Geography 411 - Field Techniques

Topographic Surveying

Topographic Surveying

• determining the relative locations of points (places) on the earth's surface by measuring horizontal distances, differences in elevation and directions
• topos (Gr.): place; topographic maps give the locations of places (observable features); they serve as base maps

Methods

1. horizontal distance
   • tachymetry: a rapid optical means of measuring distance using a telescope with cross hairs and a stadia rod (one stadium = about 607 feet)
   • measuring slopes distance with a tape and reducing it to horizontal distance using the cosine of the slope gradient
     
2. difference in elevation
   • leveling with a level telescope and a stadia rod, or
   • measuring a vertical angles and a slope distance (height is the product of the distance and sine of the angle)

   Leveling is more accurate since elevation differences are measured not calculated. Two readings are taken at each position of the automatic level: a backsight towards a station located before the level on the traverse and a foresight to the next station on the traverse. Thus the stadia rod occupies two stations, before and after the level on the survey.  The difference in elevation between successive stations if the difference between the backsight and the foresight read from the stadia rod.  For each position of the level, the lengths of the foresights and backsights should be approximately the same since accuracy is a function of the distance of a sighting.  The level operator should anticipate the distance to the next station and set up the level midway along this distance (note: the distance that can be sighted decreases with increase in slope, since the stadia rod will disappear above or below the level line of sight).  The lengths of backsight and foresight can be paced by the rod person or measured by the interval between the upper and lower cross hairs (tachymetry).
   

3. direction

• horizontal angle measured with a compass
• precise measuring devices use vernier scales
• direction is expressed relative to a reference line or meridian

• true meridian: a north-south line
• magnetic meridian: a line parallel with the earth's magnetic lines of force
• assumed meridian: an arbitrary line

Types of horizontal angles

1. bearing: angles expressed relative to a meridian using the quadrant and an acute angle, e.g. N37^oE, S62^oE, N20^oW
2. azimuth: the clockwise angle from the north branch of a meridian, e.g. 45^o (northeast), 180^o (south)
3. deflection angle: the angle between a line and the prolongation of a preceding line; it is a right or left angle depending on whether the new line is right (clockwise) or left (counterclockwise) of the preceding line
4. interior angle: an angle inside a closed polygon

Types of Traverses

1. azimuth: along a single direction (azimuth); common for slope profile surveys where the profile is always perpendicular to contours (i.e. maximum slope angle)
2. closed traverse: begins and ends at fixed control points of known location; permits calculation and adjustment for closure error
3. closed-loop traverse: begins and ends at the same station; permits calculation and adjustment for closure error and use of interior and deflection angles
4. open traverse: surveying from a known position to a point of unknown position; does not enable computational checks for error, rather all measurements must be repeated to check for error

Shape Of The Earth And Error: The two basic problems in topographic surveying

The shape of the earth (the geoid) is a consideration only in geodetic surveying, where over long distances flat surfaces are not level, plumb lines are not parallel and the sum of the angles in a triangle is greater than 180^o; thus precise surveys over large areas employ the principles of geodesy (the mathematical properties of an ellipsoid that emulates the earth); however with most surveys, including virtually all topographic surveying in geography, the departure of horizontal lines from level and plane angles from spherical angles are negligible and can be ignored

plane surveying: where the earth's surface is regarded as a plane; level lines are considered straight, angles are considered to be plane angles and plumb lines are considered to be parallel within the survey

with these assumptions, the relative locations of points can be calculated using the principles of plane trigonometry:

1. for right angles: sinA = a/c, sinB = b/c, and c² = (a² + b²), where A & B are the acute angles, C is the right angle and a, b & c are the sides opposite the angles designated with the same letter; thus the sides and angles can be calculated with a knowledge of 1 side and 1 angle or 2 sides
2. for oblique triangles: a/sinA = b/sinB = c/sinC; thus given 2 angles and one side or 2 sides and 1 angle are the other sides and angles can be calculated

Error

Distances and angles can never be determined exactly; measurements are subject to error. Error can be controlled through procedure and instrumentation. Surveys are conducted according to standard levels of accuracy (first order, second order, etc.). The desired level of accuracy depends on the intended us of the survey data (e.g. locating permanent stations or surveying bridges and dams versus surveying for terrain analysis or orienteering).

For topographic mapping, the desired level accuracy is the plottable error, the shortest distance that can be depicted on a map at a given scale. The drafting of lines generally is accurate to within 0.25 mm. At 1:1000, 0.25 = 250 mm or 0.25 m on the ground. Optical measuring devices will provide this level of accuracy. At 1:25,000, 0.25 mm = 6.25 m on the ground. Pacing of distance will provide this level of accuracy, although in practise accuracy is greater than the plottable error by as much as one-third (e.g. 80 mm rather than 250 mm at a scale of 1:250,000) so that plotting and surveying errors are not compounded.

Adjusting For Closure Error

Horizontal angles

• In a closed polygon, the sum of the interior angles = 180^o (n-2), where n is the number of sides in the polygon, thus the sum of the horizontal angles in a triangle (n = 3) is 180^o; an equal angle is subtracted or added to each measurement to satisfy the equation for interior angles; if the closure error is not equally divisible by n, make the largest adjustments to the largest angles.
• The sum of deflection angles for any closed polygon is always 360^o; this provides for another means of determining and adjusting for closure error.

Difference in elevation

Closure error can be determined for closed and closed-loop traverses. The closure error can be divided by the number of stations on the traverse or the correction at each station can be calculated according to the distance from the origin of the survey:

C_i = d_i/L * E_c, where

• C_i = the correction applied to station I
• d_i = the distance to station i from the origin of the
  traverse
• L = the total length of the traverse
• E_c = the closure error

This method accounts for the propagation of error with distance.

Horizontal distances

As with leveling, closure error can be determined for closed and closed-loop traverses, where the coordinates of the end points are identical or known. Location in the horizontal plane are given by x and y coordinates (e.g. northing and easting). Using the measured horizontal distances and adjusted angles, calculate the coordinates of each station. The difference between the calculated and known coordinates of the end control point is dx and dy, the closure error in x and y. As with leveling, the adjustment is a function of the distance traversed (Li) relative to the total length of the traverse (L):

Cdx_i = dx * L_i/L Cdy_i = dy * Li_i/L,   where Cdx_i and Cdy_i are the adjustments in x and y coordinates at station i

The relative accuracy of distance measurements can be expressed as ((dx² + dy²)^1/2)/L. An angular error of one minute is equivalent to a distance measurement error of 3 cm over a distance of 100 m, since the sine of 1/60^o is .00029.