Development and testing of a new ray-tracing approach to GNSS carrier-phase multipath modelling

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Development and Testing of a New Ray-Tracing Approach to GNSS Carrier- Phase Multipath Modelling

Article in Journal of Geodesy · November 2007

DOI: 10.1007/s00190-007-0139-z

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DOI 10.1007/s00190-007-0139-z O R I G I NA L A RT I C L E

Development and testing of a new ray-tracing approach to GNSS carrier-phase multipath modelling

Lawrence Lau · Paul Cross

Received: 2 February 2006 / Accepted: 27 January 2007 / Published online: 13 March 2007

Abstract Multipath is one of the most important error sources in Global Navigation Satellite System (GNSS) carrier-phase-based precise relative positioning. Its theoretical maximum is a quarter of the carrier wavelength (about 4.8 cm for the Global Positioning System (GPS) L1 carrier) and, although it rarely reaches this size, it must clearly be mitigated if millimetre-accuracy positioning is to be achieved. In most static applications, this may be accomplished by averaging over a sufficiently long period of observation, but in kinematic applications, a modelling approach must be used. This paper is concerned with one such approach: the use of ray-tracing to reconstruct the error and therefore remove it. In order to apply such an approach, it is necessary to have a detailed understanding of the signal transmitted from the satellite, the reflection process, the antenna characteristics and the way that the reflected and direct signal are processed within the receiver. This paper reviews all of these and introduces a formal ray-tracing method for multipath estimation based on precise knowledge of the satellite–reflector–antenna geometry and of the reflector material and antenna characteristics. It is validated experimentally using GPS signals reflected from metal, water and a brick building, and is shown to be able to model most of the main multipath characteristics.

The method will have important practical applications for correcting for multipath in well-constrained environments (such as at base stations for local area GPS L. Lau (

B

⁾^·^{P. Cross}

Department of Geomatic Engineering, University College London, Gower Street, London, WC1E 6BT, UK

e-mail: [email protected] P. Cross

e-mail: [email protected]

networks, at International GNSS Service (IGS) reference stations, and on spacecraft), and it can be used to simulate realistic multipath errors for various performance analyses in high-precision positioning.

Keywords GNSS·Phase multipath·Multipath simulation·Ray-tracing·Multipath characteristics· Reflectivity

1 Introduction

Relative carrier-phase-based Global Navigation Satel- lite System (GNSS) precise positioning is subject to a number of error sources. This paper is concerned with one of these: carrier-phase multipath, and is especially relevant to applications that require very high accuracy positioning over short and medium distances when many of the other GNSS errors exhibit a very high degree of spatial correlation. In these cases, the errors that remain are primarily site-dependent and include multipath, diffraction and receiver noise; multipath is generally considered to be the most significant. It is also worth mentioning that as models for other error sources improve, multipath is becoming the increasingly limiting factor for many applications (e.g., in network real-time kinematic (RTK) applications) (Kim and Langley 2000).

In static GNSS applications, and in other situations where the satellite–reflector–antenna-geometry changes smoothly, carrier-phase multipath errors typically dis- play sinusoidal characteristics with the theoretical maximum amplitude being of a quarter of the observed wavelength. The size of the multipath error in any particular GNSS carrier-phase measurement depends primarily on four factors: the reflecting environment (including

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material), the satellite–reflector–antenna-geometry, the type of antenna used, and the receiver hardware and firmware.

The reflecting environment is the main driver: highly reflective (to GNSS signals) surfaces lead to strong multipath signals (i.e., large amplitude), and objects close to the antenna cause multipath errors with long wavelengths (conversely, distant objects usually cause multipath errors with short wavelengths). The key geometrical factor is the satellite elevation angle, with most reflected signals coming from nearby structures or the ground for a low elevation angle.

Multipath is also a function of the receiving antenna.

For example, GPS antennas are usually designed to have lower gain for low-elevation incoming signals. Of course, such a design can lead to problems when reflections are caused by objects significantly higher than the antenna and when a particular application requires direct signals from low elevation satellites (e.g., for shipborne applications and tropospheric modelling).

Since the design of the phase lock loops (PLLs) must be able to distinguish clock dynamics from user motion in order to prevent cycle slips, phase multipath at a certain level may always remain undetectable, leading to errors in phase measurement. Of course for a static receiver, multipath can be significantly reduced during data processing by averaging the error over a long enough time (Langley 1998)—but such an option is not available in dynamic applications.

Multipath is therefore likely to remain as a serious error in many GNSS carrier-phase-based positioning applications for the foreseeable future: we cannot usually alter the reflecting environment, we cannot design antennas that fully mitigate multipath whilst still receiving the signals we require, and the design of a receiver to fully mitigate phase multipath appears to be very difficult (e.g.,Stewart 2003). Therefore, multipath miti- gation strategies that can be applied as part of carrier- phase data processing need to be developed.

Comp and Axelrad (1996) andWieser and Brunner (2000) report results using signal-to-noise ratios, on the one hand to predict multipath errors directly and on the other to down-weight measurements with multipath or diffraction components.Bétaille et al. (2006) and Lau and Cross(2005) have further developed the ideas in Comp and Axelrad (1996) and provide some deeper insights into the potential use of signal-to-noise ratios.

Also, there is much other work in this field, e.g., that based on a wavelet analysis of continuous GPS data to identify common signals in data separated by a sidereal day (e.g.,Satirapod and Rizos 2005), and that based on spatial analysis of multipath errors in antenna arrays (e.g.,Ray et al. 2000;Lau and Cross 2006).

In this paper, we report our work based on ray-tracing, i.e., using known satellite–reflector–antenna geometries to estimate the amount of multipath so that GNSS carrier-phase data can be directly corrected for its multipath component. We believe this method has enor- mous potential, especially in highly constrained multipath environments (such as on board spacecraft where the attitude is known or determined by other devices such as two or more GNSS antennas, at RTK base stations and especially at International GNSS Service (IGS) reference stations) and for predicting multipath from very close objects (e.g., antenna-carrying platforms) when using tracking loops that suppress multipath from more distant objects (e.g., Bétaille et al. 2003). More- over, it can be used to simulate realistic multipath errors for various performance analyses in high precision positioning.

In order to support this work, especially for future GNSS signals, it has been necessary to develop a realistic phase data simulator, i.e. a software package that can generate data that has realistic multipath (and other) error characteristics. To do this, a detailed understanding of the whole multipath process (including geometry, the refection process, and the impact of antenna type and receiver tracking loops) is necessary. The main pur- poses of this paper therefore are to review the relevant background theory and to describe the practical imple- mentation and testing (also with real GPS data) of our simulator.

