8.3 Triangulation, Trilateration, Baseline Measurements, and National Geodetic Control Networks

Triangulation, Trilateration, Baseline Measurements, and Geodetic Control Networks

To establish accurate horizontal positional frameworks across large regions, national geodetic surveys rely on control networks. Traditionally executed via triangulation and trilateration, these networks are now integrated with modern satellite-based Active Geodetic Networks (CORS/PAGENET) maintained by NAMRIA.

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1. Triangulation and Trilateration Networks

Methods of Network Extension

1. Triangulation: A method of establishing horizontal control by measuring all internal angles of a chain or grid of connected triangles using high-precision direction theodolites. Scale is introduced by directly measuring the distance of at least one baseline.
2. Trilateration: A method where all triangle side lengths are directly measured using Electronic Distance Measurement (EDM) or invar tapes, and internal angles are calculated geometrically.
3. Triangulateration: A combined methodology where both all angles and all side lengths are observed, providing high redundancy for precision engineering networks.

Triangulation Figures and Geometry

• Single Chain of Triangles: Simple but lacks geometric checks (weakest structure).
• Center-Point Polygons: Strong structure suitable for covering wide areas.
• Braced Quadrilaterals: The strongest geodetic figure, providing internal cross-checks through overlapping triangles.

Braced Quadrilateral:                      Center-Point Polygon:
    A ----------- B                                B ------ C
    | \       / |                                 / \      / \
    |  \     /  |                                /   \    /   \
    |   \   /   |                               A ---- O ---- D
    |    \ /    |                                \   /    \   /
    |     X     |                                 \ /      \ /
    |    / \    |                                  F ------ E
    C ----------- D

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2. Strength of Figure Formula ($R$)

The Strength of Figure ($R$) is a quantitative metric developed by the U.S. Coast and Geodetic Survey (USCGS) to evaluate the geometric strength of a triangulation figure or chain. It measures how effectively a figure limits error accumulation when computing side lengths through sines of observed angles.

The Strength of Figure Equation

$$R = \frac{D - C}{D} \sum \left( \delta_A^2 + \delta_A \delta_B + \delta_B^2 \right)$$

Where:

• $D$ = Number of direction observations (excluding fixed baseline observations). $D = 2 \times (\text{lines observed in both directions})$.
• $C$ = Number of geometric conditions to be satisfied in the figure.
• $\delta_A$ = Difference per 1 second of arc in the log sine of distance angle $A$ (in units of the 6th decimal place).
• $\delta_B$ = Difference per 1 second of arc in the log sine of distance angle $B$ (in units of the 6th decimal place).

Calculating Geometric Conditions ($C$)

$$C = (n' - s' + 1) + (n - 2s + 3)$$

• $n$ = Total number of lines in the figure (including baseline).
• $n'$ = Number of lines observed in both directions.
• $s$ = Total number of stations.
• $s'$ = Number of occupied stations.
• Expression $(n' - s' + 1)$ represents the number of angle conditions.
• Expression $(n - 2s + 3)$ represents the number of side conditions.

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3. Baseline Measurements and Reductions to the Ellipsoid

Measured distances must be corrected for instrument/tape factors and reduced to horizontal distance at the reference ellipsoid (Mean Sea Level).

Baseline Tape Corrections

1. Temperature Correction ($C_t$): $C_t = \alpha L (T - T_0)$
2. Tension Correction ($C_p$): $C_p = \frac{(P - P_0) L}{A E}$
3. Sag Correction ($C_s$): $C_s = -\frac{W^2 L}{24 P^2} = -\frac{w^2 L^3}{24 P^2}$
4. Slope Correction ($C_h$): $C_h = -\frac{h^2}{2 L}$

Reduction to Mean Sea Level (Reference Ellipsoid)

To compute geodetic positions on the reference ellipsoid, a measured horizontal baseline length $S$ at mean elevation $H_m$ must be reduced to sea level length $S_0$:

$$\frac{S_0}{R_m} = \frac{S}{R_m + H_m}$$ $$S_0 = S \left( \frac{R_m}{R_m + H_m} \right) = S \left( 1 + \frac{H_m}{R_m} \right)^{-1}$$

Using binomial expansion for $H_m \ll R_m$:

$$S_0 \approx S \left( 1 - \frac{H_m}{R_m} \right) = S - S \left( \frac{H_m}{R_m} \right)$$ $$C_{msl} = -S \left( \frac{H_m}{R_m} \right)$$

Where:

• $S$ = Measured horizontal baseline length at elevation $H_m$.
• $S_0$ = Reduced baseline length at the reference ellipsoid (MSL).
• $H_m$ = Mean orthometric elevation of the baseline above sea level.
• $R_m$ = Mean radius of curvature of the ellipsoid at the baseline location ($R_m = \sqrt{M N} \approx 6,371,000\text{ m}$).

