GNSS and RTK Error Models

Visual: satellite/base/rover RTK geometry with carrier phase, integer ambiguity, atmospheric/multipath errors, corrections, covariance, and outage behavior.

GNSS provides global position and time. RTK turns carrier-phase measurements and base-station corrections into centimeter-class relative positioning when the integer ambiguities are correctly resolved. In autonomy, GNSS is both a sensor and a map-registration authority. Its errors are structured, environment dependent, and often non-Gaussian.

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1. Sensor Model Impact

Task	Why the model matters
Perception	GNSS/RTK pose and covariance decide whether detections can be placed into map context, geofenced, or compared with surveyed zones. A false fix can make correct object detections appear in the wrong lane, stand, or safety area.
SLAM	GNSS factors anchor local odometry and loop closures to a global frame, but overconfident multipath updates can warp the whole graph.
Mapping	RTK defines georeferencing for point clouds, camera semantics, surveyed boundaries, and airport map alignment. Datum, projection, and lever-arm mistakes become map errors.
Validation	GNSS quality must be replayed with fix state, covariance, correction age, satellite geometry, and multipath context so localization and perception metrics are interpreted correctly.

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2. GNSS Observables

Pseudorange

Pseudorange is code-phase range from satellite to receiver antenna:

P_i = rho_i + c*(dt_r - dt_s) + T_i + I_i
      + b_code + multipath_P + epsilon_P

where:

• rho_i: geometric range from satellite to receiver
• dt_r, dt_s: receiver and satellite clock offsets
• T_i: tropospheric delay
• I_i: ionospheric delay for code
• b_code: hardware/code biases
• multipath_P, epsilon_P: multipath and measurement noise

Pseudorange noise is meter-level for standalone GNSS, depending on signal, receiver, multipath, satellite geometry, and corrections.

Carrier Phase

Carrier phase measures the phase of the RF carrier:

Phi_i * lambda = rho_i + c*(dt_r - dt_s) + T_i - I_i
                 + lambda*N_i + b_phase
                 + multipath_Phi + epsilon_Phi

Carrier phase has millimeter-to-centimeter precision but includes an unknown integer ambiguity N_i. The ionosphere sign is opposite to code because phase is advanced while code is delayed.

Doppler

Doppler measures range rate:

D_i * lambda ~= -rho_dot_i + c*(dt_dot_r - dt_dot_s) + noise

It is useful for velocity, cycle-slip detection, and tight GNSS/IMU fusion.

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3. RTK and Ambiguity Resolution

RTK uses a base receiver at known position and a rover on the vehicle. By differencing measurements from common satellites, many satellite and receiver clock terms cancel.

Double-differenced carrier phase:

delta_delta_Phi * lambda =
  delta_delta_rho + delta_delta_T - delta_delta_I
  + lambda * delta_delta_N + noise

The estimator first solves a float solution where ambiguities are real-valued, then attempts integer ambiguity resolution:

N_fixed = integer_argmin_N (N - N_float)^T Q_N^-1 (N - N_float)

The LAMBDA method is the canonical approach for efficient integer least-squares ambiguity resolution.

RTK states:

State	Meaning	Typical autonomy handling
Single	standalone code solution	meter-level, low weight or degraded mode
DGPS/DGNSS	code differential corrections	sub-meter to meter-level
Float RTK	carrier ambiguities estimated but not fixed	decimeter-level if geometry and corrections are good
Fixed RTK	integer ambiguities resolved	centimeter-level, but false fix is dangerous

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4. Error Sources

Error source	Behavior	Mitigation
Satellite clock/orbit	common-mode for nearby base/rover	broadcast/precise corrections, differencing
Ionosphere	dispersive, frequency-dependent, spatially varying	dual frequency, short baselines, iono models
Troposphere	non-dispersive, elevation and weather dependent	mapping functions, estimate zenith delay
Multipath	reflected signals, site-specific, non-Gaussian	antenna placement, choke/ring ground plane, elevation masks, robust gating
Receiver noise	signal and hardware dependent	quality receiver/antenna, SNR weighting
Cycle slip	carrier lock interruption	Doppler/code checks, reset ambiguity
Satellite geometry	DOP amplification	multi-constellation, satellite selection
Base-rover baseline	residual atmosphere grows with distance	local base or network RTK/VRS
Interference/spoofing	corrupted measurements	RF monitoring, consistency with IMU/LiDAR/map

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5. Atmospheric Modeling

Ionosphere

First-order ionospheric delay is frequency dependent:

I_f proportional_to TEC / f^2

Dual-frequency receivers can form ionosphere-free combinations, but RTK often relies on short-baseline differencing so rover and base see similar ionosphere. Residual ionosphere grows with baseline length and ionospheric activity.

Troposphere

Tropospheric delay is commonly split:

T(elevation) = m_h(e) * ZHD + m_w(e) * ZWD

where ZHD is zenith hydrostatic delay, ZWD is zenith wet delay, and m(e) are elevation mapping functions. Low-elevation satellites have longer paths and larger atmospheric and multipath errors.

Practical weighting:

sigma_sat_i = sigma0 / sin(elevation_i)

Many receivers internally weight observations by elevation and carrier-to-noise density C/N0.

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6. Multipath and Non-Gaussian Errors

Multipath occurs when direct and reflected GNSS signals reach the antenna. Airside environments are multipath-heavy:

• aircraft fuselage and tails
• terminal glass and metal structures
• jet bridges and service vehicles
• wet apron surfaces
• hangars and fuel trucks

Multipath can bias pseudorange by meters and carrier phase by centimeters or more. It is not well modeled as zero-mean Gaussian. In a factor graph or ESKF, GNSS updates should use robust kernels, innovation gating, and quality-based covariance inflation.