This paper describes the multipath problem in signal transmission order. Section2describes the signal transmitted from satellite antenna to the receiving antenna and reflector. The ways in which the signal changes during reflection and the properties of the reflected signal are discussed in Sect.3. Section4describes the response of GNSS receiving antenna to line-of-sight (LOS) and reflected signals and the resulting multipath errors in phase measurements are described in Sect.5. Details of the multipath simulator are given in Sect.6, and its veri- fication in Sect.7. Section8describes the characteristics of multipath in different scenarios and, finally, the conclusions are given in Sect.9. Note that GPS signals are used here as examples of GNSS signals throughout but the methods are fully applicable to modernised GPS, GLONASS and Galileo.

2 GNSS signal transmission

Signals transmitted from GPS satellites are right-hand circularly polarized (RCP) (Fig.1). The time-phase angle by which E_Y leads E_X of RCP is −90^◦ as shown in Fig.1a. The ratio of the major to the minor axes of the

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Fig. 1 Right-hand circularly polarized wave approaching. a The field vector E along the direction of propagation and its components in E_Xand E_Y. b The rotational direction of the field vector and its amplitudes E₁and E2in X and Y respectively

polarization is called the axial ratio (AR) and for circu- lar polarization E₁=E₂and AR = 1 as shown in Fig.1b, where E₁is the amplitude of wave linearly polarized in the x direction and E₂is the amplitude of wave linearly polarized in the Y direction.

Multipath occurs when an antenna not only receives a LOS signal but also an indirect signal that has been reflected by an object(s). Long distance multipath (i.e., that due to reflectors that lead to a large additional path- length for the reflected signal) can often be filtered out by the receiver correlator, so most multipath errors in high-precision GPS measurements are due to reflectors that are only short distances from the receiving antenna (or more strictly those that only introduce additional path lengths of a few metres) (Bétaille et al. 2003;Fenton and Jones 2005).

It is also relevant to note that the GPS signal transmitted from a satellite (i.e., before arrival at a receiving antenna or a reflector) has a phase wind-up error (Wu et al. 1993) as shown in Fig.2. Phase wind-up error is due to the change of satellite antenna reference orientation

Medium 1: ε₁,µ₁,σ₁,Ζ₁,η₁ ^{Medium 2:}ε2,µ2,σ2,Ζ2,η2

Angle of incidenceθ_i

Angle of reflectionθ_r Angle of refractionθ_t Incident

Reflected Transmitted

RCP

LEP

⎟⎟⎠⎞

⎜⎜⎝⎛

1

, 2

ε θ ε ρ i

Phase wind-up error

Fig. 2 Reflection and transmission between two media

during its orbit, but the magnitudes of phase wind-up errors in the direct and reflected signals are very similar as the distance between receiving antenna and reflector is rather small—so the effect is usually ignored in multipath studies.

3 GNSS reflection signal

When a GPS signal arrives at a reflector, it—or part of it—may be reflected towards the antenna. The power of this reflected signal depends on the Fresnel reflection coefficientρand the relative permittivity (or dielectric constant)ε2/ε1of the media. In the case of GPS multipath, the first medium is air and the second medium is the material of the reflector. The Fresnel reflection coefficient is a function of the incidence angleθiand the relative permittivity of the material, which varies with the signal frequency.

The electric field of an incident signal can be resolved into perpendicular and parallel components; therefore the Fresnel reflection coefficient can also be resolved into perpendicular ρ⊥ and parallel ρ components.

According toKraus and Fleisch (1999), if both media are lossless nonmagnetic dielectrics, the perpendicular Fresnel reflection coefficientρ⊥is

ρ_⊥= cos θi− ε2

ε1−sin² θi

cos θi+ ε2

ε1−sin² θi

(1)

and the parallel Fresnel reflection coefficientρis

ρ= − ε2

ε1

cosθi+ ε2

ε1−sin²θi

ε2 ε1

cosθi+ ε2

ε1−sin²θi

(2)

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Z X Y

Minor axis Major axis

Polarization ellipse

E1

E2

EX

EY

E

Fig. 3 Polarization ellipse at tilt angleτshowing instantaneous components of E_Xand E_Yand amplitudes E₁and E₂and relation of anglesε,τ, andγ

For parallel polarization, it is possible to find an incidence angle θi so that ρ= 0 and the signal is totally transmitted into medium 2. This is the Brewster angle θiBand is determined by:

θiB=tan⁻¹ ε2

ε1

(3) The Brewster angle is also sometimes called the polar- izing angle, since a signal composed of both perpendicular and parallel components and incident at the Brewster angle produces a reflected signal with only a perpendicular component. Thus, a circularly polarized signal incident at the Brewster angle becomes linearly polarized after reflection.

If the incidence angle is not equal to the Brewster angle, then the total reflection coefficientρis (Born and Wolf 1999)

ρ=ρcos²θi+ρ⊥sin²θi (4) If medium 2 is a more dense dielectric than medium 1, i.e.,ε2 > ε1, the quantity under the square root in Eq. (1) will be positive andρ_⊥will be real. Ifε1 > ε2, and if sin²θi ε1/ε2, then ρ_⊥ becomes complex and

|ρ⊥| =1. Under these conditions, total internal reflection occurs that the incident signal is reflected back into the denser medium. This is true for both the parallel and perpendicular components. In this case, the incidence angle is called critical angle. However, this case does not occur in GNSS applications because GNSS signals always travel from less dense air to the denser reflector if they reflect.

When a signal is reflected, its polarization state will be changed. Firstly, definitions of the terms needed to describe polarization state are given. At a fixed value of Z-component (see Fig.1), the electric vector E rotates as a function of time, the tip of the vector describing an ellipse, called the polarization ellipse, as shown in Fig. 3. Figure 3 also shows the relationship between

Fig. 4 Poincaré sphere showing relation of angleε,τ,δ,γ(Kraus and Fleisch 1999)

the different angles describing polarization (so-called parameters of polarization): the tilt angleτ (0^◦ τ 180^◦) is the angle of the major axis from X-axis,εis an auxiliary angle defined in Eq. (5) which characterizes the ellipticity of polarization, andγ =tan⁻¹(E₂/E₁).

The polarization state of a signal can be represented geometrically by a Poincaré sphere (Fig.4). The definition of Poincaré sphere is inKraus and Fleisch(1999) and its derivation is inBorn and Wolf(1999). A Poincaré sphere describes the polarization state as a point on a sphere where the latitude (2ε) and longitude (2τ) (see Fig. 4) of the point are related to parameters of the polarization ellipse by

ε=cot⁻¹(∓AR) (5)

where−45^◦ε+45^◦. The AR and angleεare neg- ative for RHP and positive for left-handed polarization (LHP). The polarization state can also be expressed in terms of the angle subtended by the great circle drawn from a reference point on the equator (2γ) and the angle between the great circle and the equator (δ) (see Fig.4).

