LECTURES 5 AND 6: JANUARY 15, 2004

ELECTRONIC THEODOLITE An electronic theodolite (Figure 1d) is very similar to the optical theodolite in construction except that in this case the horizontal and vertical circles are not graduated in the conventional sense but coded like a circular barcode and are read using arrays of Light Emitting Diodes (LEDs) and Light Detecting Diodes (LDDs). The coding on a given location of the circles is unique. Consequently, manner in which the light generated by the LEDs is transmitted to the LDDs is unique to a particular position on the circle. Depending on the pattern of light transmitted to the LDDs the angular position on the circle is electronically read and recorded digitally. The principle is illustrated with Figure 7. In addition, the instrument is equipped with electronically sensors that can detect deviation of the plate from horizontal and apply appropriate corrections to the digital records of angle measurement without human intervention. Some of the very recent variants of the electronic theodolite are equipped with Automatic Target Recognition (ATR) capability obviating the necessity of the presence of a surveyor at the equipment station.

Figure 7. Coded Circles of an Electronic Theodolite

DEFINITIONS Vertical Axis is the axis about which the telescope is rotated in a horizontal plane. This is the axis about which the two plates rotate.

Horizontal or trunnion axis is the axis about which the telescope rotates in the vertical plane.

The Line of Site is the line passing through the intersection of the cross-hairs and the optical center of the object glass and its extension. The Line of Collimation is the line connecting the optical centers of the object glass and the eyepiece.

Transiting or plunging or reversing is turning the telescope in a vertical plane through 180o. Swinging (right or left) on the other hand is turning the axis in horizontal plane.

Face-left observation is the reading taken with the vertical circle on the left of the surveyor. Face-right observation is similarly defined. Flipping from face-left to face-right mode is called changing face.

The horizontal angle is the angle measured in the horizontal plane, i.e., the angle about the horizontal axis of a well-adjusted theodolite. The vertical angle is the one measured in the vertical plane, i.e., the angle measured about the trunnion axis typically from the zenith of the observer.

PROCEDURE Observations

Center the instrument accurately over the station.

Level the instrument with plate bubble vials. Once measurements are made, check the level of the instrument again.

Minimize sighting parallax during each pointing operation by focusing the reticule (eye-piece) first and then the objective lens.

Measure both horizontal and zenith angles in at least four sets. Each reading is taken as follows:

• Begin with both upper and lower clamps locked. Unclamp the lower clamp and point the telescope to the first target and bisect it approximately. Clamp the lower clamp and use the tangent screw of the upper clamp to bisect the first target accurately.

• Unclamp the upper clamp and point the telescope to the second target and bisect it approximately. Clamp the lower clamp and use the tangent screw of the upper clamp to bisect the first target accurately. Read the horizontal angle using the vernier.

Re-center and re-level the instrument between each set (repetition). With well-designed targets and proper methodology, an angle measurement accuracy of 1" is possible with precision electronic theodolites if four sets of observations each comprised of face-left and face-right readings (double centering). The requirement of two positions must always be followed in order to eliminate errors caused by mechanical misalignment of the theodolite axes.

Read the horizontal and vertical circles to the least count of the instrument (usually 0.1 arc-second in modern instruments).

For each station pair (i.e., angle between the backsight and foresight), a full set will consist of a direct angle measurement and a separate horizon closure angle measurement. Their sum will be taken to find the closure to 360o.

Observing conditions: Avoid measurements close to any surface that has a different temperature than the surrounding air. If any suspicion of refraction influence arises, the surveys should be repeated in different conditions in order to randomize its effect. Ideally, observations should be limited to days when the weather conditions are fairly neutral (e.g., cloudy day with a light breeze). Days with temperature extremes should

be avoided. If the instrument must be used when the temperature is hot, then it should be protected from the sun by an umbrella.

Data Reduction

For each sighted direction in the set, determine the face-left and face-right mean direction value, starting with the backsight observation. Subtract the initial or backsight circle reading from the mean direction value of the foresight in the measurement set (backsight value will then be reduced to zero). Repeat the above procedures for each of the four independent direct angle sets and calculate the mean value for each direction, Repeat each of the above steps for each corresponding horizon closure angle set. Difference the sum of the means of the direct and closure sets from 360o. Distribute the misclosure equally to correct the final mean reduced value from each set.

ERRORS IN ANGLE MEASUREMENTS WITH THE THEODOLITE Errors are usually measured with the standard deviation of the measurement.

Random Errors Random error in horizontal angle measurement using an optical theodolite may arise from the following (USACE EM 110-2-1009, 2002). • Pointing (or Targeting) Error: The pointing error in an optical angle measurement,

pσ , approximately relates to the magnification, M, of the telescope according to:

( )NMp ×′′≈ 54σ (1)

where N is the number of repeated readings.

• Leveling Error: The leveling error, lσ , is approximately one-fifth of the bubble sensitivity. For instance, for a 30˝ bubble, the leveling error is expected to be about 6˝.

• Centering Error: Error in centering, cσ , a theodolite using the plumb bob is less accurate than that with an optical plummet. The forced centering method, in which the survey marker is fitted with a tribrach upon which the instrument (theodolite or other surveying tool), yields even more precise centering. The centering error can be approximated as 5×10-4 times the height of instrument in case of centering with an optical plummet and 1×10-4 times the height of instrument in case of forces centering.

• Reading Error: The accuracy of the instrument, rσ , is typically quoted by the manufacturer.

Assuming the components of random error listed above to be mutually exclusive, the random error in horizontal angle measurement can then be estimated using:

2222rclpp σσσσσ +++= (2)

Systematic Errors Systematic error in horizontal angle measurement using an optical theodolite may arise from the following.

• Non-horizontal Plate Level: Find the error in a horizontal angle measurement as a function of non-horizontal plate bubble as a function of the angle between the horizontal and plate level as an assignment. If the angular measurement is obtained by taking the face-left and face-right readings and averaging the two, this error cancels out.

• Non-horizontal Trunnion Axis: Find the error in a horizontal angle measurement as a function of non-horizontal plate bubble as a function of the angle between the horizontal and plate level as an assignment. If the angular measurement is obtained by taking the face-left and face-right readings and averaging the two, this error cancels out.

• Collimation Error: Collimation error is the systematic error that arises in angle measurements when the line connecting the crosshair and the target is not identical to the optical centerline of the telescope. This error becomes pronounced when the elevation of the targets differ significantly in comparison with their horizontal distance from the instrument station. If the angular measurement is obtained by taking the face-left and face-right readings and averaging the two, the collimation error cancels out.

• Refraction Error: Atmospheric conditions affect propagation of electromagnetic signals. The error can be estimated from the knowledge of temperature gradient, atmospheric pressure and relative humidity along the line of sight. The procedure for estimation of refraction error will be discussed later.