Angles, total stations and traverses

Distances and heights are not enough to locate a point. To place a building corner you also need direction: which way from a known point, measured as an angle from a known line. The total station, which combines an electronic theodolite for angles with an EDM for distances and a computer for coordinates, is the main tool for measuring and setting out directions on a construction site. This lesson explains angles, bearings and azimuths, how a total station is set up and oriented, and how a traverse establishes and checks control points around a site.

Horizontal and vertical angles

A horizontal angle is measured in the horizontal plane between two lines of sight from the instrument. It is usually measured clockwise from a backsight, the reference direction to a known point, to a foresight, the direction to the new point. A vertical angle is measured in the vertical plane. Most total stations report the zenith angle, measured downward from straight up, so a horizontal sight reads 90°. A vertical angle measured from the horizontal is 90° minus the zenith angle.

Angles are recorded in degrees, minutes and seconds. One second of arc subtends about 0.5 millimeters at 100 meters, so an instrument with a 5-second angular accuracy can place a point at 100 meters within about 2.5 millimeters of direction error. Construction total stations commonly have angular accuracies of 1 to 5 seconds.

Bearings and azimuths

The direction of a line is expressed relative to a reference meridian, usually grid north on the project coordinate system. There are two conventions.

• An azimuth is measured clockwise from north, from 0° to 360°. A line running due east has an azimuth of 90°; due south, 180°; due west, 270°. Azimuths are what total stations and coordinate calculations use.
• A bearing is measured from north or south toward east or west, from 0° to 90°, and is written with the quadrant: N 38°15′ E, S 32°10′ W. Bearings are traditional on legal plans and property descriptions.

The back azimuth of a line, the direction looking back from its end to its start, differs by 180°. If line AB has an azimuth of 38°15′, line BA has an azimuth of 218°15′. Carrying azimuths around a traverse uses this rule repeatedly: the azimuth of the next line equals the back azimuth of the previous line plus the angle turned clockwise at the station, minus 360° if the result exceeds 360°.

The total station

A total station measures horizontal angles, vertical angles and slope distances, and its onboard software converts them into horizontal distances, height differences and coordinates. Several kinds are common on construction sites.

• Manual total stations are pointed by the operator, with a second person holding the prism pole.
• Robotic total stations track the prism automatically and are controlled from a data collector at the pole, so one person can lay out points. They are now the standard for building layout.
• Scanning total stations add a laser scanner for capturing as-built surfaces.

Setting up a total station follows a fixed sequence. The tripod is set over the station with the head roughly level. The instrument is centered over the point using the optical or laser plummet and leveled with the foot screws and electronic bubble. The height of instrument above the mark is measured. The operator then orients the instrument by sighting a backsight on a known control point, entering both coordinates so the instrument knows which way is north, and checks the orientation by measuring to a third control point. A check shot that does not agree within a few millimeters means something is wrong: the wrong point, a disturbed control mark, a bumped tripod or a coordinate entry error.

When it is not convenient to set up over a known point, the operator can use a resection, or free station. The instrument is set up anywhere with good visibility and measures angles and distances to two or more control points, usually three or more for a check. The software computes the instrument's position and orientation from those measurements. Resection is standard practice on building sites, where control points are on columns, walls or targets around the perimeter, and the instrument is set up wherever there is a clear view of the work.

Instrument errors and measuring in two faces

Total stations have small mechanical errors that make the line of sight not quite perpendicular to the horizontal axis (collimation error), or the horizontal axis not quite perpendicular to the vertical axis (trunnion axis error), or the vertical circle index not quite correct. Most of these errors are canceled by measuring each angle twice: once in face left, then again after rotating the telescope through 180° in both axes, in face right. The average of the two faces is free of most instrument errors.

Control surveys use multiple sets of two-face observations. Layout is normally done in one face, so the instrument must be checked and calibrated regularly, and important points should be checked from a second setup.

A total station also depends on the tripod and the setup. Sun on one leg, soft ground, frost heave or a worker bumping a leg can rotate or tilt the instrument during the day. Robotic stations can monitor their own tilt and orientation, but the operator should still re-check the backsight frequently and always before setting critical points.

Traverses

A traverse is a series of connected lines whose lengths and angles are measured, used to establish the positions of control points. A closed traverse starts and ends on known points, either returning to its starting point as a loop or ending on a different known point, so its accuracy can be checked. An open traverse does not close and cannot be checked, which is why it should never be used for control.

Construction projects use traverses to set up the site control network: a ring of stable points around the building or along a road that will be used for all later layout. GNSS has replaced traversing for much primary control on open sites, but traverses remain essential in dense urban areas, under tree cover, inside buildings and tunnels, and wherever high precision is needed.

Step 1: angular closure

For a closed loop traverse with n sides, the interior angles must add up to (n − 2) × 180°. The difference between the measured sum and this value is the angular misclosure. If it is within the allowable value, it is distributed equally among the angles, because each angle was measured with similar care. Specifications express the allowable angular misclosure as a constant times the square root of the number of angles, for example 10 seconds × √n for construction control.

Step 2: azimuths, latitudes and departures

With balanced angles and a known starting azimuth, the azimuth of each line is computed in turn. Each line is then resolved into its north-south component, the latitude, and its east-west component, the departure. North and east are positive; south and west are negative.

If the traverse were perfect, the latitudes would sum to zero and the departures would sum to zero, because the traverse returns to its start. In practice they do not quite, and the resulting gap is the linear misclosure. Dividing the total length of the traverse by the linear misclosure gives the relative precision.

Step 3: adjusting the traverse and computing coordinates

A traverse that passes is adjusted so that it closes exactly. The common method on construction surveys is the compass rule, also called the Bowditch rule. It assumes that errors are proportional to the length of each line, so each latitude and departure is corrected in proportion to the length of its line divided by the total length. Survey software also offers least squares adjustment, which weights each measurement by its expected precision and is the standard for control networks, but the compass rule is easy to do by hand and illustrates the idea.

Once all coordinates are adjusted, the control points are published in a control sheet showing each point's number, description, northing, easting, elevation and how it is marked, often with a sketch or photograph. Every layout crew on the project then works from the same control sheet.

Area from coordinates

Coordinates also make it easy to calculate areas, for example of a lot, a paved area or a stockpile footprint. The coordinate method, sometimes called the shoelace formula, multiplies each northing by the difference of its neighbors' eastings, or equivalently cross-multiplies around the polygon.

Survey software and CAD do this automatically, but the formula is worth knowing because it shows that areas computed from coordinates are only as good as the coordinates themselves.

Practical control on a building site

On a typical building project, the surveyor establishes four or more primary control points around the site, far enough from the excavation that they will not be disturbed, with clear sight lines to the building. Additional control may be set on adjacent buildings or on monuments. As the structure rises, control is transferred to each floor by plumbing up through openings in the slab or by resection from external targets. The construction manager's responsibilities include protecting control points with barriers and paint, keeping equipment and stockpiles away from them, reporting any disturbance, and making sure that control is re-checked against each other at regular intervals.

A disturbed control point is dangerous because everything set from it is wrong in the same way, and the error may not be noticed until a precast panel does not fit or a steel beam is short. If a control point might have been hit, stop using it until the surveyor has checked it from at least two other points.