2.6.3. Precise Trig Leveling Calculator

2.6.3.1. Overview

The Precise Trig Leveling calculator computes height differences from reciprocal (simultaneous) trigonometric leveling: each line is observed from both ends — forward (From → To) and reverse (To → From). Averaging the two one-way results cancels the earth-curvature and atmospheric-refraction error, so no refraction coefficient has to be assumed. The discrepancy between the two directions is reported, along with the refraction coefficient k it implies — a powerful quality check for every line.

The tool is found under Calculations > Survey Calculator > Precise Trig Leveling.

../../_images/Precise_trig_leveling.jpg

The Precise Trig Leveling calculator.

2.6.3.2. Inputs

Parameters (top editor):

Key

Default

Meaning

Earth radius (m)

6371000

Used only for the implied-k diagnostic; the mean height difference does not depend on it.

Benchmark point

A

Optional: a point name whose height is known. Leave blank to skip height propagation.

Benchmark height

0.000

The benchmark’s height (m).

Reciprocal pairs grid — one row per leveling line, ten columns:

From | To | Zen-> (gon) | Slope-> | Hi-> | Ht-> | Zen<- (gon) | Slope<- | Hi<- | Ht<-

  • -> columns are the forward observation (instrument at From, target at To): zenith angle (gon), slope distance (m), instrument height, target height.

  • <- columns are the reverse observation (instrument at To, target at From) — usually a different instrument/target setup on the same line.

  • Zenith angles are in gon; 100 gon = level sight.

  • A row is used when both From and To are filled.

2.6.3.3. Mathematics

For each direction the raw one-way vertical component is

dh = Slope * cos(zenith) + Hi - Ht

Curvature and refraction would add d²·(1−k)/(2R) to each one-way value (d = horizontal distance) — but because the error is (nearly) the same in both directions, the reciprocal mean cancels it:

mean dH     = ( dh_forward − dh_reverse ) / 2      = H(To) − H(From)
discrepancy =   dh_forward + dh_reverse            ≈ −2 · curvature/refraction term
implied k   =   1 + R · discrepancy / d²           (diagnostic only)

A discrepancy near zero means the pair was well balanced (near-simultaneous, comparable refraction, similar distances). A large value flags a bad reading or unbalanced conditions on that line.

If a Benchmark point is given, heights are propagated from it through the chain of computed lines (each pair is a bidirectional link worth mean dH), and the derived height of every reachable point is reported. The benchmark name must match a From/To name in the grid.

2.6.3.4. Results

One block per pair:

A -> B
  mean dH   = 6.3419
  fwd dH    = 6.4482
  rev dH    = -6.2357
  discrep   = 0.2125
  implied k = 0.303
  H(B)      = 106.3419

mean dH is H(To) − H(From): positive means To is higher. The sketch pane draws the first pair’s geometry (From at the origin, To at its horizontal distance and mean height difference).

2.6.3.5. Import

Import… reads a plain CSV/semicolon/tab file, one pair per line, no header, exactly this field order:

From,To,ZenF,SlopeF,HiF,HtF,ZenR,SlopeR,HiR,HtR
A,B,99.5244,850.0237,1.600,1.500,100.4868,850.0248,1.550,1.600

Lines with fewer than ten fields are skipped. Imported rows are appended to the grid (existing rows are kept). A sample file is installed as testfiles/surveycalculator/ptl_pairs.csv. Use Copy to place the results on the clipboard; there is no file export in this calculator.

2.6.3.6. Notes

  • The refraction coefficient is never an input: the reciprocal method eliminates it from the mean. implied k merely tells you what refraction was in effect — typical values are around 0.13; values far outside ~0–0.3 deserve suspicion.

  • Changing the Earth radius affects only implied k.

  • Hi/Ht must be the actual heights of the instrument and target for that direction’s setup — they generally differ between the forward and reverse observations.