Stadia Surveying

Plane Surveying Surveyi ng – Stadia Civil Engineeri Engineering ng 235

Departme Department nt of Civil Engineering, Engineering, UBC UBC

STADIA  A rapid and efficient way of indirectly measuring distances and elevation differences.  

Measurements can be taken with theodolites, transits and levels. New and portable GPS receivers designed for rapid and extremely accurate real-time positioning are gradually relegating stadia to the background.

Principle of Stadia Stadia Equipment  A theodolite or transit reticle for stadia has two additional horizontal cross hairs, called stadia hairs, spaced equidistant from the center one as illustrated in Figure 10-5 (b) and (c).



A graduated rod called stadia rod.

Procedure  The stadia method is based on the principle that in similar triangles, corresponding sides are proportional. 

With the line of sight horizontal and directed towards a graduated rod held vertically at a point some distance away, the interval between the two stadia hairs of most instruments is precisely 1/100 of the distance to the rod. (Figure 15-1) Thus, the intercept on a rod held 100 feet away will be 1 foot.

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Plane Surveying – Stadia Civil Engineering 235 

Department of Civil Engineering, UBC

The distance on the rod AB = I is called rod intercept or stadia interval, while the distance ab = i is the spacing between stadia hairs.

Standard symbols used in stadia measurements and their definitions are as follows: f i f

= focal length of lens (a constant for any particular compound objective lens) = Spacing between stadia hairs (ab in Figure 15-1) = stadia interval factor , usually 100 and denoted by K

i

I c

C d D

= rod intercept ( AB in Figure 15-1), also called stadia interval = Distance from instrument center (vertical axis) to objective lens center (varies slightly when focusing the objective lens for different sight lengths but is generally considered to be a constant) = stadia constant = c + f = distance from focal point F in front of telescope to face of rod = distance from instrument center to face = C + d

From similar triangles of Figures 15-1, f d I = or d = I = KI f i i

Thus D = KI + C

(15-1)

The geometry illustrated in figure 15-1 pertains to a simplified type of external focusing telescope. These telescopes are now obsolete, and only internal focusing telescope instruments are used these days. The objective lens of an internal telescope remains in fixed position and therefore the stadia constant C is so small that it can be assumed equal to zero and drops of the equation (15-1), so it reduces to: D=KI The stadia interval factor is K is given by the manufacturer of the instrument and is usually 100. If the factor is not given or there is a need to determine it, a rod intercept I for horizontal sight of known distance D is read. Accuracy in determining K is increased by averaging values from several lines whose measured lengths vary from about 100 to 500 feet by 100 feet increment.

Stadia Measurements on Incl ined Sites Most stadia shots are inclined because of varying topography, but the intercept is read on plumbed (vertical) rod and the slope length is reduced to horizontal and vertical distances. Principles on inclined stadia measurements are shown in figure 15-2.

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Plane Surveying – Stadia Civil Engineering 235


The instrument is set over point M and rod held at O. With the middle cross hair set on point R on the rod to make RO equal to the height of the instrument EM = hi, the vertical angle (angle of inclination) is α. If L represents the slope length ER, H the horizontal distance EG = MN, and V the vertical distance RG = ON, than: H = L cos α

(1)

V = L sin α

(2)

If the rod could be held normal to the line of sight at point O, a reading A’B’ or I’ would be obtained, making L = K L’ (3) Since it is not practical to hold the rod at an inclination angle α, it is plumbed and reading AB or I is taken. For the small angle at R, it is sufficiently accurate to consider angle AA ’R as a right angle. Therefore I’ = I cos α (4) L = K I cos α

(5)

Finally, substituting the equation (5) into (1) and (2) we can express the horizontal and vertical distance as: H = K I cos2 α (6) V = K I sin α cos α

(7)

If zenith angles z are read rather than vertical angles, than the distances are given as: H = K I sin2 z

(8)

V = K I sin z cos z

(9)

where the zenith angle z = 90o - α If the trigonometric identity ½ sin 2 α is substituted for sin α cos α, the formulas for vertical distance become: V = ½ K I sin 2α (10)

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Department of Civil Engineering, UBC V = ½ K I sin 2z

(11)

The elevation at point O can be calculated from elevation at point M as: elev. O = elev. M + hi + V – R

(12)

From the last equation the advantage of sighting an R-value that is equal to hi when reading the zenith angle is evident. Since both values are equal and opposite in sign they cancel each other and can be omitted from calculations. If hi can not be seen because of obstructions, any rod reading can be sighted and equation (12) used. Setting the middle cross wire on a full-foot mark just above or below the hi simplifies the arithmetic.

