Surveying Mathematics and Error Analysis
This domain tests computation fluency and statistically defensible treatment of error: azimuths and bearings from coordinates, traverse adjustment, area by coordinates, precision ratios, and propagation of uncertainty. Expect questions that mix geometry with a decision about whether a closure is acceptable.
Core computations
- Azimuth from coordinate differences: azimuth = arctangent of delta E divided by delta N, placed in the correct quadrant from the signs of both differences, expressed clockwise from north.
- Latitudes and departures: delta N = d cos(azimuth), delta E = d sin(azimuth). Sums should close on a closed figure.
- Area by coordinates: twice the area equals the sum of X(i) times (Y(i+1) minus Y(i-1)) around the figure; take the absolute value and divide by two.
- Traverse adjustment: the compass (Bowditch) rule distributes the latitude and departure misclosures in proportion to course length; the transit rule distributes them in proportion to the absolute latitudes and departures, appropriate when angles are stronger than distances. Least squares handles weighted networks.
- Precision ratio = total traverse length divided by linear misclosure. Error propagation: independent errors add in quadrature, so sigma of a sum is the square root of the sum of the variances, and a constant multiplier scales the standard deviation by its absolute value.
Worked example
A closed traverse has perimeter 2,500.00 ft, sum of latitudes = +0.12 ft, and sum of departures = -0.08 ft. Linear misclosure = square root of (0.12 squared + 0.08 squared) = square root of (0.0144 + 0.0064) = 0.1442 ft. Precision ratio = 2,500.00 / 0.1442 = 17,337, reported as 1:17,300. Under the compass rule, the latitude correction for a 500.00-ft course is -0.12 times (500/2,500) = -0.024 ft, and its departure correction is +0.08 times (500/2,500) = +0.016 ft. Check that corrections sum to the negatives of the misclosures. For an azimuth check: delta N = -300.00 and delta E = +500.00 gives a bearing angle of arctan(500/300) = 59 degrees 02 minutes, so the azimuth is 180 - 59 degrees 02 minutes = 120 degrees 58 minutes.
Traps that cost marks
Applying corrections with the same sign as the misclosure; dividing misclosure by the wrong length; reporting precision as a decimal instead of a ratio; rounding intermediate values before the final step; averaging angles without converting from degrees-minutes-seconds; using the transit rule on a compass-rule traverse or vice versa.