Jones vector representation for the polarization state of characteristic cases.
1. Introduction
GNSS signals may arrive at the receiving antenna not only through the direct path, i.e. the lineofsight (LOS) path, but also on multiple indirect paths, due to different electromagnetic effects as signal reflection or diffraction. These signal components arrive with a certain delay, phase, and amplitude difference relative to the LOS component. We will call these signal components multipath components (MPCs) and the phenomena multipath propagation.
Multipath propagation degrades the positioning accuracy. Moreover, in precise applications, multipath errors dominate the total error budget. Despite the different approaches developed, several aspects of multipath propagation are still not fully understood. The generally unknown number of MPCs and their path geometry, the signal characteristics, the diffraction and reflection effects as well as their changing nature together with a complex antenna and receiver design make multipath mitigation very challenging. Furthermore, the sitedependent characteristics of multipath decorrelate the errors caused by multipath propagation at different antenna locations and thus, differential techniques, like e.g. double differences (DD), cannot mitigate it.
The superposition of the MPCs and the LOS signals yields a compound signal at the receiving antenna. Depending on the relative phase between the MPCs and the LOS signal, constructive or destructive interference appears. As a result, during signal tracking the correlation output between the received signal and the local pseudorandom noise (PRN) code replica generated by the receiver is deformed. Since MPCs arrive generally at the receiving antenna with small extra paths, up to 20 m, relative to the LOS signal, the correlation output is biased and the receiver is not able to discriminate between MPC and the LOS signals. This correlation output is the fundamental input for the next iteration of the code and phase tracking loops of the receiver as well as for C/N_{0} estimation algorithms. As a result, the three GNSS observables codephase, carrierphase, and C/N_{0} are biased by multipath propagation. In this text, errors in codephase and carrierphase observations caused by multipath propagation are referred to as code multipath and carrierphase multipath, respectively, and in general as multipath errors.
In the observation domain, multipath errors are not constant in time. They show a sinusoidal behavior which can be noticed in carrierphase residuals from Precise Point Positioning (PPP), double differences (DD) or C/N_{0} time series. This behavior is due to the change of the relative phase between the direct and indirect signals as the satellite vehicle moves above the local horizon of the antenna. The magnitude of these oscillations depends on the relative amplitude of the MPC which varies as geometry changes. The C/N_{0} observable is the only GNSS observation type in which multipath propagation effects are directly visible without any sophisticated data preprocessing. In contrast, in the phase or code domain, residuals should be analyzed or differences should be formed in order to eliminate all other errors sources. This is one of the main reasons why signal strength measures have attracted much attention in GNSS multipath studies. Since the relative signal amplitude between the LOS and MPC signals plays a key role for the understanding of multipath propagation and also for the magnitude of the multipath error in the GNSS observables, the following contribution focuses on an extended description and proposes an analytical model for modeling GNSS signal amplitudes.
This chapter is structured as follows. A compact overview on different approaches for multipath mitigation or characterization will be presented next. The approaches are categorized into techniques in the observation domain, receiverinternal as well as antennarelated techniques and further methods. Cornerstone methods of each category will be highlighted. In the third section, the multipath phenomenon and its impact on GNSS code, carrier phase and C/N_{0} will be summarized. Special emphasize is given to the reflection process including signal polarization. An analytical model for GNSS signal amplitude is proposed. The equations for phase and code errors due to multipath propagation are extended so that the signal amplitude can be analytically calculated for each epoch. Finally, results from a dedicated experiment are shown in order to highlight the key features of multipath propagation.
2. Overview of multipath related studies in GNSS
In the beginning of the 1970s multipath effects on L1 frequency were first studied by [1] and the fundamental relationships between code error due to multipath and the driving parameters were derived. In [2], it was shown that the presence of multipath can be identified by using double differenced phase observations. Since then and during the last 4 decades many researchers have been involved in the characterization and modeling of this propagation phenomenon. A large number of scientific papers have been published on this topic, where different approaches and aspects of the problem have been investigated under certain predefined assumptions. Consequently, the scientific literature on multipath propagation for GNSS positioning can appear very rich for scientists. Despite the large number of different approaches developed, a universal solution of this problem is not achieved until the time of writing. Nevertheless, different promising approaches are under consideration and development.
In almost all textbooks on geodesy or navigation, with very few exceptions (e.g. [3]), multipath propagation is presented in short texts of a couple of pages (e.g. [4, 5, 6]). Most of the time, the phenomenon is explained geometrically, while other physical or electromagnetic properties of the reflected signals are not discussed. The progress in multipathrelated studies is documented in various PhD thesis, we cite here exemplarily [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].
