Geomatics Engineering

Leveling and fore sight

Leveling is the process of finding the difference in elevation between points by reading a levelling staff held on them from a level. Each setup of the instrument produces staff readings that are classified by purpose. The first reading taken after the level is set up on a station is a backsight (BS), taken on a point of known elevation such as a benchmark or a previously fixed change point. It is used to find the height of the instrument, HI = RL + BS. The last reading taken before the instrument is shifted is a foresight (FS), taken on a point whose elevation is wanted. It is used to find the RL of that point, RL = HI - FS. Readings taken between these two are intermediate sights, and a change point is a point on which a foresight is taken before the instrument moves and a backsight is taken after it moves. A first reading is therefore never a foresight, which is one of the most common errors in this topic.

The rise-and-fall relation gives an arithmetic check. The difference between the sum of backsights and the sum of foresights equals the change in RL from the first point to the last, so RL_{last} = RL_{first} + \sum BS - \sum FS.

When a staff is held inverted, as when sighting up to a point under a bridge or a ceiling, its readings are recorded as negative values, and the same formulas are applied with the negative sign carried through. For example, if a backsight of -1.20 m is taken on an inverted staff at a benchmark of 50.000 m, the height of instrument is 50.000 + (-1.20) = 48.800 m.

Reciprocal levelling is used when a river, valley or long sight makes it impossible to balance backsight and foresight distances. The level is set up near each bank in turn, the height difference is observed with the same sign convention, and the mean of the two observed differences is taken as the true difference. Errors from earth curvature and atmospheric refraction are nearly equal in both directions under the same conditions, so they cancel in the average. For example, observed differences of +0.80 m and +0.60 m give a true difference of +0.70 m.

If the height of the plane of collimation is known, the reduced level of any point is the height of the plane of collimation minus the staff reading held on that point.

The gradient between two points is the difference in RL divided by the horizontal distance between them, expressed as a percentage. A fall is a negative gradient. In a series of readings taken at a uniform interval, the change point is counted only once in the horizontal distance.

A level surface is an equipotential surface of the gravity field. At every point it is perpendicular to the plumb line, and two level surfaces never intersect, because gravity has only one potential value at each point. The geoid is the irregular equipotential surface that approximates mean sea level, while the ellipsoid is the regular mathematical surface that approximates the geoid. Contours of different elevations can meet only at an overhanging cliff, surveying proceeds from the whole to the parts, and a planimeter is an instrument for measuring the area of a plan.

Closed traverse bearing computation

A closed traverse is a series of connected survey lines that starts and ends at the same station, so its geometry must satisfy closure conditions. For a closed polygon with n sides, the sum of the interior angles is (n-2)\times 180^\circ. Adding one side to a polygon increases this sum by 180^\circ, so a triangle with 180^\circ becomes a quadrilateral with 360^\circ.

The angular misclosure is the difference between the measured sum of interior angles and this theoretical sum. For a four-sided traverse, the theoretical sum is 360^\circ, and any excess or deficit is distributed as an angular correction. The excess is expressed in degrees, minutes and seconds, so it must be converted to decimal degrees before rounding, for example 10'\,55'' = 10.9167'/60 \approx 0.182^\circ.

Bearings propagate around a traverse. The back bearing of a line equals its fore bearing plus or minus 180^\circ. For a clockwise traverse with interior angles, the fore bearing of the next line equals the back bearing of the previous line plus the interior angle at the common station, with 360^\circ subtracted if the result exceeds 360^\circ.

For each leg of length d and whole circle bearing \theta, the latitude is L = d\cos\theta and the departure is D = d\sin\theta. The coordinates of a point reached by a path are obtained by summing the latitudes and departures of its legs. In a closed traverse, the sums of latitudes and departures should be zero. Any residual is the closing error, whose length is e = \sqrt{(\sum L)^2 + (\sum D)^2}. Its direction is found from \tan\phi = |\sum D|/|\sum L| measured from north or south toward east or west. For example, if \sum L = 1.5 m and \sum D = -2.0 m, then e = 2.5 m and the error lies in the north-west quadrant.

The straight-line distance from the starting point to the end of a path is found from its summed coordinates by the same Pythagorean relation, so the closing distance of a path that returns in a different direction can be computed directly.

Bearing and azimuth conversion

A bearing can be expressed as a whole circle bearing (WCB), also called an azimuth, which runs from 0^\circ to 360^\circ measured clockwise from north. It can also be expressed as a quadrantal bearing, written in the form N \theta E, N \theta W, S \theta E or S \theta W, with \theta from 0^\circ to 90^\circ.

