Duran, Rachel, Managing Editor has reference to this Academic Journal, PHwiki organized this Journal CE 220 ADVANCED SURVEYING TOPICS ELECTROMAGNETIC DISTANCE MEASUREMENT (EDM) First introduced by Swedish physicist Erik Bergstr in addition to (Geodimeter) in 1948. Used visible light at night to accurately measure distances of up to 40km. In 1957, the first Tellurometer, designed by South African, Dr. T.L. Wadley, was launched. The Tellurometer used microwaves to measure distances up to 80km during day or night. Although the first models were very bulky in addition to power hungry, they revolutionized survey industry which, until their arrival, relied on tape measurements as long as accurate distance determinations. The picture above shows the slave (or remote) unit of the Tellurometer CA1000, a model which was extensively used in the 70’s in addition to 80’s.

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INITIAL IMPACTS OF EDM EDM Traverses ( in addition to Trilateration) Propagation of Electromagnetic Energy Velocity of EM energy, V = ƒ ƒ is the frequency in hertz (cycles/second) is the wavelength In vacuum the velocity of electromagnetic waves equals the speed of light. V = c/n n >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 = c/n or = cf/n (i.e. varies with n !) Note that n in any homogeneous medium varies with the wavelength . White light consists of a combination of wavelengths in addition to hence n as long as 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 there as long as e necessary to determine n of the atmosphere at the time in addition to location at which the measurement is conducted. Propagation of Electromagnetic Energy The refractive index of air varies with air density in addition to is derived from measurements of air temperature in addition to atmospheric pressure at the time in addition to site of a distance measurement. For an average wavelength : na= 1 + ( ng-1 ) x p – 5.5e x 10-8 1 + 0.003661T 760 1 + 0.003661T Where ng is the group index of refraction in a st in addition to ard atmosphere (T=0°C, p=760mm of mercury, 0.03% carbon dioxide) ng = 1+ ( 2876.04 + 48.864/2 +0.680/ 4 ) x 10-7 p is the atmospheric pressure in mm of mercury (torr) T is the dry bulb temperature in °C in addition to e is the vapor pressure Where e= e’+de in addition to e’=4.58 x 10a, a=(7.5T’)/(237.3+T’), de=-(0.000660p (1+0.000115T’) (T-T’) in addition to T’ is the wet-bulb temperature Hence measuring p, T in addition to T’ will allow as long as the computation of n as long as a specific

