Undifferenced and single differenced GNSS noise analysis through a constrained baseline vector

Undifferenced and Single Differenced GNSS

noise analysis through a constrained

baseline vector

P.J. Buist

and D. Imparato

1_{National Aerospace Laboratory (NLR), the}

Netherlands, Peter.Buist@nlr.nl 2

Delft University of Technology, t h e N e t h e r l a n d s , Davide.Imparato@tudelft.nl

Abstract: GNSS noise characteristics can be obtained by short and zero baseline analysis, for both pseudorange and carrier phase measurements. Differencing observations from a GNSS satellite tracked by two GNSS receivers (connected to the same or different antennas) will explicitly eliminate common errors.

In this contribution, the known baseline vector between the antennas will be used to adjust (“constrain”) the observation vectors. The advantages of a noise analysis through a constrained baseline vector over an ordinary baseline processing will be discussed and demonstrated using undifferenced and single differenced residuals obtained from dedicated experiments.

The influence of multipath, known to be time correlated for sampling rates of 1 Hz or higher, will be also investigated. For static baselines, multipath effects of each GNSS satellite are repeated with a period close to a sidereal day due to the baseline-satellite geometry. In this contribution, we will demonstrate how this effect can be utilized to compensate for multipath errors on a constrained baseline vector.

BIOGRAPHIES

Peter Buist is a Senior R&D Engineer at the National Aerospace Laboratory NLR. He holds a MSc and a PhD degree from Delft University of Technology, the Netherlands.

He worked in the Japanese aerospace industry for years, specializing in particular in Global Navigation Satellite System. In Japan, he developed GPS receivers for the –among others- SERVIS-1, USERS, ALOS satellites and the H2A rocket. His current research interest includes all expects of navigation and onboard data processing.

In 2011 he was one of the winners of the European Satellite Navigation Competition with a project on sensor integration. He is a board member of the Netherlands Space Society NVR and editor-in-chief of their magazine.

Davide Imparato is a Ph.D. student at Delft University of Technology, The Netherlands, at the Department of Geoscience and Remote Sensing. He holds a M. Sc. Degree from the faculty of aerospace engineering of the University of Pisa, Italy. His research focuses on Aircraft Navigation Integrity, statistical hypothesis testing for RAIM algorithms and future GNSS integrity concepts.

1 INTRODUCTION

GNSS noise characteristics can be obtained by short and zero baseline analysis, for both pseudorange and carrier phase measurements (Amiri-Simkooei, A.R. and C.C.J.M Tiberius, 2007), (Bakker, de P.F., C.C.J.M. Tiberius, H. van der Marel, R.J.P. van Bree, 2011).

In this contribution, we will use a different approach by constraining the observation vectors using the known baseline vector. The advantages of a noise analysis through a constrained baseline vector over an ordinary baseline processing will be discussed in section 3 and demonstrated using undifferenced and single differenced residuals obtained from dedicated experiments in section 4.

The influence of multipath will also be investigated: multipath is known to be strongly time correlated for sampling rates of 1 Hz or higher. For static baselines, multipath effects of each

GNSS satellite are also repeated with a period close to a sidereal day due to the baseline-satellite geometry. In section 4, we will demonstrate how this effect can be utilized to compensate for multipath errors on a constrained baseline vector.

We will start in section 2 with deriving the undifferenced, single and double difference equations for both short and zero baselines, following the approach described in (Buist, P.J., 2013).

2 GNSS OBSERVATION MODEL

2.1 Short baseline Observation Model

In a short baseline experiment, two GNSS receivers are connected to their own antenna system. These two antennas form a baseline with a length of, typically, a few meters. The single frequency code and phase observation can be expressed by

𝑷𝒓𝒔(𝒕) = 𝝆𝑷,𝒓𝒔 (𝒕, 𝒕 − 𝝉𝒓𝒔) + 𝑰𝒓𝒔+ 𝑻𝒓𝒔+ 𝒄�𝜹𝒓(𝒕) − 𝜹𝒔(𝒕 − 𝝉𝒓𝒔)� + 𝒄 �𝒅𝑷,𝒓(𝒕) + 𝒅𝑷𝒔(𝒕 − 𝝉𝒓𝒔)� + 𝑴𝑷,𝒓𝒔 + 𝝐𝒓𝒔 𝝋𝒓𝒔(𝒕) = 𝝆𝝋,𝒓𝒔 (𝒕, 𝒕 − 𝝉𝒓𝒔) − 𝑰𝒓𝒔+ 𝑻𝒔𝒓+ 𝒄�𝜹𝒓(𝒕) − 𝜹𝒔(𝒕 − 𝝉𝒓𝒔)� + 𝒄 �𝒅𝝋,𝒓(𝒕) + 𝒅𝝋𝒔(𝒕 − 𝝉𝒓𝒔)�

