Global Navigation Satellite System (GNSS)

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This section covers the GNSS constellations and segments, the principle of satellite positioning, the GNSS observables — code pseudorange, carrier phase and Doppler — and the characteristics of GNSS antennas and receivers.

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Systems and Segments System Owner and main facts GPS (NAVSTAR) USA; nominally 24+ satellites in 6 orbital planes, inclination 55°, altitude ≈ 20 200 km, period ≈ 11 h 58 min (half a sidereal day);

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CDMA signals; reference system WGS84 GLONASS Russia;

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24 satellites in 3 planes, inclination 64.8°, altitude ≈ 19 100 km; originally FDMA (each satellite its own frequency), now adding CDMA; reference system PZ-90 Galileo European Union;

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24+ satellites in 3 planes, inclination 56°, altitude ≈ 23 222 km; civil system with a high-accuracy service; frame GTRF BeiDou (BDS) China; a mixed constellation of MEO, IGSO and GEO satellites; frame CGCS2000 Regional and augmentation NavIC/IRNSS (India) and QZSS (Japan) regional systems;

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SBAS — WAAS (USA), EGNOS (Europe), GAGAN (India, covering the region including Nepal), MSAS (Japan) broadcast corrections and integrity from geostationary satellites • Segments: the space segment (the satellites, each with atomic clocks and signal generators); the control segment — a master control station, worldwide monitor stations and ground antennas that track the satellites, compute the orbits and clock corrections and upload the navigation message; and the user segment (receivers and antennas). • Principle: the receiver measures the travel time of signals from several satellites whose positions are known from the broadcast ephemeris, and solves by trilateration (resection in space).

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Because the receiver clock is not synchronised with satellite time, the measured range contains the receiver clock error and is called a pseudorange; the clock error is treated as a fourth unknown, so a minimum of four satellites is required for a three-dimensional fix (three suffice if the height is known).

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GNSS Observables Observable Model and characteristics Code pseudorange P = ρ + c(dtr − dts) + I + T + multipath + noise, obtained by correlating the received PRN code with a receiver replica; unambiguous and easy to use, but noisy — about 3 m with the C/A code and 0.3 m with the P code (roughly 1% of the chip length); the basis of navigation and DGNSS Carrier phase Φ = ρ + c(dtr − dts) − I + T + λN + noise; measured by tracking the phase of the carrier, with noise of only 1–2 mm (about 1% of the 19 cm wavelength) — but it contains the unknown integer ambiguity N and is lost on a cycle slip.

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Resolving the ambiguities is what makes millimetre-to-centimetre positioning possible.

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Note that the ionospheric term has the opposite sign to that on the code (code delay, phase advance) Doppler (range rate) The frequency shift of the received carrier gives the instantaneous rate of change of range; used for velocity determination, receiver clock-drift estimation, cycle-slip detection and signal acquisition Antennas • GNSS signals are right-hand circularly polarised (RHCP) and very weak, so the antenna must have a suitable gain pattern (hemispherical, with reduced gain at low elevations to limit multipath) and a low-noise pre-amplifier. • Types: microstrip (patch) antennas in handheld and mapping receivers; helical/quadrifilar; and geodetic antennas with a ground plane or choke ring, which strongly suppress multipath from below. • Phase centre: the electrical point to which the measurements refer.

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It does not coincide with a physical point and varies with the elevation and azimuth of the satellite and with frequency, so the phase-centre offset (PCO) and variation (PCV) are calibrated and published (IGS ANTEX files) and applied in precise work; measurements are referred to the antenna reference point (ARP). • Field practice: measure the antenna height carefully (vertical or slant to the ARP, recorded with the method), keep the antenna level, orient geodetic antennas consistently (usually to north) for the highest accuracy, and choose a site clear of reflecting surfaces and radio transmitters.

