Survey Computations, Adjustments, and Least Squares
Traverse computation and adjustment
Latitude = d cos(azimuth), departure = d sin(azimuth). Summing a closed traverse gives the closures in latitude and departure, from which linear misclosure and the relative precision (1:N) are computed before any adjustment. The Compass (Bowditch) rule distributes closures in proportion to the length of each course; the transit rule distributes them in proportion to the summed latitudes and departures; Crandall's method and least squares are used where weights are known.
Least squares in outline
Write observation equations in the form A x = b + v, form normal equations N x = A-transpose W A with weights w = 1/variance, and solve. The reference variance is (v-transpose W v)/(n - u), with n observations and u unknowns, so degrees of freedom = redundancy must be positive. The covariance matrix is the inverse of N scaled by the reference variance, which enables precision of any derived quantity and blunder detection through standardized residuals.
Traps
Computing precision ratio after adjustment, applying the Compass rule to angular errors, misusing degrees of freedom, and confusing standard deviation with standard error of the mean.
Worked example (error propagation). A distance of 100 m is measured with a direction uncertainty of 10 seconds. Convert: 10 s = 10/206 265 = 4.85 x 10 to the -5 rad. Transverse error = 100 x 4.85 x 10 to the -5 = 0.005 m. Independent errors add in quadrature: total = square root of the sum of squares.