The trigonometric relationships between the polarization parametersε,τ,γ, andδare as follows [the deri- vations are based on spherical trigonometry and can be found inBorn and Wolf(1999)]:

cos 2γ =cos 2εcos 2τ (6)

tan δ = tan 2ε

sin 2τ (7)

tan2τ =tan2γcosδ (8)

sin2ε=sin2γsinδ (9)

Using the above definition of polarization parameters, the polarization state of the reflected signal can be determined. For a RCP incident signal (GPS LOS signal),γi=45^◦,δi= −90^◦, the phase angle of reflected

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signalδr by which electric field in parallel component leads the perpendicular component is (Kraus and Fleisch 1999):

δr=δi+π+(φ_⊥−φ) (10)

whereφandφ_⊥are the phase angles of the parallel and perpendicular reflection coefficientsρandρ⊥, respectively. The great circle component of reflected signal, γr, is

γr=tan⁻¹ ρ

ρ_⊥tanγi

(11) Substitutingδrandγrinto Eqs. (8), (9), and (5), enables τr and AR_r of the reflected signal to be determined.

If AR_r > 0 and AR_r = 1, the reflected signal is left- handed ellipse polarized (LEP). A reflected (multipath) GPS signal is always LEP because the incidence angle at the reflector is always less than the Brewster angle.

Similar arguments can be found inBraasch(1996) and Rao et al.(2000).

4 GNSS signal reception

When there is multipath, the receiving antenna receives both LOS and reflected signals. GPS receivers usually use antennas designed to receive RCP signals with a high gain. This is because the received LOS signal is RCP; therefore, the polarization efficiency of the LOS signal at the antennas is high with some polarization mismatch loss (ICD-GPS-200C 2000).

As described in Sect.3, the polarization state of GPS signal will usually be changed after reflection and depends on the incidence angle and the relative permittivity of the reflector. Therefore, the polarization efficiency of a reflected signal at a GPS RCP antenna is always less than 100% since the reflected signal is no longer RCP. The polarization efficiency (sometimes called the polarization factor F) of a reflected signal at a RCP antenna can be determined from (Kraus and Marhefka 2002)

F=cos²MM_a

2 (12)

where MM_adenotes the angle subtended by the great- circle from M to M_a, M is the polarization state of the incident signal, and Mais the polarization state of receiving antenna (see Fig.5).

The polarization efficiency can also be determined by using the axial ratio AR and the tilt angleτ as:

F=(AR1AR2−1)²+(AR1−AR2)²+(AR²₁−1)(AR²₂−1)cos2(τ1+τ2)

2(AR²₁+1)(AR²₂+1) (13)

Fig. 5 The match angle MMabetween the polarization state of wave (M) and receiving antenna (Ma). For MMa=0^◦, the match is perfect. For MMa=180^◦, the match is zero (Kraus and Marhefka 2002)

where AR₁denotes the AR of the incident signal (either LOS or reflected), AR₂denotes the AR of the receiving antenna, andτ1andτ2denote the tilt angles of the incident and reflected signals, respectively. Using Eq. (12) or (13), the polarization efficiency of the reflected signal can be determined no matter whether the reflected signal is LCP or LEP. Note that Eq. (13) is used in the computations described in Sect.6because it is easier to computer-program than Eq. (12).

Additionally, because of the different incidence angles of the LOS and reflected signals at a receiving antenna, they have different antenna gains. How different they are depends on the type of antenna and the two incidence angles (as an example, the antenna gain pattern for the Leica Geosystems AT502 antenna is shown in Fig.6). In order to compute the effect (i.e., multipath phase error) of interference from a particular reflected signal, it is necessary to know its power attenuation relative to the LOS signal at the antenna, but before ana- logue to digital conversion. The antenna gain (attenuation) ratio between the LOS and reflected signalsηacan be determined by:

ηa= gs

g_m (14)

where gs denotes the gain of LOS signal (in dB) and g_mdenotes the gain of reflected signal (in dB). For the

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Fig. 6 Antenna gain pattern of the Leica AT502 antenna (provided by Leica Geosystems)

Leica AT502 antenna, if a LOS signal enters the antenna at an elevation angle of 60^◦, its gain will be−6.5 dB and if a reflected signal enters at an elevation angle of 20^◦, its gain will be−12 dB (Fig.6). Thus, the antenna gain (attenuation) ratioηais 0.542, i.e., the antenna gain of the reflected signal is about 1.85 times smaller than that of the LOS signal.

5 Multipath error in carrier-phase measurement It follows from the discussion in Sects.2,3and4, that the damping factorαof the power of a reflected signal relative to a LOS signal can be determined by:

α=ρFηa (15)

The phase multipath errorψin the PLL discriminator can then be calculated using (Braasch 1996)

ψ=tan⁻¹

αA(τ)sinθm

1+αA(τ)cosθm

(16) where A denotes the PRN code correlation function,τ is the time-delay of the reflected signal relative to the direct signal, and A(τ)is

A(τ)=1−|τ|

T, |τ|T

=0, |τ|>T (17)

where T is the PRN code bit period, or correlator spacing, andθmin Eq. (16) denotes the phase shift due to the extra distance travelled by the multipath signal relative to the LOS signal (Fig.7) plus a phase wind-up error.θm

is given by θm= L_m−L_d

λ +ϕw (18)

where L_m is the length of the multipath signal from the satellite antenna via the reflector to the receiving antenna, L_d is the length of the LOS signal from the satellite antenna to the receiving antenna,λis the wavelength of the signal, andϕwis the phase wind-up error, which is insignificant for reasons explained in Sect. 2 (although it is included in Eq. (18) for completeness).

(L_m−L_d)is the differential path delay of the reflected signal.

We can summarise by stating that the characteristics of multipath error in GNSS carrier-phase measurements depend on the following factors:

(i) relative permittivity of the reflector—Eqs. (1)–

(4), (10), and (11),

(ii) incidence angles—Eqs. (1), (2), (4), and (14), (iii) polarization efficiency state—Eq. (12) or (13), (iv) correlator spacing—Eq. (17),

(v) distance between the receiving antenna and the reflector—Eqs. (17) and (18), and

(vi) wavelength of the carrier—Eq. (18).