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4. Laplace Stations and Azimuth Correction

Because baseline angle observations are referenced to the local gravity vector (plumb line), accumulated observations along long triangulation chains tend to twist systematically. To control this "azimuth drift," Laplace stations are established at regular intervals.

Deflection of the Vertical

The difference between the physical gravity vector (plumb line) and the ellipsoid normal is the deflection of the vertical, having components:

• $\xi = \Phi - \phi$ (Meridian component: astronomical latitude $\Phi$ minus geodetic latitude $\phi$).
• $\eta = (\Lambda - \lambda) \cos \phi$ (Prime vertical component: astronomical longitude $\Lambda$ minus geodetic longitude $\lambda$).

The Laplace Equation

At a Laplace station, both astronomical position ($\Phi, \Lambda$) and astronomical azimuth ($\alpha_A$) are measured. The true geodetic azimuth ($\alpha_G$) is obtained using the Laplace Equation:

$$\alpha_G - \alpha_A = (\lambda_G - \lambda_A) \sin \phi = -\eta \tan \phi$$ $$\alpha_G = \alpha_A + (\lambda_G - \lambda_A) \sin \phi$$

Where:

• $\alpha_G$ = Geodetic azimuth.
• $\alpha_A$ = Astronomical azimuth observed with astronomical instruments (e.g., star observations).
• $\lambda_G$ = Geodetic longitude.
• $\lambda_A$ = Astronomical longitude.
• $\phi$ = Geodetic latitude.

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5. NAMRIA National Geodetic Control Network Standards

The National Mapping and Resource Information Authority (NAMRIA) under the DENR maintains the Philippine national geodetic network. Control points are classified into accuracy tiers:

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6. Worked Examples

Worked Example 8.3.1: Reduction of Baseline to Sea Level

Problem: A baseline measured across a highland plain has a horizontal length $S = 8,450.620\text{ m}$ at a mean elevation $H_m = 620.00\text{ m}$. Assuming mean Earth radius $R_m = 6,371,000\text{ m}$, compute the sea-level reduced length $S_0$.

Solution:

1. Use the sea-level reduction formula: $$S_0 = S \left( 1 - \frac{H_m}{R_m} \right)$$
2. Calculate the ratio $\frac{H_m}{R_m}$: $$\frac{620.00}{6,371,000} = 0.000097316$$
3. Compute sea level reduction correction $C_{msl}$: $$C_{msl} = -8,450.620 \times 0.000097316 = -0.8224\text{ m}$$
4. Subtract from measured horizontal length: $$S_0 = 8,450.620\text{ m} - 0.8224\text{ m} = 8,449.798\text{ m}$$
5. The baseline length reduced to sea level is $8,449.798\text{ m}$.

Worked Example 8.3.2: Laplace Azimuth Correction

Problem: At a Laplace control station in Central Luzon (latitude $\phi = 15^\circ 30' 00''\text{ N}$), an astronomical azimuth $\alpha_A = 142^\circ 45' 20.00''$ is measured to a target hub. The observed astronomical longitude is $\lambda_A = 120^\circ 40' 15.00''\text{ E}$, and the geodetic longitude is $\lambda_G = 120^\circ 40' 21.00''\text{ E}$. Calculate the corrected geodetic azimuth $\alpha_G$.

Solution:

1. Determine longitude discrepancy $(\lambda_G - \lambda_A)$: $$\lambda_G - \lambda_A = 120^\circ 40' 21.00'' - 120^\circ 40' 15.00'' = +6.00''$$
2. Apply the Laplace correction term: $$\text{Correction} = (\lambda_G - \lambda_A) \sin \phi = +6.00'' \times \sin(15.5^\circ)$$ $$\sin(15.5^\circ) = 0.267238$$ $$\text{Correction} = +6.00'' \times 0.267238 = +1.6034''$$
3. Calculate geodetic azimuth $\alpha_G$: $$\alpha_G = \alpha_A + \text{Correction} = 142^\circ 45' 20.00'' + 1.6034'' = 142^\circ 45' 21.60''$$
4. The corrected geodetic azimuth $\alpha_G$ is $142^\circ 45' 21.60''$.