Lecture EDM Theory Lecture Notes

Lecture EDM Theory Lecture Notes Text Coverage: same as other lecture outline

How an EDM works:

Carrier wavelength: Modulation Frequency: Modulation Wavelength: Speed of light in a vacuum: Speed of light in air:

Basic formula for distance: 2D  N1m  a1

Systematic Errors in EDM work: A. Constant Errors 1. Glass (prism) constant. (GC) all expressed as corrections 2. Instrument Offset constant (IOC) 3. Total Offset constant (TOC)

B. Scale Errors 4. Frequency count errors 5. PPM correction a. Standard atmosphere b. Formula for the refractive index: Given: Wavelength modulation frequency, f0 = 14985.4 kHz Can be found Wavelength of infrared carrier beam, 1 = 0.900m in Modulation Wavelength, 0 = 20 m Manufacturer Velocity of light in a vacuum, V = 299,792.458 km/sec 0 Manual Observe the Following at Time of Observations: p = Atmospheric pressure (Use average of 762.0 mmHg) t = Air temperature, C

1. Calculate the refractive index of air under standard conditions, ng:   4.8864   0.068  n 1 287.604       106 q   2   4    1   1 

2. Calculate the refractive index of air under the condition at time of observation, na:  0.359474n 1p   q  (Eq. 1-4) na  1    273.2  t 

3. Calculate the reference index of refraction for the instrument, n0: n0 = V0 / (0 f0) (Eq. 1-5)

4. Calculate the manufacture’s EDM reference index of refraction, N0, and the refractive index for the conditions at the time of observation, Na:

6 N0 = (n0 – 1)10 (Eq. 1-6)

6 Na = (na – 1)10 (Eq. 1-7)

5. Finally compute the Correction (ppm):

Correction (ppm) = (N0 - Na) (Eq. 1-8)

C. EDM Random Errors – After all scale and constant errors have been eliminated by correction, there is still randomness: a. RANDOMNESS OF THE ELECTRONICS – there is randomness of the EDM in measuring the phase shift. The EDM measures this many times and averages before displaying the answer, but there is still randomness. For example for a short distance in a stable environment, repeated displays of a distance are not the same down to a millimeter. A 25 reading sample can be taken and a standard deviation calculated for this. This random error is NOT dependent on distance. b. RANDOMNESS IN THE ATMOSPHERE – The actual path of the light is through cooler air (more dense) and hot rising thermals (less dense). Hopefully the average of all the air path temperature and pressure matches that used in the ppm correction. However, this is highly random because the number of thermals passed through on a particular distance is variable. Therefore, readouts of distance will vary more for long distances than for short (dependent on distance). On hot days, the best an EDM can give is about 1/30,000 because of this randomness. Reduce randomness by observing on overcast days or at night. c. RANDOMNESS OF CENTERING – The center of the EDM is never exactly over the point. This will be a random error source of a size depending on the care used and the adjustment of the optical plumbing device.

Manufacturers combine all these effects into a RMS error for a particular instrument such as:

RMS = (root mean square, a European term for standard deviation, since a standard deviation is the square root of the mean of residuals squared)

Sample statement by manufacturer: RMS = +/- (.003 m + 5 ppm). The fixed randomness accounts for the effect of the electronics and centering. The variable randomness accounts for the atmosphere effects.