Stadia Field Procedures When the stadia method is employed, it is most often for locating details in topographic mapping. However, it can also be used to run control traverses for topographic mapping or for other lover order applications. In topographic detailing by stadia, the transit is set up on a control station and oriented on in azimuth bay backsighting along a line of known directions with the line’s azimuth on the horizontal circle. Than azimuths can be read directly to all sighted points. Many shots can be taken from one instrument station, thus orientation should be checked by sighting on the control line after every 10 or 20 topographic points, and before leaving the setup. An order of field procedures well suited for topographic detailing is as follows: 1. Bisect the rod with the vertical cross hair. 2. With the middle cross hair approximately on the hi, set the lower hair at a full-foot mark. 3. Read the upper cross hair and mentally subtract the lower hair reading from it to get the rod intercept. Record only the rod intercept. 4. Move the middle cross hair to the hi by using the vertical tangent screw. 5. Release the rodperson for movement to the next point. 6. Read and record the horizontal angle (azimuth). 7. Read and record the vertical angle (zenith). When stadia is used for traversing, rod intercepts and horizontal and vertical angles are measured both forward and back at each traverse point. Average values of horizontal distances and elevation distances are used in subsequent computations. An elevation check should be secured by closing on the initial traverse point, or a nearby bench mark, and if within tolerances any misclosure distributed. The horizontal angles should also be checked for closure and again the misclosure adjusted. Departures and latitudes can then be computed and the traverse precision checked by methods described before. Stadia Field Notes  Record title, date, weather, names of party members, duties etc. 

Record check of index error and confirmation of stadia interval factor. Page 4 of 12



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Record all measurements for stadia traverse and all side shots – see example below (Plate D-7).

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Make a sketch of the traverse points including angles and elevations.

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At end (in office), give tables showing traverse closure, latitudes, departures and elevations.

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Location of Points in Stadia Measurements Objective of the Stadia Measurements is to produce a topographic map i.e. a 2-D representation of a 3-D surface. In order to do that we require: 

Locations of points on the map (two coordinates)

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Elevations of the points (3rd coordinate)

Points in 2-D space can be located by: 

Knowing the distances from two known points

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Knowing the directions from two known points

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Knowing the direction from 1 point and the distance from another point

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Knowing the direction and distance from one known point

Note that all methods involve two pieces of information. We will use the fourth method. The direction of a line is the horizontal angle between it and an arbitrarily chosen reference line called a meridian. Azimuths are horizontal angles measured clockwise from any reference meridian (Figure 8-7). Depending on the reference meridian chosen azimuths can be true north, magnetic north or assumed. Note that in figure 8-7 point O is a known point.

We will use true north as our reference meridian. Use first the compass on the transit to determine the magnetic north and then assume that magnetic north has a 23 o azimuth from our true north meridian. Traversing Very often during surveying the are to be surveyed cannot be seen from one reference point. In that case, we need a number of reference points connected by a traverse. A traverse is a series of consecutive

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straight lines whose lengths and directions have been determined from field measurements. A traverse connects a succession of established points, called traverse stations or traverse points, along the route of survey. There are two basic kinds of traverses: closed and open. Two categories of closed traverses exist: polygon and link. With the polygon type as shown in Figure 12-1(a) the lines return to the starting point, thus forming a closed figure (geometrically and mathematically closed). Link traverses finish upon another station that has positional accuracy equal or greater than that of the starting point. Closed traverses provide checks on the measured angles and distances, an extremely important consideration.

Traversing by Azimuths There are different methods of measuring traverse angles or directions such as interior angles, compass angles, deflection angles, angles to the right, azimuth angles etc. We will use azimuth-traversing method (Figure 12-3).

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The procedure of azimuth stadia traversing is the follows: 

Set up on firs station (choose arbitrarily)

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Locate true north as described previously and presented on the sketch

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Do all the work as required for topographic map from this station including:







-

Establish elevation of the point from known bench mark

-

Take side shots to as many side points as required for topographic map

-

Locate all side points by measuring their azimuths, distance and elevation

Establish the second station as follows: -

Sight second station from the first station and measure its azimuth α1 (see the sketch)

-

Measure the horizontal distance to the second station and its elevation

Move instrument setup at the second station -

Establish True North (TN) by setting HCR to the back azimuth α1 + 180o

-

Sight station 1 and release the “upper clamp”

-

Do work as required for topographic map from this station

Establish the third station (and all subsequent stations) -

From second station sight the third station and measure its azimuth α2

-

Do the stadia measurements as for the previous point

-

Backsight to the first station

-

Set HCR to zero

-

Foresight to third station and measure β. Check that 180 - α1 + α2 = β (see the sketch below)

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Move the instrument and continue as before

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Finish the traverse on the same station as you started

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Check for closure of elevations on the benchmark


Checking for closure of the traverse  The azimuth of the first line as observed at closure should equal the azimuth of the same line at the beginning of the survey 

Check internal angles

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Summation of interior angles should be (n-2) 180 o, where n is number of sides of the traverse. Determine interior angles as differences between azimuths.

