6.3 Differential, Trigonometric, Barometric, and Profile Leveling Methods

6.3 Differential, Trigonometric, Barometric, and Profile Leveling Methods

Leveling is the branch of surveying performed to determine the relative elevations of points above or below a reference datum (usually Mean Sea Level / MSL). In Philippine geodetic practice, leveling provides essential vertical control for topographic mapping, infrastructure engineering, and cadastral boundary control. This section details differential leveling math, field ledger validation, earth curvature and atmospheric refraction dynamics, trigonometric leveling formulations, and profile/cross-sectional earthwork calculations.

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1. Differential Leveling Operations and Field Ledger Mechanics

Differential leveling establishes elevations by taking staff/rod readings through a leveled line of sight.

Core Terminology and Governing Formulas

• Benchmark (BM): A permanent, fixed reference mark of known elevation.
• Backsight (BS or +S): A staff reading taken on a point of known elevation to determine instrument height.
• Height of Instrument (HI): Elevation of the instrument's line of sight above the datum.

$$\text{HI} = \text{Elevation}_{\text{point}} + \text{BS}$$

• Foresight (FS or -S): A staff reading taken on a point of unknown elevation (or Turning Point) to establish its elevation.

$$\text{Elevation}_{\text{new}} = \text{HI} - \text{FS}$$

• Turning Point (TP): An intermediate point upon which both a foresight (to establish elevation) and a backsight (to move instrument forward) are taken.

Complete Worked Differential Leveling Field Ledger

Below is a complete field book table for a level loop starting at $\text{BM}_1$ (known elevation $125.450\text{ m}$) and closing on $\text{BM}_2$:

Full Arithmetic Check

The field ledger calculations must be verified using the standard Arithmetic Check:

$$\sum \text{BS} - \sum \text{FS} = \text{Final Elevation} - \text{Initial Elevation}$$ $$\text{Check Left Side: } 7.925\text{ m} - 5.689\text{ m} = +2.236\text{ m}$$ $$\text{Check Right Side: } 127.686\text{ m} - 125.450\text{ m} = +2.236\text{ m} \quad \checkmark \text{ (Exact Match)}$$

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2. Earth Curvature and Atmospheric Refraction Corrections

Over long sight distances, the horizontal line of sight departs from a level surface due to Earth's curvature, while atmospheric refraction bends light rays downward toward denser air layers near the ground.

Mathematical Derivation

1. Curvature Correction ($h_c$): Using Earth radius $R \approx 6,371\text{ km}$:

   $$h_c = \frac{K^2}{2R}$$

   where $K$ is the sight distance in kilometers. Curvature causes rod readings to read too high by $+0.0785 K^2$ meters.

2. Refraction Correction ($h_r$): Refraction bends the sight line downward, making rod readings read too low by approximately $k \cdot h_c \approx 0.14 h_c = 0.0110 K^2$ meters (where $k \approx 0.14$ is the coefficient of refraction).

3. Combined Correction ($h_{cr}$): The combined effect of curvature and refraction is:

   $$h_{cr} = h_c - h_r = 0.0785 K^2 - 0.0110 K^2 = 0.0675 \cdot K^2 \quad (\text{in meters, } K \text{ in km})$$ $$h_{cr} = 0.0206 \cdot k^2 \quad (\text{in feet, } k \text{ in thousands of feet})$$

Practical Rule for Leveling: Equalizing backsight and foresight distances ($\text{Distance}_{\text{BS}} = \text{Distance}_{\text{FS}}$) completely eliminates combined curvature and refraction error $h_{cr}$ as well as line-of-sight collimation error!

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3. Trigonometric Leveling

Trigonometric leveling determines elevation differences by measuring inclined slope distances ($S$) or horizontal distances ($H$) and zenith angles ($Z$) or vertical angles ($\alpha$).

Governing Equations

For a single sight from Station $A$ to Station $B$:

$$\Delta h_{AB} = H \cdot \tan \alpha + h_i - h_r + h_{cr}$$ $$\Delta h_{AB} = S \cdot \sin \alpha + h_i - h_r + 0.0675 \cdot \left(\frac{S \cos \alpha}{1000}\right)^2$$

Where:

• $S = \text{Measured slope distance}$.
• $\alpha = \text{Vertical angle} = 90^\circ - Z$.
• $h_i = \text{Height of instrument above station } A$.
• $h_r = \text{Height of target/reflector above station } B$.
• $h_{cr} = \text{Combined curvature and refraction correction}$.

Reciprocal Trigonometric Leveling

To eliminate atmospheric refraction and curvature uncertainty over long spans (e.g., river crossings or canyon spans), simultaneous reciprocal vertical angle measurements are taken from both ends ($A$ to $B$, and $B$ to $A$):

$$\Delta h_{AB} = \frac{H}{2} (\tan \alpha_A - \tan \alpha_B) + \frac{h_{iA} - h_{rA} + h_{iB} - h_{rB}}{2}$$

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4. Barometric, Profile, and Cross-Section Leveling for Earthwork

Barometric Leveling

Barometric leveling measures atmospheric pressure differences to estimate relative elevations using hypsometric formulas.

• Near sea level, atmospheric pressure decreases by approximately $1\text{ hPa}$ ($0.75\text{ mmHg}$) per $8.5\text{ meters}$ ($30\text{ feet}$) of altitude gain.
• Used primarily in reconnaissance surveys, mountainous terrain exploration, and regional geology.

Profile and Cross-Section Leveling

1. Profile Leveling: Determines ground elevations along a fixed line (road centerline, pipeline route, railway axis).

   • Stations are established at standard $20\text{-m}$ intervals (e.g., Sta $0+000$, Sta $0+020$, Sta $0+040$).
   • Readings on intermediate ground points are called Intermediate Foresights (IFS).
   $$\text{Elevation}_{\text{ground}} = \text{HI} - \text{IFS}$$

2. Cross-Section Leveling: Takes elevations along lines perpendicular to the centerline at each station to determine lateral ground profiles.

3. Earthwork Volume Computation (End-Area Formula): Between two cross-sectional areas $A_1$ and $A_2$ separated by distance $L$:

   $$V_e = \left( \frac{A_1 + A_2}{2} \right) \cdot L \quad (\text{Trapezoidal / End-Area Formula})$$ $$V_p = \frac{L}{6} \left( A_1 + 4 A_m + A_2 \right) \cdot (\text{Prismoidal Formula})$$

   Where $A_m$ is the cross-sectional area at the mid-station ($L/2$).