1. Electronic Distance Measurement Faculty of Applied Engineering and Urban Planning Civil Engineering Department Lecture 9 - Week 7 2 nd Semester 2007/2008 UP Copyrights 2007 Engineering Control and Surveying

6. Propagation of Electromagnetic Energy Velocity of EM energy V = ƒ λ ƒ is the frequency in hertz (cycles/second) λ is the wavelength In vacuum the velocity of electromagnetic waves equals the speed of light. V = c/nn >1, n is the refractive index of the medium through which the wave propagates c is the speed of light = 299 792 458 m/sec f λ = c/n or λ = cf/n Note that n in any homogeneous medium varies with the wavelength λ. White light consists of a combination of wavelengths and hence n for visible light is referred to as a group index of refraction. For EDM purposes the medium through which electromagnetic energy is propagated is the earths atmosphere along the line being measured. It is therefore necessary to determine n of the atmosphere at the time and location at which the measurement is conducted.

8. THE FRACTION OF A WAVELENGTH AND THE PHASE ANGLE Amplitude θ + r - r 0° 90° 180° 270° ¼λ¼λ ¼λ¼λ ¼λ¼λ ¼λ¼λ λ ½λ½λ ½λ½λ θ λ 360 A fraction of a wavelength can be determined from a corresponding phase angle θ Note: For θ = 0° the fraction is 0 For θ = 90° the fraction is ¼ For θ = 180° the fraction is ½ For θ = 270° the fraction is ¾ For θ = 360° the fraction is 1 EDM INSTRUMENTS CAN MEASURE PHASE ANGLES

9. Principles of Electronic Distance Measurement If an object moves at a constant speed of V over a straight distance L in a time interval ∆t, then L= V∆t = (c/n)∆t Knowing the speed of light c and being able to determine the refractive index, we could measure the time interval it takes for an electromagnetic wave to move from A to B to determine the distance L between A and B. But since the speed of light (c) is very high, the time interval ∆t would need to be measured extremely accurately. Instead, the principle of EDM is based on the following relationship: L = (m + p) λ 1 23 4 56 7 8 9 10 11 12 p λ λ λλ λ λ λλλλλ λ A B L m is an integer number of whole wavelengths, p is a fraction of a wavelength So L can be determined from λ, m and p

10. 1 23 4 56 7 8 9 10 11 12 p1p1 λ λ λλ λ λ λλλλλ λ A B L λ2λ2 p2p2 Solving for the integer number (m) of whole wavelengths (Resolving the ambiguity in the number of whole wave lengths) Additional waves of known lengths λ 3 = kλ 2 and λ 2 = k λ 1 (k is a constant), are introduced to measure the same distance L: L = (m 3 + p 3 ) λ 3 L =(m 2 + p 2 ) λ 2 L =(m 1 + p 1 ) λ 1 Determining p 1 p 2 and p 3 by measuring phase angles θ 1 θ 2 and θ 3 and solving the above equations simultaneously yields L ( Note: For L < λ 3, m 3 = 0 ). p3p3

11. For example, if λ 1 = 10.000 meters, k = 10.000 and p 1 = 0.2562, p 2 = 0.2620 and p 3 = 0.0125 (measured) Then λ 2 = 10.000m x 10.000 = 100.000 and λ 3 = 100.000 x 10.000 = 1000.000 L= (m 3 + p 3 ) λ 3 = (0+0.1250)x 1000.000 = 125.000m approximately M 2 = 125/ λ 2 = 125/100= 1 and hence L = (1+ 0.2620)x100.000 = 126.200m approximately M 1 = 126.2/ λ 1 = 126.2/10 = 12 and hence L = (12+ 0.2562)x10 = 122.562m [m i = whole wavelengths; p i = fractional parts of a wavelength; k = constant] USING DIFFERENT WAVELENGTHS

12. Propagation of Electromagnetic Energy Velocity of EM energy V = ƒ λ ƒ is the frequency in hertz (cycles/second) λ is the wavelength In vacuum the velocity of electromagnetic waves equals the speed of light. V = c/nn >1, n is the refractive index of the medium through which the wave propagates c is the speed of light = 299 792 458 m/sec f λ = c/n or λ = cf/n Note that n in any homogeneous medium varies with the wavelength λ. White light consists of a combination of wavelengths and hence n for visible light is referred to as a group index of refraction. For EDM purposes the medium through which electromagnetic energy is propagated is the earths atmosphere along the line being measured. It is therefore necessary to determine n of the atmosphere at the time and location at which the measurement is conducted.

13. Propagation of Electromagnetic Energy The refractive index of air varies with air density and is derived from measurements of air temperature and atmospheric pressure at the time and site of a distance measurement. For an average wavelength λ: n a = 1 + ( n g -1 ) x p - 5.5e x 10 -8 1 + 0.003661T 760 1 + 0.003661T Where n g is the group index of refraction in a standard atmosphere (T=0°C, p=760mm of mercury, 0.03% carbon dioxide) n g = 1+ ( 2876.04 + 48.864/ λ 2 +0.680/ λ 4 ) x 10 -7 p is the atmospheric pressure in mm of mercury (torr) T is the dry bulb temperature in °C and e is the vapor pressure Where e= e’+de and e’=4.58 x 10a, a=(7.5T’)/(237.3+T’), de=-(0.000660p (1+0.000115T’) (T-T’) and T’ is the wet-bulb temperature So measuring p, T and T’ will allow for the computation of n for a specific λ