we propose a method for generating statistically robust site-specific multipath stacking (MPS) maps using congruent cells. It focuses on high-precision static GNSS positioning and is applicable to continuously operating GNSS networks. In “Advanced stacking approach”, the proposed stacking approach is presented considering both geometrical and statistical aspects. “Study area and GNSS data analysis” describes the area of a case study and the GNSS data analysis carried out in precise point positioning (PPP) mode. In “Results and discussion”, based on representative PPP residuals, the advantages of the suggested multipath characterization are demonstrated in comparison with conventional approaches employing a fixed azimuthal resolution. Finally, “Conclusions and outlook” provides concluding remarks and an outlook on future research.

Advanced stacking approach

The main objective of this study was to develop a sophisticated approach for generating MPS maps to efficiently assess and mitigate the limiting site-specific effects in static GNSS positioning. Apart from achieving multipath reduction, the resulting MPS maps can also be used to correct the remaining errors in receiver antenna models. Since the various GNSS have different orbit repeat times, e.g., GPS: approx. 1 sidereal day, GLONASS: approx. 8 sidereal days, Galileo: approx. 10 sidereal days (Eissfeller et al. 2007), the spatial stacking variant is chosen to ensure applicability to all GNSS. If multipath errors are not considered during the course of GNSS data analysis, they remain in residuals and bias parameter estimates, e.g., site coordinates. The advanced stacking process makes use of carrier phase residuals from PPP (Zumberge et al. 1997) and enables—in contrast to differential GNSS—an individual site-satellite multipath detection, which is easily interpretable with respect to signal direction. Being superior to conventional methods using stacking cells with a fixed azimuthal resolution, the proposed MPS method applies congruent cells, along with rigorous statistical assessments. Assuming that all remaining effects, e.g., atmosphere, vary randomly over time the proposed stacking technique is described considering both geometrical and statistical aspects.

Geometrical aspects

Most common approaches to spatially mapping and stacking of site-specific multipath effects employ a constant resolution in azimuth and elevation direction, e.g., 5° × 2° (Wanninger and May 2000; Lidberg et al. 2009) or 1° × 1° (Iwabuchi et al. 2004; Bilich and Larson 2007; Bender et al. 2008). The main disadvantage of specifying a fixed azimuthal resolution is that the arc of the horizontal circle decreases with increasing elevation angle, schematically illustrated in Fig. 1, i.e., b 0 > b 1 > b 2 for dA 0 = dA 1 = dA 2. This leads to a decreasing size of the stacking cell as the elevation angle increases. Accordingly, fewer residuals are included in a stacking element at higher elevation angles. Considering that GNSS signals from high-elevation satellites may be distorted significantly by near-field multipath effects, appropriate cell sizes are required to obtain reliable stacking results over the entire antenna hemisphere.

Fig. 1
figure 1

Variation of azimuthal increments b i with constant azimuth angles depending on elevation angles E i

Full size image

In this study, MPS maps are generated using almost congruent cells, which have a similar shape and size. Since the increment in elevation direction is kept constant, the congruence of stacking cells is realized by means of a variable azimuthal resolution. As Fig. 1 shows, the arc of the horizontal circle b i can be expressed depending on the elevation angle E i as

$$b_{i} = dA_{i} \cdot r_{i} = dA \cdot r_{0} \cdot \,\cos E_{i}$$
(1)

where dA i is the azimuthal increment in degrees and r 0 is the radius of the maximum horizontal circle. For a constant azimuthal resolution, i.e., dA i  = dA 0, b i decreases with increasing E i , since the cosine function is monotonically decreasing in the interval [0°, 90°]. In order to satisfy the congruence condition, namely b i  = b 0, the variable azimuthal increment dA i and the associated integer number of cells C i can be derived as

$$dA_{i} = \frac{{dA_{0} }}{{\cos E_{i} }}\,{\text{and}}\,C_{i} = {\text{ROUND}}\left( {\frac{360^\circ }{{dA_{i} }}} \right)$$
(2)