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Table 1 Phase centre variation of Leica AT502 antenna

Elevation angle (^◦) 90 85 80 75 70 65 60 55 50 45

L1 phase (mm) 0.0 1.3 2.0 2.2 2.0 1.7 1.4 1.0 0.8 0.8

Elevation angle (^◦) 40 35 30 25 20 15 10 5 0

L1 phase (mm) 1.0 1.4 2.0 2.5 3.3 4.1 5.1 0.0 0.0

2θ_i θi

θd θⁱ

d antenna

image reflector

signal from a satellite

Fig. 7 Multipath: geometry of a signal, reflector, and a receiving antenna

Given antenna phase centre

Phase centre at the elevation angle of direct signal

Phase centre at the elevation angle of reflected signal

Direct signal

Reflected signal

Phase centre variation of reflected signal

Phase centre variation of direct signal Zenith

Fig. 8 Phase centre variations in direct and reflected signal

Note that factors (ii) and (v) are directly related to the satellite antenna, reflector and receiving antenna geometry (Fig. 7). Additionally, multipath errors are affected by the fact that the position of the antenna phase centre varies with different signal incidence angles (Schupler and Clark 2001). As an example, the phase centre variations of the Leica AT502 antenna for the GPS L1 frequency are shown in Table1(from the IGS http://www.igscb.jpl.nasa.gov). As explained in Sect.4, the elevation angles of direct and reflected signals are usually different. Therefore, the phase centre for the direct and reflected signals will usually be different as shown in Fig.8.

Note that the directions of the LOS signal at both roving and reference antennas are very similar for short baseline, and phase centre variations will have no impact on relative positions if the same antennas are used at both stations. The elevation angles of reflected signals are, however, usually not the same at nearby stations and, as a result, errors due to phase centre variations in reflected signals are usually completely absorbed in the form of sinusoidal phase multipath errors. The ray-tracing model takes this into account by using

an antenna phase centre offset and variation table (Sect.6.2).

6 Multipath estimation by ray-tracing

6.1 Geometry of ray-tracing

The relevant mathematical background of ray-tracing is presented as the geometrical basis of multipath estimation (it was explained in Sect. 5 that two of the six factors affecting multipath are dependent on the satellite–reflector–antenna geometry).

Section6.1.1describes the determination of the equation of a plane using a vector approach (needed to define a reflecting surface) and Sect.6.1.2describes how to find the intersection point of a line and a plane (needed to trace a ray from a satellite to an antenna via a reflecting surface).

6.1.1 The vector equation of a plane

A way of checking whether a given point, with position vector m lies in the plane or not, is as follows (see Fig.9).

Let a be the position vector of the given point in the plane, and let n be the given normal vector to the plane.

Then n is in the plane only if the dot product

(m−a)·n=0 (19)

This is because (m – a) is the vector from points A to M, and any vector that lies completely in the plane must

Y

X

Z

n

a

b

c c-a b-a

m

Fig. 9 The vector equation of a plane

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be orthogonal to n, which implies their dot product must equal 0.

The equation of plane may also be defined by using three points. Let a, b, and c be the position vectors of three points in the plane (see Fig.9). The two vectors of (b−a) and (c−a) must both lie in the plane, so their normal vector is found by taking their cross product:

n=(b−a)×(c−a) (20)

So, as before, our equation for the plane consists of all vectors, m satisfying

(m−a)·n=0 (21)

which is the same as Eq. (19). Therefore

(m−a)·((b−a)×(c−a))=0 (22) Equation (20) is used with any three of the input four corners’ coordinates of a reflector to determine the unique normal vector of the reflector.

6.1.2 The intersection of a line and a plane

In order to assess the possibility of multipath with a particular satellite–reflector–antenna geometry (Fig. 10), the following steps may be used:

(i) Find the closest point r on the reflector to the antenna p and hence compute the position difference vector

r−p

from the antenna p to this point.

(ii) Find the antenna image position q (behind the reflector).

n c

q

r

X

Y Z

antenna satellite

antenna image

planar reflector p g

s

Fig. 10 Geometry of the intersection of a line and a plane

(iii) Determine the intersection point s of the reflector and the line joining satellite g and the antenna image q.

(iv) Check whether or not the intersection point s is located within the reflector using the known positions its four corners. If so, multipath will occur.

In step (i) above, a real number parameter t is first obtained from

t=((c−p)·n)/(n·n) (23) where c is any point on the plane (in our case, one of the four input corner positions of the reflector is used). The closest point is then found from

r=p+tn (24)

The image position q in step (ii) is then obtained from the parametric equation:

q=p+2(r−p) (25)

In step (iii), we obtain s from the parametric equation:

s=g+t(q−g) (26)

where t is a real number parameter that describes the intersection point in the line joining points g and q. We know s must satisfy [according to Eq. (19)]:

(s−c)·n=0 (27)

where c is any point on the planar reflector (e.g., any one of the four input corners). Thus, we can combine Eqs.

(26) and (27) to find t:

(g+t(q−g)−c)·n=0 (28)

Using the distributive property, Eq. (28) can be developed as follows,

g·n+t(q−g)·n−c·n=0 (29) t(q−g)·n=c·n−g·n (30) t=((c−g)·n)/((q−g)·n) (31) Putting t into Eq. (26), allows s to be obtained.

Step (iv) involves carrying out a geometrical bound- ary test using the knowledge of the four input corners’

positions of the reflector. If the intersection point s lies on the reflector, the differential path delay is computed and used in Eq. (18).

6.2 Multipath modelling

In order to test the ray-tracing multipath correction model, a software-based GNSS carrier-phase data simulator has been written (Lau and Cross 2003). The

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simulator can generate realistic data with models for ion- ospheric and tropospheric errors, receiver clock biases, satellite clock biases, orbital errors, integer ambiguities, measurement noise, and multipath.

This section concentrates on the multipath modelling aspects of our simulator. The following assumptions have been made:

(i) Multipath is from a single quadrilateral reflector (single reflection for each reflector).

(ii) The reflection is specular (i.e., from a smooth surface) and the law of reflection applies.

(iii) No bending of the signal path occurs in the iono- sphere.

(iv) Reflection occurs on the surface of the reflector, i.e. the depth of penetration is negligible.

(v) The reflector is a lossless (or negligible loss) nonmagnetic dielectric material.

(vi) Interference between the wave-fronts of the reflected and direct signals is negligible.

These assumptions are justified as follows:

(i) Multipath signals arriving at the antenna after multiple reflections are generally too weak to be of concern to GPS systems (Ray et al. 2000).

Therefore, the contributions of the multiple- reflection multipath signals to the resultant multipath errors can be considered to be negligible.