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Receivers • A receiver contains an antenna and pre-amplifier, a radio-frequency section, multiple channels each tracking one satellite signal with a delay-lock loop (code) and phase-lock loop (carrier), a microprocessor, memory, display/controller and power supply. • Classification: by signal used — code-only, code + carrier; by frequency — single-frequency (L1) or dual/multi-frequency (L1/L2/L5, multi-constellation), the latter needed to remove the ionosphere and for long baselines; by application — navigation grade (3–10 m), mapping/GIS grade (sub-metre to metre with DGNSS) and geodetic/survey grade (mm–cm with carrier phase); also timing receivers and OEM boards for machine control and UAVs. • Important characteristics: number of channels and constellations tracked, frequencies, measurement noise and C/N0, sampling (data) rate, multipath mitigation, RTK capability and correction formats (RTCM, NTRIP), internal memory, battery life, environmental protection, and the ability to output standard RINEX (Receiver Independent Exchange) data for post-processing and NMEA for navigation.

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This section covers the coordinate reference systems used for GNSS, point (absolute) positioning, relative positioning, and the mathematical model of the GNSS satellite orbit.

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Coordinate Reference Systems for GNSS • GNSS coordinates are computed in an earth-centred, earth-fixed (ECEF) Cartesian system and then converted to geodetic φ, λ, h on the corresponding ellipsoid.

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Each system has its own frame — WGS84 (GPS), PZ-90 (GLONASS), GTRF (Galileo) and CGCS2000 (BeiDou) — all now agreeing with the ITRF at the centimetre level. • Because the plates move, precise coordinates must carry an epoch and a velocity; results are transformed to the national datum (for Nepal, to the local Everest-based system and the MUTM grid) with published transformation parameters, and ellipsoidal heights are converted to orthometric heights with a geoid model (H = h − N).

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Point (Absolute) Positioning • A single receiver observes code pseudoranges to several satellites.

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For each satellite j the observation equation is Pj = ρj(X, Y, Z) + c·dtr + corrections + ε, where ρj = √[(Xj − X)² + (Yj − Y)² + (Zj − Z)²].

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The four unknowns (X, Y, Z and the receiver clock error dtr) are solved by linearising about approximate coordinates and applying least squares; with more than four satellites the solution is over-determined and gives residuals for quality control. • Accuracy: single-point positioning (SPP) with broadcast orbits and clocks gives about 3–10 m (better with multi-constellation and dual frequency).

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Precise Point Positioning (PPP) uses dual-frequency carrier phase with precise IGS orbits and clocks and detailed models, giving centimetre accuracy from a single receiver after a convergence period of tens of minutes (much shorter with PPP-RTK and ambiguity fixing).

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Relative (Differential) Positioning • Two or more receivers observe the same satellites simultaneously; the result is the baseline vector ΔX, ΔY, ΔZ between them.

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Errors that are common to both stations — satellite clock and orbit errors and much of the atmospheric delay — cancel or are greatly reduced by differencing, so the relative accuracy is far better than the absolute accuracy of either point.

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The baseline is then added to the known coordinates of the reference station, so the result is only as good as that control. • Code differential (DGNSS) gives sub-metre to metre accuracy in real time; carrier-phase relative positioning gives millimetres to centimetres, in post-processing (static, rapid static) or in real time (RTK, network RTK).

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Residual errors grow with baseline length, which is expressed as the ppm term in the accuracy specification. • Networks of baselines are combined and adjusted by least squares (Chapter 1.6), with the covariance matrices from the baseline processing as weights, and are constrained to existing control.

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Satellite Orbit Model • To first order a satellite follows a Keplerian orbit described by six elements: semi-major axis a, eccentricity e, inclination i, right ascension of the ascending node Ω, argument of perigee ω and mean anomaly M (or true anomaly). • Real orbits are perturbed by the earth's oblateness (J2, the largest effect), the attraction of the sun and moon, solar radiation pressure, earth and ocean tides, albedo and relativistic effects. • Broadcast ephemeris: the navigation message contains a set of Keplerian elements plus harmonic correction terms (Δn, Cuc, Cus, Crc, Crs, Cic, Cis, Ω̇, IDOT) and a reference time toe, from which the satellite position is computed for any epoch; it is valid for a couple of hours and has an accuracy of about 1–2 m, with satellite clock corrections (af0, af1, af2) and a relativistic correction term. • Precise ephemeris: the IGS computes post-processed orbits and clocks in SP3 format at 15-minute intervals with an accuracy of about 2.5 cm (final products), together with rapid and ultra-rapid (predicted) products for near-real-time work.