+ 𝝀𝒇�𝝋𝒓�𝒕𝒓,𝟎� − 𝝋𝒔(𝒕𝟎𝒔)� + 𝝀𝒇𝒛𝒓𝒔+ 𝑴𝝋,𝒓𝒔 + 𝜺𝒓𝒔

where

t is the

are the range satellite 𝑠 -receiver 𝑟 for code and phase respectively, 𝐼𝑟𝑠 is the ionospheric delay,

𝑇𝑟𝑠 the tropospheric delay, c is the speed of light, 𝛿𝑟 and 𝛿𝑠 the clock errors of receiver and

satellite, 𝑑,𝑟 and 𝑑𝑠 instrumental (code and carrier phase) delays for the receiver and the satellite

respectively, 𝑀,𝑟𝑠 is the multipath error and 𝜖𝑟𝑠 and 𝜀𝑟𝑠 the thermal noise on the code and phase

observation respectively. 𝜆𝑓 is the wavelength of frequency f, 𝜑𝑟�𝑡𝑟,0� is the initial phase for r and

𝜑𝑠_(𝑡

0𝑠) initial phase for s.

From here on the time dependence is discarded to simplify the notation. If the two receivers are sufficiently close, the delays due to ionosphere and troposphere would cancel out, but for the moment we will keep those terms. We will collect the set of code and phase observations at the receiver r into a vector of dimension 2(n+1), with n+1 is the number of locked GNSS satellites:

𝒚𝒓= �𝑷_𝝋𝒓𝒓� = (𝑷𝒓𝟏 … 𝑷𝒓𝒏+𝟏, 𝝋𝒓𝟏 … 𝝋𝒓𝒏+𝟏)𝑻

The single differences (SD) can be obtained by subtracting the measurements collected at two different receivers:

𝑷𝟏𝟐𝒔 = 𝝆𝑷,𝟏𝟐𝒔 + 𝑰𝟏𝟐𝒔 + 𝑻𝟏𝒓𝒔 + 𝒄𝜹𝟏𝟐+ 𝒄𝒅𝑷,𝟏𝟐+ 𝑴𝑷,𝟏𝟐𝒔 + 𝝐𝟏𝟐𝒔

𝝋𝟏𝟐𝒔 = 𝝆𝝋,𝟏𝟐𝒔 − 𝑰𝟏𝟐𝒔 + 𝑻𝒔𝟏𝟐+ 𝒄𝜹𝟏𝟐+ 𝒄𝒅𝝋,𝟏𝟐+ 𝝀𝒇𝝋𝒓(𝒕𝟎) + 𝝀𝒇𝒛𝟏𝟐𝒔 + 𝑴𝝋,𝒓𝒔 + 𝜺𝟏𝟐𝒔

Thus this SD operation will explicitly eliminate satellite clock 𝛿𝑠, instrumental delays 𝑑𝑠 and initial

phase 𝜑𝑠(𝑡0𝑠). The double differences (DD) will be obtained by subtracting the single differences from

two satellites:

𝑷𝟏𝟐𝟏𝟐= 𝝆𝑷,𝟏𝟐𝟏𝟐 + 𝑰𝟏𝟐𝟏𝟐+ 𝑻𝟏𝒓𝟏𝟐+ 𝑴𝑷,𝟏𝟐𝒔 + 𝝐𝟏𝟐𝟏𝟐 𝝋𝟏𝟐𝟏𝟐= 𝝆𝝋,𝟏𝟐𝟏𝟐 − 𝑰𝟏𝟐𝟏𝟐+ 𝑻𝟏𝟐𝟏𝟐+ 𝝀𝒇𝒛𝟏𝟐𝟏𝟐+ 𝑴𝝋,𝒓𝒔 + 𝜺𝟏𝟐𝟏𝟐

Thus the DD operation will explicitly eliminate the receiver clock 𝛿12, instrumental delay 𝑑,12 and initial

phase 𝜑𝑟(𝑡0). Now the ionospheric and the tropospheric delay can be removed by differencing as

the differential delays can be considered negligible for (such) short baselines, thus 𝑷𝟏𝟐𝟏𝟐= 𝝆𝑷,𝟏𝟐𝟏𝟐 + 𝑴𝑷,𝟏𝟐𝟏𝟐 + 𝝐𝟏𝟐𝟏𝟐

𝝋𝟏𝟐𝟏𝟐= 𝝆𝝋,𝟏𝟐𝟏𝟐 + 𝝀𝒇𝒛𝟏𝟐𝟏𝟐+ 𝑴𝝋,𝒓𝒔 + 𝜺𝟏𝟐𝟏𝟐 After linearization, the single difference equations become

𝜟𝑷𝟏𝟐𝒔 _{= (−𝒖𝟏}𝒔₎𝑻_{𝒃 + 𝒄𝜹}

𝟏𝟐+ 𝒄𝒅𝑷,𝟏𝟐

𝜟𝝋𝟏𝟐𝒔 _{= (−𝒖𝟏}𝒔₎𝑻_{𝒃 + 𝒄𝜹}

𝟏𝟐+ 𝒄𝒅𝝋,𝟏𝟐+ 𝝀𝒇𝝋𝒓(𝒕𝟎) + 𝝀𝒇𝒛𝟏𝟐𝒔

𝜟𝑷𝟏𝟐𝟏𝟐= �−𝒖𝟏𝟏𝟐�𝑻𝒃 𝜟𝝋𝟏𝟐𝟏𝟐= �−𝒖𝟏𝟏𝟐�𝑻𝒃 + 𝝀𝒇𝒛𝟏𝟐𝟏𝟐

Where 𝛥𝑃 and 𝛥𝜑 are the ‘observed minus computed’ observations, 𝒖r𝑠 is the line-of-sight vector

between r and s and b is the increment vector of the baseline coordinates.