A prominent example for the successful reduction of
One of the most popular methods to characterize and quantify code multipath, are the socalled multipath linear combinations [20]. The original code and carrierphase observations from dual frequency receivers are combined in such a way, that the code multipath can be isolated. Due to its computational simplicity this approach is often used to assess the overall multipath contamination at continuously operating reference stations (CORS), for example, the IGS network, or to characterize the performance of new receivers or new satellite signals, like e.g., the upcoming Galileo or GPS L5. However, it should be noted that the characterization is only valid if no code smoothing was applied.
Contrary to the code observations, the
In postprocessing, sidereal filtering or sidereal differencing is often applied. Taking advantage of the sidereal repetition of the GPS orbits, observation or coordinate time series of subsequent days can be subtracted in order to reduce the impact of multipath. Genrich and Bock in [22] showed that a reduction of about 80% can be obtained in this way. However, strictly speaking, each GPS satellite has its own, timevarying orbital period differing up to 10 sec with respect to the nominal sidereal period. Different approaches have been proposed to find the correct individual repeat times cf. [23, 24]. In addition, due to the nodal precession, the satellite ground track deviates over time, so that different reflectors will be illuminated, cf. for exemplary ground track variations [21, 16, 24]. Finally, changing weather conditions like rain or snow will influence the reflection properties of the antenna vicinity, so that the similarity of multipath errors is reduced. All three effects restrict the power of multipath mitigation by sidereal differencing, especially over longer time intervals.
Finally, the analysis of undifferenced carrierphase residuals from PPP is a useful tool for accessing the impact of multipath effects [25]. For this purpose, the residuals are colorcoded and depicted in a skyplot. The variations of the residuals translate into a sequence of concentric rings in the skyplot. But, since PPP residuals contain further remaining systematic effects, like e.g., varying tropospheric refraction, averaging strategies may be necessary [26].
Observations of the
A lot of effort was given on the
Closelyspaced antennas [12, 51] are also developed for the reduction of multipath errors. They form a type of small antenna arrays. Based on a leastsquares adjustment, the multipath relative amplitude α, the multipath relative phase (ΔΦ) as well as azimuth and elevation of the assumed reflector can be estimated [9]. Further developments leads to beam steering and adaptive beam forming antennas or antennareceiver combination, like e.g., DLR’s GALANT receiver [52]. However, this interesting technology seems to be not mature enough to be installed at GNSS reference stations.
In [53] the concept of
In order to get a better understanding of the physical processes involved in the multipath phenomenon, models from
In recent years, the concept of GNSS reflectometry and scatterometry has strongly evolved, cf. e.g., [63, 64, 65, 66, 67, 68, 69]. Here multipath propagation is not considered as bias but as basic information. In the context of GNSSR, much progress has been made, especially in the mathematical and physical modeling of the wave propagation. Hence, these publications can be a valuable source for mitigating multipath in positioning.
3. Multipath characterization
3.1. Basic considerations
After reaching the receiving antenna, the GNSS signals are downconverted from radiofrequency (RF) to an intermediate frequency (IF) by the RF frontend of the receiver. Afterwards, the signal is digitized, downconverted to baseband and correlated with locallygenerated replicas of the PRN codes. Then, the result of the correlation is accumulated for a certain time interval. In a typical GNSS receiver architecture, three replicas for each PRN code are generated, the socalled prompt, early and late replicas. The resulting correlation outputs of the prompt replica of the PRN code in the presence of a number of k MPCs are written as in [34, 17]:
where A is the amplitude of the direct signal, D is the navigation bit, R is the correlation function, τ is the code tracking error and
Based on the correlation outputs (I_{P} and Q_{P}) of the prompt correlator, a GNSS receiver is able to calculate the amplitude (
Assuming a perfect tracking (i.e. R=1) and the relationship
The compound signal amplitude can be derived as a function of α, ΔΦ and
By using trigonometric identities, the expression for the compound signal amplitude is rearranged as:
An equation for pseudorange error (ρ) is given in [13]:
with δ being the extra path length of the MPC expressed in meters.
