Levelling loops
Trigonometric levelling with the curvature and refraction correction
A single long total-station shot is reduced to an elevation, carrying instrument and target heights and the combined correction for earth curvature and atmospheric refraction.
Given
- A total station is set over control station STA 14, whose elevation is 1,286.31 ft (US survey feet). The height of the horizontal axis above the mark is 5.42 ft.
- A prism is set over point P. The height of the prism center above the mark at P is 6.15 ft.
- Observed slope distance from instrument to prism = 6,480.22 ft.
- Observed zenith angle = 87°14′35″. Angles are read to 1″.
- Combined curvature and refraction correction, in feet, is 0.0206 times the square of the horizontal distance expressed in thousands of feet.
- Elevations are reported to 0.01 ft.
Required
- The horizontal distance and the vertical component of the observed line.
- The combined curvature and refraction correction.
- The elevation of point P.
Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.
Worked solution
Reduce the zenith angle to decimal degrees
The zenith angle is measured from the upward vertical, so a reading less than 90° means the target is above the horizontal plane through the instrument. Converting to decimal degrees before taking trigonometric functions avoids the commonest arithmetic slip, which is dividing the seconds by 60 instead of 3600.
Carrying eight decimal places here is not fussiness. At a slope distance of nearly 6,500 ft, one second of zenith angle is worth 0.03 ft of elevation, so the angle has to survive the conversion intact.
Z = 87° + 14′/60 + 35″/3600
Z = 87 + 0.23333333 + 0.00972222 = 87.24305556°
cos Z = 0.048099192
sin Z = 0.998842564Vertical and horizontal components
The slope distance resolves into a vertical component, which is the slope distance times the cosine of the zenith angle, and a horizontal component, which is the slope distance times the sine. The vertical component is the rise from the instrument's horizontal axis to the prism center, along the line actually observed; it is not yet a difference in elevation between the two ground marks.
The horizontal component is what the curvature and refraction correction is a function of, so it has to be computed before that correction can be applied.
dV = 6480.22 x cos Z = 6480.22 x 0.048099192 = 311.693 ft
H = 6480.22 x sin Z = 6480.22 x 0.998842564 = 6472.720 ftCurvature and refraction
Two effects act on a long line of sight and they act in opposite directions. The earth's surface curves away from the level line of sight, which makes a distant target appear lower than it is, and the atmosphere refracts the ray downward, which partly offsets the curvature. Refraction typically recovers about one seventh of the curvature, and the two are combined into a single coefficient.
The correction is always added when reducing an observation to a distant point, because the net effect of the pair makes the target read too low. It grows with the square of the distance, so it is negligible on a 300 ft shot and decisive on a shot of more than a mile like this one.
M = H expressed in thousands of feet = 6472.720 / 1000 = 6.47272
M squared = 41.8961
c + r = 0.0206 x 41.8961 = 0.863 ftAssemble the elevation of P
The chain runs from the mark at STA 14 up to the horizontal axis of the instrument, along the observed line to the prism center, up by the curvature and refraction correction, and finally down from the prism center to the mark at P. Each term enters once and with its own sign, and the two setup heights are the terms most often dropped.
The height of instrument here is the height of the trunnion axis above the ground mark, which is a different quantity from the height of instrument on a level page. Both are called HI, and confusing them is a standing hazard in mixed level and total-station work.
elev P = elev STA 14 + hi + dV + (c + r) - ht
elev P = 1286.31 + 5.42 + 311.693 + 0.863 - 6.15
elev P = 1598.136 -> 1,598.14 ftSummary sheet
Laid out as a computation sheet, the reduction reads down a single column of signed terms.
| Term | Value (ft) | Sign | Running total (ft) |
|---|---|---|---|
| Elevation of STA 14 | 1286.31 | start | 1286.31 |
| Height of instrument axis | 5.42 | + | 1291.73 |
| Vertical component of the observed line | 311.693 | + | 1603.423 |
| Curvature and refraction | 0.863 | + | 1604.286 |
| Height of prism above the mark at P | 6.15 | - | 1598.136 |
| Elevation of P, reported | - | - | 1598.14 |
Two independent checks
The first check runs the computation backwards: take the reported elevation of P, strip off the target height, the instrument height and the curvature and refraction term, and see what vertical component and therefore what zenith angle it implies. The recovered angle must reproduce the field reading to the second.
The second check computes the curvature and refraction term in metric units with the independent metric coefficient. Converting 6,472.720 ft to 1.97288 km and applying 0.0675 times the square of the distance in kilometers gives 0.2627 m, which is 0.862 ft. The two coefficients are independent statements of the same physics and they agree to 0.001 ft, so neither has been misapplied.
back-solve: dV = 1598.14 - 1286.31 - 5.42 + 6.15 - 0.863 = 311.697 ft
cos Z = 311.697 / 6480.22 = 0.0480998 -> Z = 87°14′35″ (matches the field reading)
metric check: 6472.720 ft = 1.97288 km
0.0675 x 1.97288 squared = 0.2627 m = 0.862 ft (against 0.863 ft)Answer
- Horizontal distance = 6,472.72 ft
- Vertical component of the observed line = +311.69 ft
- Combined curvature and refraction correction = +0.86 ft
- Elevation of P = 1,598.14 ft
Check
Reversing the computation from the reported elevation recovers a zenith angle of 87°14′35″, identical to the field reading.
The metric form of the correction, 0.0675 times the square of the distance in kilometers, gives 0.2627 m = 0.862 ft against the 0.863 ft used, confirming the coefficient was applied correctly.
Magnitude check: neglecting curvature and refraction entirely would have put P at 1,597.27 ft, 0.86 ft low. On a shot of 1.2 miles that error is far larger than any reasonable tolerance, which is the practical reason the correction is never optional on long trigonometric shots.
More levelling loops
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