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The almanac is a coarse, long-validity orbit description used for satellite acquisition and mission planning. • Computing a satellite position from the broadcast message involves solving Kepler's equation for the eccentric anomaly, applying the harmonic corrections, computing the position in the orbital plane and rotating it into the ECEF frame, including the earth's rotation during the signal travel time (≈ 0.07 s).

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This section covers the fundamentals of GNSS signals, linear carrier-phase combinations — single, double and triple differencing and carrier smoothing of the code — and the system biases and errors: multipath, timing and orbital biases, and the troposphere and ionosphere.

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Fundamentals of GNSS Signals • Each satellite transmits carriers derived from a fundamental frequency of 10.23 MHz:

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L1 = 1 575.42 MHz (154 × 10.23, λ ≈ 19.0 cm), L2 = 1 227.60 MHz (120 × 10.23, λ ≈ 24.4 cm) and L5 = 1 176.45 MHz (115 × 10.23, λ ≈ 25.5 cm). • The carriers are modulated with pseudo-random noise (PRN) codes: the C/A code (1.023 Mchips/s, repeating every 1 ms, on L1 — civil), the P(Y) code (10.23 Mchips/s, 7-day period — authorised users) and the modern civil signals L2C, L5 and L1C;

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Galileo and BeiDou use similar BPSK/BOC modulations.

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GPS separates satellites by code division (CDMA), while legacy GLONASS used frequency division (FDMA). • The navigation message is modulated at 50 bps in frames of 1 500 bits (five subframes, 30 s per frame) carrying the satellite clock corrections, the broadcast ephemeris, ionospheric model coefficients, health and the almanac of the whole constellation (12.5 minutes for the complete almanac).

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Linear Combinations and Differencing Combination What it removes / its use Single difference (between receivers) Two receivers, one satellite: removes the satellite clock error and most of the orbital and atmospheric error on short baselines Single difference (between satellites) One receiver, two satellites: removes the receiver clock error Double difference Two receivers and two satellites: removes both receiver and satellite clock errors while preserving the integer nature of the ambiguity — the fundamental observable of static processing and RTK Triple difference Double differences between two epochs: removes the ambiguities (provided there is no cycle slip) — used to detect and repair cycle slips and to obtain a robust approximate solution, but it is noisy Ionosphere-free (L3) Combination of L1 and L2 that removes the first-order ionospheric delay (≈ 99%); essential for long baselines and PPP, but the ambiguity is no longer an integer and the noise is amplified Wide-lane (L1 − L2) and narrow-lane Wide-lane has λ ≈ 86 cm, which makes ambiguity resolution much easier; narrow-lane (λ ≈ 10.7 cm) is used afterwards for precision;

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Melbourne-Wübbena combines code and phase for wide-lane ambiguities Geometry-free (L4) L1 − L2 phase difference removes the geometry and leaves the ionosphere — used for ionospheric studies and cycle-slip detection Carrier smoothing of the code The precise but ambiguous carrier phase is used to smooth the noisy but unambiguous code (Hatch filter): the smoothed code has far less noise and multipath — the basis of good DGNSS/SBAS performance System Biases and Errors Error source Magnitude and treatment Satellite orbit (ephemeris) error ≈ 1–2 m with broadcast orbits (centimetres with IGS products); largely cancels in relative positioning over short baselines Satellite clock error Corrected by the broadcast polynomial; residual error of a few nanoseconds (≈ 1 m); eliminated by differencing between receivers Ionospheric delay Dispersive: delay ∝ TEC/f²; from about 2–10 m at the zenith up to tens of metres at low elevation and during high solar activity; removed by the dual-frequency ionosphere-free combination, reduced by the Klobuchar/NeQuick models or by differencing on short baselines.

