Error Sources

Overview#

Every GNSS position is corrupted by errors from multiple sources. Understanding these errors -- their magnitudes, their behavior, and how to mitigate them -- is what separates a surveyor who achieves reliable centimeter-level results from one who does not. The difference between a 5-meter autonomous position and a 1-centimeter RTK position is not better hardware alone; it is the systematic elimination or cancellation of errors through differential techniques, modeling, and proper field procedures.

GNSS error sources fall into four broad categories: satellite-based errors, atmospheric errors, receiver-based errors, and site-dependent errors. Some are systematic and predictable; others are random and must be managed through observation design. Some cancel in differential processing; others do not.

"The key to high accuracy with GPS is the elimination or significant reduction of the biases and errors that affect the observations. Most precise GPS techniques achieve their accuracy not because the raw observations are inherently precise, but because the error sources can be modeled, differenced, or otherwise mitigated." -- Van Sickle, GPS for Land Surveyors (4th Ed.), Ch. 7, p. 145

Satellite-Based Errors#

Satellite Clock Errors

Each GPS satellite carries multiple atomic clocks (rubidium and cesium), but even atomic clocks drift. The control segment monitors satellite clocks and uploads correction parameters (polynomial coefficients $a_0$, $a_1$, $a_2$) in the navigation message. After applying these corrections, residual satellite clock errors are typically less than a few nanoseconds, corresponding to range errors of approximately 1--2 meters.

For survey-grade work, satellite clock errors are effectively eliminated through differential processing. When both the base and rover observe the same satellite simultaneously, the satellite clock error cancels in the between-receiver difference.

For Precise Point Positioning (PPP), which uses a single receiver, satellite clock corrections from IGS or other analysis centers (accurate to ~0.1 ns, or ~3 cm) replace the broadcast corrections.

Satellite Orbit Errors

The broadcast ephemeris provides satellite positions that are typically accurate to 1--2 meters. This orbit error translates directly into a range error of similar magnitude for single-point positioning. However, orbit errors are largely spatially correlated over short to moderate baselines.

The effect of orbit error on a baseline of length $b$ is approximately:

$\Delta b \approx b \times \frac{\Delta \rho}{\rho_s}$

where $\Delta \rho$ is the orbit error and $\rho_s$ is the satellite range (~20,200 km). For a 10 km baseline with a 2-meter orbit error:

$\Delta b \approx 10{,}000 \times \frac{2}{20{,}200{,}000} \approx 0.001 \text{ m} = 1 \text{ mm}$

This spatial decorrelation is why orbit errors have minimal effect on short baselines in differential processing but become significant as baseline length increases.

"For baselines shorter than about 10 km, broadcast orbit errors are essentially negligible in differential processing. For longer baselines, precise ephemerides should be used." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 14, p. 401

Atmospheric Errors#

Ionospheric Delay

The ionosphere is a region of the atmosphere (~60--1000 km altitude) where solar radiation ionizes gas molecules, creating free electrons. These free electrons affect the propagation of radio signals in a frequency-dependent manner. The ionospheric delay on the pseudorange (group delay) is:

$d_{\text{iono}} = \frac{40.3 \times \text{TEC}}{f^2}$

where:

• $d_{\text{iono}}$ is the ionospheric delay in meters
• $\text{TEC}$ is the Total Electron Content along the signal path (in electrons per square meter, or TECU where $1 \text{ TECU} = 10^{16} \text{ el/m}^2$)
• $f$ is the signal frequency in Hz

The ionospheric delay on the carrier phase has the same magnitude but opposite sign -- the ionosphere advances the phase while delaying the code. This is a consequence of the phase velocity exceeding the group velocity in a dispersive medium.

Key characteristics of ionospheric delay:

• Magnitude. Typically 2--15 meters at L1, but can exceed 30 meters during periods of high solar activity (solar maximum) or during ionospheric storms.
• Frequency dependence. The $1/f^2$ relationship means L2 experiences approximately 65% more delay than L1 for the same path.
• Diurnal variation. TEC is highest in the early afternoon (local time) and lowest around midnight. The delay varies by a factor of 5--10 over a day.
• Solar cycle. The 11-year solar cycle drives long-term variation in ionospheric activity. Solar maximum conditions can increase TEC by a factor of 3--5 compared to solar minimum.
• Spatial correlation. Ionospheric conditions are similar for nearby receivers, which is why the error largely cancels in differential processing over short baselines. Over long baselines (>50 km), the ionospheric gradient may not cancel effectively.

