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CE 220

Learn about EDM, a surveying technique using electromagnetic waves to accurately measure distances, its history, principles, and components.

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CE 220

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  1. CE 220 ADVANCED SURVEYING

  2. TOPICS

  3. ELECTROMAGNETIC DISTANCE MEASUREMENT (EDM) First introduced by Swedish physicist Erik Bergstrand (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 and power hungry, they revolutionized survey industry which, until their arrival, relied on tape measurements for 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 and 80’s.

  4. SCALE DETERMINATION IN TRIGONOMETRICAL CONTROL NETWORKS Trigonometric Base Line Extension TRAVERSES TO EXPAND AND DENSIFY NATIONAL CONTROL NETWORKS INITIAL IMPACTS OF EDM EDM Traverses (and Trilateration)

  5. Propagation of Electromagnetic Energy Velocity of EM energy, V = ƒ λƒ is the frequency in hertz (cycles/second) λis 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 λ = c/n or λ = cf/n (i.e. λvaries with 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.

  6. 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: 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 standard atmosphere (T=0°C, p=760mm of mercury, 0.03% carbon dioxide) ng = 1+ ( 2876.04 + 48.864/λ2 +0.680/ λ4 ) 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 Hence measuring p, T and T’ will allow for the computation of n for a specific λ

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

  8. Principles of Electronic Distance Measurement 10 11 12 9 8 1 4 7 2 3 5 6 B A p λ λ λ λ λ λ λ λ λ λ λ λ L 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 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) λ m is an integer number of whole wavelengths, p is a fraction of a wavelength Thus L can be determined from λ, m and p

  9. Solving for the integer number (m) of whole wavelengths(Resolving the ambiguity in the number of whole wave lengths) p3 λ2 p2 10 11 12 9 8 1 4 7 2 3 5 6 B A p1 λ λ λ λ λ λ λ λ λ λ λ λ L Additional waves of known lengths λ3= kλ2 and λ2= k λ1(k is a constant), are introduced to measure the same distance L: L = (m3 + p3) λ3 L =(m2 + p2) λ2 L =(m1 + p1) λ1 Determiningp1 p2andp3by measuring phase anglesθ1θ2andθ3and solving the above equations simultaneously yields L ( Note: For L < λ3 , m3 = 0). For example, if λ1= 10.000m, k= 10.000 and p1 = 0.2562, p2 = 0.2620 and p3 = 0.0125 Then λ2= 10.000m x 10.000 = 100.000m and λ3= 100.000 x 10.000 = 1000.000 L= (m3 + p3) λ3= (0+0.1250)x 1000.000 = 125.000m approximately m2= 125/ λ2=125/100=1 and hence L = (1+0.2620)x100.000 = 126.200m approximately m1= 126.2/ λ1=126.2/10=12 and hence L = (12+ 0.2562)x10 = 122.562m

  10. Basic Components of an EDM Instrument Reflector Reflector L Beam Splitter Length of measured path is 2xL Variable Filter Interference Filter Transmitter L F4 F1 Receiver Optics and phase-difference circuits F2 F3 Measurement signal Reference signal Frequency Generator Phase Meter Display 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.

  11. General Remarks on EDM • The original Tellurometer models, using microwaves, consisted of two units, the master and 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 and the wet- and 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 for convenient simultaneous measurements of distances as well as directions. Nowadays EDM components are completely integrated into total stations. • Total stations allow for the direct input of temperature and pressure and automatic application of meteorological corrections. • Most of the current EDM instruments use LASER beams and passive optical reflectors, thus reducing the possibility of multipathing considerably. • The latest models provide for reflector-less measurements, thus improving efficiency for certain applications drastically.

  12. Sources of Error in EDM: • Instrumental • Instrument not calibrated • Electrical center • Prism Constant (see next • slide) • Personal: • Careless centering of instr. and/or reflector • Faulty temperature and pressure measurements • Incorrect input of T and p • Natural • Varying ‘met’ along line • Turbulence in air Remember: L = (m + p) λ

  13. Sources of Error in EDM: Determination of System Measuring Constant C A B • Mistakes: • Incorrect ‘met’ settings • Incorrect scale settings • Prism constants ignored • Incorrect recording settings • (e.g. horizontal vs. slope) • Measure AB, BC and AC • AC + K = (AB + K) + (BC + K) • K = AC- (AB + BC) • If electrical center is calibrated, K rep- • resents the prism constant. Good Practice: Never mix prism types and makes on same project!!! Calibrate regularly !!!