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Scintillation is strong at low latitudes (including South Asia) around sunset Tropospheric delay Non-dispersive (affects all frequencies equally, so dual frequency does not help): ≈ 2.3 m of dry (hydrostatic) delay at the zenith plus 0–0.4 m of wet delay, increasing roughly as 1/sin(elevation); modelled by Saastamoinen or Hopfield with mapping functions, and the residual wet delay is estimated as an unknown in precise processing Multipath Reflected signals from ground, water, buildings and vehicles distort the measurement — up to several metres on code and a few centimetres on phase; reduced by site selection, choke-ring/ground-plane antennas, an elevation mask, receiver correlator design and longer observation; it does not cancel by differencing because it is site-specific Receiver noise and hardware biases A few decimetres on code, millimetres on phase; differential code biases (DCB) and antenna phase-centre variations are calibrated Cycle slips Loss of lock breaks the integer count — detected by triple differences or geometry-free combinations and repaired in processing Other Relativistic effects, earth and ocean tide loading, phase wind-up, and formerly Selective Availability (SA) — deliberate degradation, switched off in May 2000 • The combined effect at one satellite is the user equivalent range error (UERE); the resulting position error is approximately UERE × DOP (5.4).

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This section covers the fundamentals of satellite geometry, survey planning, dilution of precision, and GNSS survey specifications and quality assurance.

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Satellite Geometry and DOP • The accuracy of a GNSS position depends not only on the measurement quality but also on the geometry of the satellites as seen from the receiver: position error ≈ DOP × UERE. • Dilution of precision (DOP) is a dimensionless number computed from the cofactor matrix of the solution:

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GDOP (geometric — position and time), PDOP (3-D position), HDOP (horizontal), VDOP (vertical) and TDOP (time), with GDOP² = PDOP² + TDOP² and PDOP² = HDOP² + VDOP². • Interpretation: the DOP is small when the satellites are widely spread over the sky (geometrically, the volume of the tetrahedron formed by the unit vectors to four satellites is large) and large when they are clustered or confined to one part of the sky.

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PDOP < 2 excellent, 2–4 good, 4–6 fair, > 6 poor (avoid).

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VDOP is always worse than HDOP because all satellites are above the horizon, which is why GNSS heights are about 1.5–2 times less accurate than horizontal positions. • More satellites, well distributed, and a lower elevation mask improve the DOP; but low satellites suffer more atmospheric delay and multipath, so a mask of 10°–15° is normally used.

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Survey Planning • Reconnaissance and station selection: open sky above the elevation mask, away from buildings, trees, water and metal surfaces (multipath), away from high-power transmitters and radar (interference), stable and safe monumentation, vehicle access, and good connection to existing control. • Mission planning: using a current almanac, software predicts satellite visibility, the number of satellites and the DOP through the day for the site's latitude, with an obstruction (sky) diagram for each station — sessions are then scheduled to avoid periods of poor geometry or few satellites. • Session planning: choose the observation mode (static, rapid static, RTK), occupation time from the baseline length and receiver type, number of receivers and the network design (independent baselines, closed loops, repeat baselines, ties to at least two known control points), logistics, communication and personnel. • Field procedure: centre and level the antenna, measure and record the antenna height twice (before and after) with the method stated, record the station name, receiver/antenna serial numbers, start and end times, weather, and note any obstructions or events.

