Leveling Notes

Leveling

Definition

Leveling is the process of determining differences in elevation.

Benchmark

Originates from chiseled horizontal marks made by surveyors in stone structures.
Ensures accurate repositioning of the leveling rod.
Usually indicated with a chiseled arrow below the horizontal line.
A survey marker used as an elevation reference.
Symbolized as BM, meaning Benchmark.

Basic Rules for Leveling

Start and finish on a benchmark and close the loops.
Maintain equal foresight and backside distances.
Keep lines of sight short (less than 50 meters).
Never read below 0.5 meters on a staff to avoid refraction issues.
Use stable and well-defined change points.
Beware of shadowing effects and crossing waters.

Methods for Determining Difference in Elevation

Differential Leveling

Employs a series of setups of a leveling instrument along a selected route.

Barometric Leveling

Uses a barometer to measure atmospheric pressure.
Higher elevation = lower atmospheric pressure.
Lower elevation = higher atmospheric pressure.

Trigonometric Leveling

Utilizes tachiometry and vertical angles.
Vertical angle is measured from the horizontal to the line of sight.

Profile Leveling

Determines differences in elevation between points at designated short measured intervals along an established line.
Provides data for plotting a vertical section of the ground surface.
Includes plan view and elevation view measurements at specific points.
Turning points are intermediate elevation points between benchmarks.

Reciprocal Leveling

Used for bodies of water where short and equal sights are difficult to maintain.

Grid Leveling

Locates contours by staking an area in squares and determining corner elevations by differential leveling.
Grid size depends on project extent, ground roughness, and accuracy requirements.

Cross Section or Borrow Pit Leveling

Used on construction jobs to ascertain quantities of earth, gravel, rock, or other materials to be excavated or filled.

Sources of Errors in Leveling

Instrumental Errors

Instrument out of adjustment.
Standard length rod discrepancy.
Defective tripod.
Air bubble not centered.
Parallax: Misalignment in the line of sight due to positioning relative to an object. Optical instruments are dependent on horizontal distances and the line of sight.

Personal Errors

Faulty rod readings.
Rod not held plumb.

Natural Errors

Curvature of the Earth.
Atmospheric refraction.
Wind.
Settlement of instrument.
Faulty turning points.

Effects of Earth's Curvature

The horizontal line departs from the Earth's surface, which is a level surface.
Radius of Earth is approximately $6,371$ kilometers.
$r^2 + k^2 = (r + h_c)^2$
Where:
- $r$ = radius of the Earth
- $k$ = horizontal distance
- $h_c$ = curvature correction
$r^2 + k^2 = r^2 + 2hcr + hc^2$
$k^2 = 2h_cr$
$h_c = \frac{k^2}{2r}$
$h_c = 0.0785k^2$
- Where $h_c$ is in meters and $k$ is in kilometers.

Correction for Refraction

Rays of light passing through Earth's atmosphere are refracted or bent from a straight path.
Bending usually occurs towards the Earth's surface.
Refraction diminishes the effect of curvature by about one-seventh or 14%.
Refraction correction equals one seventh of the curve effect of curvature.
$h{cr} = hc - \frac{1}{7}hc = \frac{6}{7}hc$
$h_{cr} = \frac{6}{7} * 0.0785k^2$
$h_{cr} \approx 0.0675k^2$

Example Problem 1

Two points A and B are $525.85$ meters apart. A level is set upon a line between A and B at a distance of $240.5$ meters from A. If the rod reading on A is $3.455$ meters and that on B is $2.806$ meters, determine the difference in elevation between the points, taking account of the effects of curvature and atmospheric refraction.
Curvature and refraction correction formula: $0.0675k^2$
- For point A:
  - $0.0675 * (240.5/1000)^2 = 0.0039$ meters
- For point B:
  - $0.0675 * (285.35/1000)^2 = 0.00555$ meters
Corrected rod reading at A:
- $3.455 - 0.0039 = 3.4511$ meters
Corrected rod reading at B:
- $2.806 - 0.0055 = 2.8005$ meters
Difference in elevation:
- $3.4511 - 2.8005 = 0.6506$ meters

Example Problem 2

A woman standing on a beach can just see the top of a lighthouse $24.14$ kilometers away. If her eye height above sea level is $1.738$ meters, determine the height of the lighthouse above the sea level.
Correction formula: $0.0675k^2$
$1.738 = 0.0675 * k_w^2$
$k_w = \sqrt{1.738/0.0675} = 5.074$ kilometers
$k_l = 24.14 - 5.074 = 19.066$ kilometers
Height of the lighthouse: $0.0675 * 19.066^2 = 24.537$ meters

Example Problem 3

The elevation of triangulation station A is 250 meters, while that of B is 685 meters. In between stations A and B is a mountain C with an elevation of 325 meters. Height down transit, which is placed at A, is 1.2 meters. The distance AC is 30 kilometers and BC is 50 kilometers. Determine the height of the tower that is to be constructed at B such that the line of sight will just pass through the mountain C with a clearance of 1.5 meters.
Correction due to refraction and curvature at points A and B.
Correction formula: $0.0675k^2$
- Point A:
  - $0.0675 * 30^2 = 60.75$ meters
- Point B:
  - $0.0675 * 50^2 = 168.75$ meters
Simplified trapezoid:
- A: $1.2 + 250 - 60.75 = 190.45$ meters
- B: $685 - 168.75 + x = 516.25 + x$ meters
- C: $1.5 + 325 = 326.5$ meters
Similar triangles:
- Triangle ABO (big triangle):
 - $y2 /(80 km) = y1 /(30 km)$
- Vertical height from C to point on AB $y_1 = 326.5 - 190.45 = 136.05$ meters.
- Vertical height from B to point on AB $y_2 = 516.25 + x - 190.45 = 325.8 + x$
Plugging to similar triangles proportion:
- $(325.8 + x)/(80 km) = 136.05/(30 km)$
- $x= ((136.05 * 80)/30) - 325.8 = 37$ meters

Differential Leveling

Used to determine difference in elevations between two or more points that are a distance apart.
Useful for comparing the elevation of several points or objects, requiring a series of setups of instruments along a general route.

Benchmark

A fixed point of reference whose elevation is either known or assumed.

Backsight

The reading taken on a rod held on a point of known or assumed elevation.

Foresight

Reading from instrument on the leveling rod.

Turning Point

An intervening point between two benchmarks upon which point foresight and backside Rodriguez are taken to enable leveling operation to continue from a new instrument position.

Height of Instrument

The elevation of the line of sight of an instrument above or below the reference date.
HI = BS + elevation of A
Elevation of B = HI - FS
Difference of elevation = LFB - LFE or backside - foresight

Differential Leveling Example

HI = Elevation + Backsight
Elevation = HI - Foresight

Example Problem 3

Station 1 is a turning point, then BM or benchmark.
Try identify values elevation backside, foresight.
The leveling rod readings are given in the order in which they were taken on benchmark 1, and the last reading is taken on benchmark 2.
Thus, the point whose elevation is desired, set up and complete the differential level notes and include the customary math check. The elevation is given under each problem number.
For the benchmark one, p p one and p p two. So benchmark one. P p one turning point two benchmark benchmark
Given, readings is elevation. And then, of course, backside for the benchmark one
Complete table given left side height of instrument equal elevation plus backside.
Elevation equal height of instrument minus foresight.