Traverses: closure and the compass rule

A traverse is a series of connected lines whose lengths and directions are measured, running from point to point across a site. It is the classic method of establishing horizontal control, and even in the age of GNSS it remains essential where satellite signals are blocked by trees, buildings or terrain, inside tunnels and buildings, and wherever the highest relative accuracy between nearby points is needed. This lesson covers the kinds of traverses, field procedure, the angular and linear checks that closed traverses provide, the computation of latitudes and departures, and the compass rule adjustment that produces final coordinates.

For the Coulee Ridge subdivision, Grant's crew runs a closed loop traverse around the site from control point CR-1, whose coordinates and the azimuth to CR-5 were established with GNSS, through four new points, CR-5 to CR-8, and back to CR-1. The traverse uses a project coordinate system with CR-1 at northing 5000.000 m and easting 1000.000 m. Leah computes and adjusts it; her work is the worked example of this lesson.

Kinds of traverses

• A closed loop traverse starts and ends at the same point. It provides checks on the angles and on the computed coordinates, but a loop alone cannot reveal a systematic error that scales all distances equally, such as a wrong prism constant applied consistently, or a wrong starting azimuth, because the shape still closes.
• A connecting or link traverse starts at one known point and ends at a different known point, ideally with known azimuths at both ends. It checks the angles, the distances and the scale and orientation of the whole survey, and it is the preferred form for route surveys and for tying projects to existing control.
• An open traverse ends at an unknown point and provides no check at all. Open traverses should be avoided except for rough or temporary work, and even then the lines should be measured twice.

Field procedure

At each traverse station, the total station is set up and centred, and the angle right from the previous station (the backsight) to the next station (the foresight) is measured on both faces, usually in several sets. Distances to the backsight and foresight are measured, so each line is measured from both ends and the two values can be compared. Targets are set on tripods with tribrachs over the points, and forced centring is used where possible, as described in lesson 6. Short lines are avoided, because centring errors produce large angular errors on short sights.

For a closed polygon, crews usually measure interior angles. With the traverse run counterclockwise around the figure, the angles right measured from backsight to foresight are the interior angles; with the traverse run clockwise, they are the exterior angles, which sum to \((n + 2) \times 180°\). Either way works, but the computation must follow consistently.

Angular closure

The interior angles of any closed polygon with \(n\) sides must sum to \((n - 2) \times 180°\). The difference between the sum of the measured angles and this value is the angular misclosure. Specifications limit it to a value that grows with the square root of the number of angles, for example \(10''\sqrt{n}\) for good engineering control or larger values for lower-order work.

If the misclosure is acceptable, it is usually distributed equally among the angles, since each was measured with similar care. If one angle was measured under poor conditions, for example with a very short sight, it can receive more of the correction. After adjustment, azimuths are computed around the traverse, and the last computed azimuth must return exactly to the starting azimuth, which checks the arithmetic.

Latitudes and departures

Each traverse line is resolved into components along the north and east axes. The latitude is the north component, positive northward, and the departure is the east component, positive eastward:

For a closed loop, the latitudes and departures must each sum to zero, because the traverse returns to its starting point. For a link traverse, they must sum to the difference between the known coordinates of the end and start points. The amounts by which they fail are the misclosures in latitude and departure, \(e_L\) and \(e_D\). The linear misclosure and the relative precision are:

Relative precision is usually written as a ratio such as 1:20,000, meaning one unit of misclosure for every 20,000 units of distance travelled. Typical requirements range from about 1:5,000 for rough construction work to 1:20,000 or better for engineering control, and much better for geodetic work. Modern total stations routinely achieve 1:50,000 or better on carefully run traverses.

The compass rule

The compass rule, also called the Bowditch rule, distributes the misclosures in latitude and departure in proportion to the lengths of the lines. It assumes that angles and distances are measured with similar relative precision, which is reasonable for total station traverses. The corrections to line \(i\) are:

The corrected latitudes and departures are then accumulated from the starting coordinates to give the coordinates of each point. The final computed coordinates of the closing point must equal its known coordinates, which is a check on the arithmetic. Other rules exist, such as the transit rule, which distributes corrections in proportion to the latitudes and departures themselves, but the compass rule is the most common simple method. For important control, least squares adjustment is now standard, because it handles any network shape, weights each measurement according to its precision and gives the precision of every adjusted coordinate.