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Specifications and Quality Assurance • Specifications for a GNSS survey define the required accuracy class or order (as mm + ppm, or as a relative accuracy such as 1:100 000), and prescribe: minimum number of satellites (≥ 4, preferably 5–6), maximum PDOP, elevation mask, sampling interval (typically 15 or 30 s for static and 1 s for RTK), minimum occupation time as a function of baseline length and receiver frequency, dual frequency for long baselines, number of independent occupations/repeat baselines, and the ties to existing control. • Quality assurance and control: in the field — checking DOP, satellite count, C/N0, and whether the RTK solution is 'fixed' rather than 'float', checking into a known point at the start and end, and re-observing points in a different session with a different satellite configuration; in processing — the RMS and ratio test of ambiguity resolution, repeatability of repeated baselines, loop misclosures of independent baselines, and a least-squares network adjustment with a chi-square/variance-factor test, residual (data-snooping) analysis and error ellipses. • Deliverables should include the metadata — datum, epoch, geoid model, processing software and parameters, antenna calibration used, and the statistics of the adjustment.

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This section covers the fundamentals, performance and applications of static, rapid static, kinematic, pseudo-kinematic and stop-and-go positioning, real-time positioning, and continuously operating reference stations (CORS).

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Mode Procedure, performance and applications Static positioning Two or more receivers occupy the stations simultaneously for a long period (from ≈ 30 minutes on short lines to several hours or days on long ones), recording carrier phase at 15–30 s; processed as baselines and adjusted as a network.

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Accuracy ≈ 3–5 mm + 0.5–1 ppm (millimetres over long lines with precise orbits and long sessions).

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Used for geodetic control networks, CORS, densification, deformation and tectonic monitoring and for all long baselines Rapid (fast) static Short occupations of 5–20 minutes on baselines usually under 15–20 km, with dual-frequency receivers and fast (on-the-fly) ambiguity resolution.

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Accuracy ≈ 5–10 mm + 1 ppm.

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Used for control densification, cadastral corners and photo control where many points must be fixed quickly Kinematic positioning One receiver stays on a known point while the rover moves continuously, recording at 1 s or faster; ambiguities are resolved by an initialisation (static period, known baseline or on-the-fly) and lock must be maintained.

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Used for topographic detail, road and rail profiling, hydrographic survey and the trajectory of airborne photogrammetric/LiDAR sensors and mobile mapping systems Pseudo-kinematic (pseudo-static, re-occupation) Each point is occupied twice for a few minutes, separated by about an hour, so that the change in satellite geometry strengthens the solution without continuous tracking; lock need not be kept between occupations.

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Accuracy at the centimetre level; useful where the sky is partly obstructed or few satellites are available, but slow in logistics Semi-kinematic (stop-and-go) After an initialisation, the rover stops briefly (a few epochs) on each point and moves between them keeping lock on the satellites.

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Very productive for detail survey in open ground; if lock is lost, re-initialisation is needed Real-time positioning DGNSS — code corrections broadcast from a base give sub-metre to metre accuracy for GIS and navigation.

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RTK — carrier-phase corrections over radio or internet (NTRIP) give 1–3 cm horizontally (2–5 cm vertically) in real time, typically within 10–20 km of the base.

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Network RTK (VRS, FKP, MAC) uses a network of CORS to model the errors and extends full accuracy to 50–70 km.

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PPP and PPP-RTK deliver corrections from global services without a local base Continuously Operating Reference Stations (CORS) • A CORS is a permanently installed GNSS station on a stable monument with a geodetic antenna, continuous power and communication, logging data 24 hours a day. • Functions: realising and maintaining the national reference frame (and linking it to ITRF); providing RINEX data for post-processing so that a single field receiver can be processed against it; broadcasting RTK/network-RTK corrections; supporting PPP; monitoring crustal deformation and tectonics (in Nepal, the Himalayan convergence and co-seismic displacements of the 2015 Gorkha earthquake were measured this way); and meteorological applications through estimated water vapour. • Requirements: a stable, deep-founded monument free of multipath, a calibrated geodetic antenna, uninterruptible power, reliable communication, continuous quality monitoring (data completeness, cycle slips, multipath indices), documented site logs and coordinates with velocities. • Nepal's Survey Department operates a national network of permanent GNSS/CORS stations for the reference frame, RTK services and deformation studies.

