GELE Geodesy — Satellite Geodesy and GNSSDetailed Explanation

GNSS Positioning Principle: Trilateration and the Four-Satellite Requirement

A GNSS receiver determines its three-dimensional position by measuring its distance (range) to multiple satellites whose positions are precisely known from their broadcast ephemeris data. This process is called trilateration — not triangulation, which involves angles. Each satellite defines a sphere of possible receiver positions; the intersection of multiple spheres yields a unique point. The critical complication is that a GNSS receiver uses an inexpensive quartz clock, not an atomic clock (which costs ~USD 50,000 and weighs kilograms). Because the receiver clock is imperfect, the measured travel time contains a clock bias Δt_receiver, making the computed distance slightly wrong. This wrong distance is called a pseudorange (ρ), because it is not the true geometric range but is contaminated by the receiver clock error. The four unknowns are: X (receiver ECEF coordinate), Y (receiver ECEF coordinate), Z (receiver ECEF coordinate), and the receiver clock bias Δt. Four equations (one per satellite) are therefore needed, requiring a minimum of FOUR satellites for a 3-D fix. With only three satellites, the clock bias cannot be separated from the position unknowns, and the solution is indeterminate in 3-D. If the receiver is constrained to a known ellipsoidal height (2-D mode), only three unknowns (X, Y, Δt) remain and three satellites suffice — but this is not standard geodetic practice. In the Philippine context, all GNSS control surveys for cadastral and geodetic purposes must ultimately be referenced to PRS92 (Philippine Reference System 1992), which is the national horizontal datum established by NAMRIA (National Mapping and Resource Information Authority) under RA 8560. GNSS observations natively produce coordinates in WGS84/ITRF; a seven-parameter Helmert transformation is applied to convert to PRS92.

GNSS Observables: Pseudorange and Carrier Phase

GNSS receivers extract position information from two fundamentally different types of measurements called observables. **1. Pseudorange (Code-Phase Observable)** The receiver correlates the incoming satellite signal with an internally generated replica of the satellite's PRN (Pseudo-Random Noise) code. The time shift required for correlation gives the signal travel time Δt. Multiplying by the speed of light c gives the pseudorange: ρ = c·Δt. The GPS L1 C/A code chip length is ~293 m (300 μs chip period), yielding a pseudorange noise of roughly 1–3 m under normal conditions. This is the observable used by standard GPS receivers (smartphones, handheld units). Accuracy: metre-level (typically 3–10 m for civilian single-frequency). **2. Carrier Phase Observable** The carrier wave itself (L1 = 1575.42 MHz, wavelength λ₁ ≈ 19.0 cm; L2 = 1227.60 MHz, λ₂ ≈ 24.4 cm) is used as the measuring scale. The receiver tracks the phase of the incoming carrier and counts the number of complete wavelengths plus a fractional part. The phase measurement has a noise of ~1–2 mm (about 1% of a wavelength). However, when lock is first acquired, the receiver does not know how many complete cycles N lie between the satellite and receiver — this unknown integer is called the integer ambiguity or integer cycle ambiguity N. Once the ambiguity is resolved (fixed), carrier-phase positioning achieves millimetre-to-centimetre accuracy. The carrier-phase observation equation: Φ = ρ/λ + N + (clock terms) + (atmospheric terms) + noise. Resolving N is the key challenge of precise GNSS — techniques include LAMBDA method, FARA, and wide-laning. **Comparison:** | Feature | Pseudorange | Carrier Phase | |---|---|---| | Measurement unit | metres | cycles (then × λ → metres) | | Noise level | 1–3 m | 1–2 mm | | Ambiguity | None | Integer N (must be resolved) | | Application | Navigation, DGPS | RTK, static geodetic surveys | | Accuracy | Metre-level | Centimetre/millimetre-level | For Philippine geodetic control surveys (establishing new PRS92 control points), static carrier-phase GNSS is mandatory. NAMRIA guidelines specify minimum observation times and GDOP thresholds.

GNSS Error Sources and Dilution of Precision (DOP)