2.2 Zero baseline Observation Model

In a zero baseline set-up, two GNSS receivers are connected to the same antenna. Differencing observations from a GNSS satellite tracked by both receivers will explicitly eliminate all common errors (ionospheric, tropospheric, antenna, front-end and multipath errors):

𝑷𝟏𝟐𝒔 = 𝝆𝑷,𝟏𝟐𝒔 + 𝒄𝜹𝟏𝟐+ 𝝐𝟏𝟐𝒔

𝝋𝟏𝟐𝒔 = 𝝆𝝋,𝟏𝟐𝒔 + 𝒄𝜹𝟏𝟐+ 𝝀𝒇𝝋𝒓(𝒕𝟎) + 𝝀𝒇𝒛𝟏𝟐𝒔 + 𝜺𝟏𝟐𝒔

As the observations collected at both receivers are sharing the same antenna and front-end, most of

the instrumental delays 𝑑𝑃,12 and 𝑑𝜑,12 are removed by differencing, although strictly speaking some

residual receiver related hardware delays might remain. Double differencing will result in:

𝑷𝟏𝟐𝟏𝟐= 𝝆𝑷,𝟏𝟐𝟏𝟐 + 𝝐𝟏𝟐𝟏𝟐 𝝋𝟏𝟐𝟏𝟐= 𝝆𝝋,𝟏𝟐𝟏𝟐 + 𝝀𝒇𝒛𝟏𝟐𝟏𝟐+𝜺𝟏𝟐𝟏𝟐

2.3 Multipath combination

The Mc combination is obtained from code and dual frequency phase observations: 𝑴𝒄= 𝑷𝒓𝒔₋𝒇𝟏𝟐+ 𝒇𝟐𝟐

𝒇𝟏𝟐− 𝒇𝟐𝟐𝝋𝒓,𝟏

𝒔 ₋ 𝟐𝒇𝟐𝟐

𝒇𝟏𝟐− 𝒇𝟐𝟐𝝋𝒓,𝟐 𝒔

where 𝑃_𝑟𝑠 is for example the code C/A measurement while 𝜑_𝑟,1𝑠 and 𝜑_𝑟,2𝑠 are the carrier phase

measurements on the frequencies L1 and L2. The standard deviation of the Mc combination is the same as the code observation.

3 GNSS NOISE ANALYSIS THROUGH A CONSTRAINED BASELINE VECTOR

In this section, the known baseline vector between the antennas will be used to adjust (“constrain”) the observation vectors: in section 3.1 for undifferenced code noise and in section 3.2 for single difference code and carrier phase noise. The advantages of a noise analysis through a constrained baseline vector over an ordinary baseline processing will be discussed.

3.1 Undifferenced code noise

Under the assumption that the code instrumental delay 𝑑𝑃,12 is lumped in the term

𝛿

, and the

in the term 𝜖12𝑠 , the single difference code observations can be written as:

𝜟𝑷𝟏𝟐= 𝑮𝑺𝑫_{𝒃 + 𝒄𝜹�𝟏𝟐+ 𝒆𝟏𝟐}

where 𝐺𝑆𝐷 is the (n+1)x3 matrix of normalized SD line-of-sight vectors −𝒖1𝑠 and 𝒆𝟏𝟐𝑠 is the single

difference residual vector. A measurement 𝑃₁₂𝑠 is available from each satellite in view, whereas the

relative position 𝒃 (and thus the relative range 𝜌_𝑃,12𝑠 ) is known a-priori with high accuracy. Therefore at

each epoch, n+1 measurements are available (with n+1 is the number of locked GNSS satellites) and

only one parameter (𝛿̅𝟏𝟐) has to be solved for. The system of n+1 equations can be written in the

standard form:

𝜟𝑷𝟏𝟐− 𝑮𝑺𝑫_{𝒃 = 𝒄𝜹�𝟏𝟐+ 𝒆𝟏𝟐}

From this equation it is clear that this is a much stronger model for residuals estimation than the standard model of undifferenced and single differenced baseline processing, as there is only one

To obtain the relationship between undifferenced and single difference code residuals, we can

apply the theory of free variates (Teunissen, 2003). As 𝑷1is the undifferenced code observations

at receiver 1, 𝑷2 the undifferenced code observations at receiver 2 and 𝑷12 the single difference

code observations𝑷12= 𝑷1− 𝑷2, we can write:

⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝜟𝑷𝟏𝟐𝟏 𝜟𝑷𝟏𝟐𝟐 ⋮ 𝜟𝑷𝟏𝟐𝒏+𝟏 𝜟𝑷𝟏𝟏 𝜟𝑷𝟏𝟐 ⋮ 𝜟𝑷𝟏𝒏+𝟏_⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ − � 𝟎_𝑮𝑺𝑫_𝒓𝟐� = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ �−𝒖𝟏𝟏�𝑻 𝟏 𝟎 𝟎 … 𝟎 �−𝒖𝟏𝟐�𝑻 𝟏 𝟎 𝟎 … 𝟎 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ �−𝒖𝟏𝒏+𝟏�𝑻 𝟏 𝟎 𝟎 … 𝟎 �−𝒖𝟏𝟏�𝑻 𝟏 𝟏 𝟎 … 𝟎 �−𝒖𝟏𝟐�𝑻 𝟏 𝟎 𝟏 … 𝟎 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ �−𝒖𝟏𝒏+𝟏�𝑻 𝟏 𝟎 𝟎 … 𝟏⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎢ ⎡_{𝒄𝜹𝟏𝟐}𝒃 𝒄𝜹𝟏 𝒄𝜹𝟐 ⋮ 𝒄𝜹𝒏+𝟏_⎦⎥ ⎥ ⎥ ⎥ ⎤

With this formulation, the 𝑷1 vector contains the so called free variates, since to each of them

corresponds a free unknown parameter (in this case the satellite clock error). For the free variates residuals, it holds:

𝒆�𝑷𝟏 = −𝑸𝑷𝟏𝑷𝟏𝟐 𝑸𝑷−𝟏𝟏𝟐𝒆�𝑷𝟏𝟐

Now, by applying the propagation law of the variance, it is straightforward that 𝑄𝑃_1𝑃12 = 𝑄𝑃₁, and

𝑄𝑃₁₂= 𝑄𝑃₁+ 𝑄𝑃₂. This means:

𝒆�𝑷_𝟏 = −𝑸𝑷_𝟏� 𝑸𝑷_𝟏+ 𝑸𝑷_𝟐�−𝟏𝒆�𝑷_𝟏𝟐

With the assumption, that the measurements variance is the same at the two receivers (a reasonable assumption if the same type of receiver is used for both receivers of the baseline, and both are placed in the same environment), the expression simplifies to:

𝒆�𝑷_𝟏 = −𝟏_𝟐𝒆�𝑷_𝟏𝟐

Hence, the relationship between the undifferenced and the single difference code residuals is linear.

3.2 Single difference code and carrier noise

Next, we will develop a model that can be applied to analyse both code and carrier phase noise characteristics. It will also make use of the known baseline vector and thus has the same advantage as found in the previous section: only one unknown (the clock error) is remaining in the residuals estimation, instead of the full relative position and time vector of an ordinary undifferenced and single differenced baseline processing. The advantage of this approach over the one proposed in section 3.1, is that it utilizes the more precise carrier phase observations. In order to be able to analyse the carrier phase observations, we will introduce ambiguity resolution and start the processing with the double difference equations. The standard DD model for short baselines will be applied:

𝐸(𝒚𝐷𝐷_{) = 𝐴𝒛 + 𝐺}𝐷𝐷_{𝒃, 𝐸(𝒚}𝐷𝐷_{) = 𝑄𝑦𝑦}

where 𝐸(𝒚𝐷𝐷) is the expectation and 𝐷(𝒚𝐷𝐷) is the dispersion. The following 7 step procedure as

described in (Buist, P.J., 2013) can be applied to evaluate noise characteristics of GNSS signals, receiver, etc. The first 3 steps are called preprocessing, step 4-7 are referred to as analysis. The output of the preprocessing steps can be replaced by -for example- an average baseline vector calculated over all the correctly fixed epochs or the a-priori known baseline vector (as was done in

section 3.1). For zero baseline analysis, the user could utilize 𝒃 = [0 0 0]𝑇 . The default processing is

Preprocessing

1) The float solution �𝒛�𝜖𝑅𝑛, 𝒃�𝜖𝑅3� is obtained from

�𝒛�𝒃�� = �𝑄𝑧̂𝑧̂ 𝑄𝑧̂𝑏�

𝑄𝑏�𝑧̂ 𝑄𝑏�𝑏�� �[𝐴 𝐺]

𝑇_𝑄

𝑦𝑦−1𝒚𝐷𝐷�

For which the variance-covariance matrix is derived as �𝑄𝑧̂𝑧̂ 𝑄𝑧̂𝑏�

𝑄𝑏�𝑧̂ 𝑄𝑏�𝑏�� = �[𝐴 𝐺]𝑇𝑄𝑦𝑦−1[𝐴 𝐺]�

−1

With this float solution, the solution of the linear system of observations becomes the squared norm of the least-squares DD residual vector:

‖𝒆�𝐷𝐷_‖

𝑄𝑦𝑦

2 _{= �𝒚}𝐷𝐷_{− 𝐴𝒛� − 𝐺}𝐷𝐷_𝒃��

𝑄𝑦𝑦

2

2)

The vector of integer least-squares estimates of the ambiguities 𝒛� is solved using standard

LAMBDA or constrained LAMBDA (for baselines whose length

‖𝒃‖ = 𝒍 is known), where 𝒛� is the

vector of integers that minimizes the term

min

‖𝒛� − 𝒛‖

.