(ii) A rough surface causes more reflections than a smooth surface. In modelling reflective surfaces, it is impractical to measure the actual roughness of the surfaces and trace the all reflection points on the rough surfaces in the multipath modelling. Therefore, generalisation of the surfaces is required for multipath modelling.

Reflections from a rough surface will cause bias to the modelled multipath error. The magnitude of the bias depends on the location of the point of reflection on the rough surface (diffuse reflection). If the point of the diffuse reflection is close to the point of specular reflection, the bias con- tributed to the differential path delay is small because the roughness (peak and trough) is seldom more than few millimetres.

If the point of diffuse reflection is far from the point of specular reflection, the differential path delay of the diffuse reflection is greater than that of the specular reflection and the bias to the multipath modelling is great. However, our experimental results (Sect.7.2) show that specular reflection is the dominant multipath error source, even if the reflective surface is rough.

(iii) The bending of the signal path due to the ion- osphere is negligible in the multipath problem because the effect on the LOS signal and reflected signal are identical for nearby reflectors.

(iv) High-frequency GNSS signals have <1 mm depth of penetration (reflected signal only) for most reflective materials (Born and Wolf 1999;Kraus and Fleisch 1999;Lau 2005).

(v) If the reflectors are lossy dielectrics, the predicted multipath errors would have greater amplitudes than the measured multipath errors. Some experimental results in Sect.7displayed this property, possibly indicating a limitation of this assumption.

(vi) The impact of interference between LOS and reflected signals before arriving at the antenna (in air) on multipath errors is seldom discussed in the literature. This effect is unknown or may be insignificant. We believe that this effect should be insignificant because the polarizations of the LOS and the reflected signals are always different.

The inputs to the simulation model are as follows:

(i) Cartesian coordinates of the four corners of the quadrilateral reflectors (complex and irregular surfaces are differentiated and modelled as very small quadrilaterals).

(ii) Relative permittivity of the reflector.

(iii) Polarization state of the receiving antenna such as AR and RCP (in preparation for future a GNSS that may not transmit a RCP signal).

(iv) Antenna gain pattern.

(v) Correlator spacing of the receiver.

(vi) Antenna phase centre offset and variation table such as IGS_01.pcv or ant_info.003, which can be obtained from the IGS (http://www.igscb.jpl.nasa.

gov) and the US National Geodetic Survey (NGS http://www.ngs.noaa.gov), respectively.

The steps in the carrier-phase multipath modelling pro- cedure are:

(i) Determine whether or not multipath would occur if the reflector was of infinite size by using the following sequence of ray-tracing equations to find the reflection point: Eqs. (23), (24), (25), (31) and (26).

(ii) Decide if the reflection point is within the bound- ary of the reflector as described by the coordinates of its four corners—only then will the reflector cause multipath.

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(iii) Compute the damping factor of the reflected signal using the sequence of Eqs. (1), (2), (4), (10), (11), (8), (9), (5) and (13)–(15).

(iv) Simulate the phase multipath error using Eq. (16), having computed the multipath phase shift and correlation function from Eqs. (18) and (17), respectively.

(v) Apply the phase centre variations using the incidence angles of the LOS and reflected signals and the input phase centre variation table.

The multipath computed in the above sequence is simply subtracted from the measured phase in order to apply the correction. In the case of data simulation, it needs to be added to the simulated phase of the direct signal, i.e., that obtained by division of the geometric distance between the satellite phase centre and the receiving antenna phase centre by the carrier wavelength (either with or without the addition of other GNSS biases and errors).

7 Validation of carrier-phase multipath modelling The multipath errors generated by the methodology described in Sect. 6 have been validated by comparing them with the errors in real GPS measurements collected in three experiments with known (and care- fully controlled) reflector geometry. When processing real data, pre-determined integer ambiguities (obtained from known coordinates) are applied to the measurements, which are then processed by the double difference (DD) method.

7.1 Modelling of reflections from a steel panel

In our steel panel experiment, two Leica Geosystems System 530 receivers attached to AT502 antennas were used in tests carried out at the Laboratoire Central des Ponts et Chaussées (LCPC) near Nantes, France, during May 2002. In order to create a sufficiently strong multipath signal, a 5 m by 2.5 m steel panel was constructed and placed about 5 m to one of the receiving antennas (Fig.11). Full details of the experiment can be found in Bétaille et al.(2003).

DD carrier-phases computed from the known coordinates were subtracted from those computed from the observations in order to obtain DD residuals. Due to the short length of the baseline (∼86 m), these DD residuals comprised mostly multipath errors and measurement noise, and since the random measurement noise was very small (when compared with multipath errors), we effectively obtained a time-series of DD multipath

Fig. 11 Experimental setup for collection of multipath data in LCPC

Fig. 12 Double difference residual (observed minus computed) in L1 of raw (red) and simulated (blue) data. It shows multipath error in L1

errors. In parallel, DD errors were computed following Sect.6using the measured geometry of the experimental set-up and a value of 3.9 for the relative permittivity of steel [sand casting carbon steel containing dielectric SiO2(Nalwa 1999;Samsonov 1968)]. The phase centre offset and variation table used in the modelling were obtained from the antenna calibration results in the ant_info.003 file from the NGS.

Satellite PRN02 was identified (by residual analysis of real data) as a satellite with significant multipath and comparisons have been made between the real and modelled DD residuals for this satellite, using the highest elevation satellite (PRN03) as the reference.

Figures12and13show computed DD residuals along with the observed DD residuals for the GPS L1 and L2 data, respectively. The computed DD residuals for both GPS L1 and L2 are also plotted in Fig. 14so that the

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Fig. 13 Double difference residual (observed minus computed) in L2 of raw (red) and simulated (blue) data. It shows multipath error in L2

Fig. 14 Double difference simulated multipath data minus DD true range in L1 (red) and L2 (blue)

frequency-dependence of multipath can be more clearly seen.

Whilst, on initial inspection, the amplitudes of the multipath signal in the computed DD data (Figs. 12 and 13) do not agree well with those of the real DD data, the phases of both the computed and real DD

data are coherent. The L1 multipath error, the damping factor, and its components (reflection coefficient, polarization efficiency, antenna gain ratio of direct and reflected signals) are shown in Fig.15as time-series to enable their relationships to be seen. Figure16shows the elevation angles of the direct and reflected signals during the observation period.

Theoretically, multipath error at a static antenna should be a sinusoid with zero mean (Wieser and Brunner 2000; Ray and Cannon 1999). However, the results in Fig. 12 indicate a non-zero mean approxi- mately between epochs 201830 and 203248 due to more than one multipath source in the real data (reflections over the limited time period from other nearby objects will lead to multipath errors with a non-zero mean).