From equations (3), (4) and (5) the 90° phase shift between errors in code, amplitude domains and phase domain can be noticed. When ΔΦ=0°/180° and ψ=0, then ρ,
The relative phase of each multipath component can be expressed as a function of the extra path length (δ) and the wavelength of the carrier signal: ΔΦ=2πδ/λ, where δ is expressed in meters. Moreover, δ is a function of station height and reflection angle, which in turns depends on the satellite elevation and the orientation of the reflector in space. In the special case of ground multipath, which we will investigate in the following parts of this chapter, the extra path delay is a function of the station height (h) and the satellite elevation (el):
In conclusion, it can be stated that some of the introduced equations can be calculated based on the geometry of the scenario. The signal amplitudes for both LOS and MPC are not directly accessible. However, this information or at least the relative amplitude is a crucial part of all introduced formulas. Therefore, in the following an analytical model of the relative amplitude α is developed.
3.2. Polarization state of the GNSS Signals
In order to understand the amplitude relation between the LOS and the MPCs, in a first step, the polarization state of a signal is introduced and linked to the righthand circular polarization (RHCP) of the GNSS signal. According to [70], there are many ways to represent the polarization state of a signal or of an antenna. Some are graphic in nature (e.g. Poincaré sphere) and can be easily visualized, while others can be more appropriate for specific applications, like e.g., investigation of antenna wave interaction (e.g. Stokes parameters and/or complex vector representation).
In the case of a circularly polarized (CP) planar wave, the electric field vector is propagating in a helical way (see Figure 2a), where the projection of the tip of this vector forms a circle in a fixed plane normal to the direction of propagation (z). The clockwise or counter clockwise sense of rotation of the electric field vector looking towards the direction of propagation defines whether the signal is left or righthand circularly polarized, respectively. At each instant in time, the electric field vector is a combination of two components (see Figure 2 b, c):
where E_{1} and E_{2} are the amplitudes of the instantaneous electric field [V/m] in x and y directions, respectively, ω is the angular frequency in [radians/sec] and ζ is the relative phase shift [radians] by which the y component leads the x component.
Each of the two components represents a linearly polarized wave. When
Another decomposition of the electric field vector yields orthogonal circular polarization states (LHCP and RHCP). Furthermore, the electric field phasor expressed in terms of CP phasors can be written as [70]:
where
Finally, the Jones vector is a further way for the representation of signal polarization. According to [72], the Jones vector can be used only for completely polarized signals, e.g. GNSS signals. Equation (10) can be rewritten in complex vector notation as:
where the relative phase shift between the two electric field components is ζ=ζ_{x} – ζ_{y} (see Figure 3). According to [72], this can be further factorized as:
which is the socalled Jones vector (
where A^{2}+B^{2}=1 and the
Eq. (13) can be written in a column vector form as:
Eq. (14) completely describes the polarization state of GNSS signals since information of the amplitude of both electric field components as well as their relative phase can be extracted. Particular cases for the Jones vector can be seen in Table 1:










3.3. Reflection process
In multipath propagation, one or multiple reflections as well as diffraction of the transmitted signal may occur. In the following, we focus on the description of the reflection process. In the case of reflection of the incident field, the reflection coefficients will indicate how much the reflected field will be attenuated and how the polarization state of the incident field will be deformed. The reflection coefficients used in this investigation are as in [7]:
With
where




3  2*10^{5} 

20  4 

30  2*10^{1} 
The reflection coefficients can also be expressed as circular components of the two orthogonal polarizations (
For the RHCP satellite signals, R_{co} can be associated with the RHCP component, while R_{cross} represents the orthogonal LHCP component.
In Figure 3a), the circular reflection coefficients are plotted for the L1 carrier. The copolarized components are plotted with solid lines and the crosspolarized with dashed. The material properties of the exemplary materials (see Table 2) were used for the calculation of the reflection coefficients. The reflection coefficients for a concrete reflector are plotted in black, for a sea water reflector in blue, and for a wet ground reflector in red. They indicate how the electrical field components of the incident field will be changed upon reflection, in terms of magnitude and relative phase.