As the constrained model is much stronger than the unconstrained model, it might be preferred for

epochwise processing. For more information on standard and constrained LAMBDA, see (Teunissen,

P.J.G., 1995)(Buist, P.J., 2007), (Teunissen, P.J.G., 2010), (Teunissen, P.J.G., G. Giorgi, and P.J. Buist,

2010).

As we use the constraint that the solution of 𝒛 has to be an integer �𝒛�𝜖𝑍𝑛, 𝒃� = 𝒃�(𝒛�)𝜖𝑅3�, the minimum

solution becomes: ‖𝒆�𝐷𝐷_‖ 𝑄𝑦𝑦 2 _{= �𝒚}𝐷𝐷_{− 𝐴𝒛� − 𝐺}𝐷𝐷_𝒃�� 𝑄𝑦𝑦 2 = ‖𝒆�𝐷𝐷_‖ 𝑄𝑦𝑦 2 _{+ ‖𝒛� − 𝒛�‖𝑄} 𝑧𝑧 2

The minimum of this equation is clearly equal to or larger than the float solution obtained in step 1.

3) The fixed baseline solution is obtained using the estimated integer ambiguities. The residual

(𝒛� − 𝒛�) is used to adjust the float solution 𝒃� of the first step. Analysis

4) Next, the observation vector 𝒚𝐷𝐷 is adjusted with the baseline solution 𝒃 using the original

non-linear model with the assumption that 𝑄𝑦�𝑦�= 𝑄𝑦𝑦

𝒚�𝐷𝐷 _{= 𝒚}𝑫𝑫_{− 𝐺}𝐷𝐷_𝑏

5) The least-squares criterion for the new unconstrained problem reads as:

min 𝑧𝜖𝑍𝑛‖𝒚�𝐷𝐷− 𝐴𝒛‖𝑄2𝑦�𝑦�= �𝒆��𝐷𝐷�𝑄𝑦�𝑦� 2 + min_{𝒛𝜖𝑍𝑛}�𝒛�� − 𝒛�_𝑄 𝑧��𝑧�� 2

With the float solution (𝒛�� 𝜖𝑅𝑛_{), the solution of the linear system of adjusted observations becomes the}

squared norm of the least-squares DD residual vector: min 𝒛𝜖𝑍𝑛�𝒚�𝐷𝐷− 𝐴𝒛���_𝑄 𝑦�𝑦� 2 = �𝒆��𝐷𝐷_� 𝑄𝑦�𝑦� 2

6) As we use the constraint that the solution of 𝒛 has to be an integer 𝒛�𝜖𝑅𝑛, the minimum fixed

solution for the adjusted problem becomes: �𝒚�𝐷𝐷_{− 𝐴𝒛���} 𝑄𝑦�𝑦� 2 = �𝒆��𝐷𝐷_� 𝑄𝑦�𝑦� 2 = �𝒆��𝐷𝐷_� 𝑄𝑦�𝑦� 2 + �𝒛�� − 𝒛���_𝑄 𝑧��𝑧�� 2

The minimum of this adjusted problem is different than the minimum of original problem if 𝒃 is

7) As a last step, the SD residual vector is calculated from the DD residuals. The following transformation is used for every observation type between SD and DD residuals as described in (van der Marel and Gȕndlich, 2006):

𝐷𝑆𝐷→𝐷𝐷𝒆�𝑆𝐷_{= �𝑤}1 𝑤2 𝑤3 … 𝑤𝑛+1 𝐷� � ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑒̌_𝑒̌12 ⋮ 𝑒̌𝑗 ⋮ 𝑒̌𝑛+1_⎦⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡_𝑒̂012 ⋮ 𝑒̂1,𝑖 ⋮ 𝑒̂1,𝑛_⎦⎥ ⎥ ⎥ ⎥ ⎤ = 𝒆�𝐷𝐷

Here 𝐷� is the matrix that forms the difference between GNSS satellites at the same frequency and

observation type. The zero-mean property is applied that states that the weighted sum of SD residuals for a baseline is zero, i.e.:

� 𝑤𝑗_𝑒̌𝑗 𝑛+1 𝑗=1

= 0

with 𝑤𝒋 is the weight of the SD observation. The equation states that the weighted sum of all the SD

residuals to all tracked GNSS satellites for each baseline at each epoch should be zero. If no

weighting is used for the observations than all the entries of 𝑤𝒋 are one. The matrix 𝐷� itself is

non-invertible, by adding the vector [𝑤1 _𝑤2 _𝑤3 … 𝑤𝑛+1] an invertible matrix 𝐷𝑆𝐷→𝐷𝐷 is created.

