9.3 Global Navigation Satellite Systems (GNSS/GPS), Carrier-Phase Positioning, RTK, and Static Baselines

9.3 Global Navigation Satellite Systems (GNSS/GPS), Carrier-Phase Positioning, RTK, and Static Baselines

Global Navigation Satellite Systems (GNSS) have revolutionized geodetic surveying, replacing traditional optical triangulation networks with satellite-based baseline vectors referencing global geodetic datums (such as WGS84 and ITRF/PRS92).

---

1. GNSS Constellations & System Segments

Modern multi-constellation GNSS integrates signals from four major global satellite networks:

Functional Segments

Every GNSS operates via three interconnected segments:

1. Space Segment: The constellation of orbiting satellites transmitting PRN codes, carrier signals, and navigation messages (ephemerides, clock bias parameters).
2. Control Segment: Master Control Station (MCS), ground antennas, and tracking stations monitoring orbits, calculating ephemerides, and uploading clock/orbital corrections.
3. User Segment: GNSS antennas, field survey receivers, and post-processing software used by geodetic engineers.

---

2. Pseudorange vs. Carrier-Phase Measurements

GNSS receivers determine distance to satellites using two fundamental measurement types:

Pseudorange (Code) Measurement

Pseudorange ($P$) measures time of arrival ($\Delta t$) of the satellite's Pseudo-Random Noise (PRN) code:

$$P = \rho + c \cdot (dt - dT) + I + T + \epsilon_P$$

where $\rho$ is true geometric range, $c$ is speed of light, $dt$ is receiver clock error, $dT$ is satellite clock error, $I$ is ionospheric delay, $T$ is tropospheric delay, and $\epsilon_P$ is code noise.

• Accuracy: Code-based positioning (standalone GNSS) yields accuracies of $0.5 \text{ m}$ to $3.0 \text{ m}$.

Carrier-Phase Measurement

Carrier-phase ($\Phi$) measures the phase of the continuous sinusoidal carrier wave (e.g. L1 wavelength $\lambda_1 \approx 19.05 \text{ cm}$):

$$\Phi = \frac{1}{\lambda} \rho + N + \frac{c}{\lambda}(dt - dT) - \frac{I}{\lambda} + \frac{T}{\lambda} + \epsilon_\Phi$$

where $N$ is the Integer Phase Ambiguity—the unknown whole number of carrier cycles between satellite and receiver at signal lock-on.

• Accuracy: Resolving $N$ to correct integers (Fixed Solution) yields millimeter-to-centimeter precision ($1 \text{ mm} - 10 \text{ mm}$).

---

3. Atmospheric Delays and Error Sources

---

4. Dilution of Precision (DOP)

Dilution of Precision (DOP) is a dimensionless scalar factor describing the impact of satellite constellation geometry on positioning accuracy:

$$\text{Positioning Error } \sigma_{\text{pos}} = \text{DOP} \times \sigma_{\text{range}}$$

Types of DOP

• GDOP (Geometric DOP): Overall geometry combining 3D position and time ($\text{GDOP} = \sqrt{\text{PDOP}^2 + \text{TDOP}^2}$).
• PDOP (Position DOP): 3D spatial geometry ($\text{PDOP} = \sqrt{\text{HDOP}^2 + \text{VDOP}^2}$).
• HDOP (Horizontal DOP): 2D horizontal positional strength.
• VDOP (Vertical DOP): Vertical height determination strength (typically $1.5\times$ to $2\times$ higher than HDOP due to satellites only being visible above horizon).

Rule of Thumb: Ideal surveys require $\text{PDOP} < 3$. If $\text{PDOP} > 6$, satellite geometry is weak, and precision survey observations must be suspended.

---

5. GNSS Survey Positioning Modes

---

6. Baseline Processing, Differencing, and Loop Closure

Differencing Techniques

In differential GNSS processing, carrier-phase observations are differenced to cancel systematic errors:

1. Single Difference: Formed between two receivers tracking the same satellite. Cancels satellite clock error ($dT$).
2. Double Difference: Formed by differencing two single-differences between two receivers and two satellites. Cancels both receiver and satellite clock biases, leaving integer phase ambiguities ($N$) as solveable integers.
3. Triple Difference: Formed by differencing double-differences between consecutive time epochs. Cancels integer phase ambiguities; used primarily for cycle slip detection.

Independent vs. Dependent Baselines

For a session with $n$ GNSS receivers observing simultaneously:

• Total Baselines Formed: $N_{\text{total}} = \frac{n(n - 1)}{2}$
• Independent (Non-Trivial) Baselines: $N_{\text{ind}} = n - 1$

Critical Exam Rule: Processing software must only include independent baselines ($n-1$) in network adjustments. Including trivial (dependent) baselines artificially inflates statistical degrees of freedom.

Loop Closure Analysis

A closed loop formed by GNSS baseline vectors must sum to zero:

$$\sum \Delta X = 0, \quad \sum \Delta Y = 0, \quad \sum \Delta Z = 0$$

Misclosure vector magnitude $W$:

$$W = \sqrt{(\sum \Delta X)^2 + (\sum \Delta Y)^2 + (\sum \Delta Z)^2}$$

According to DENR standards (DAO 2007-29), primary control GNSS baseline loops must satisfy relative error tolerances $\le 1 : 100,000$ or maximum misclosure limit $W \le 10 \text{ mm} + 2 \text{ ppm} \times L$ (where $L$ is total loop length).