Therefore, a time fast Fourier transform (TFFT) was applied to trace the change of the multipath signal’s frequency components through time and the result is shown in Fig.17a. A relatively small peak frequency, close to the maximum peak frequency can be seen, justifying the assertion that there is at least one other multipath source (Fig.17a has been rotated into Fig.17b for better illustration).

After removal of this bias (the mean of the DD residuals in that period), the L1 DD residuals are plotted in Fig.18. The results in Figs. 18and12show the agreements between real and modelled DD residuals using a variable damping factor for the L1 and L2 frequencies, respectively. The differences are now mainly due to measurement noise and for most of the time are within

∼3 mm (the assumed nominal DD measurement noise (Kim et al. 2004;Park et al. 1998), which is consistent with the expected level of noise for modern geodetic GPS receivers (Brown and Wang 1999;Gratton et al.

2004).

We can therefore conclude that our model also pre- dicts the amplitude of the multipath accurately, as well as its phase and frequency. For those periods where the

Fig. 15 Relationships of the simulated multipath (MP), damping factor (DF), reflection coefficient (RC), polarization efficiency (PE), and antenna gain ratio (GR) of the simulated multipath L1 data in the observation time series of PRN02

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Fig. 16 Elevation angle of the simulated multipath L1 data; red line represents the elevation angle of direct signal and blue line represents the elevation angle of reflected signal

Fig. 17 Time fast Fourier transform of L1 DD residual to analyse change of signal’s frequency components through time. a 3D view, b X−Z axes view

discrepancy is >3 mm, we consider it most likely that either the models for the damping factor and reflection are still incomplete (possibly due to the assumptions (ii), (iv)—(vi) not being completely valid), or that additional multipath occurred in that period.

A possible factor leading to model incompleteness is that the antenna gain pattern shown in Fig.6might

Fig. 18 Double difference residual showing multipath error in L1 corrected to the bias from GPS second 201830 to 203248 in Fig.12

not truly represent the real variation in antenna gain for the particular instrument used. However, overall, the model described in Sects. 2 to5 leads to a reasonably good approximation of the real damping factor in this validation.

7.2 Modelling reflections from irregular surfaces 7.2.1 Modelling reflections from a water surface

In this experiment, two Leica Geosystems System 530 receivers and AT502 antennas were again used, but tests were carried out beside the Serpentine (a small lake in Hyde Park, London) on 4 and 5 March 2004 using the water surface [its dimensions are about 850 m (North- South) by 200 m (East–West)] as a reflector to generate a multipath signal (Fig.19).

Fig. 19 Experimental setup for collection of multipath data in Hyde Park; it also shows approximate directions of multipath for satellites PRN11 and PRN16

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Table 2 Table of relative permittivities for some common media (Kraus and Fleisch 1999)

Medium Relative permittivity (dimensionless)

Copper 1

Sea water 80

Rural ground 14

Urban ground 3

Fresh water 80

Wood (dry) 2–4

Lead glass 6

Mica 6

Marble 8

Flint glass 10

The rover antenna was set up on the north side of the lake and the reference antenna was set up about 60 m further north. The experimental periods (expressed in GPS seconds) were from 384717 to 399146 for day 1 (4 March 2004) and from 469609 to 483398 for day 2 (5 March 2004), i.e.∼4 h for each session. The height difference between the rover antenna and water surface was measured by spirit levelling (repeated measurements were made before and after GPS observations for each day) and the height difference between the water surface and a temporary benchmark (TBM) was measured at least three times for each day during the data collection (the rover station was tied to TBM as well). The mean height difference between the rover station and water surface was then computed.

The reference antenna coordinates in Hyde Park were tied to a continuous GPS station (∼3 km away) in London operated by the Ordnance Survey using the Leica Geosystems Ski-Pro software. In this way, we ensured that the reference frame for the real GPS data was the same as that for the modelled data. GPS DD errors were then computed following the methods described in Sects. 2 to 5 using the measured geometry of the experimental set-up and a value of 80 for the relative permittivity of water (Table2).

Satellites PRN11 and PRN16 (both are Block IIR satellites) have been identified (by DD residual analysis of real data) as having multipath during the period of observation. The following investigations are therefore based on measurements from those satellites and PRN03 is used as the reference satellite for differencing.

The DD residuals from real and modelled L1 data for satellites PRN11 and PRN16 are shown in Figs.20 and21, respectively. The elevation angles of the two satellites in the time-series vary from 15^◦to 51^◦(PRN11 in Fig.20), and 75^◦to 15^◦(PRN16 in Fig.21). Note that no measurement noise is generated in the modelled data and that results obtained using L2 data are not shown here because they show similar patterns to the L1 data.

Fig. 20 Double difference residuals in L1 of real (red) and simulated (blue) data for PRN11 in Day 1

Fig. 21 Double difference residuals in L1 of real (red) and simulated (blue) data for PRN16 in day 1

The results (Figs.20and21) show good agreement in the amplitudes of modelled and real DD multipath data for both PRN11 and PRN16, but rather poor agreements in their phases, especially when the elevation angle of the direct signal is <∼50^◦. In both satellites’ data, the frequencies of the computed multipath signals increase with decrease in elevation angles. However, this trend is not so clear in the real data with low elevation satellites.

Also, the expected sinusoidal property of multipath signal is not very obvious in the real data, especially at low elevation angles. A sidereal day repeatability check was carried out for PRNs 11 and 16 (Figs.22and23). The results show generally good agreements in the observed multipath on two consecutive sidereal days. However, discrepancies greater than the nominal measurement noise∼3 mm can still be seen, especially at low elevation angles for PRN16.

Nevertheless, the agreements prove the presence of multipath in carrier-phase measurements on both satellites and the loss of obvious sinusoidal properties may be due to other reflectors contributing towards the total multipath effect. Also, there may be multipath on signals

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Fig. 22 Day-to-day repeatability of double difference residuals in L1 of real data for PRN11 for day 1 (red) and day 2 (blue)

Fig. 23 Day-to-day repeatability of double difference residuals in L1 of real data for PRN16 for day 1 (red) and day 2 (blue)

Fig. 24 3D view of TFFT analysis of the frequencies of raw multipath signal of the satellite PRN11 for day 1

from the reference satellite and multiple reflections from the water surface caused by wave motion.