respectively, where ROUND(x) denotes the rounding of x to the nearest integer. For representative elevation angles of the cell center E i , Table 2 compares the azimuthal cell numbers C i and arc lengths b i determined using the conventional and proposed approaches, where dA 0 = 1°, r 0 = 180/π, and the increment in elevation direction is 1°. In contrast to the proposed MPS approach, using a fixed azimuthal resolution of 1°, each horizontal circle has the same number of stacking elements, i.e., C i  = 360. However, as E i increases from 0.5° to 80.5°, the arc length of the azimuthal increment b i significantly decreases from 1 to 0.17 rad. This decrease is particularly observable for E i  > 50°. Such an effect is usually neglected in the conventional multipath analysis by assuming that site-specific errors are dominant at low elevation angles of up to 50° (Wanninger and May 2000). Making use of a variable azimuthal resolution, the proposed solution produces different cell numbers C i ranging between 59 and 360 for E i between 80.5° and 0.5°. Despite increasing elevation angles, the resulting arc lengths b i exhibit similar values, producing almost congruent cells for accurately characterizing site-specific effects at both low and high elevation angles.

Table 2 Comparison of the azimuthal cell numbers C i and arc lengths b i based on fixed and congruent cell solutions
Full size table

Statistical aspects

The site-specific MPS map is generated by computing the arithmetic mean of the PPP residuals being located in a certain azimuth-elevation interval. Since the cell mean value depends on the sample data, e.g., sample size, and is sensitive to outliers, statistical assessments are performed during the stacking procedure to derive representative and reliable cell means. This marks a significant improvement in the proposed approach in comparison with previous attempts. Under the assumption that the residuals {x 1, x 2,…, x n } are independent and normally distributed as \(N (\mu ,\,\sigma^{2} )\) then the expected value of the mean \(\bar{X}\) is \(E(\bar{X}) = \mu\), the SD is

$$\sigma_{{\bar{X}}} = \sigma /\sqrt n$$
(3)

Applying the standardization to the mean \(\bar{X}\), it follows according to Snedecor and Cochran (1980, p. 45) that

$$Y{ \sim } = \frac{{\bar{X} - \mu }}{\sigma /\sqrt n } \sim N\left( {0,1} \right)$$
(4)

As a result, the (1 − α) confidence interval for μ can be written as

$$\left[ {\bar{X} - \underbrace {{z_{1 - \alpha /2} \cdot \sigma /\sqrt n }}_{e},\quad \bar{X} + \underbrace {{z_{1 - \alpha /2} \cdot \sigma /\sqrt n }}_{e}} \right]$$
(5)

where z q denotes the q-quantile of the standard normal distribution, e is referred to as the absolute error, and α is the significance level corresponding to probability of committing a Type I error. For a given e in (5), one can compute the minimum number of residuals in one stacking cell as

$$n_{\hbox{min} } = {\text{ROUND}}\left( {\frac{{z_{1 - \alpha /2}^{2} \cdot \sigma^{2} }}{{e^{2} }}} \right)$$
(6)

Taking α = 5 % and σ = 2 e as an example, n min is equal to 15. It should be noted that for multipath mitigation using cell-based stacking methods, the choice of an appropriate n min depends on various factors such as the total number of residuals to be stacked, the cell resolution in azimuth and elevation direction, as well as the specified significance level and error tolerance. In this context, it can be seen that the geometrical and statistical aspects are correlated with each other.

Prior to conducting the stacking process, residual outliers are handled in a statistically rigorous manner by means of a three-step iterative procedure consisting of identification, verification, and elimination. Assuming that GNSS observations are normally distributed, outlying residuals in a single cell are first identified using the well-known 3-sigma rule. More precisely, the residual x i I from cell I is considered as an outlier if

$$|x_{I}^{i} - \bar{x}_{I} | \, > 3 \cdot s_{I}$$
(7)

where \(\bar{x}_{I}\) and s I are the corresponding sample mean and SD, respectively. Since the sample variance is more sensitive to outliers than the sample mean, the F-test is used in the verification step to evaluate the influence of the detected outliers on the sample variance (Snedecor and Cochran 1980, p. 98). Thereby, the residual data sets with and without outliers are denoted as x I,1 and x I,2, respectively. Since the sample variance s 2 I,1 is generally larger than s 2 I,2 , the one-sided F-test for equality of variances is carried out based on the test statistic

$$T_{F} = \frac{{s_{I,1}^{2} }}{{s_{I,2}^{2} }}\sim F(f_{I,1} ,f_{I,2} )$$
(8)

with f I,1 = n I,1 − 1 and f I,2 = n I,2 − 1 degrees of freedom, where n I,1 and n I,2 are the sample sizes of x I,1 and x I,2, respectively. If s 2 I,1 is significantly larger than s 2 I,2 , i.e., rejecting the null hypothesis of equal variances, the residual x i I with the greatest distance to the mean \(\bar{x}_{I}\) is removed in the elimination step. Given that there are still enough residuals in cell I (i.e., n I  ≥ n min), this three-step procedure is iterated until all significant outliers are eliminated. The final stacked values are actually the arithmetic means of the outlier-free residuals.