The accuracy of a GNSS fix is the product of two factors: (1) the quality of the range measurements (user equivalent range error, UERE or σ), and (2) the geometric strength of the satellite constellation (DOP). The relationship is: **Accuracy ≈ DOP × σ (range error)** **ERROR SOURCES (ranked by typical magnitude):** 1. **Ionospheric Delay:** The ionosphere (50–1000 km altitude) slows the code signal and advances the carrier phase. The delay depends on Total Electron Content (TEC) and is frequency-dependent. Single-frequency receivers use broadcast ionospheric models (Klobuchar model); dual-frequency receivers compute the actual delay from the difference between L1 and L2 measurements. In the Philippines, the ionosphere is particularly active due to the equatorial location — GNSS surveys during solar maximum require careful ionospheric correction. 2. **Tropospheric Delay:** The neutral atmosphere (0–50 km) delays both code and carrier equally (non-dispersive). Separated into a dry component (~2.3 m at zenith, modeled from surface pressure) and a wet component (~10–35 cm, highly variable, related to water vapor). Models: Saastamoinen, Hopfield, Niell Mapping Functions. Critical for long baselines and low-elevation satellites. 3. **Satellite Clock Error:** Satellite atomic clocks drift slightly despite periodic USNO corrections. Broadcast ephemeris includes clock correction coefficients; SP3 precise ephemerides (IGS) give sub-centimetre satellite positions and sub-nanosecond clock corrections. 4. **Satellite Orbit Error:** Broadcast ephemeris has orbit errors of ~1 m; IGS precise ephemerides reduce this to ~2 cm, with a latency of ~2 weeks for final products. 5. **Multipath:** Signals reflected from buildings, terrain, or water before reaching the antenna cause multipath error (up to several metres for code; cm-level for phase). Mitigated by: choke-ring antennas, antenna height, site selection away from reflective surfaces, signal-processing algorithms. 6. **Receiver Noise:** Thermal noise in the receiver electronics; typically 1–3 mm for carrier phase, 1–3 m for pseudorange. **DILUTION OF PRECISION (DOP):** DOP quantifies the geometric effect of satellite distribution on positioning accuracy. It is derived from the trace of the cofactor matrix of the least-squares solution. Satellites spread evenly across the sky give low (good) DOP; satellites clustered together give high (bad) DOP. DOP Components: - **PDOP (Position DOP):** 3-D position (X, Y, Z) - **HDOP (Horizontal DOP):** Horizontal position (E, N) - **VDOP (Vertical DOP):** Vertical position (U) - **TDOP (Time DOP):** Receiver clock - **GDOP (Geometric DOP):** All four unknowns (X, Y, Z, clock) Relationship: GDOP² = PDOP² + TDOP²; PDOP² = HDOP² + VDOP² DOP rating scale for geodetic surveys: - < 1: Ideal - 1–2: Excellent - 2–5: Good (acceptable for most surveys) - 5–10: Moderate (marginal for precise work) - > 10: Poor (avoid for geodetic surveys) The accuracy formula Accuracy ≈ DOP × σ is fundamental: if PDOP = 2 and σ = 3 m, then positional accuracy ≈ 6 m. To achieve 2 m accuracy with the same σ = 3 m, you need PDOP < 0.67 — not physically realistic, showing that reducing σ (better receivers, differential corrections) is the path to higher accuracy.

Differential GNSS and RTK Positioning

Differential GNSS (DGNSS) is a technique that dramatically improves positioning accuracy by using a GNSS receiver at a known point (the base or reference station) to compute range corrections that are then applied to the measurements of a rover receiver in the field. The key insight is that errors in GNSS signals — particularly satellite clock errors, orbit errors, and (for short baselines) ionospheric and tropospheric delays — are spatially correlated: nearby receivers experience very similar errors. By differencing the measurements, common errors cancel. **How DGPS Works (Code/Pseudorange):** 1. The base receiver at a known point computes what the pseudorange to each satellite should be (from the known coordinates and satellite ephemeris). 2. It compares this computed value to the measured pseudorange → the difference is the pseudorange correction (PRC). 3. PRCs are broadcast to the rover (via UHF radio, cellular, or RTCM-over-internet). 4. The rover applies the PRCs to its own pseudoranges before computing position. 5. Result: sub-metre to decimetre accuracy (vs. ~3–10 m standalone). **DGPS Baseline Rule:** Accuracy degrades as baseline length increases (typically ~1 ppm degradation) because errors become less correlated over distance. At >100 km, ionospheric and tropospheric differences between base and rover become significant. **RTK (Real-Time Kinematic) — Carrier Phase Differential:** RTK applies differential corrections to carrier-phase observations in real time, achieving centimetre-level accuracy. Process: 1. Base station streams carrier-phase and pseudorange observations (plus known coordinates) to the rover via radio link. 2. Rover forms between-receiver, between-satellite double differences to eliminate most errors. 3. Software resolves the integer ambiguities (typically in seconds to minutes). 4. Once ambiguities are fixed (not float), position accuracy reaches 1–3 cm horizontally, 2–5 cm vertically. 5. If ambiguities are not fixed (float solution), accuracy degrades to decimetre-level. **Double Differencing:** The mathematical workhorse of RTK: ∇ΔΦ = (ΦₐⁱⱼA−ΦₐⁱⱼB) — formed between two receivers (A, B) and two satellites (i, j). This eliminates receiver and satellite clock errors, leaving geometry + integer ambiguity + noise. **RTK vs. Static GNSS:** | Feature | Static GNSS | RTK | |---|---|---| | Observation time | 1 hour to days | Seconds to minutes | | Accuracy | mm to cm | 1–3 cm horizontal | | Ambiguity | Post-processed | Real-time fixed | | Application | Control networks | Stakeout, detail survey | | RA 8560 compliance | Mandatory for control | Accepted for detail | **Network RTK (NRTK):** Uses a network of CORS (Continuously Operating Reference Stations) — like NAMRIA's CORS network — to model spatial errors across the coverage area. A virtual reference station (VRS) is created near the rover, providing corrections without a physical base station. Reduces baseline-length accuracy degradation. **Philippine Application:** Under NAMRIA's technical specifications, RTK surveys are accepted for cadastral surveys (lot boundaries) provided ambiguities are fixed, PDOP < 6, and results are transformed from WGS84 to PRS92 using NAMRIA-approved transformation parameters before submission.

