6.5 Surveying

Measurements and Differential Leveling

Surveying establishes positions in three dimensions: horizontal distance, angles/directions, and elevation. Distances come from EDM/total stations or taping (with corrections for temperature, tension, and sag); directions are given as bearings (N/S, angle, E/W, 0–90°) or azimuths (0–360° clockwise from north).

Differential leveling transfers elevation from a benchmark using the height-of-instrument (HI) method:

• HI = Elevation(known point) + Backsight (BS)
• Elevation(new point) = HI − Foresight (FS)

A backsight (BS) is the rod reading on a point of known elevation (a 'plus' sight); a foresight (FS) is the reading on a point of unknown elevation (a 'minus' sight). Turning points (TP) carry the line forward.

Worked leveling example. Benchmark elevation = 100.00 ft. BS on BM = 4.20 ft → HI = 104.20 ft. FS on point X = 6.50 ft → Elev_X = 104.20 − 6.50 = 97.70 ft. The loop misclosure (returning to the start) is checked against allowable error, often C = 0.05·√M ft for M miles of leveling.

Traverse Computation and Adjustment

A traverse is a series of connected lines with measured lengths and directions. For each leg of length L and bearing/azimuth angle α:

• Latitude = L·cos α (north +, south −)
• Departure = L·sin α (east +, west −)

For a closed traverse, the latitudes and departures must each sum to zero. The residuals are the misclosure:

• Linear misclosure = √[(ΣLat)² + (ΣDep)²]
• Precision (relative) = misclosure / total perimeter, expressed as 1:N.

The Compass (Bowditch) rule adjusts each leg's latitude and departure proportional to its length:

• Correction to a leg's latitude = −(ΣLat)·(L_leg / ΣL)
• Correction to a leg's departure = −(ΣDep)·(L_leg / ΣL)

Worked misclosure example. If ΣLat = +0.30 ft and ΣDep = −0.40 ft over a 2000-ft perimeter, linear misclosure = √(0.30² + 0.40²) = √(0.09 + 0.16) = √0.25 = 0.50 ft, giving a precision of 0.50/2000 = 1:4000. The Transit rule is an alternative that weights by the latitude/departure magnitudes instead of leg length.

Area, Control, GNSS, and Error Theory

Area by coordinates (the coordinate/shoelace method) uses adjusted station coordinates (X_i, Y_i):

• 2A = |Σ X_i·(Y_{i+1} − Y_{i−1})|, then A = that /2

This is the standard way the FE asks for parcel area once a traverse is adjusted.

Control provides the reference framework: horizontal control (NAD 83 coordinates, monuments) fixes plan position; vertical control (NAVD 88 benchmarks) fixes elevation. GPS/GNSS uses satellite ranging; key terms: differential/RTK (real-time kinematic) correction for centimeter accuracy, geoid vs ellipsoid height (orthometric height H = h − N, where N is geoid undulation), and DOP (dilution of precision) describing satellite geometry.

Error theory — a frequent FE distinction:

Random errors propagate as the root-sum-of-squares: for n equal measurements, the error of the sum is E·√n and of the mean is E/√n.

Angles, Azimuths, and a Worked Area Example

Angle and direction conventions must be handled carefully. An azimuth runs 0°–360° clockwise from north; a bearing is an acute angle (0°–90°) measured from north or south toward east or west. Converting between them is a common FE step: an azimuth of 135° is bearing S 45° E; an azimuth of 200° is S 20° W. The back azimuth = forward azimuth ± 180°.

In an open or connecting traverse, interior angles must satisfy Σ(interior angles) = (n − 2)·180° for an n-sided closed polygon; any angular misclosure is distributed equally among the angles before computing latitudes and departures.

Worked area-by-coordinates example. A triangular parcel has adjusted coordinates A(0, 0), B(100, 0), C(100, 80) (ft). Using the shoelace formula:

• 2A = |x_A(y_B − y_C) + x_B(y_C − y_A) + x_C(y_A − y_B)|
• 2A = |0(0 − 80) + 100(80 − 0) + 100(0 − 0)| = |0 + 8000 + 0| = 8000
• A = 8000/2 = 4000 ft² (≈ 0.092 acre, since 1 acre = 43,560 ft²)

This matches the simple right-triangle area ½·base·height = ½·100·80 = 4000 ft², confirming the method. Always use adjusted (balanced) coordinates from the Bowditch step; using raw, unbalanced coordinates carries the traverse misclosure into the area. Acre conversions (43,560 ft²) and station-to-foot conversions are the usual unit traps.

The full surveying workflow the FE rewards runs in order: measure angles and distances, balance the angular misclosure, compute latitudes and departures, check the linear misclosure and relative precision, adjust each leg by the compass (Bowditch) rule, then compute coordinates and finally area by the coordinate method. Skipping the balancing steps and computing area from raw field data carries the misclosure straight into the result. Likewise, leveling is checked by looping back to the starting benchmark: the loop misclosure must fall within the allowable C = 0.05·√M ft (for M miles run) before the elevations are accepted and distributed.