The last component of the statistical assessments is to check whether the cell mean residual \(\bar{x}_{I}\) differs significantly from zero. This is accomplished by calculating the corresponding (1 − α) confidence interval as

$$\left[ { - t_{{n_{I} - 1, \, 1 - \alpha /2}} \cdot s_{I} /\sqrt {n_{I} } , \, t_{{n_{I} - 1, \, 1 - \alpha /2}} \cdot s_{I} /\sqrt {n_{I} } } \right]$$
(9)

where t f,q denotes the q-quantile of Student’s t-distribution with f degrees of freedom. The stacked value \(\bar{x}_{I}\) is significantly different from zero if \(\bar{x}_{I}\) lies outside this interval.

Stacking process

The input data of the stacking process, see Fig. 2, are residuals of the ionosphere-free (L3) phase observations, obtained from a static GPS data analysis in PPP mode. Based on the user-specified resolution in azimuth and elevation direction, the congruent stacking cells are constructed for allocating the residuals according to their azimuth and elevation angles. If the minimum number of residuals given by (6) is fulfilled, the cell sample mean and SD are computed, which are then used for the three-step iterative outlier detection, see (7) and (8). After all significant outliers are removed, the stacked values are determined for each cell by calculating the arithmetic means of the outlier-free residuals. Next, the statistical significance of each stacked value is verified by means of (9), and only those that are significantly different from zero are used to generate site-specific MPS maps. In comparison with conventional spatial stacking approaches that assume a fixed azimuthal resolution, the proposed variant enables an advanced cell construction and a rigorous statistical evaluation.

Fig. 2
figure 2

Flowchart of the proposed stacking approach using congruent cells

Full size image

Study area and GNSS data analysis

To verify the performance of the suggested stacking method, a case study is carried out in the area of the GNSS Upper Rhine Graben Network (GURN; Mayer et al. 2012), located in Central Europe. The current GURN consists of about 80 permanently operating reference sites for high-precision determination of recent crustal movements (Fuhrmann et al. 2012) and atmospheric water vapor fields (Fuhrmann et al. 2010; Alshawaf et al. 2012). The data quality of all GURN sites was previously evaluated (Knöpfler et al. 2010), thereby detecting significant multipath effects at both low and high elevation angles, which has to be handled carefully so that they neither propagate into coordinate estimates nor create velocity biases (King and Watson 2010).

Using the Bernese GPS Software Version 5.0 (Dach et al. 2007), 30 days (DOY2008:276-305; October 2–31, 2008) of 30-s GPS measurements are analyzed in PPP mode. Table 3 lists the most important specifications of the GPS data analysis. In terms of external data and correction models, the precise and up-to-date products from International GNSS Service (IGS) and Center for Orbit Determination in Europe (CODE) are incorporated. The resulting residuals are then used to generate site-specific MPS maps.

Table 3 Specifications of the PPP data analysis
Full size table

Results and discussion

This section first compares the fixed azimuth and congruent cell solutions under geometrical aspects. Next, representative examples of MPS maps produced by the advanced stacking approach are presented. Finally, the MPS maps are applied to residuals to illustrate the differences to stacking with a fixed azimuthal resolution.

Geometrical aspects

In order to obtain statistically reliable stacked values, the minimum number of residuals per cell \(n_{\hbox{min} }\) given by (6) is set to 15. Taking the almost obstacle-free GURN site TÜBINGEN as an representative example, the stacking of 10 consecutive days (DOY2008:276-285) of 239,550 PPP residuals yields 1,051 cells (7.3 %) with a number of residuals less than \( n_{\hbox{min} }\) if a fixed elevation and azimuth resolution of 1° × 2° is used (14,400 cells in total). In this case, a total of 7,658 residuals (3.2 %) are excluded from the stacking process. In contrast, employing the proposed approach using congruent cells, where dA 0 = 2° and the azimuth increment dA i increases with increasing elevation angle, see Fig. 1 and 2, only 324 cells (3.8 %) of 8,525 cells in total do not fulfill the criterion of \(n_{\hbox{min} } = 15\). Accordingly, a total of 2,198 residuals (0.9 %) do not contribute to generating site-specific MPS maps. This actually represents a significant reduction of unused residuals by about 70 %.