• Quadrantal to WCB — for N \theta E the WCB is \theta; for S \theta E it is 180^\circ - \theta; for S \theta W it is 180^\circ + \theta; for N \theta W it is 360^\circ - \theta. For example, N 40^\circ W gives 320^\circ.
• Minutes to decimal degrees — divide the minutes by 60 and add to the degrees before applying the conversion. For example, N 40^\circ 30' W gives 360^\circ - 40.5^\circ = 319.50^\circ.

The magnetic bearing is measured from magnetic north and the true bearing from geographic north. The magnetic declination is \delta = \text{true bearing} - \text{magnetic bearing}, taken positive when east. When the sun lies exactly on the geographic meridian at noon, its true bearing is 180^\circ, so the declination is found by subtracting the observed magnetic bearing from 180^\circ. A negative result is a westerly declination.

The included angle at a station between an incoming and an outgoing line can be found from their bearings. It is the back bearing of the incoming line measured at the station minus the fore bearing of the outgoing line. The result must be checked against a sketch, because it may need to be adjusted by 360^\circ to give the angle on the required side of the traverse.

Aerial photograph scale

The scale of a vertical aerial photograph is the ratio of the focal length of the camera to the height of the camera above the ground. It is written S = f/(H - h), where f is the focal length, H is the flying height above mean sea level or datum, and h is the average ground elevation above the same datum. Both heights must be in the same reference and the same units, so kilometres and millimetres must be converted to metres before dividing.

The scale can also be found from measured lengths: the ratio of a distance on the photograph to the corresponding distance on the ground gives the same value. For example, a photo distance of 9 cm corresponding to a ground distance of 900 m gives S = 0.09/900 = 1/10000.

Rearranging gives the focal length required for a given scale, f = S\times(H - h). This is used in reconnaissance planning, where the flying height is fixed by the permissible limits and the camera must be selected to achieve a specified scale. For example, a scale of 1:10000 with a height above terrain of 1500 m requires f = 1500/10000 = 0.15 m, that is 150 mm.

Most probable value of observations

The most probable value (MPV) of a set of observations is the value that is most likely given the data. It depends on the weights attached to the observations, and the weight of an observation is inversely proportional to its variance.

When all observations are of equal precision, they have equal weights, and the MPV is the arithmetic mean of the observations. For example, three readings of 45^\circ 10', 45^\circ 12' and 45^\circ 11' taken with the same instrument give an MPV of 45^\circ 11'.

When the mean of each set is based on a different number of observations, the weight of each mean is taken as proportional to the number of observations n it contains, so the MPV is the weighted mean \theta = \dfrac{\sum n\theta_i}{\sum n}. For example, a value of 30^\circ 00' from 4 observations and a value of 30^\circ 06' from 2 observations give a weighted mean of (4\times 30^\circ 00' + 2 \times 30^\circ 06')/6 = 30^\circ 04'.

Tape length and slope corrections

A tape is standardized at a reference temperature and tension, and its actual length may differ from its nominal length. If the tape is actually shorter than its nominal length, the true length of a measured line is the measured length multiplied by the ratio of actual to nominal length. Because the tape counts too many full lengths, the measured line is too long and must be reduced. For example, a 50 m tape that is actually 49.9 m long measures a line of 350 m as 350 \times 49.9/50 = 349.3 m.

The temperature correction for a tape is C_t = \alpha (T - T_0) L, where \alpha is the coefficient of linear expansion, T is the field temperature, T_0 is the standard temperature and L is the measured length. If the field temperature is above the standard temperature, the tape is longer than its standard length and the correction is positive. For example, with \alpha = 11\times 10^{-6}/^\circC, a 10 ^\circC excess over standard on a 200 m line gives C_t = 0.022 m, or 22 mm.

When a line is measured along a slope, its horizontal length is L_h = L_s\cos\theta. For a gradient of 1 in n, \tan\theta = 1/n, so \cos\theta = n/\sqrt{n^2+1}. The standardization and temperature corrections are applied to the slope length first, and the slope reduction is applied afterwards.

Plan scale distance correction

A distance taken from a plan is correct only if the scale used to convert it is correct. When a plan was drawn at one scale but the distance was later converted with a different scale, the distance on the plan is recovered first from the recorded value and then multiplied by the true scale. Equivalently, the recorded distance is multiplied by the ratio of the true scale to the scale that was used. For example, if a plan drawn at 1 cm = 40 m was converted with 1 cm = 20 m and gave 300 m, the true distance is 300\times 40/20 = 600 m.