A TFFT was performed to analyse the frequencies of the multipath signals for both satellites. Three- dimensional views of the results for satellite PRN11 are shown in Figs.24and25for days 1 and 2, respectively. Similarly, results for PRN16 are given in Figs.26

and27. Several signals, some with different frequencies, are present in the raw data (Figs.24,25,26,27). Some peaks show the expected sidereal day repeatability of

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Fig. 28 a Relationships of the simulated multipath (MP), damping factor (DF), reflection coefficient (RC), polarization efficiency (PE), and antenna gain ratio (GR) of the simulated multipath L1 data in the observation time-series of PRN11 for day 1; b change of differential path delay (DPD) of reflected signal for PRN11 in the observation time-series of day 1

multipath, but many small random peaks exist, which are probably due to multiple reflections from the water surface caused by wave motion.

Through analysis of the modelled data with a TFFT using the same precisions, number of segments and steps as for the raw data, the simulated multipath signals show good agreements with raw multipath data in the dominant peak frequencies, but agreements with the power spectrum are not always so good. The most likely causes of spectrum discrepancies are described in Sect.7.1.

Note that the relationships between the L1 multipath errors, the damping factors, and other factors (e.g., reflection coefficient, polarization efficiency, antenna gain ratio of direct and reflected signals) for satellites PRN11 and PRN16 of this validation are shown in Figs.28a and29a respectively, and the differential path delays of reflected signals for satellites PRN11 and PRN16 are shown in Figs.28b and29b, respectively.

Finally, we remark that uncertainties occur due to the assumption of specular reflection from the water surface, but it was impractical to measure the water wave motion during the observation period and, anyway, it would be very difficult to incorporate this into practical multipath modelling. The results nevertheless give a good indica- tion of the impact of reflections from water surfaces on GPS carrier-phase data.

7.2.2 Modelling reflections from a brick building Moore et al. (2005) use the multipath model described in this paper to predict GPS carrier-phase multipath errors from a brick building (Fig. 30) and compare the predicted multipath errors with the real multipath errors.

Generalised areas of a side of the building are used for the validation; the areas are the two rectangles in Fig. 30. A value of 5.361 for the relative permittivity of brick (http://www.ofcom.org. uk/research/technology/

spectrum_efficiency_scheme/ses2003-04/ay24462a/

package4.pdf) was used in the modelling.

Two Leica Geosystems AT504 choke-ring antennas were used for the data collection; therefore, significant multipath errors can only arrive at the rover antenna from above the horizontal plane of the antenna.

DD residuals obtained from known coordinates were computed as in the steel panel (Sect. 7.1) and water surface (Sect. 7.2.1) validations. Satellite PRN01 was identified (by DD residual analysis of real data) as a satellite with significant multipath and comparisons have been made between the real and modelled DD residuals for this satellite, using the highest elevation satellite (PRN11) as the reference.

Figures31and32show computed DD residuals along with the observed DD residuals for the GPS L1 and L2

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Fig. 29 a Relationships of the simulated multipath (MP), damping factor (DF), reflection coefficient (RC), polarization efficiency (PE), and antenna gain ratio (GR) of the simulated multipath L1 data in the observation time series of PRN16 in day 1;

b change of differential path delay (DPD) of reflected signal for PRN16 in the observation time-series of day 1

Fig. 30 Generalised areas (the two yellow rectangles) of the wall are used for validation of the multipath modelling (Moore et al.

2005)

data, respectively. The right-hand sides of the green dot- ted lines in Figs.31and32are the un-modelled epochs, when no reflection was found by the ray-tracing in the generalised areas of the brick building. The elevation angle of the satellite varied from 30^◦to 45^◦during the time in which there were reflections.

Fig. 31 Double difference residuals for PRN01 L1 data; raw DD residuals are in red and simulated DD residuals are in blue (Moore et al. 2005)

The results in Figs.31and32show that, apart from disagreements when the reflection point is near the edge of the building, the multipath model can predict very good amplitude and reasonably good phase for multipath errors. Possible reasons for the disagreement in the phase at some epochs were described in the discussion of the steel panel (Sect.7.1) and water surface (Sect.7.2.1) validations. This good agreement suggests that specular reflection is the dominant multipath error source even if the surface is rough.

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Fig. 32 Double difference residuals for PRN01 L2 data; raw DD residuals are in red and simulated DD residuals are in blue (Moore et al. 2005)

8 Influence of factors contributing to the damping factor used in carrier-phase multipath modelling The main factors contributing to carrier-phase multipath were described in Sect.5. In this section, we report on tests carried out with our model to investigate the influence of some of these in more detail. One of the reasons for doing this is to discover whether or not errors in these factors may explain the discrepancies between the modelled and experimental results in Sect.7. A total of four tests were conducted. In the first three, satellite–

reflector–antenna geometry from the steel panel experiment (Sect.7.1) was used, and in the second and fourth tests DD data collected for the water surface experiment (Sect.7.2) was used.

8.1 Influence of the distance between the receiving antenna and reflector

Multipath errors were generated using distances of 5 and 10 m between the receiving antenna and reflector and a relative permittivity of 3.9 for the reflector (Figs.33 and34).

Increasing the distance between the receiving antenna and reflector increases the frequency of the phase multipath error. The frequency of phase multipath errors increased from 0.0033 to 0.0069 Hz when the receiving antenna–reflector distance was doubled. This is because the increase in distance leads to an increase in the rate of change of the time delay of the reflected signal. This characteristic is well known (Ray and Cannon 1999) and easily predicted from theory, see Eqs. (16) and (17). This test was been carried out mainly to demonstrate the cor- rectness of the modelling process.

Fig. 33 Simulated multipath error in L1 with the tilting reflector about 5 m away from the receiving antenna and a variable damping factor with a relative permittivity of 3.9

Fig. 34 Simulated multipath error in L1 with the tilting reflector about 10 m away from the receiving antenna and a variable damping factor with a relative permittivity of 3.9

8.2 Influence of the permittivity of the reflector

Table2shows the relative permittivities of some common media that might cause reflections of GPS signals. As explained in Sects.3and5, and from Eqs. (1) and (2), the amplitude of multipath error arising from such reflections is highly dependent on the relative permittivity of the medium. If flint glass rather than steel had been used in Sect. 7.1, the multipath effect would be more serious (relative permittivity of 10 compared with∼4).

This point can be seen in detail by comparing Figs.35 and 33. The amplitudes of multipath errors in Fig.33 are <4 mm [except when the change of incidence angle leads to a significant change in the reflection coefficients, which in turn significantly changes the polarisation of the reflected signal, see Eqs. (1), (2), (10), (11), and (13)], while those in Fig.35are always between∼6 and 7 mm.