Considering the same example, Fig. 3 compares the number of residuals per cell and its sum with respect to the applied stacking approach. For a representative fixed azimuth of A i  = 146° and variable elevation angles E i , Fig. 3, top, clearly illustrates the benefit of using congruent cells at E i  > 35°. Taking all azimuthal sections with \(A_{i} = 0^\circ , \, 2^\circ , \, 4^\circ , \ldots , \, 358^\circ\) into account, Fig. 3, bottom, shows that a fixed azimuthal resolution leads to a strong decrease in the total number of residuals, particularly at higher elevation angles. Applying the congruent cell approach, the sum of number of residuals per cell varies at a relatively constant level for E i  < 75°, and significantly larger amounts of high-elevation residuals are available for stacking. In addition, a statistical testing of residuals as proposed in “Statistical aspects” is more valid using the congruent cell approach, which represents the spatial and statistical distribution of residuals superiorly.

Fig. 3
figure 3

Number of residuals per cell and its sum for an elevation and azimuth resolution of 1° × 2° (site TÜBINGEN, DOY2008:276-285): top panel single azimuthal section with A i  = 146°, bottom panel all azimuthal sections with A i  = 0°, 2°, 4°,…, 358°

Full size image

Multipath stacking maps

The reliability of the stacked values or MPS maps strongly depends on the number of residuals per cell, which is related to the user-specified resolution in azimuth and elevation direction. Generally speaking, for the same database, a higher (lower) resolution results in a smaller (larger) number of residuals allocated to cells. To demonstrate the influence of cell resolution on multipath characterization, Fig. 4 shows the 3D MPS maps generated using different resolutions for the GURN site FRETTE. As can be seen from Fig. 4a, a fine resolution of 1° × 1° (elevation × dA 0; see Fig. 1) allows for the detection of the most significant multipath effects, but leaves a large amount of cells unstacked. In contrast, coarse resolutions, such as 2° × 4° and 4° × 4° depicted in Fig. 4d and e, respectively, provide inaccurate multipath characterization due to the fact that a stacked value is not representative for the whole cell. By comparing Fig. 4b and c with each other, the resolution 1° × 2° seems to be a good compromise between geometrical representation of multipath effects and statistical reliability. This resolution is used below for the validation of the generated MPS maps.

Fig. 4
figure 4

3D visualization of site-specific MPS maps using congruent cells. Site FRETTE, DOY2008:276-283, white cells no stacked values, grey cells “north hole” area in which no satellites are visible. Resolutions: a 1° × 1° (elevation × azimuthal resolution dA 0; see Fig. 1), b 1° × 2°, c 2° × 2°, d 2° × 4°, e  4° × 4°

Full size image

Validation of multipath stacking maps

After subtracting the stacked values (see Fig. 4) from the corresponding residuals, the congruent cell approach for generating MPS maps is first verified in the space and time domains. Taking the GURN site SELESTAT for example, Fig. 5 compares the residuals on DOY2008:278 before and after correcting multipath effects, where a total of approx. 25,000 PPP residuals from 10 consecutive days (DOY2008:276-285) contribute to the associated MPS map. In Fig. 5a, an oscillating pattern is visible at low elevation angles (E < 50°), indicating strong multipath effects (Rost and Wanninger 2009). However, after applying multipath corrections in form of a MPS map, the oscillations are significantly reduced (see Fig. 5, bottom). In addition, the large residual values displayed in red and blue colors in Fig. 5, top, almost disappear in Fig. 5, bottom.

Fig. 5
figure 5

Residuals before and after multipath correction (site SELESTAT, DOY2008:278). Top panel uncorrected residuals, bottom panel corrected residuals

Full size image

The impact of multipath reduction on residuals is also studied in the time domain. For the site SELESTAT and GPS satellite PRN 6, Fig. 6 compares the residual time series on DOY2008:285 before and after employing the MPS map. One can easily discern that significant multipath reduction is achieved at low elevation angles (E < 50°), particularly in the descending part of the satellite arc from 19 to 20 h. Furthermore, noticeable corrections are also obtained at higher elevations, which can be explained by the more sophisticated congruent cell construction proposed in this study. These promising results of multipath mitigation at the residual level motivate the application of MPS maps at the observation level in future research.

