GNSS and positioning

Draft

Satellite navigation is a distributed clock comparison dressed up as a positioning system. Understanding it properly means starting from the clock, because almost every design feature follows from the difficulty of comparing time across 20 000 km.

The measurement

A GNSS satellite continuously broadcasts a signal stamped with the time of transmission and its own orbital position. A receiver notes the time of arrival by its own clock and computes

\rho = c\,(t_{\text{received}} - t_{\text{transmitted}})

If both clocks were perfectly synchronised, $\rho$ would be the true geometric range and three satellites would suffice to trilaterate a position in three dimensions.

They are not synchronised. The satellite carries a caesium or rubidium standard, disciplined by the ground control segment; the receiver carries a quartz oscillator costing a few cents. A receiver clock error of one microsecond corresponds to 300 metres of range error, and cheap quartz drifts far more than that within seconds.

The measurement is therefore not a range but a pseudorange, contaminated by an unknown receiver clock offset $\delta t$ that is common to all satellites simultaneously observed:

Pseudorange

\rho_i = \sqrt{(x_i-x)^2 + (y_i-y)^2 + (z_i-z)^2} + c\,\delta t + \epsilon_i

There are four unknowns: three position coordinates and the clock offset. Four satellites give four equations, and the system is solvable.

This is the single most important idea in GNSS and it is elegant. The receiver does not need an accurate clock, because the clock error is solved for as an output. A consequence is that a GNSS receiver is also a precision timing device, and that turns out to be at least as economically important as the positioning: telecommunications networks, power grids and financial exchanges are synchronised by GNSS, and their dependency is a systemic vulnerability discussed below.

The constellations

System	Operator	Satellites	Planes	Inclination	Altitude	Period
GPS	United States	24 baseline, ~31 operational	6	55°	20 180 km	11 h 58 m
GLONASS	Russia	24	3	64.8°	19 100 km	11 h 16 m
Galileo	European Union	24 plus spares	3	56°	23 222 km	14 h 5 m
BeiDou-3	China	24 MEO plus 3 GEO and 3 IGSO	3 MEO	55°	21 528 km	12 h 53 m
NavIC	India	7, regional	—	Mixed GEO and IGSO	35 786 km	24 h
QZSS	Japan	4, regional augmentation	—	IGSO and GEO	~35 786 km	24 h

Three design choices are common to all the global systems and each has a reason.

Medium Earth orbit. The requirement is that at least four satellites, and preferably six to ten, are visible from anywhere at once with good angular spread. LEO would need hundreds of satellites; GEO cannot cover high latitudes and gives poor vertical geometry. MEO delivers global visibility from a few dozen satellites, and its slow orbital motion means the geometry changes gradually, which suits a receiver tracking continuously.

Moderate inclination near 55°. High enough to cover populated latitudes with good geometry, low enough that satellites are not concentrated near the poles where they would give poor coverage at the equator.

Orbital periods near half a sidereal day. GPS repeats its ground track daily after exactly two revolutions, so the geometry at a given site repeats each day. This makes system performance predictable and simplifies multipath characterisation, since the same reflections recur.

Note the radiation cost of that MEO choice. GNSS satellites sit inside the outer belt, in the harshest radiation environment of any operational regime, which is part of why they are expensive and long-lived rather than cheap and replaceable. See Space radiation.

The regional systems, NavIC and QZSS, use inclined geosynchronous orbits so that satellites remain continuously high in the sky over their service area. For Japan in particular this matters: in urban canyons a satellite near the horizon is blocked by buildings, and QZSS is designed so at least one satellite is always near zenith over Tokyo.

The error budget

The residual $\epsilon_i$ in the pseudorange equation is the sum of several independent contributions, collectively called the user equivalent range error.

Source	Magnitude, single frequency	With dual frequency	Character
Satellite clock	~1.1 m	~1.1 m	Corrected by broadcast parameters; residual from prediction age
Ephemeris	~0.8 m	~0.8 m	Error in the broadcast satellite position
Ionosphere	5 to 15 m	~0.1 m	The dominant term, and removable
Troposphere	0.2 to 0.7 m	0.2 to 0.7 m	Modelled, not measured; wet component is the hard part
Multipath	0.5 to 2 m	0.5 to 2 m	Site-specific reflections; worst in urban environments
Receiver noise	0.1 to 0.5 m	0.1 to 0.5 m	Thermal and quantisation
Total UERE	~5 to 7 m	~1.5 m

The ionosphere and why two frequencies matter

The ionosphere is a dispersive plasma: it delays the signal by an amount proportional to the total electron content along the path and inversely proportional to the square of the frequency.

\delta_{\text{ion}} \approx \frac{40.3\,\text{TEC}}{f^2}

At a moderate 50 TEC units:

Signal	Frequency	Delay
L1	1575.42 MHz	8.1 m
L2	1227.60 MHz	13.4 m
L5	1176.45 MHz	14.6 m

That frequency dependence is not a nuisance, it is the solution. Because the delay is a known function of frequency, measuring the same signal at two frequencies gives two equations in the unknown TEC, and forming the appropriate linear combination cancels the first-order ionospheric delay almost entirely. This is why every precision GNSS receiver is dual-frequency and why the civil L5 signal was added: it converts the largest error source into a small residual.

Single-frequency receivers instead apply a broadcast model, the Klobuchar model for GPS, which removes roughly half the error on average and considerably less during ionospheric disturbances.

Dilution of precision

Position error is not simply the range error. It is the range error amplified by the geometry of the visible satellites:

Dilution of precision

\sigma_{\text{position}} = \text{PDOP} \times \sigma_{\text{UERE}}

The intuition is straightforward. If all visible satellites lie in the same direction, their range measurements constrain position strongly along that direction and weakly across it, so the error ellipse is enormously elongated. Satellites spread across the whole sky constrain all directions comparably.