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Balance the angles to the proper geometric total by applying an average correction to each angle where observing conditions were approximately the same at all stations. The correction is found by dividing the total angular error (misclosure) by the number of angles.

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Calculate the departures and latitudes of each line in the traverse (Figure 13-2).

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Departure (ΔXAB) = L sin α Latitude (ΔYAB) = L cos α Where L is the horizontal length and α is the azimuth of the course. 

Calculate the total traverse departure misclosure (DM) and total Latitude misclosure (LM).

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Check the linear error (misclosure) of the traverse (E) as follows: E=



(

2

2

DM ) + ( LM)

Adjust the departures and latitudes of traverse lines using their corrections in proportion to their lengths (LAB). Corrections in departures for AB (CD AB) and corrections in latitudes (CL AB) can be expressed as: CD

= AB −

CL

= AB −

DM TP LM TP

⋅ L

⋅ L

AB

AB

where: TP is the traverse perimeter. 

Calculate the final coordinates of the points (B for example) by adding the adjusted departures or adjusted latitudes of line AB to the X and Y coordinates of A (method of rectangular coordinates). XB = XA + CDAB YB = YA + CLAB

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TRANSIT-STADIA TRAVERSE EXERCISE Introduction This exercise requires a closed transit-stadia traverse and side shots to enable you calculate the location and elevation of certain control points.

Equipment Transit, rod, small equipment. Before arriving at the site, determine the vertical circle index error, the stadia interval factor and, depending on whether the instrument is internal or external focussing, the stadia constants for the transit. Index Error This is determined by sighting on a distance object at some vertical angle. Read the VCR with the telescope direct and then with it inverted. The correct VCR is the average of these two. Determine index constant and whether it should be added or subtracted from all direct VCR readings. Perform at least two such tests. Corrections less than 30” can be ignored. But note that the VCR is not the correct vertical angle unless the horizontal plate is level, thus when taking a VCR on the job ensure that the horizontal plate is level. Stadia Interval Factor Set out a course on level ground on Main Mall with taped distances of 30 m and 60 m from the transit. For each distance, determine the stadia interval factor and check that it is 100. Determine the stadia constants if the telescope is external focussing. Show index error, stadia constants and computations in field book. Accuracy Read: Stations, control points and object locations to nearest 5 min. of Az and stadia interval to nearest 0.002 m. Ground elevation shots to the nearest 5 min or 10 min of Az and stadia interval to nearest 0.005 m. VCR to nearest 1 min. on very long shots. After taking all the desired readings at one station including the foresight to the next station, you must make an independent check of the enclosed angle between backsight and foresight. This is done by setting zero on the HCR, sighting on the B.S. and reading clockwise to the F.S. This angle must agree within 5 min with the difference between back and forward azimuths. If the two techniques don’t agree within 5 min, then the entire procedure (setting back azimuth, etc.) must be repeated until they do. Do not leave station until this check is completed. Show this check at bottom of page. Before recording sightings indicate the station at which instrument is set up, the height of instrument, and the elevation of inst. station.

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Each student must perform all the instrument functions at, at least, one station. Suggested Procedure at Traverse Station (1)

Set up directly over station and level instrument with plate bubbles.

(2)

Improve the level of the instrument by using the telescope bubble after using the plate bubbles. Eliminate parallax.

(3)

Set back azimuth on horizontal circle A vernier and sight on back-sight. Set lower clamp and release upper clamp.

(4)

Find stadia interval by setting upper or lower stadia-hair to nearest whole metre in vicinity of height of instrument, then read and record.

(5)

Set centre cross-hair to height of instrument, if possible, and then wave rod-person on.

(6)

Read VCR and Azimuth while rod-person moves to next position.

As a final check, when you are set-up at your last station in the closed traverse, the forward azimuth should agree with the back azimuth when you started. Also, close your level circuit back to the original B.M. Fieldbook Each person’s fieldbook should show the field notes for the readings taken when that person was the instrument-person. In addition, the notes needed to determine the other closed traverse stations should be copied from the other people’s fieldbooks, i.e., copy the notes that would enable each person to determine the coordinates of the stations in the closed traverse. Also, the notes for the closed level circuit should be copied. Calculations in each fieldbook should contain: (i)

Traverse calculations for the closed traverse including corrections.

(ii)

Level calculations including check of closed circuit. If your error of closure is more than 0.1 m you must adjust your levels.

(iii)

Elevation and distances to all side shots.

The above calculations should be made and handed in on the following morning in the usual manner.

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Stadia Surveying

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