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(Verify the current number and service status.) • Benefits to the user: one receiver instead of two, no base to set up or guard, consistent datum for all users, and immediate quality control — which is why CORS-based RTK has become the standard method for cadastral, engineering and topographic survey.

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This section covers the fundamentals of inertial navigation, its mathematical model, Kalman filtering and the integration of INS with GNSS.

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Fundamentals of INS • An inertial navigation system determines position, velocity and attitude by dead reckoning from measurements made entirely on board — it receives no external signals.

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Its core is the inertial measurement unit (IMU): three accelerometers measuring specific force along three orthogonal axes and three gyroscopes measuring angular rate about them. • Principle: the gyroscopes maintain (or compute) the orientation of the sensor axes; the accelerometer output is corrected for gravity and rotated into the navigation frame, then integrated once to give velocity and twice to give position, starting from a known initial position, velocity and attitude (initialisation and alignment). • Mechanisation: gimballed (stable platform) systems physically keep the sensors level; strapdown systems fix the sensors to the vehicle body and perform the rotations in software — lighter, cheaper and now universal. • Sensor technologies: ring-laser gyros (RLG), fibre-optic gyros (FOG), spinning-mass and MEMS sensors; grades range from navigation grade (drift ≈ 0.01°/h) through tactical (1–10°/h) to low-cost MEMS (> 10°/h). • Characteristics: very high data rate (100–1 000 Hz), excellent short-term accuracy, direct measurement of attitude (roll, pitch, heading), complete autonomy and immunity to jamming and obstruction — but the errors of the sensors are integrated, so the position error grows with time (drift), which is the fundamental limitation.

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Mathematical Model • The mechanisation (navigation) equations in a local-level frame update, at every IMU epoch: the attitude (from the gyro angular rates, using a direction-cosine matrix or quaternions, corrected for earth rotation and transport rate); the velocity (from the specific force rotated into the navigation frame, minus gravity and the Coriolis terms); and the position (by integrating the velocity with the ellipsoidal radii of curvature M and N). • For estimation the system is linearised into an error-state model, typically with 15 states: three position errors, three velocity errors, three attitude errors and the biases of the three accelerometers and three gyroscopes (further states may model scale factors and lever arms).

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Sensor errors are described as bias, scale-factor and misalignment errors plus random walk and drift; an uncorrected accelerometer bias produces a position error growing roughly with t², and a gyro drift with t³.

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Kalman Filtering and GNSS/INS Integration • The Kalman filter is a recursive, optimal (minimum-variance) estimator for a linear system with Gaussian noise.

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Each cycle has two steps: prediction — propagate the state and its covariance with the dynamic model and the process noise Q; and update — combine the prediction with a new measurement, weighted by the Kalman gain K = P HT(H P HT + R)−1, where R is the measurement-noise covariance; the covariance P is then reduced accordingly.

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The extended Kalman filter (EKF) linearises non-linear models about the current estimate. • GNSS and INS are complementary:

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GNSS has bounded long-term accuracy but a low data rate and outages (under bridges, in tunnels, among buildings and trees), while INS has high rate, continuity and attitude but drifts.

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A Kalman filter uses the GNSS positions/velocities to estimate and correct the INS errors and sensor biases, while the INS bridges GNSS gaps and aids re-acquisition and ambiguity resolution. • Coupling: loose (GNSS position/velocity solutions update the INS — simple, but needs ≥ 4 satellites), tight (raw pseudoranges and Doppler are used, so the filter works even with fewer than four satellites) and ultra-tight/deep (the INS aids the receiver's tracking loops — best in high dynamics and jamming). • Applications: direct georeferencing of airborne photogrammetric and LiDAR sensors (providing the exterior orientation without ground control), mobile mapping systems and UAVs, hydrographic survey (heave, roll, pitch), machine guidance, vehicle and pedestrian navigation, and pipeline and tunnel surveying where GNSS is unavailable.