DOP term	Constrains
GDOP	Position and time together
PDOP	Three-dimensional position
HDOP	Horizontal position
VDOP	Vertical position
TDOP	Time

PDOP	Quality
Below 1	Ideal, rarely achieved
1 to 2	Excellent
2 to 5	Good, typical open-sky
5 to 10	Marginal
Above 10	Poor; positions should be treated with suspicion

Combining the two: a dual-frequency receiver with 1.5 m UERE and PDOP of 1.8 achieves about 2.7 m; a single-frequency receiver with 7 m UERE in a partially obstructed site with PDOP 2.5 gets 17.5 m.

Vertical accuracy is always worse than horizontal, typically by a factor of 1.5 to 2.5. The reason is geometric and unavoidable: all satellites are above the receiver, none below, so the vertical direction is constrained from one side only. Terrestrial systems that need vertical precision use barometric augmentation for exactly this reason.

Augmentation

Raw GNSS is not accurate enough for many applications, and a layered set of augmentation techniques exists.

Technique	Accuracy	Coverage	How it works
SBAS: WAAS, EGNOS, GAGAN, MSAS	1 to 3 m	Regional, by satellite broadcast	A network of reference stations computes corrections, uplinked and rebroadcast from geostationary satellites
DGPS	Sub-metre	Local, within ~100 km of a base	A base station at a known location broadcasts its own pseudorange errors
RTK	1 to 2 cm	Local, within ~20 km of a base	Uses the carrier phase rather than the code, resolving integer cycle ambiguities
PPP	2 to 10 cm	Global	Precise satellite orbit and clock products, no local base, but 10 to 30 minutes to converge

RTK deserves explanation because the accuracy jump is so large. The code signal has a chip length of hundreds of metres, so code-based ranging resolves to a fraction of that. The carrier wave has a wavelength of about 19 cm at L1, and its phase can be tracked to a hundredth of a cycle, or about 2 mm.

The difficulty is that phase measurement gives the fractional part of the cycle count, not the integer number of whole wavelengths between satellite and receiver. Resolving those integer ambiguities is the central problem of precise GNSS, and it is solvable given enough measurements over enough time or a nearby base station whose errors are nearly identical. This is what makes centimetre agriculture, machine control and surveying possible.

PPP achieves comparable accuracy without a local base, using globally computed precise orbit and clock corrections, at the cost of a long convergence time. The two techniques are increasingly blended as PPP-RTK.

Integrity, jamming and spoofing

Integrity is a distinct requirement from accuracy: it is the ability to detect when the system is giving a wrong answer and to warn the user within a specified time. Aviation requires it, because an undetected 50 m error during approach is far more dangerous than a known 50 m uncertainty. Receiver autonomous integrity monitoring uses redundant satellites to cross-check consistency, and SBAS broadcasts integrity messages alongside its corrections.

The vulnerability that matters most is a consequence of link budget arithmetic. A GNSS signal arrives at Earth’s surface at roughly −158.5 dBW, which is below the thermal noise floor. It is recovered only by correlating against the known spreading code, which provides substantial processing gain.

Jamming: A transmitter of a few watts can deny GNSS over many kilometres, because it needs only to raise the noise floor above what the correlator can overcome. Cheap illegal jammers are widely available and routinely disrupt airports.
Spoofing: Broadcasting counterfeit signals that a receiver accepts as genuine, gradually walking its computed position or time away from the truth. Far more dangerous than jamming, because the receiver reports a confident wrong answer rather than no answer.
Meaconing: Recording and rebroadcasting genuine signals with a delay. Simple to execute and hard to distinguish from multipath.
Systemic exposure: Power grids, telecom networks and financial timestamping depend on GNSS time. A timing attack has consequences well beyond navigation.

Countermeasures include controlled reception pattern antennas that null out signals from the horizon where jammers usually are, cryptographic authentication of the navigation message, which Galileo has deployed as OSNMA, inertial coupling so a receiver can detect implausible position jumps, and terrestrial backup timing sources.

Multi-constellation receivers

Modern receivers track several constellations at once, and the benefits compound in ways worth stating separately.

More satellites visible means better geometry, so PDOP falls. It means better performance in obstructed environments, since urban canyons and forest canopy block different parts of the sky and more satellites raise the chance that enough remain visible. It means redundancy against one system being degraded or denied. And it improves integrity, since inconsistency between constellations is itself evidence that something is wrong.

The practical complication is that each system uses its own reference frame and its own time scale, so a multi-constellation solution must estimate inter-system time offsets as additional unknowns, requiring one extra satellite per additional constellation. With thirty or more satellites visible this is a small price.

References

Misra, P. and Enge, P. Global Positioning System: Signals, Measurements and Performance, 2nd ed., Ganga-Jamuna Press, 2011.
Teunissen, P. J. G. and Montenbruck, O. (eds.) Springer Handbook of Global Navigation Satellite Systems, Springer, 2017.
Kaplan, E. D. and Hegarty, C. J. (eds.) Understanding GPS/GNSS: Principles and Applications, 3rd ed., Artech House, 2017.
Global Positioning System Standard Positioning Service Performance Standard, US Department of Defense, current edition.
Humphreys, T. E. et al. “Assessing the Spoofing Threat: Development of a Portable GPS Civilian Spoofer”, ION GNSS 2008.
Ashby, N. “Relativity in the Global Positioning System”, Living Reviews in Relativity, 6:1, 2003. See also Relativity for space systems.