Fig. 6
figure 6

Time series of residuals before and after multipath correction (site SELESTAT, DOY2008:285, GPS satellite PRN 6)

Full size image

In addition, the influence of applying MPS maps is statistically analyzed by computing per satellite the arithmetic means and SDs of residuals for each site. As an example, Fig. 7 illustrates the statistical characteristics obtained for the site SELESTAT, where 10 days of PPP residuals are considered. In Fig. 7, top, the mean values of the corrected residuals exhibit smaller variations and are closer to zero than those of the uncorrected residuals. This implies a great potential of MPS maps for reducing biases in GNSS parameter estimates. As can be seen from Fig. 7, bottom, the employment of the MPS map significantly reduces the average SD from about 11 mm to 7 mm, corresponding to a relative enhancement of approx. 35 %. Therefore, less variable residuals can be expected if the multipath correction is performed at the observation level to improve the functional model of GNSS measurements.

Fig. 7
figure 7

Satellite-related statistical characteristics of residuals before and after multipath correction (site SELESTAT, DOY2008:276-285) top panel arithmetic mean, bottom panel SD

Full size image

Considering different cell resolutions, Fig. 8 compares the relative improvement in the SD of residuals, achieved by means of the conventional and advanced stacking approaches. Taking all 61 GURN sites analyzed into account, Fig. 8, top, illustrates a considerable enhancement of up to about 25 % in the SD of the corrected residuals. This benefit is larger than 35 % for the site BINGEN, see Fig. 8, bottom, a station which is strongly affected by multipath effects even at high elevation angles (E > 50°; Knöpfler et al. 2010). Within the framework of the presented case study, the optimal results are achieved using an elevation and azimuth resolution of 0.5° × 1°. The advanced stacking approach seems to be superior to the conventional one if the specified cell resolution is better than 1° × 1°. In addition, the congruent cell solution is particularly applicable when capturing site-specific effects at high elevations. Although within our approach strict statistical assessments are made before a correction is applied onto the residuals of the cells, the reduction in SDs only differs by about 1 % from a solution without statistical assessment, e.g., subtracting cell mean values even if only two residuals are within the cell. On the other hand, the application of statistical conditions offers the possibility to apply the cell mean values on observations of different days enabling a reliable multipath correction at permanent GNSS sites in real-time.

Fig. 8
figure 8

Relative improvement in the SD of residuals before and after multipath correction (DOY2008:276-285) top panel all analyzed 61 GURN sites, bottom panel site BINGEN with strong multipath effects

Full size image

Conclusions and outlook

We presented a new approach for the generation of residual-based MPS maps focusing on continuously operating GNSS sites. The proposed MPS approach is characterized by two major advantages. First, cell mean values are computed using congruent cells. Second, statistical testing and outlier detection are performed for the calculation of reliable and representative stacked mean values for the cells. As MPS is applied to residuals in the space domain, the approach is easily applicable to all GNSS. In addition, the proposed approach is independent of satellite maneuvers and data tracking rates. The use of congruent cells enables the detection and reduction of far-field multipath effects as well as near-field multipath effects from low- and high-elevation satellites with a high spatial resolution. A validation of MPS maps at 61 GPS sites proofed a significant reduction of multipath effects in PPP residuals. A comparison with standard MPS approaches, which make use of a fixed azimuthal cell resolution, shows that the MPS approach presented is able to mitigate site-specific errors more appropriate. After the correction of residuals using cell mean values, the SD of residuals was reduced by 38 % at sites with a strong multipath environment. Within our case study, different cell resolutions were applied to PPP residuals of 10 consecutive days at a sample interval of 30 s. The highest reduction of SDs of residuals is achieved using a cell resolution of 0.5° × 1° (elevation × azimuth at horizontal circle).

The generation of MPS maps is a helpful tool to visualize and interpret multipath signals in the surrounding of a GNSS antenna. In the future, the MPS method developed will be used for the correction of GNSS signals at observation level and for modified observation data evaluation strategies. Because of their statistical robustness, the cell mean values computed from residuals of the past days can be used for real-time correction of observations. As the stacked values comprise all site-specific effects affecting the GNSS antenna, MPS maps also correct for the remaining errors in receiver antenna models. Such corrections of residuals are required, especially if residuals itself are to be exploited, e.g., within the calculation of temporally highly resolved water vapor fields.