Mitigation strategies:

Tropospheric Delay

The troposphere (0--40 km altitude) delays GNSS signals due to the refractivity of the neutral atmosphere. Unlike the ionosphere, the tropospheric delay is not frequency-dependent and therefore cannot be eliminated by dual-frequency combinations.

The tropospheric delay has two components:

Dry (hydrostatic) component:

• Accounts for approximately 90% of the total tropospheric delay (~2.3 m at zenith)
• Highly predictable from surface pressure measurements
• Can be modeled to ~1 cm accuracy using models such as Saastamoinen or Hopfield

Wet component:

• Accounts for approximately 10% (~0.1--0.4 m at zenith, but highly variable)
• Depends on water vapor content, which varies rapidly in space and time
• Much harder to model; residual errors of 1--5 cm are common
• The primary limiting factor in tropospheric correction for long baselines

The total tropospheric delay at an elevation angle $E$ is approximately:

$d_{\text{tropo}} = d_{\text{zenith}} \times m(E)$

where $d_{\text{zenith}}$ is the zenith delay and $m(E)$ is the mapping function that accounts for the increased path length through the atmosphere at lower elevation angles:

$m(E) \approx \frac{1}{\sin E}$

This is a simplified approximation; operational mapping functions (e.g., Niell, VMF1, GMF) use more sophisticated formulations that account for the curvature of the atmosphere.

At an elevation angle of 10 degrees, the mapping function amplifies the zenith delay by a factor of approximately 5.7, which is why observations below 10--15 degrees elevation are often excluded or down-weighted.

"The wet component of the tropospheric delay is the most troublesome atmospheric error source in GPS surveying, because it cannot be removed by dual-frequency observations and is difficult to model accurately." -- GEOG 862, GPS and GNSS for Geospatial Professionals, Penn State, Lesson 6

Mitigation strategies:

Receiver-Based Errors#

Receiver Clock Bias

Receiver clocks use inexpensive quartz oscillators with stability orders of magnitude worse than the satellite atomic clocks. The receiver clock error can be milliseconds or more -- equivalent to hundreds of kilometers of range error. However, this is not a problem in practice because the receiver clock bias is estimated as a parameter in the position solution (which is why four satellites are needed rather than three) and cancels completely in between-receiver differencing.

Receiver Noise

Thermal noise in the receiver's electronics introduces random errors into both pseudorange and carrier phase measurements. The magnitude depends on receiver quality and signal strength:

Antenna Phase Center Variation

The electrical phase center of a GNSS antenna -- the point to which measurements are actually referenced -- does not coincide exactly with the physical antenna reference point (ARP). Furthermore, the phase center varies with the elevation and azimuth of the incoming signal. This phase center variation (PCV) introduces systematic errors that can reach several centimeters vertically if uncalibrated.

For survey-grade work:

• Use absolute antenna calibration models (e.g., from NGS or IfE/Geo++ in ANTEX format)
• Apply PCV corrections in processing software
• When mixing antenna types on a baseline, ensure both antennas have proper calibration models applied

"Antenna phase center corrections are essential for achieving centimeter-level accuracy. Ignoring phase center variations can introduce vertical errors of 5--10 cm, depending on the antenna and the elevation distribution of observed satellites." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 14, p. 406

Site-Dependent Errors#

Multipath

Multipath occurs when the satellite signal reaches the receiver via indirect paths -- reflected from buildings, vehicles, fences, the ground, water bodies, or other surfaces -- in addition to the direct path. The reflected signal interferes with the direct signal, corrupting both the pseudorange and carrier phase measurements.

Multipath characteristics:

The most effective multipath mitigation strategies are:

1. Site selection. Avoid setting up near reflective surfaces (buildings, vehicles, chain-link fences, bodies of water).
2. Antenna design. Ground planes and choke ring antennas attenuate signals arriving from below the antenna. These are standard for static control surveys.
3. Observation duration. For static surveys, observing for multiples of the sidereal repeat period (~23h 56m) allows multipath signatures to be averaged out.
4. Elevation mask. Higher elevation masks reduce low-angle multipath at the cost of fewer observations.

Signal Obstruction

Obstructions (buildings, terrain, dense tree canopy) can block satellite signals entirely, reducing the number of available satellites, degrading geometry (higher PDOP), and potentially causing cycle slips as signals are intermittently blocked.

Dense tree canopy is particularly problematic because it can attenuate the signal enough to cause tracking loss and cycle slips without completely blocking the signal. In forested environments, static sessions should be extended to compensate for reduced satellite availability and degraded data quality.

Near-Field Effects

Objects in the immediate vicinity of the antenna -- the tripod, the observer, equipment cases, vehicles -- can diffract and reflect signals, introducing systematic errors. These near-field effects are difficult to model and are best mitigated by maintaining a clean antenna environment.

