Electronic Distance Measuring Equipments - Problem 5

A 750m. baseline distance is measured with an electronic total station. If the vendor's accuracy specification of the instrument is ±(3mm. + 2ppm). Determine the precision of the measured distance to the nearest thousand.

Solution:

\begin{equation} RA = \frac{e}{b} \end{equation}

\begin{equation} e = \pm A + (B \times C) \end{equation}

\begin{equation} e = \pm 3mm. + [(750mm. \times \frac{1000mm.}{1m.}) \times (2 \times 10^{-6})] \end{equation}

\begin{equation} e = \pm 4.5mm. \end{equation}

\begin{equation} RA = \frac{4.5mm.}{750m. \times \frac{1000mm.}{1m.}} \end{equation}

\begin{equation} RA = \frac{1}{166666.667} \end{equation}

\begin{equation} RA \approx \pm \frac{1}{166000} \end{equation}

Answer: \begin{equation} RA = \pm \frac{1}{166000} \end{equation}

Electronic Distance Measuring Equipments - Problem 4

If the error in EDM measurements is ±0.016ft in 10ft. distance. What is the relative accuracy of the EDM?

Solution:

\begin{equation} RA = \frac{e}{B} \end{equation}

\begin{equation} RA = \frac{0.016ft.}{10ft.} \end{equation}

\begin{equation} RA = \pm \frac{1}{625} \end{equation}

Answer: \begin{equation} RA = \pm \frac{1}{625} \end{equation}

Electronic Distance Measuring Equipments - Problem 3

If the error in EDM measurements is ±0.021ft in 1000ft. distance. What is the relative accuracy of the EDM?

Solution:

\begin{equation} RA = \frac{e}{B} \end{equation}

\begin{equation} RA = \frac{0.021ft.}{1000ft.} \end{equation}

\begin{equation} RA = \pm \frac{1}{47619} \end{equation}

Answer: \begin{equation} RA = \pm \frac{1}{47619} \end{equation}

Electronic Distance Measuring Equipments - Problem 2

A distance of 10ft. was measured by an EDM with ±(5mm. + 5ppm). What is the error in this measurement?

Solution:

\begin{equation} e = \pm A + (B \times C) \end{equation}

\begin{equation} e = \pm 5mm. + [(10ft. \times \frac{12in.}{1ft.} \times \frac {25.40mm.}{1in.}) \times (5 \times 10^{-6})] \end{equation}

\begin{equation} e = \pm 5.015mm. \times \frac{1in.}{25.40mm.} \times \frac{1ft.}{12in.} \end{equation}

\begin{equation} e = \pm 0.016ft. \end{equation}

Answer: \begin{equation} e = \pm 0.016ft. \end{equation}

Electronic Distance Measuring Equipments - Problem 1

A distance of 1000ft. was measured by an EDM with ±(5mm. + 5ppm). What is the error in this measurement? 

Solution:

\begin{equation} e = \pm A + (B \times C) \end{equation}

\begin{equation} e = \pm 5mm. + [(1000ft. \times \frac{12in.}{1ft.} \times \frac{25.40mm.}{1in.}) \times (5\times 10^{-6}) ] \end{equation}

\begin{equation} e = \pm 6.524mm. \times \frac{1in.}{25.40mm.} \times \frac{1ft.}{12in.} \end{equation}

\begin{equation} e = \pm 0.021ft. \end{equation}

Answer: \begin{equation} e = \pm 0.021ft. \end{equation}

Electronic Distance Measuring Equipments

EDM Instruments 

• EDM as a component of a total station (laser light or infrared light) 
• Handheld laser EDM (no reflector is used) 

Types of EDM Instruments 

1. Handheld lasers 
• < 200m. 
2. Light waves (laser or infrared) 
• Short range (0.5 - 3km.) 
• Medium range (3 - 10km.) 
• Long range (10 - 20km.) 
3. Micro waves 
• Typical range: -50km. 
• Requires two identical EDM units at the two ends of the line 
• Not common anymore and was replaced by GPS 

How do Light Wave EDM works? 

• EDM instrument sends a laser light or infrared light beam to a prism 
• The signal is reflected by a prism at the other end of the line back to the instrument (The prism is designed so that the light is reflected back in the exact opposite direction) 

Scientific Principle of EDM - (The Wave Theory)

\begin{equation} \lambda = \frac{c}{f} \end{equation}

Where:

λ = Wavelength (in meters, m) 

c = Velocity (in meter per second, m/sec) 

f = Frequency (in cycle per second, Hz) 

\begin{equation} L = \frac{n \lambda + \phi}{2} \end{equation}

Where:

L = Distance (in meters, m)

n = Number of waves

λ = Wavelength (in meters, m)

ϕ = Partial wavelength (in meters, m)

Errors in EDM Measurements

\begin{equation} e = \pm A + (B \times C) \end{equation}

Where:

e = Error in EDM measurements

A = Absolute error constant (depends on instrument type)

B = Measured distance

C = Relative error constant (distance-dependent)

EDM Relative Accuracy/Relative Precision

\begin{equation} RA = \frac{e}{B} \end{equation}

Where:

RA = Relative Accuracy; Relative Precision of the EDM

e = Error in EDM measurements

B = Measured distance

EDM Prism/Reflector Constant

• Manufactures usually calibrate their EDMs to a particular prism/reflector 
• If a different prism is used, a prism/reflector constant should be determined and entered into the EDM settings 
• Use EDM and prism/reflector of the same brand 
• Prism/reflector constant is issentially a correcton factor 

The figure below shows the use of prism/reflector constant in determining a distance

\begin{equation} AC + K = (AB + K) + (BC + K) \end{equation}

\begin{equation} K = AC - (AB + BC) \end{equation}

Where:

K = Prism/reflector constant