# Closed loop traverse: angular adjustment, latitudes and departures, compass rule
import math

def dms(d, m=0, s=0): return d + m / 60 + s / 3600
def fmt(x):
    x %= 360; d = int(x); m = int((x - d) * 60); s = round(((x - d) * 60 - m) * 60)
    if s == 60: s, m = 0, m + 1
    if m == 60: m, d = 0, d + 1
    return f"{d:3d}°{m:02d}'{s:02d}\""

names = ["CR-1", "CR-5", "CR-6", "CR-7", "CR-8"]
angles = [dms(84, 7, 32), dms(120, 33, 58), dms(105, 18, 35), dms(93, 58, 57), dms(136, 1, 10)]
dist = [373.537, 498.496, 365.687, 431.553, 415.638]
az0, N0, E0 = dms(146, 23, 31), 5000.000, 1000.000
n = len(names)

mis = (sum(angles) - (n - 2) * 180) * 3600
print(f"angular misclosure {mis:+.1f}\" (allowable {10 * math.sqrt(n):.0f}\")")
adj = [a - mis / n / 3600 for a in angles]

az = [az0]
for i in range(1, n):
    az.append((az[-1] + 180 + adj[i]) % 360)
print("closing azimuth check:", fmt((az[-1] + 180 + adj[0]) % 360), "=", fmt(az0))

lat = [L * math.cos(math.radians(a)) for L, a in zip(dist, az)]
dep = [L * math.sin(math.radians(a)) for L, a in zip(dist, az)]
eL, eD, P = sum(lat), sum(dep), sum(dist)
e = math.hypot(eL, eD)
print(f"eL = {eL:+.4f} m, eD = {eD:+.4f} m, e = {e:.4f} m, precision 1:{P / e:,.0f}")

N, E = N0, E0
for i in range(n):
    N += lat[i] - eL * dist[i] / P
    E += dep[i] - eD * dist[i] / P
    print(f"{names[i]}-{names[(i + 1) % n]}  az {fmt(az[i])}  lat {lat[i]:+9.3f}  dep {dep[i]:+9.3f}  -> {names[(i + 1) % n]} N {N:9.3f} E {E:9.3f}")

angular misclosure +12.0" (allowable 22")
closing azimuth check: 146°23'31" = 146°23'31"
eL = +0.0311 m, eD = +0.0343 m, e = 0.0463 m, precision 1:45,024
CR-1-CR-5  az 146°23'31"  lat  -311.098  dep  +206.756  -> CR-5 N  4688.897 E  1206.750
CR-5-CR-6  az  86°57'27"  lat   +26.459  dep  +497.793  -> CR-6 N  4715.349 E  1704.535
CR-6-CR-7  az  12°15'59"  lat  +357.338  dep   +77.693  -> CR-7 N  5072.682 E  1782.222
CR-7-CR-8  az 286°14'54"  lat  +120.749  dep  -414.316  -> CR-8 N  5193.424 E  1367.899
CR-8-CR-1  az 242°16'01"  lat  -193.418  dep  -367.892  -> CR-1 N  5000.000 E  1000.000

Finding blunders

If a traverse fails to close within tolerance, the cause is usually a single blunder rather than an accumulation of random errors. Two simple tricks help locate it before the crew returns to the field.

• An angle blunder: compute the coordinates forward from the start and backward from the end, using the unadjusted angles. The station where the two sets of coordinates agree is likely the one with the wrong angle, because a wrong angle rotates the rest of the traverse about that point.
• A distance blunder: compute the direction of the linear misclosure, \(\tan^{-1}(e_D / e_L)\). A single wrong distance shifts the traverse along the direction of that line, so the misclosure will be nearly parallel to the line containing the blunder, and its size will be close to the error.

Common blunders include a backsight on the wrong point, a misread or mistyped angle, a transposed digit in a distance, and a wrong instrument or target height entered in a three-dimensional traverse. Once found, the suspect measurement is remeasured, not simply corrected on paper.

Tying traverses to the reference system

A loop traverse closed on itself proves internal consistency, but it does not prove that its orientation and scale match the national or state reference system. For that, the traverse must start and end on control points of known coordinates, or include GNSS-positioned points, as described in lessons 9 and 10. The starting azimuth on the Coulee Ridge traverse came from GNSS positions of CR-1 and CR-5, which ties the traverse to the reference system in orientation. If the traverse distances are ground distances and the coordinates are meant to be grid coordinates, the distances must also be scaled to grid before computation; on the project coordinate system used here, which is scaled to ground, that step is not needed.

Three-dimensional traverses

Modern total stations measure zenith angles along with horizontal angles and distances, so a traverse can carry elevations as well as horizontal coordinates. The elevation difference on each line is computed by trigonometric levelling, as described in lesson 3, measured in both directions and averaged to reduce the effects of refraction and curvature. The sum of elevation differences around a loop should be zero, giving an elevation misclosure that can be adjusted much like a level loop. Traverse elevations are adequate for many purposes, but benchmarks for drainage design are still established by differential levelling.