Error sources

The assumption that all error sources can be neglected is seldom an appropriate one for precise applications, but for some use cases these can be reduced. Sources of error include:

ˆ Tropospheric refraction, where the signal is delayed while passing through the troposphere. This can be divided into a dry and a wet portion; the dry portion causes more refraction, but it's easier to model and compensate for, so the wet portion has more impact on the nal position solution.

∗_{Some literature measures the carrier pseudorange in metres, multiplying the integer ambiguity by}

the wavelength; other literature measures the carrier pseudorange in cycles, dividing the true distance and receiver clock error by the wavelength. Sources such as Chang et al. (2005) even measure the code pseudorange in wavelengths.

CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 44 ˆ Ionospheric refraction, where the signal is delayed while passing through the ionosphere. The data broadcast by GPS satellites includes Klobuchar model parameters to model the state of the ionosphere, allowing some mitigation. ˆ Ephemeris errors (as mentioned in section 3.1 on page 33) where, because there is

only a limited bandwidth for broadcasting satellite orbit information, imperfect but succinct information is sent.

ˆ Measurement noise, where receivers cannot measure signals with complete pre- cision. Carrier wave signals, which have a much shorter wavelength, encounter much less measurement noise than C\A code measurements.

ˆ Multipath, where a receiver detects a reected signal (which has taken a longer path) rather than a direct signal from a satellite. Signals can reect o buildings or the ground. High-cost receivers use special antennas with choke ring ground planes; quarter-wavelength grooves attenuate signals from low altitudes as the half-cycle-oset conducted signal interferes destructively. Attenuating signals from low altitudes helps with reections from the ground, but doesn't oer the same benets with signals reected from above the ground, e.g. from buildings. The highest precision GPS applications demand an unobstructed, reection-free view of the sky for this reason.

ˆ Poor satellite geometry, where other errors have a magnied inuence on the nal position error, as shown in gure 3.10 . As this depends on the position of satellites in the sky it can be predicted in advance, making it possible to choose times when the eects will be at their lowest.

ˆ Poor satellite visibility, where at certain times of the day fewer satellites are in view, reducing redundancy and the ability to take averages. As this depends on the position of satellites in the sky it can be predicted in advance, making it possible to choose times when the eects will be at their lowest.

CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 45

Figure 3.10: Eects of satellite position on accuracy: In the left image, with satellites spread out, receiver movement of 1 metre causes a range change of 90 centimetres; in the right image, with satellites grouped together the same movement changes the ranges by only 30 centimetres. Hence, in the former case a range measurement error of 1 metre would lead to a position error of 1.11 metres, while in the latter case the same error would lead to a position error of 3.33 metres. GPS receivers report Dilution Of Precision (DOP) to numerically represent this eect. For more information, see section 3.4.5.4 on page 73.

Error source Means of mitigation

Tropospheric refraction Modelling (IGS data); dierential corrections. Ionospheric refraction Modelling (broadcast or IGS data); dierential

corrections.

Satellite ephemeris errors Modelling (IGS data); dierential corrections. Measurement noise Low-noise carrier phase measurement.

Multipath Antenna selection and placement. Poor satellite geometry Select times for best satellite geometry.

Poor satellite visibility Select times for best satellite visibility. Operator error, software bugs Detect and avoid!

CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 46 Error sources are summarised in table 3.1 . Our application allows some of these to be mitigated through dierential corrections and carrier phase measurement, as noted in the table.

Chang et al. (2005) takes some of these errors into account, using this formulation for code pseudorange:

Rs_r =%s_r−I_rs+T_rs+cδr+dr−ds+νrs (3.6)

whereIs

r is the error caused by the ionosphere (in metres); Trs is the error caused by the

troposphere (in metres); dr and ds are the receiver and satellite hardware code delays

(in metres); and νs

r is the code measurement noise and multipath noise (in metres).

Similarly, the formulation for the carrier pseudorange is:

λΦs_r=%s_r+I_rs+T_rs+cδr+λNrs+Dr−Ds+ϕs0 +ϕr0 +µ

s

r (3.7)

where Dr and Ds are the receiver and satellite hardware carrier delay (in metres); ϕs

0 and ϕr0 are the satellite and receiver initial phase

∗_{; and µ}s

r is the carrier measu-

rement noise and multipath noise (in metres). You might wonder why Is

r is negative

in equation 3.6 but positive in equation 3.7. This is because the carrier is a single frequency signal (and hence propagates at the phase velocity), while the C/A code is comprised of several signals of slightly dierent frequencies (and hence propagates at the group velocity). Although the phase velocity and the group velocity would be the same in a vacuum, the frequency-dependent eects of the ionosphere have opposite eects on the phase velocity and the group velocity. For more information, see section 5.3 of Hofmann-Wellenhof et al. (2008).

3.3.3.1 Precision and accuracy

In discussing errors the terms precise and accurate are often used; as shown in - gure 3.11 , measurements are precise if they are tightly grouped together and accurate if the grouping is centred around the true value. Geographical coordinates make this

CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 47

a. Not precise, not accurate

b. Accurate, not precise

c. Precise, not accurate

d. Precise and accurate

slightly more complicated due to dierent frames of reference - true values change as tectonic plates move relative to one another, unless measurements are taken relative to a xed point on the same tectonic plate. Further complicating matters, some systems aren't precise enough to notice tectonic plate motion and don't take it into account. For the purposes of our application, if measurements are inaccurate due to a syste- matic error like incorrect ambiguity resolution that is a dierent matter to if they are inaccurate due to tectonic plate motion. For the avoidance of doubt, unless otherwise specied the term accurate is used to refer to accuracy relative to a nearby xed base station.

CHAPTER 3. INTRODUCTION TO SATELLITE NAVIGATION 48