Survey Measurement and Computation
Survey Measurement and Computation
This domain tests the error theory and computation mechanics behind every other domain: closure, adjustment, precision statements, and coordinate geometry.
Error budget
- Random errors are reduced by redundancy; systematic errors (prism constant, scale error, temperature, collimation) bias every observation and must be eliminated, not averaged; blunders must be found and rejected.
- Weights are proportional to 1/σ². Unequal-precision observations are combined by weighted mean, never by simple averaging.
- Closure standards: angular misclosure for a closed traverse, linear closure as a ratio, and level loops on the order of 0.05√M ft for M miles of ordinary work.
Adjustment methods
- Compass (Bowditch) rule distributes misclosure proportionally to line length; the transit rule distributes by latitude and departure magnitude; Crandall method adjusts angles and lengths separately; least squares minimizes the sum of weighted residuals and yields the most probable values.
- Latitudes and departures: distance × cos(azimuth) and distance × sin(azimuth); closure = √(Σlat² + Σdep²), expressed as a ratio to total length.
- Area by coordinates (shoelace formula), closing the figure back to the first point and taking the absolute value of half the cross-sum.
Worked example
A distance is measured twice: 100.00 ft with σ = 0.02 ft and 100.06 ft with σ = 0.04 ft.
Weights: w1 = 1/0.02² = 2,500; w2 = 1/0.04² = 625. Weighted mean = (2,500 × 100.00 + 625 × 100.06) / 3,125 = (250,000 + 62,537.5) / 3,125 = 100.01 ft. Standard deviation of the mean = √(1/3,125) = 0.018 ft.
The second observation carries one-quarter the weight of the first because its variance is four times larger. Report 100.01 ft with a standard deviation near 0.02 ft, not a simple 100.03 ft average.
Traps
- Averaging observations of unequal precision arithmetically.
- Sign errors in departures or latitudes; east and north are positive.
- Reporting 0.001 ft precision from data good to 0.02 ft.
- Applying compass rule to angles instead of coordinate misclosures, or adjusting angles twice.