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Fig. 35 Simulated multipath error in L1 with the tilting reflector about 5 m away from the receiving antenna and a variable damping factor with a relative permittivity of 10

Fig. 36 Simulated multipath error in L1 with the horizontal reflector about 29 cm below the receiving antenna and a variable damping factor with a relative permittivity of 3.9

In the water surface validation (Sect. 7.2.1), some discrepancies were found between the modelled and real DD multipath data (Figs.20and21). The question therefore arises as to whether or not those discrepancies could have come from the variation of the relative permittivity of water due to different temperature and impurities of the water (i.e., different to the standard environment when measuring the standard relative permittivity). In order to address this, the impacts of chang- ing the relative permittivity by±10% on the agreement between the simulated and real DD multipath data were investigated:

The relative permittivity of 80 (see Table2) for water was changed to 88 and 72 respectively and input to multipath modeller, whilst keeping the other input parameters unchanged. A slight increment (of the order 0.5 to 1 mm) in multipath amplitude was found for the two multipathing satellites. Similarly a 10% reduction in the relative permittivity to 72 leads to slight reductions (of the order 0.5 mm) in amplitude. This shows that a change

of relative permittivity by 10% does not affect multipath errors significantly.

8.3 Influence of reflectors just the below antenna If a receiving GNSS antenna is set on a platform such as on top of a vehicle or placed just above ground, multipath may come from very close reflectors below the antenna’s horizon. This is a very common situation in practice (e.g., mobile mapping). Therefore, a test was carried out in which the platform was assumed to be horizontal, smooth (so causing specular reflection), static (compared with the antenna), and to be 29 cm below the L1 phase centre of the receiving antenna’s position.

The reflector was assumed to have a relative permittivity of 3.9 (i.e., the same as that for the steel panel), and all other factors are as described in Sect.7.1.

The resulting DD multipath errors are shown in Fig.36. It can be seen that both the amplitude and frequency of the errors although small (∼3 mm in amplitude and 2.5×10⁻⁴Hz in frequency), could be significant in high-accuracy dynamic environments where there is no opportunity for averaging. On the other hand, with a detailed knowledge of the materials and geometry, it should be possible to calibrate a particular antenna sce- nario for such errors, especially in situations where the antenna, satellite and reflector geometry is known.

Note that if the reflecting surface is rough, such as normal ground, it can be considered as causing many specular reflections from reflectors with different normal vectors. The resultant multipath error would be then the superposition of the multipath error from each reflector (Lau and Mok 1999;Lau 2005).

8.4 Sensitivity to the height difference between the antenna and reflector

In practice, it is often difficult to know the exact height difference between a reflecting surface and a GNSS antenna, leading to the possibility of error in implement- ing the modelling method suggested in this paper, especially when dealing with reflections from water surfaces.

In order to test the sensitivity of the method to this error, experiments were carried out in which the positions of the antenna and water surfaces described in Sect.7.2were each raised and lowered by 1 cm (which is much greater than any possible levelling measuring error over such a small distance, <30 m).

When the water level was raised or lowered by 1 cm, the phase of modelled multipath errors (expressed in terms of length) changed by amounts that were between 1 and 10 mm. However, the modelled errors showed

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no significant improvements on the agreement with the real multipath contaminated DD residuals. In the case of raising the antenna position, the results showed a slightly worse overall agreement in amplitude. Also, the modelled multipath error in satellite PRN11, and the computed DD residuals of satellite PRN16, lost their zero mean properties.

On the other hand, whilst lowering the antenna position had no significant effect on the frequencies of multipath errors, it did lead to a slightly worse agreement in amplitude for most of time. In some cases, however, a better agreement was obtained for some parts of the time-series.

8.5 Additional characteristics

Carrier-phase multipath errors also showed the following characteristics:

• In general, the maximum amplitude of the multipath error increased when the incidence angle at the reflector decreased. This effect can be seen in Figs.20 and21and predicted from Eqs. (1), (2) and (4). It is one of the reasons for assigning a lower weight to measurements from low elevation satellites in GNSS stochastic modelling.

• Multipath errors displayed a non-sinusoidal pattern with a non-zero mean when the multipath signal was strong (the relative amplitude of the indirect signal was large). This occurred in the Hyde Park dataset when the multipath errors were greater than 10 mm, e.g., Figs. 20(the first ∼5 cycles) and 21 (the final

∼10 cycles).Braasch 1996 also describes this non- zero mean property. It is important because it shows that multipath errors may not always average to zero in static applications.

8.6 Analysis of the sensitivity tests on the water surface experiment

The sensitivity test results for the water surface experiment in Sects.8.2and 8.4indicate that there is some sensitivity to the height difference between the reflector and the antenna and that another antenna position might give a better overall agreement. However, the effect is small and we believe that multipath from other reflectors (ground and/or water waves) are the main reason for the discrepancies between real and modelled multipath (as confirmed by the TFFT analyses).

Since this experiment is not well-controlled due to the difficulty of modelling the reflective water surface, the results of this experiment is less conclusive on the performance of the ray-tracing method. However, the

results reproduce some of the key characteristics of the real multipath such as the change of amplitude and frequency with time.

9 Summary and conclusions

This paper describes the physical and mathematical basis of a model of the GNSS carrier-phase multipath process using ray-tracing, and identifies the key factors that can contribute to multipath errors. It presents a complete set of formulae to predict GNSS carrier-phase multipath when the following are known (cf. Sect.8).

• The relative positions of the receiving antenna and reflectors.

• The relative permittivities of the reflecting surfaces.

• The correlator spacing of the receiver.

• The RCP gain pattern of the receiving antenna.

• The phase centre offset and variation table of the receiving antenna.

The method was verified by comparing modelled multipath errors with observed multipath errors in three experiments; one based on reflections from a smooth steel panel and the others on reflections from a fresh water surface and a brick building. The real and modelled results from the steel panel experiments are in excellent agreement—perfectly in phase with only small disagreements in amplitude (always less than 5 mm and usually less than 2 mm). If the model had been used to correct for this multipath the results would always be improved.

The results from the water surface experiments were less conclusive, probably due to difficulties in modelling the exact position of the water surface at any time and due to other unknown reflections. Nevertheless they reproduce some of the key characteristics of the real multipath such as the change of amplitude and frequency with time. The real and modelled results from the brick building experiments are in good agreement in amplitude. There are, however, some disagreements in phase at some epochs.

Sensitivity tests with our model showed that the accuracy of the predicted multipath errors was highly dependent on the precise knowledge of the relative antenna-reflector geometry. For instance errors of up to 1 cm in their relative height difference can cause errors in the modelled multipath from reflectors below the antenna by up to 1 cm. Also, it was shown that an error of up to 10% in the assumed permittivity of the reflector would not have a noticeable effect on the modelled multipath.