Relativistic Effects#

Einstein's theory of relativity introduces two effects on GPS satellite clocks:

Special relativity (time dilation): Because GPS satellites are moving at approximately 3.87 km/s, their clocks run slower than clocks on the ground by about 7 microseconds per day.

General relativity (gravitational time dilation): Because GPS satellites are at higher altitude (weaker gravitational field), their clocks run faster than ground clocks by about 45 microseconds per day.

The net effect is that satellite clocks run approximately 38 microseconds per day faster than identical clocks on the ground. This corresponds to a range error of about 11.4 km/day if uncorrected. The correction is applied by adjusting the satellite clock frequency before launch:

$f_{\text{adjusted}} = f_0 \times \left(1 - 4.4647 \times 10^{-10}\right) = 10.22999999543 \text{ MHz}$

instead of the nominal $f_0 = 10.23 \text{ MHz}$. A smaller periodic relativistic correction due to orbital eccentricity is applied by the receiver using parameters from the navigation message.

"The relativistic correction is not optional -- without it, GPS would accumulate roughly 11 km of position error per day. It is one of the most striking practical confirmations of general relativity." -- Van Sickle, GPS for Land Surveyors (4th Ed.), Ch. 7, p. 156

Error Budget Summary#

The following table summarizes the major error sources, their approximate magnitudes, and how they are mitigated in different positioning modes:

Net Accuracy by Method

The Power of Differencing#

The foundation of survey-grade GNSS accuracy is the systematic cancellation of errors through differencing:

Single difference (between receivers): Eliminates satellite clock error.

Double difference (between receivers and between satellites): Eliminates both satellite and receiver clock errors.

Triple difference (double differences at two epochs): Eliminates integer ambiguities (but at the cost of precision). Useful for cycle slip detection and initial position estimation.

The double-difference carrier phase observation is:

$\nabla\Delta\Phi_{AB}^{jk} = \nabla\Delta\rho_{AB}^{jk} - \nabla\Delta d_{\text{iono}} + \nabla\Delta d_{\text{tropo}} + \lambda \nabla\Delta N_{AB}^{jk} + \nabla\Delta\epsilon$

where superscripts $j,k$ denote satellites and subscripts $A,B$ denote receivers. Note that all clock terms have been eliminated. For short baselines, the atmospheric residuals ($\nabla\Delta d_{\text{iono}}$ and $\nabla\Delta d_{\text{tropo}}$) are small because both receivers observe through essentially the same atmosphere.

Key Takeaways#

• GNSS errors fall into four categories: satellite-based (clock, orbit), atmospheric (ionosphere, troposphere), receiver-based (clock bias, noise, antenna PCV), and site-dependent (multipath, obstruction).
• Ionospheric delay is the largest atmospheric error (2--30 m), is frequency-dependent, and is eliminated by dual-frequency combinations or short-baseline differencing.
• Tropospheric delay cannot be removed by dual-frequency combinations. The wet component is the most difficult atmospheric error to model accurately.
• Multipath is the dominant remaining error source in differential positioning because it does not cancel between receivers. It is best mitigated by antenna design and careful site selection.
• Differential processing (single, double, triple differences) is the mechanism by which most systematic errors are eliminated. Double differencing removes both satellite and receiver clock errors.
• For short baselines (<10 km), atmospheric and orbit errors are highly correlated between receivers and largely cancel. For long baselines, these errors decorrelate and become the limiting factor.
• Relativistic effects would cause 11.4 km/day of drift if uncorrected; they are handled by pre-launch frequency adjustment and a real-time eccentricity correction.
• The choice of positioning method determines which errors dominate the error budget and therefore the achievable accuracy.

References#

1. Van Sickle, J. GPS for Land Surveyors (4th Ed.). CRC Press, 2015. Chapters 7--8.
2. Ghilani, C.D. & Wolf, P.R. Elementary Surveying: An Introduction to Geomatics (13th Ed.). Pearson, 2012. Chapter 14.
3. GEOG 862: GPS and GNSS for Geospatial Professionals. Penn State College of Earth and Mineral Sciences. Lessons 6--7.
4. Hofmann-Wellenhof, B., Lichtenegger, H. & Wasle, E. GNSS: Global Navigation Satellite Systems. Springer, 2008. Chapters 5--6.
5. International GNSS Service (IGS). "IGS Products: Orbits, Clocks, and Atmospheric Parameters." https://igs.org/products/
6. Seeber, G. Satellite Geodesy (2nd Ed.). Walter de Gruyter, 2003. Chapters 7--8.