Traverse adjustment
Transit-rule adjustment compared with the compass rule
The same five-sided loop is readjusted by the transit rule, which distributes the closure in proportion to each latitude and departure rather than to course length, and the two sets of coordinates are compared.
Given
- The same closed-loop traverse A-B-C-D-E-A, already reduced to adjusted azimuths and observed distances.
- Courses: A-B 114°30′00″ 434.24 ft; B-C 52°07′37″ 398.95 ft; C-D 335°13′39″ 358.09 ft; D-E 261°03′46″ 450.30 ft; E-A 199°46′03″ 340.14 ft.
- Latitudes: −180.076, 244.921, 325.138, −69.955, −320.096 ft. Departure: 395.142, 314.920, −150.046, −444.833, −115.037 ft.
- Latitude closure −0.069 ft, departure closure +0.147 ft, perimeter 1981.72 ft.
- Station A held at N 5000.00, E 5000.00.
- The crew used a 5 second instrument on tripods but taped the distances with a hand-held tape, so the directions are believed to be markedly better than the lengths.
Required
- Transit-rule corrections to each latitude and departure.
- Adjusted latitudes, departures and coordinates under the transit rule.
- A comparison with the compass-rule coordinates, and a statement of when each rule is the right choice.
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
What the transit rule assumes
The compass rule spreads the closure in proportion to course length, which is the right model when angular and linear errors contribute about equally. The transit rule instead spreads each closure in proportion to the size of the component being corrected: a course with a large latitude absorbs a large share of the latitude closure.
That model corresponds to error that lives almost entirely in the distances, with the directions taken as effectively exact. Here the crew turned angles with a 5 second instrument and taped with a hand-held tape, so that is the case at hand.
One consequence is that the transit rule is not invariant to the choice of north. Rotate the coordinate system and the corrections change, because the latitude and departure of each course change. The compass rule has no such defect. That is the main theoretical objection to the transit rule, and the reason most modern practice defaults to the compass rule or to a least-squares adjustment.
correction to latᵢ = −ΣLat × (|latᵢ| / Σ|lat|)
correction to depᵢ = −ΣDep × (|depᵢ| / Σ|dep|)
Σ|lat| = 180.076 + 244.921 + 325.138 + 69.955 + 320.096 = 1140.187 ft
Σ|dep| = 395.142 + 314.920 + 150.046 + 444.833 + 115.037 = 1419.977 ftTransit-rule corrections
Each latitude correction is +0.069 ft times that course's share of 1140.187 ft; each departure correction is −0.147 ft times that course's share of 1419.977 ft. The two columns are handled entirely separately, which is the structural difference from the compass rule.
Watch course C-D. It carries the largest latitude, 325.138 ft, and one of the smallest departures, −150.046 ft, so it takes the largest latitude correction and one of the smallest departure corrections. Under the compass rule the two corrections for that course would have been in the same proportion as each other on every course.
| Course | Latitude (ft) | Departure (ft) | Lat correction (ft) | Dep correction (ft) | Adjusted latitude (ft) | Adjusted departure (ft) |
|---|---|---|---|---|---|---|
| A-B | −180.076 | 395.142 | +0.011 | −0.041 | −180.066 | 395.101 |
| B-C | 244.921 | 314.920 | +0.015 | −0.033 | 244.936 | 314.888 |
| C-D | 325.138 | −150.046 | +0.020 | −0.016 | 325.158 | −150.061 |
| D-E | −69.955 | −444.833 | +0.004 | −0.046 | −69.951 | −444.879 |
| E-A | −320.096 | −115.037 | +0.019 | −0.012 | −320.077 | −115.049 |
| Sums | −0.069 | 0.147 | +0.069 | −0.147 | 0.000 | 0.000 |
Coordinates under the transit rule
Accumulate from station A exactly as before. The traverse must again close exactly on A.
The precision figure is untouched: 0.162 ft over 1981.72 ft, or 1:12,235. Adjustment redistributes a misclosure, it never reduces it, and quoting a better precision after adjustment is a misstatement of the survey.
| Station | Compass N (ft) | Compass E (ft) | Transit N (ft) | Transit E (ft) | Difference (ft) |
|---|---|---|---|---|---|
| A | 5000.00 | 5000.00 | 5000.00 | 5000.00 | 0.00 |
| B | 4819.94 | 5395.11 | 4819.93 | 5395.10 | 0.01 |
| C | 5064.87 | 5710.00 | 5064.87 | 5709.99 | 0.01 |
| D | 5390.02 | 5559.93 | 5390.03 | 5559.93 | 0.01 |
| E | 5320.08 | 5115.06 | 5320.08 | 5115.05 | 0.01 |
How to choose
On this traverse the choice is worth at most 0.01 ft in any coordinate. That is the usual outcome on a well closed traverse, and it is the honest answer to give a client who asks which rule was used: on good work, it does not matter.
The choice starts to matter when the misclosure is large, when the courses differ greatly in length, or when one course runs nearly due north while another runs nearly due east. In those cases run both and look at the spread; if the two rules disagree by more than the survey's tolerance, the traverse is not good enough for either of them to rescue it.
Use the compass rule as the default. Use the transit rule only when the directions are genuinely of a much higher order than the distances, and record on the computation sheet which was applied, because a later surveyor cannot reverse-engineer the choice from the coordinates alone.
Answer
- Transit-rule latitude corrections: +0.011, +0.015, +0.020, +0.004, +0.019 ft on courses A-B through E-A.
- Transit-rule departure corrections: −0.041, −0.033, −0.016, −0.046, −0.012 ft.
- Transit-rule coordinates: A 5000.00 / 5000.00; B 4819.93 / 5395.10; C 5064.87 / 5709.99; D 5390.03 / 5559.93; E 5320.08 / 5115.05.
- The two rules differ by at most 0.01 ft at any station on this traverse. The precision is 1:12,235 under either rule, because adjustment does not change the misclosure.
- Area from the transit-rule coordinates: 264,208 sq ft, or 6.0654 acres, the same to four decimal places as the compass-rule area.
Check
Both correction columns must sum to the negative of their closures: +0.069 ft in latitude and −0.147 ft in departure. They do, and the adjusted columns therefore sum to 0.000.
The transit correction to the departure of course D-E, the largest departure in the traverse, is −0.147 × (444.833 / 1419.977) = −0.046 ft, the largest in its column, as the rule requires.
Compare the two areas. The compass-rule adjusted figure is 264,207.96 sq ft and the transit-rule figure is 264,207.68 sq ft, a difference of 0.28 sq ft on 6 acres. A large disagreement here would mean an arithmetic error in one of the two adjustments, not a real difference between the rules.
Neither adjustment changes the linear misclosure of 0.162 ft, so both must report 1:12,235.
More traverse adjustment
All in this category- Closed-loop traverse from field angles to compass-rule coordinatesA five-sided loop traverse is taken from raw interior angles and taped distances through angular closure, azimuths, latitudes and departures, misclosure and precision, to compass-rule adjusted coordinates.
- Closed-link traverse between two pairs of control monumentsA four-course traverse runs from one control pair to another. Azimuth closure comes from the two published control azimuths and position closure from the published coordinates of the far monument.
- A traverse that fails its precision specification, and what to do about itA four-sided traverse closes at 1:2,538 against a 1:10,000 requirement. The direction of the closure line points straight at the course carrying the blunder, which is remeasured and the traverse recomputed.