Indicators:

low C/N0
high residuals on a subset of satellites
low elevation
rapid position jumps
float/fix toggling
fixed solution inconsistent with inertial prediction

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7. Covariance and Fusion

A GNSS receiver may output position covariance, HDOP/VDOP, fix status, and protection/integrity fields. Treat these as useful but not infallible.

Approximate position covariance:

Sigma_position ~= DOP_matrix * sigma_range^2

For a position factor:

r = p_gnss_ENU - (p_base_ENU + R_base * lever_arm_base_to_antenna)
r ~ N(0, Sigma_gnss)

Lever-arm correction is mandatory:

p_antenna = p_base + R_base * l_base_antenna

If orientation uncertainty is significant, lever-arm uncertainty contributes to position covariance:

delta_p ~= -R_base * skew(l_base_antenna) * delta_theta

Practical covariance policy:

Condition	Position sigma starting point
RTK fixed, open sky, short baseline	0.01 to 0.03 m horizontal, 0.02 to 0.06 m vertical
RTK float	0.1 to 0.5 m, quality dependent
DGNSS	0.3 to 2.0 m
standalone GNSS	1.5 to 10 m
terminal/aircraft multipath	inflate by 5x to 50x or reject
suspected false fix	reject fixed status and gate as outlier

Use receiver-provided covariance when available, but cap minimum covariance to avoid overconfident fixes in multipath.

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8. Outage and Degradation Behavior

GNSS degradation is not binary. Modes include:

• satellite count falls gradually
• geometry worsens while fix remains valid
• corrections are delayed or lost
• RTK fixed drops to float
• receiver reports fixed but position jumps
• carrier cycle slips reset ambiguities
• complete outage under aircraft, terminal, or hangar

State-estimation policy:

if fixed and innovation passes gate:
    use tight covariance
elif float and innovation passes gate:
    use inflated covariance
elif stale corrections or poor C/N0/DOP:
    inflate or reject
else:
    rely on IMU + wheel + LiDAR/radar/vision localization

During outage, uncertainty should grow. If the fused output covariance remains small while GNSS is unavailable and LiDAR/map constraints are weak, the estimator is overconfident.

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9. Map Georeferencing

GNSS ties local maps to Earth frames. Common frame chain:

WGS84 lat/lon/height
  -> ECEF
  -> local ENU or UTM
  -> map
  -> odom
  -> base_link

Mapping implications:

• The map origin, datum, geoid/ellipsoid height convention, and projection must be versioned.
• UTM zone and local tangent-plane origin must be fixed for a site map.
• Base station coordinates must be surveyed in the same datum as the map.
• Antenna phase center offsets and vehicle lever arm must be documented.
• Do not mix ellipsoidal height, geoid height, and local surveyed elevation silently.

For airside mapping, centimeter local consistency matters near aircraft stands, but absolute georeferencing matters for integration with airport maps, asset management, and incident reconstruction.

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10. Validation

Validation should include:

• open-sky baseline runs
• terminal and hangar multipath runs
• aircraft occlusion and service-vehicle occlusion
• correction outage and delayed correction replay
• repeated map alignment passes
• false-fix detection tests
• lever-arm and heading sensitivity tests

Useful metrics:

GNSS innovation NIS
fix state duration and transition count
correction age
satellite count by constellation
HDOP/VDOP/PDOP
C/N0 distribution
cycle slip count
position jump rate
map residual versus GNSS residual

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11. Failure Modes

Failure mode	Cause	Mitigation
False RTK fix corrupts trajectory	wrong integer ambiguities accepted	innovation gating, inertial/map consistency, hold fixed minima
Meter jump near terminal	multipath and satellite blockage	robust kernels, covariance inflation, reject low-quality epochs
Wrong map offset	datum/projection/base coordinate mismatch	survey control and georeference audits
Heading error from GNSS course	low speed or multipath	do not use course heading below speed threshold
Lever-arm bias	antenna offset ignored or wrong frame	calibrate and apply R * lever_arm
Outage overconfidence	process noise too small or stale covariance	grow covariance and trigger degraded mode
Vertical inconsistency	ellipsoid/geoid/local height mix	explicit height convention in map metadata

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12. Sources

• RTKLIB ver. 2.4.2 Manual. https://www.rtklib.com/prog/manual_2.4.2.pdf
• Navipedia, "GNSS Measurements Modelling." https://www.navipedia.org/navipedia/index.php/GNSS_Measurements_Modelling
• Teunissen, "The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation." Journal of Geodesy, 1995.
• Teunissen, "Performance of the LAMBDA method for fast GPS ambiguity resolution." ION GPS, 1997. https://www.ion.org/publications/abstract.cfm?articleID=100034
• Misra and Enge, "Global Positioning System: Signals, Measurements, and Performance." Ganga-Jamuna Press.
• Hofmann-Wellenhof, Lichtenegger, Wasle, "GNSS: Global Navigation Satellite Systems." Springer.
• Takasu and Yasuda, "Development of the Low-cost RTK-GPS Receiver with an Open Source Program Package RTKLIB." 2009.
• Seepersad and Bisnath, "Reduction of PPP convergence period through pseudorange multipath and noise mitigation." GPS Solutions, 2015.