THE FRACTION OF A WAVELENGTH AND THE PHASE ANGLE Amplitude + r – r 0° 90° 180° 270° ¼ ¼ ¼ ¼ ½ ½ 360 A fraction of a wavelength can be determined from a corresponding phase angle Note: For = 0° the fraction is 0 For = 90° the fraction is ¼ For = 180° the fraction is ½ For = 270° the fraction is ¾ For = 360° the fraction is 1 EDM INSTRUMENTS CAN MEASURE PHASE ANGLES 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= Vt = (c/n)t Knowing the speed of light c in addition to being able to determine the refractive index, we could measure the time interval it takes as long as an electromagnetic wave to move from A to B to determine the distance L between A in addition to B. But since c, the speed of light, 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) m is an integer number of whole wavelengths, p is a fraction of a wavelength Thus L can be determined from , m in addition to p 1 2 3 4 5 6 7 8 9 10 11 12 p1 A B L 2 p2 Solving as long as the integer number (m) of whole wavelengths (Resolving the ambiguity in the number of whole wave lengths) Additional waves of known lengths 3= k2 in addition to 2= k 1 (k is a constant), are introduced to measure the same distance L: L = (m3 + p3) 3 L =(m2 + p2) 2 L =(m1 + p1) 1 Determining p1 p2 in addition to p3 by measuring phase angles 1 2 in addition to 3 in addition to solving the above equations simultaneously yields L ( Note: For L < 3 , m3 = 0). For example, if 1 = 10.000m, k= 10.000 in addition to p1 = 0.2562, p2 = 0.2620 in addition to p3 = 0.0125 Then 2= 10.000m x 10.000 = 100.000m in addition to 3= 100.000 x 10.000 = 1000.000 L= (m3 + p3) 3 = (0+0.1250)x 1000.000 = 125.000m approximately m2= 125/ 2=125/100=1 in addition to hence L = (1+0.2620)x100.000 = 126.200m approximately m1= 126.2/ 1=126.2/10=12 in addition to hence L = (12+ 0.2562)x10 = 122.562m p3 Basic Components of an EDM Instrument L Reflector Display Frequency Generator Transmitter Beam Splitter Interference Filter Receiver Optics in addition to phase-difference circuits Phase Meter F1 F2 F3 F4 Variable Filter Measurement signal Reference signal Reflector To obtain the phase angle the reflected signal phase is compared to the reference signal phase. Note also that the measured distance equals 2 x L. Length of measured path is 2xL L General Remarks on EDM The original Tellurometer models, using microwaves, consisted of two units, the master in addition to the remote, both of which had to be manned. The carrier wave was used to establish a voice channel between the operators who had to coordinate the manual switching of the frequencies. The measuring signals were directed (bundled) by means of metal cones. For long lines careful measurements of pressure in addition to the wet- in addition to dry-bulb temperatures were made at each end of the line. Measurements were very susceptible to multipath reflections (ground swing). Developments in electronics reduced the size of the components drastically so that EDM instruments could be mounted on top of theodolites as long as convenient simultaneous measurements of distances as well as directions. Nowadays EDM components are completely integrated into total stations. Total stations allow as long as the direct input of temperature in addition to pressure in addition to automatic application of meteorological corrections. Most of the current EDM instruments use LASER beams in addition to passive optical reflectors, thus reducing the possibility of multipathing considerably. The latest models provide as long as reflector-less measurements, thus improving efficiency as long as certain applications drastically. Sources of Error in EDM: Personal: Careless centering of instr. in addition to /or reflector Faulty temperature in addition to pressure measurements Incorrect input of T in addition to p Remember: L = (m + p) Instrumental Instrument not calibrated Electrical center Prism Constant (see next slide) Natural Varying ‘met’ along line Turbulence in air Sources of Error in EDM: A B C Determination of System Measuring Constant Measure AB, BC in addition to AC AC + K = (AB + K) + (BC + K) K = AC- (AB + BC) If electrical center is calibrated, K rep- resents the prism constant. Mistakes: Incorrect ‘met’ settings Incorrect scale settings Prism constants ignored Incorrect recording settings (e.g. horizontal vs. slope) Good Practice: Never mix prism types in addition to makes on same project!!! Calibrate regularly !!! CURVES SIMPLE CURVES Circular Curves TYPES OF CURVES: R R Simple Curve R R Easement or Transitional Curve spiral spiral R r Compound Curve R R Reverse Curve Definitions 100 ft R R “Degree of Curve” Central angle subtended by a circular ARC of 100 ft (highways) D/100’ = 360/ 2p r = full circle angle / circumference PI PC PT L PI = Point of Intersection PC = Point of Curvature PT = Point of Tangency L = Length of Curve Formulae I I/2 R R E M LC L LC = Long Chord M = Middle Ordinate E = External Distance T = Tangent Distance I = Intersection Angle T T I PC PT T = R Tan I/2 L = 100 I0/D0 = R I rads LC = 2 R Sin I/2 R/ (R+E) = Cos I/2 => E = R [(1/Cos (I/2)) – 1] (R – M)/R = Cos I/2 => M = R [1 – (Cos (I/2)]

Stationing (usually every 100 feet) 0+00.00 PI T 1+00.00 2+00.00 3+00.00 4+00.00 PC 4+86.75 L 5+00.00 6+00.00 7+00.00 PT 7+27.87 8+00.00 9+00.00 10+00.00 11+00.00 PC sta = PI sta – T PT sta = PC sta + L Curve Layout PC PI Need to stake at “full stations” (XX+00.00) Set up on PC, backsight PI, turn deflection angle (d), measure chord distance (c) d chord Vertical Curves Crest Curve Crest Curve Provides a smooth transition between different grades Parabola – constant rate of change of grade GRADE: 4.00’ 100’ Grade = +4.00% + rising grad – falling grade

