7.2 Traverse Computation and Adjustment

7.2 Traverse Computation and Adjustment

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1. Angular closure comes first

Never compute coordinates before checking angles. A blunder in an angle propagates into every subsequent bearing, and the linear misclosure will then be large without telling you where the fault lies.

For a closed polygon traverse of $n$ sides:

$\sum \text{interior angles} = (n-2) \times 180^\circ$ $\sum \text{exterior angles} = (n+2) \times 180^\circ$

For a link (closed-route) traverse between two lines of known bearing, the check is against the difference between the fixed opening and closing bearings.

The angular misclosure is the difference between the observed and theoretical sums. It must satisfy regulation 5(A)(iii): 3 seconds times the square root of N, with N not exceeding 30. If it does, distribute it equally among the angles - each angle was observed the same way, so each carries the same expected error - and then propagate bearings from the fixed opening bearing.

Worked example. A five-sided closed traverse gives observed interior angles summing to 540 deg 00' 25".

• Theoretical: $(5-2) \times 180 = 540$ deg 00' 00"
• Misclosure: +25"
• Tolerance: $3'' \times \sqrt{5} = 3 \times 2.236 = 6.7''$

25 seconds exceeds the tolerance. The correct response is not to distribute it but to look for a blunder - a misread angle, a wrong station, a transcription error - because 25 seconds in five stations is not random error. This is the discipline the Regulations impose, and it is the discipline a Folio must demonstrate.

Had the misclosure been, say, +5", it would be within tolerance and each angle would be corrected by $-1''$.

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2. Bearings, latitudes and departures

Propagate bearings around the traverse from the fixed opening bearing using the corrected angles, and reduce each to the nearest thirty seconds as regulation 5(E)(i) requires.

For each leg of length $L$ and bearing $\theta$ (measured clockwise from north):

$\Delta N = L\cos\theta \quad (\text{latitude}) \qquad \Delta E = L\sin\theta \quad (\text{departure})$

For a closed polygon, both sums should be zero:

$\sum\Delta N = 0, \qquad \sum\Delta E = 0$

For a link traverse, they should equal the known coordinate differences between the terminal stations.

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3. Misclosure and its ratio

$e_N = \sum\Delta N - (\Delta N)_{known}, \qquad e_E = \sum\Delta E - (\Delta E)_{known}$

$e = \sqrt{e_N^2 + e_E^2}$

$\text{Fractional linear misclosure} = \frac{e}{\sum L} = \frac{1}{(\sum L)/e}$

Worked example. A traverse of total length 1,842.60 m closes with $e_N = -0.28$ m and $e_E = +0.42$ m.

$e = \sqrt{(-0.28)^2 + (0.42)^2} = \sqrt{0.0784 + 0.1764} = \sqrt{0.2548} = 0.505\text{ m}$

$\text{Ratio} = \frac{0.505}{1842.60} = \frac{1}{3648}$

1:3,648 is better than 1:3,000, so the traverse meets the regulation 3 standard and may be adjusted. Had it come out at 1:2,400 the survey would have failed the standard and the correct answer is re-observation, not adjustment. Adjustment redistributes random error; it does not convert a substandard survey into a compliant one.

The direction of the misclosure vector is worth computing too. A misclosure that lies almost along one leg's bearing often points to a length blunder in that leg; one perpendicular to it points to an angular problem.

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4. Bowditch (compass) rule

The Bowditch rule assumes that angles and distances are of comparable precision, so error accumulates in proportion to the length of each leg:

$\text{Correction to } \Delta N_i = -e_N \times \frac{L_i}{\sum L}, \qquad \text{Correction to } \Delta E_i = -e_E \times \frac{L_i}{\sum L}$

Worked example. Continuing from above, with $e_N = -0.28$ m, $e_E = +0.42$ m and $\sum L = 1842.60$ m, a leg of $L = 421.30$ m receives:

• $\Delta N$ correction: $+0.28 \times \dfrac{421.30}{1842.60} = +0.28 \times 0.22864 = +0.064$ m
• $\Delta E$ correction: $-0.42 \times 0.22864 = -0.096$ m

Apply the corrections with the opposite sign to the misclosure, and check that the corrections sum to exactly $-e_N$ and $-e_E$ before proceeding. That check catches arithmetic slips immediately.

Bowditch is the default in Nigerian cadastral practice because ordinary traverse work does observe angles and distances to comparable relative precision.

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5. Transit rule

The transit rule assumes that angles are more precise than distances, so error is attributed to the linear components:

$\text{Correction to } \Delta N_i = -e_N \times \frac{|\Delta N_i|}{\sum|\Delta N|}, \qquad \text{Correction to } \Delta E_i = -e_E \times \frac{|\Delta E_i|}{\sum|\Delta E|}$

It is appropriate where a precise theodolite is used with less precise distance measurement. Its weakness is that the adjusted result depends on the orientation of the traverse relative to the axes: rotate the whole traverse and the transit-adjusted coordinates change shape, which Bowditch-adjusted coordinates do not. For general cadastral work that is a decisive objection.

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6. Coordinates and area

Accumulate the adjusted latitudes and departures from the fixed starting coordinates. Regulation 5(E)(iii) requires coordinates computed to the nearest tenth of a foot, by logarithms of not less than five figures or by calculating machine, and requires that the Colony coordinates of at least one beacon in the survey be computed where Colony coordinates are available for the beacon to which the survey is connected.

For area, regulation 5(E)(iv) requires the area of land bounded by straight lines to be computed arithmetically from the coordinates of the corners - the cross-product ("surveyor's") formula:

$A = \frac{1}{2}\left|\sum_{i=1}^{n}\left(E_i N_{i+1} - E_{i+1} N_i\right)\right|$

with the polygon closed by taking $(E_{n+1}, N_{n+1}) = (E_1, N_1)$.

Where a boundary is irregular, the area between it and the adjacent traverse line is computed as a series of trapezoids, consistent with regulation 9, which requires traverse lines to follow approximately the course of an irregular boundary with offsets taken at suitable points.

Finally, state the area to the accuracy prescribed by regulation 31(j) - which reduces the number of decimals as the parcel grows, because relative boundary accuracy, not arithmetic, limits how many digits are meaningful.

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7. The order of operations

  1. Angular misclosure  ->  check against 3" x sqrt(N), N <= 30
                             OUT OF TOLERANCE?  Find the blunder.
  2. Distribute angular misclosure equally; propagate bearings
     from the fixed opening bearing; reduce to nearest 30".
  3. Reduce distances: standard, slope, temperature, sag;
     then to the ellipsoid and to the grid (combined factor).
  4. Compute latitudes and departures.
  5. Linear misclosure and its RATIO  ->  check against 1:3,000.
                             SUBSTANDARD?  Re-observe. Do not adjust.
  6. Adjust (Bowditch by default); verify corrections sum to -e.
  7. Accumulate coordinates from the fixed start.
  8. Area from coordinates; irregular strips as trapezoids.
  9. State area to reg. 31(j) accuracy.