Looking at reflection coefficients for the different reflection angles in Figure 3a) and for a concrete reflector, it can be seen that when the reflection angle approaches 0°, then the crosspolarized components (LHCP component) tends to 0 and the copolarized components (RHCP component) is 1, i.e. no loss. When the reflection angle is between 0° and 30°, both components exist but still the copolarized one is larger in magnitude and the reflected signal is righthand elliptically polarized (RHEP), with the eccentricity of the polarization ellipse getting bigger. At around 30°, the magnitude of the reflection coefficients is equal. In geodetic literature, this angle is referred to as the Brewster angle [7, 9] for a concrete reflector, although this definition deviates from the common use of the Brewster angle in electromagnetic theory. For these material properties (concrete) and for reflection angle of 30°, the reflected signal is linear polarized. When the reflection angle is between 30° and 90°, the cross polarized component has a higher magnitude and this results in a change of the initial polarization from RH to LH. The eccentricity of the ellipse is then getting smaller as the reflection angle increases. When the reflection angle approaches 90°, the reflected signal is LHCP since the copolarized components is zero. When the material properties change, then the changes on the reflected field are different. For example, the angle for which the magnitude of the circular reflection coefficients is the same, may vary significantly for different reflectors. Furthermore, small magnitude changes between the two different frequencies (L1 and L2) may occur. These differences may also vary between different reflectors.
In Figure 3b), the relative phase (ζ) between the field components is illustrated for L1 for the three exemplary material properties. The relative phase between the two components is plotted for all possible reflection angles of the incident field. Without reflection (reflection angle = 0°), the relative phase (ζ) is 90°, which is the relative phase shift for RHCP signals. As the reflection angle increases, ζ is getting smaller and smaller. When the reflection angle is equal to the Brewster angle (for this particular reflector), then ζ=0°, which yields linear polarization. Finally, for reflection angles larger that the Brewster angle, ζ has an opposite sign which indicates that the reflected signal has changed polarization.
3.4. Signal amplitude modeling
According to [73, 74], the signal after the receiving antenna can be written as:
where S_{R} is the received signal,
More information on the RHCP gain pattern of the satellite antennas can be found in [75]. The signal ellipticity should not be worst that 1.2 dB for L1 and 3.2 dB for L2, [76].
The matrix
and for the LOS components as (23b)
where A_{LOS} and A_{MPC} account for the freespace loss attenuation, which for this investigation may be considered equal.
where
This expression is divided into three subsequent matrix multiplications. First, the selected base of the
The signal amplitudes for the groundreflected MPC and for the LOS component are calculated as the absolute values of eq. (25) and (26), where the transmitted signal
yielding the final formulas for the phase error, for the code error and for the compound signal amplitude (for the case of one ground MPC):
When multiple multipath components are received, the above equations can be rewritten in a similar way as:
In a similar sense, eq. (1) and (2) can be also rewritten in order to model analytically the relative amplitude factor α which is present also in these equations.
4. Example
4.1. Measurement set up
In order to illustrate the above derived formulas, various experiments were carried out. In the followings we describe one experiment in details. A dedicated experiment was conducted in a controlled environment at the PTB antenna test facility in Braunschweig, i.e. an environment where only one ground reflection would occur (Figure 5). Two antennas were set up with different heights above the reflector. The antennas were spaced about 21.3 m. The observation period lasted for about 7 hours and a cutoff angle of 0° was applied. The data rate was 1 Hz. One pair of AX1202GG LEICA antennas were used together with LEICA GRX1200+GNSS receivers.
4.2. Simulations
The following assumptions were used for the simulations: The receiving antenna gain patterns are taken from a NOV702GG antenna assumed to be similar to the AX1202 (see Figure 5). The normalized gain patterns were assumed symmetrical in azimuth. In this way the Jones vector of the receiving antenna for all possible azimuths and elevation angles of the incoming LOS and MPCs GNSS signals can be computed as in eq. (21). The satellite antenna was assumed here perfect RHCP and modeled as:
Finally, for simplicity the angles ψ_{rec} and ψ_{trans} were assumed to be equal and constant over time. Absolute and individual antennas phase center corrections from IfE Hannover were also taken into account during data processing. Furthermore, it is assumed that the ground reflector is a perfect horizontal reflector. Using eq. (27) and (29), the simulated time series for the phase error and the compound signal amplitude (dB) were computed. Under the assumption the values are independent of the satellite azimuth; one exemplary satellite (PRN 12) is plotted covering the whole elevation range. Two different antenna heights above the reflector were used according to the measuring set up. The heights of the two antennas A1, A2 were 1.358 m and 2.053 m, respectively.
In Figure 6, simulated normalized amplitudes [dB] of the MPC, the direct and compound signals are plotted for PRN12 versus satellite elevation for both antennas of the observed baseline. The direct signal’s amplitude is plotted in red, the amplitude of the compound signal in blue, and in black the difference between the direct signal and the compound one is given, thus the impact of the multipath.