Vertical Curve Geometry BVC EVC Xp Yp L/2 L/2 L = curve length Back tangent (g1) Forward tangent (g2) Y X V Constant rate of change of Grade r r = (g2 – g1) / L R should be low (long L) as long as rider com as long as t in addition to sight distance Equation of Curve (parabola): Y = YBVC + g1 X + ((g2 – g1)/2L) X2 Units: g in %, L in addition to X in stations, Y in ft/meters Or G in fractions (0.04), L, X, Y in ft/meters COORDINATE GEOMETRY

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COORDINATE GEOMETRY Except as long as Geodetic Control Surveys, most surveys are referenced to plane rectangular coordinate systems. Frequently State Plane Coordinate Systems are used. The advantage of referencing surveys to defined coordinate systems are: Spatial relations are uniquely defined. Points can be easily plotted. Coordinates provide a strong record of absolute positions of physical features in addition to can thus be used to re-construct in addition to physically re-position points that may have been physically destroyed or lost. Coordinate systems facilitate efficient computations concerning spatial relationships. In many developed countries official coordinate systems are generally defined by a national network of suitably spaced control points to which virtually all surveys in addition to maps are referenced. Such spatial reference networks as long as m an important part of the national infrastructure. They provide a uni as long as m st in addition to ard as long as all positioning in addition to mapping activities. THE TRIANGLE The geometry of triangles is extensively employed in survey calculations. A B C a b c For any triangle ABC with sides a, b in addition to c: a = b = c (LAW OF SINES) sin A sin B sin C AND a2 = b2 + c2 -2ab cosA b2 = a2 + c2 -2ac cosB (LAW OF COSINES) c2 = a2 + b2 -2ab cosC The solution of the quadratic equation ax2 + bx + c = 0 x = -b ± b2 – 4ac is also often used. 2a A + B + C = 180° THE STRAIGHT LINE A(XA,YA) P(XP,YP) B(XB,YB) XAB = XB-XA AND YAB = YB-YA LAB = XAB2 + YAB2 AzAB = atan(XAB / YAB ) + C AzAB C=0° as long as XAB >0 in addition to YAB >0 C=180° as long as YAB <0 in addition to C=360° as long as XAB <0 in addition to YAB >0 b For P on line AB: YP = mXp + b where the slope m = ((yAB / xAB ) = cot(AzAB ) AzAB = atan (1/m) + C

THE CIRCLE P(XP,YP) O(XO,YO) R2 = XOP2 + YOP2 XP2+YP2 – 2XOXP – 2YOYP + f = 0 R = XO2 + YO2 – f f R THE PERPENDICULAR OFFSET A(XA,YA) C B(XB,YB) AzAP = atan(XAP / YAP ) + C LAP = XAP2 + YAP2 AzAB = atan(XAB / YAB ) + C LAB = XAB2 + YAB2 AzAB C=0° as long as XAB >0 in addition to YAB >0 C=180° as long as YAB <0 in addition to C=360° as long as XAB <0 in addition to YAB >0 b P(XP,YP) AzAP a a = AzAB – AzAP LPC = LAB sin LAC = LAB cos a Given known points A,B in addition to P, compute distance PC (LPC) THE INTERSECTION A(XA,YA) B(XB,YB) AzAB C=0° as long as XAB >0 in addition to YAB >0 C=180° as long as YAB <0 in addition to C=360° as long as XAB <0 in addition to YAB >0 AzAP Given A(XA,YA), B(XB,YB), AzAP in addition to AzBP compute P(XP,YP) AzAB = atan(XAB / YAB ) + C LAB = XAB2 + YAB2 a = AzAB – AzAP a P(XP,YP) AzBP AzBA = AzBA – AzBP = 180° – a – LAP = LAB (sin rule) sin sin LAP = LAB (sin / sin ) XP = XA + LAP sin AzAP YP = YA + LAP cos AzAP Similarly (as a check on the calculations): LBP = LAB (sin rule) sin a sin LBP = LAB (sin / sin ) XP = XA + LBP sin AzBP YP = YA + LBP cos AzBP WARNING: USE A THIRD KNOWN POINT TO CHECK ORIENTATIONS Outside Orientation


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