GNSS Reference Frames: WGS84, ITRF, and PRS92

Every GNSS position is computed in a specific reference frame. Understanding reference frames is critical for ensuring that GNSS results are correctly integrated with existing Philippine geodetic control. **WGS84 (World Geodetic System 1984):** - The reference frame broadcast by GPS satellites and used to compute all GPS positions. - A geocentric, Earth-fixed Cartesian system (ECEF: X, Y, Z origin at Earth's center of mass). - Also defined as an ellipsoid: semi-major axis a = 6,378,137.0 m; flattening f = 1/298.257223563. - WGS84 is realized and maintained by the US National Geospatial-Intelligence Agency (NGA) and is periodically updated (G730, G873, G1150, G1674, G1762 realizations) to align with ITRF. - Modern WGS84 realizations are within ~1 cm of ITRF. **ITRF (International Terrestrial Reference Frame):** - The most precise global reference frame, maintained by the IERS (International Earth Rotation and Reference Systems Service). - Updated periodically: ITRF96, ITRF2000, ITRF2005, ITRF2008, ITRF2014, ITRF2020. - IGS (International GNSS Service) precise products are in ITRF — used for PPP and high-accuracy geodetic work. - ITRF accounts for tectonic plate motion, so coordinates have an associated epoch (e.g., 2005.0). **PRS92 (Philippine Reference System 1992):** - The national horizontal geodetic datum of the Philippines, established under RA 8560 (Philippine Geodetic Reference System Act) and maintained by NAMRIA. - Based on the GRS80 ellipsoid (a = 6,378,137.0 m, f = 1/298.257222101) — essentially the same ellipsoid as WGS84 but with slightly different defining constants. - Realized through a network of ~60 first-order and ~200 second-order control stations densified across the archipelago. - Associated map projection: PPCS (Philippine Plane Coordinate System), which is a Transverse Mercator projection with zones covering the archipelago. - PRS92 is a semi-dynamic datum — it does not account for ongoing tectonic motion, which is significant in the Philippines (Philippine Mobile Belt, ~70 mm/yr convergence rates). **Transformation: WGS84/ITRF → PRS92:** A seven-parameter Helmert (similarity) transformation is applied: (X_PRS92, Y_PRS92, Z_PRS92) = (1+s)[R](X_WGS84, Y_WGS84, Z_WGS84) + (ΔX, ΔY, ΔZ) where (ΔX, ΔY, ΔZ) are translation parameters, (ω_x, ω_y, ω_z) are rotation parameters (Euler angles), and s is a scale factor. NAMRIA provides the official transformation parameters. Software such as PROJ, Trimble Business Center, and Leica Infinity implement these parameters. Without the correct transformation, GNSS points will be offset from existing PRS92 control by meters — leading to serious errors in land registration and construction. **PPCS Zones:** The Philippine Plane Coordinate System uses 5 Transverse Mercator zones (I–V) covering the Philippines from west to east. Zone coordinates (Northing, Easting) are used for all cadastral and engineering surveys under PD 1529. **Practical Rule for Board Exam:** Whenever a GNSS survey result must be tied to existing PRS92 monuments, a datum transformation is required. This is mandatory — not optional.

Apply ρ = c·Δt directly. Use c = 299,792,458 m/s (exact SI value). The result of ~21,585 km is consistent with GPS satellite orbital altitude (~20,200 km) plus signal path through atmosphere. Always check your order of magnitude: GPS pseudoranges are in the 20,000–25,000 km range.

Use PDOP² = HDOP² + VDOP² for part (a). For accuracy estimates, match the DOP component to the direction. VDOP > HDOP confirms poorer vertical geometry. Always state units (metres or cm) and compare to specifications.

The 4-satellite minimum is a mathematical necessity, not an arbitrary rule. In Philippine urban environments, multi-constellation GNSS receivers significantly improve satellite availability.

λ = c/f is the fundamental wave equation. L1 wavelength ≈ 19 cm is a value to memorize for the board exam. The resulting range of ~16,651 km is shorter than the standard GPS altitude because the problem is illustrative with rounded numbers.

This problem tests the complete data flow from raw GNSS to usable Philippine cadastral coordinates. The board exam regularly tests knowledge of this transformation chain. Each step has a specific mathematical operation; missing any step is a technical error.

PDOP accounts for 3-D position geometry; GDOP adds the time/clock component. For most survey planning purposes, PDOP is the relevant metric. Note that GDOP > PDOP > HDOP (each successive DOP includes more components).