Using these equations, the inverse transformation of the DD residuals to the SD residuals can be calculated:

𝒆�𝑆𝐷_{= (𝐷𝑆𝐷→𝐷𝐷)}−1_𝒆�𝐷𝐷

This SD residual vector can be applied to evaluate GNSS noise characteristics.

In (van der Marel and Gȕndlich, 2006) no attempt was made to fix the ambiguities, therefore in the

applied multi-epoch solutions 𝒆�𝑺𝑫 would approach 𝒆�𝑫𝑫. In our approach, we will fix the integer

ambiguity vector. After fixing the ambiguities, our noise analysis will be driven by the much more precise carrier phase observations, whereas in section 3.1 the approach was driven by the code observations.

4 EXPERIMENTAL RESULTS

In this section, the developed methods in the previous section will be demonstrated using undifferenced and single differenced residuals obtained from dedicated experiments.

The influence of multipath is also investigated. Experimental results presented are based on observations obtained from high-end receivers on a short base-line configuration. The short base-line allows to exclude the atmospheric errors (ionospheric and tropospheric delays) in the processing and exploitation of relative positioning techniques. We will apply data collected at a short baseline, for zero baselines similar results are expected.

4.1 Undifferenced code noise (data collected in Perth)

In the first experiment, we will apply the method described in section 3.1, on data collected with GNSS receivers located at Curtin University in Perth, Australia. The configuration consists of two Trimble TRM59800.00 SCIS antennas each connected to a Trimble Net R9 receiver. The multipath mitigation was switched off. Figure 1 shows the skyplot characteristic of the site for the days of the experiment. Typically for an experiment on the southern hemisphere, there is an area in the Skyplot toward the south where GPS satellites will not pass. In Figure 2 the u n d i f f e re n c e d residuals are plotted as they vary with time for satellite PRN18 during a single day.

Multipath, which is reflection of the signal on reflective surfaces, reaching the antenna from different directions, is one of the sources of errors in GNSS measurements most difficult to confine. To assess the presence of multipath, the residuals of each distinct satellite have been considered. In Figure 2, it is possible to spot a presence of multipath: the residuals present a periodic pattern with a period of some minutes to tens of minutes. This kind of oscillations is normally attributed to multipath. Multipath is known to be strongly time correlated for sampling rates of 1 Hz or higher and can be reduced by time differencing measurements of two consecutive epochs. A possible approach to reduce the multipath errors in the measurements is to exploit the satellites repeat period: the code residuals of one sidereal day can be subtracted from the residuals of the day after (Axelrad, P., K.M. Larson and B. Jones (2005)). The precise repeat period varies slightly for each satellite, and is slightly

less than the actual sidereal day. As reported in (Axelrad, P., K.M. Larson and B. Jones, 2005), the average repeat shift for the constellation is approximately 244-246 s short of 24 hours. The actual repeat shift is different from the sidereal day depending on the true orbital period of the satellite, and can be determined based on the broadcasted ephemeris parameters.

In this study, three different pair of days observations at 1 second interval have been analyzed (8-9, 11-12 and 15-16 January 2011-12), adopting the the described approach of residuals subtraction, and the repeat interval has been computed for each satellite individually by simply finding the maximum of the auto-correlation function for the code residuals. The resulting shifts are between 239 s and 249 s less than 24 hours. As a result of the linear operation of subtraction, the variance of the resulting

residuals is doubled, but common functional errors will be cancelled.

To analyse this we will make use of the w-test, which is the residuals normalized by the expected standard deviation:

𝑤𝑖=_𝜎𝑒̂𝑒̂𝑖 𝑖

In Figure 4 to Figure 6, the effect of the linear combination (subtracting data of two consecutive days) is shown for PRN6 for the days 11-12 January. The w-tests shown in Figure 4 and Figure 5 have a variance of about 0.75. For the difference in Figure 6, a variance of about 0.8 is obtained. This is

sensibly less than the expected

2𝜎

= 2 × 0.75

c o n t r i b u t e s s ignificantly to the variance of the daily undifferenced w-tests and is also confirms the effectiveness of the multipath cancellation by linear combination.

For static baselines, multipath effects of each GNSS satellite are also repeated with a period close to a sidereal day due to the baseline-satellite geometry. In this section, we demonstrated how this effect can be utilized to compensate for multipath errors on a constrained baseline vector.

Figure 1 Skyplot at Perth site

Figure 2 Undifferenced residuals 𝒆� for satellite PRN18 from measurements at 1 second sampling, one

Figure 3 Code w-test during the first sidereal day for satellite PRN6 (𝝈𝟐∼= 𝟎. 𝟕𝟓)

Figure 4 Code w-test during the second sidereal day for satellite PRN6 (𝝈𝟐∼= 𝟎. 𝟕𝟓)

4.2 Single difference code and carrier noise (Data collected in Delft)

On 6th of September 2010, data was collected in Delft from 8h30 till 10h with two Trimble R7 receivers connected to Trimble Zepher L1/L2 antennas. Again the multipath mitigation function of the receivers was switched off. Raw data was processed at 1 Hz. Each receiver has its own antenna forming a short baseline of a few meters. Figure 6 shows the Skyplot of the visible GPS satellites during the experiment and from this figure it is clear that a number of satellites are low above the horizon.