The normalized amplitude of the LOS is a smooth curve mainly determined by the gain pattern of the receiving antenna. The resulting multipath effect (black) shows the typical amplitude variations of few dB. Finally, the compound amplitude shows a superposition of both features. Furthermore, a frequency difference of the oscillations can be noticed between the antennas, depending on the height difference. For the maximum elevation angle of 90° PRN 12 has a minimum attenuation of 0 dB. For the same satellite elevation, the amplitude of the ground MPC is almost 32 dB. The reason is that the reflected MPC, at a reflection angle of about 90° and with angle of arrival near 90° with respect to the antenna horizon, is almost LHCP (see Figure 3a) and it is strongly attenuated due to the LHCP gain pattern of the antenna for such angles of arrival (see Figure 5, right side). Thus, the LOS signal is dominating. At elevation angles near 0°, the MPC is still having most of the signal energy in the RHCP component (see Figure 3) and only a small part of it in the LHCP component. The relative amplitude α of the MPC w.r.t. LOS is about 0.6. So in this case, the MPC reflection loss is minimum, the MPC RHCP component is dominating and the antenna applies a very similar gain for both LOS and ground MPC signal components.
In a next step, the impact of different reflector material properties will be investigated. We chose exemplary material properties of perfect, concrete, sea water and wet ground reflectors (see Table 2). Since the impact will be much stronger at low elevation angles, only a part of the arc of PRN 12 is plotted. Figure 7 shows the resulting curves for the compound signal amplitude for two different antenna heights.
The change in the material properties creates mainly a change in the amplitude of the variations. A perfect reflector creates the strongest oscillation. The concrete reflector is also creating MPC with higher amplitude than water reflector and the wet ground reflector. This is because for small reflection angles the material properties of concrete are causing a smoother change of the signal polarizations (from RH to LH) in contrast to the reflections coefficients of the other two reflectors that are creating a more abrupt change (see Figures 3).
4.3. Comparison of simulations and real data
The C/N_{0} values obtained from the RINEX observation files can be compared with the simulated values. In Figure 8 for two exemplary satellites, PRN 12 and PRN 14, the corresponding time series are compared. Here, only the data for the antenna A1 at a height of 1.358 m above the reflector is considered. The reflection properties are modeled for a concrete reflector (see Table 2). Both curves show that the simulations can explain the main features present in the observations. The scale on both axes is the same. However, some disturbances occur which are due to our simplifications, e.g. neglecting a small tilting of the reflector, the approximated satellite and receiver antenna patterns.
Finally, the carrier phase observations are investigated. To this end, doubledifferences (DD) are formed between the two antennas on the short baseline. Thus, most distancedependent systematics as well as the receiver clock errors will cancel out. Again we use PRN 12 and 14. In Figure 9, the phase DD formed by PRN 12 and 14 of the short baseline are plotted together with doubledifferenced simulated phase error as in eq. (27). The time series of the observed values for an observational period of about 4.5 hours are plotted in blue in Figure 9a). In Figure 9b) the time series is overlaid by the simulated DD phase errors, where the relative amplitude of each multipath component is calculated analytical. In green no reflection loss is considered and in red the material properties of a concrete reflector are used.
5. Conclusion
Multipath propagation is still limiting the accuracy of precise GNSS applications despite the four decades of intensive research in this field. Signal amplitude, both for direct and indirect GNSS signal components that arrive at the receiving antenna with small relative delays, are crucial for the resulting phase error magnitude due to multipath propagation.
After a short introduction and a discussion on the different approaches that can be found in literature, aspects of ray propagation and signal polarization modeling are introduced. Then, an extensive description of the reflection process and its impact on the polarization state of the reflected signals is presented and all involved parameters are characterized. An analytical model for GNSS signal amplitudes is presented after. The model is then integrated into the equations for the phase error and the compound signal amplitude. Simulated and real data from a dedicated experiment are used to highlight the main properties of multipath propagation on the signal amplitude and carrierphase domains.
Acknowledgments
This work has been realized within the BERTA project (50NA1012), funded by the Federal Ministry of Economics and Technology, and based on a resolution by the German Bundestag. The authors thank the BERTA partner Prof. Thomas Kürner (Institut für Nachrichtentechnik, TU Braunschweig) for the fruitful discussions. The authors would like to thank Dr. Thorsten Schrader (PTB Braunschweig) for supporting the experiments at the antenna reference open area test site at PTB.
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