Figure 7 till Figure 8 show the multipath combination as described in section 2.3 for one of the

receivers and the Signal to Noise (SNR) values. From these multipath combination figures, it is clear that the observations collected from the GPS satellites at lower elevations angles experience in general more multipath. To confirm the presence of multipath, the SNRs for the L1 frequency has been analyzed and included in the figures, which presents a similarly oscillating behavior as the multipath combination.

In Figure 9 till Figure 10, the single difference code and carrier phase residuals for selected PRNs are

shown calculated using the procedure described in section 3.2. These plots also contain the elevations angles. The single difference residuals are only plotted if code and carrier phase are locked at both receivers of the baseline. Ambiguity resolution using standard LAMBDA was found to be sufficient for processing this data collected with geodetic grade receivers in a relatively multipath free environment. After fixing the ambiguities, the noise analysis is driven by the much more precise carrier phase observations. As a result, in the figures, we observe that the residuals do not only contain random errors but also systematic errors, most likely due to multipath. Especially carrier phase multipath for low elevation GNSS satellites is apparent. The oscillating behavior is very similar as found for the SNR and Multipath combination.

From literature it is known that code multipath to easily become a few meters, whereas carrier phase multipath is expected to remain within a few centimeters (Braasch, M.S. and A.J. Van Dierendonck, 1999). The SD carrier phase residuals show clearly oscillating behaviour with periods of a few minutes, with a maximum residual of about 2 centimeter.

The results showed that noise characteristics of both pseudo range and carrier phase measurements and multipath behaviour is much more distinctive through constrained baseline vector analysis.

Figure 7: Multipath combination and SNR for various channels of the first receiver M c1 2 [ m] M c1 2 [ m] M c1 2 [ m] M c1 2 [ m] S 1 [ d B /Hz ] S 1 [ d B /Hz ] S1 [ d B /Hz ] S1 [ d B /Hz ] PRN 12 3 PRN 12 55 2 50 45 1 40 0 35 −1 30 −2 ₂₅ −3 08:30 09:00 09:30 10:00 06−Sep−2010 20 08:30 09:00 09:30 10:00 06−Sep−2010 (a) (b) PRN 14 3 PRN 14 55 2 50 45 1 40 0 35 −1 30 −2 25 −3 08:30 09:00 09:30 10:00 06−Sep−2010 20 08:30 09:00 09:30 10:00 06−Sep−2010 (c) (d) PRN 15 3 PRN 15 55 2 50 45 1 40 0 35 −1 30 −2 25 −3 08:30 09:00 09:30 10:00 06−Sep−2010 20 08:30 09:00 09:30 10:00 06−Sep−2010 (e) (f) PRN 18 3 PRN 18 55 2 50 45 1 40 0 35 −1 30 −2 25 −3 08:30 09:00 09:30 10:00 06−Sep−2010 20 08:30 09:00 09:30 10:00 06−Sep−2010 (g) (h)

Figure 8: Multipath combination and SNR for various channels of the first receiver M c1 2 [ m] M c1 2 [ m] M c1 2 [ m] M c1 2 [ m] S1 [ d B /Hz ] S1 [ d B /Hz ] S1 [ d B /Hz ] S1 [ d B /Hz ] PRN 22 3 PRN 22 55 2 50 45 1 40 0 35 −1 30 −2 ₂₅ −3 08:30 09:00 09:30 10:00 06−Sep−2010 20 08:30 09:00 09:30 10:00 06−Sep−2010 (a) (b) PRN 26 3 PRN 26 55 2 50 45 1 40 0 35 −1 30 −2 ₂₅ −3 08:30 09:00 09:30 10:00 06−Sep−2010 20 08:30 09:00 09:30 10:00 06−Sep−2010 (c) (d) PRN 28 3 PRN 28 55 2 50 45 1 40 0 35 −1 30 −2 ₂₅ −3 08:30 09:00 09:30 10:00 06−Sep−2010 20 08:30 09:00 09:30 10:00 06−Sep−2010 (e) (f) PRN 30 3 PRN 30 55 2 50 45 1 40 0 35 −1 30 −2 ₂₅ −3 08:30 09:00 09:30 10:00 06−Sep−2010 20 08:30 09:00 09:30 10:00 06−Sep−2010 (g) (h)

Figure 9: Elevation angle, SD code and carrier residuals for the various channels SD C ode r es idu al [ m] SD C ode r es idu al [ m] S D C ode r es idu al [ m] S D C ode r es idu al [ m] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] SD Ca rri e r res idu al [ m] SD C a rri e r res idu al [ m] S D C a rri e r res idu al [ m] S D C a rri e r res idu al [ m] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] PRN12 5 4 3 2 1 0 −1 −2 −3 −4 −5 90 0.05 80 0.04 70 0.03 60 0.02 0.01 50 0 40 −0.01 30 _−0.02 20 _−0.03 10 _−0.04 0 −0.05 PRN12 90 80 70 60 50 40 30 20 10 0 8.5 9 9.5 10 8.5 9 9.5 10 (a) (b) PRN14 5 4 3 2 1 0 −1 −2 −3 −4 −5 90 0.05 80 0.04 70 0.03 60 0.02 0.01 50 0 40 −0.01 30 −0.02 20 _−0.03 10 _−0.04 0 −0.05 PRN14 90 80 70 60 50 40 30 20 10 0 8.5 9 9.5 10 8.5 9 9.5 10 (c) (d) PRN15 5 4 3 2 1 0 −1 −2 −3 −4 −5 90 0.05 80 0.04 70 0.03 60 0.02 0.01 50 0 40 −0.01 30 −0.02 20 _−0.03 10 _−0.04 0 −0.05 PRN15 90 80 70 60 50 40 30 20 10 0 8.5 9 9.5 10 8.5 9 9.5 10 (e) (f) PRN18 5 4 3 2 1 0 −1 −2 −3 −4 −5 90 0.05 80 0.04 70 0.03 60 0.02 0.01 50 0 40 −0.01 30 _−0.02 20 _−0.03 10 _−0.04 0 −0.05 PRN18 90 80 70 60 50 40 30 20 10 0 8.5 9 9.5 10 8.5 9 9.5 10 (g) (h)

Figure 10 Elevation angle, SD code and carrier residuals for the various channels SD C ode r es idu al [ m] SD C ode r es idu al [ m] S D C ode r es idu al [ m] S D C ode r es idu al [ m] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] SD C a rri e r res idu al [ m] SD C a rri e r res idu al [ m] S D C a rri e r res idu al [ m] S D C a rri e r res idu al [ m] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] E lev at ion an gl e [ d eg] PRN22 5 4 3 2 1 0 −1 −2 −3 −4 −5 90 0.05 80 0.04 70 0.03 60 0.02 0.01 50 0 40 −0.01 30 −0.02 20 _−0.03 10 _−0.04 0 −0.05 PRN22 90 80 70 60 50 40 30 20 10 0 8.5 9 9.5 10 8.5 9 9.5 10 (a) (b) PRN26 5 4 3 2 1 0 −1 −2 −3 −4 −5 90 0.05 80 0.04 70 0.03 60 0.02 0.01 50 0 40 −0.01 30 −0.02 20 _−0.03 10 _−0.04 0 −0.05 PRN26 90 80 70 60 50 40 30 20 10 0 8.5 9 9.5 10 8.5 9 9.5 10 (c) (d) PRN28 5 4 3 2 1 0 −1 −2 −3 −4 −5 90 0.05 80 0.04 70 0.03 60 0.02 0.01 50 0 40 −0.01 30 _−0.02 20 _−0.03 10 _−0.04 0 −0.05 PRN28 90 80 70 60 50 40 30 20 10 0 8.5 9 9.5 10 8.5 9 9.5 10 (e) (f) PRN30 5 4 3 2 1 0 −1 −2 −3 −4 −5 90 0.05 80 0.04 70 0.03 60 0.02 0.01 50 0 40 −0.01 30 −0.02 20 _−0.03 10 _−0.04 0 −0.05 PRN30 90 80 70 60 50 40 30 20 10 0 8.5 9 9.5 10 8.5 9 9.5 10 (g) (h)

6 CONCLUSION

GNSS noise characteristics can be obtained by short and zero baseline analysis, for both pseudorange and carrier phase measurements. Differencing observations from a GNSS satellite tracked by two GNSS receivers (connected to the same or different antennas) will explicitly eliminate common errors.

In this contribution, the known baseline vector between the antennas was used to adjust (“constrain”) the observation vectors. The advantages of a noise analysis through a constrained baseline vector over an ordinary baseline processing was discussed and demonstrated using undifferenced and single differenced residuals obtained from dedicated experiments.

Two approaches were investigated; one based on undifferenced code observations and the other on single differenced code and carrier observations. The later one includes integer ambiguity estimation. After fixing the ambiguities, it was demonstrated that the noise analysis will be driven by the much more precise carrier phase observations.

The results showed that noise characteristics of both pseudo range and carrier phase measurements and multipath behaviour is much more distinctive through constrained baseline vector analysis. This can be explained that it utilized a much stronger model where only one unknown (the clock error) is remaining in the residuals estimation, instead of the full relative position and time vector of an ordinary undifferenced and single differenced baseline processing. The multipath repeat period was exploited by differencing the measurements residuals over two sidereal days: this allowed isolating multipath effects and unveiling the residuals white noise nature.

ACKNOWLEDGEMENTS

Gabriele Giorgi, Roel van Bree, Noor Raziq, Lennard Huisman and Dennis Odijk are acknowledged for their support for the experiments described in this contribution. Peter Teunissen is acknowledged for the theory behind ambiguity resolution, and Christian Tiberius for fruitful discussion on GNSS noise analysis.

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