  14. CURVES

  15. SIMPLE CURVES

  16. Circular Curves TYPES OF CURVES: Simple Curve Compound Curve R R R r Reverse Curve spiral R R R R Easement or Transitional Curve spiral

  17. Definitions “Degree of Curve” Central angle subtended by a circular ARC of 100 ft (highways) 100 ft R R D/100’ = 360/ 2p r = full circle angle / circumference PI PC PI = Point of Intersection PC = Point of Curvature PT = Point of Tangency L = Length of Curve L PT

  18. Formulae I LC = Long Chord M = Middle Ordinate E = External Distance T = Tangent Distance I = Intersection Angle T E T L M PC PT LC R T = R Tan I/2 R I/2 L = 100 I0/D0 = R I rads I 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)]

  19. Stationing(usually every 100 feet) PC 4+86.75 4+00.00 0+00.00 1+00.00 2+00.00 3+00.00 5+00.00 T PI L 6+00.00 7+00.00 PC sta = PI sta – T PT sta = PC sta + L PT 7+27.87 8+00.00 9+00.00 10+00.00 11+00.00

  20. Curve Layout Need to stake at “full stations” (XX+00.00) Set up on PC, backsight PI, turn deflection angle (d), measure chord distance (c) PI d PC chord

  21. Vertical Curves Crest Curve Crest Curve Provides a smooth transition between different grades Parabola - constant rate of change of grade GRADE: Grade = +4.00% + rising grad - falling grade 4.00’ 100’

  22. Vertical Curve Geometry Y V Back tangent (g1) Forward tangent (g2) BVC EVC Xp Yp X L/2 L/2 L = curve length

  23. Constant rate of change of Grade r r = (g2 – g1) / L R should be low (long L) for rider comfort and sight distance Equation of Curve (parabola): Y = YBVC + g1 X + ((g2 – g1)/2L) X2 Units: g in %, L and X in stations, Y in ft/meters Or G in fractions (0.04), L, X, Y in ft/meters

  24. COORDINATE GEOMETRY

  25. COORDINATE GEOMETRY • Except for 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 and can thus be used to re-construct and 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 and maps are referenced. Such spatial reference networks form an important part of the national infrastructure. They provide a uniform standard for all positioning and mapping activities.

  26. THE TRIANGLE The geometry of triangles is extensively employed in survey calculations. B For any triangle ABC with sides a, b and 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 c a A C b A + B + C = 180° The solution of the quadratic equation ax2 + bx + c = 0 x = -b ±  b2 – 4acis also often used. 2a

  27. THE STRAIGHT LINE Y X 0 0 ∆XAB = XB-XA AND ∆YAB = YB-YA LAB =  ∆XAB2+ ∆YAB2 AzAB = atan(∆XAB / ∆YAB ) + C C=0° for∆XAB >0 and ∆YAB >0 C=180° for∆YAB <0 and C=360° for∆XAB <0 and ∆YAB >0 B(XB,YB) P(XP,YP) AzAB For P on line AB: YP = mXp + b where the slope m = ((∆yAB / ∆xAB ) = cot(AzAB ) AzAB = atan (1/m) + C A(XA,YA) b

  28. THE CIRCLE Y X 0 0 P(XP,YP) R R2 = ∆XOP2+ ∆YOP2 XP2+YP2 – 2XOXP – 2YOYP + f = 0 R =  XO2+ YO2- f O(XO,YO) f

  29. THE PERPENDICULAR OFFSET Y X 0 0 Given known points A,B and P, compute distance PC (LPC) C=0° for∆XAB >0 and ∆YAB >0 C=180° for∆YAB <0 and C=360° for∆XAB <0 and ∆YAB >0 P(XP,YP) B(XB,YB) AzAP = atan(∆XAP / ∆YAP ) + C LAP =  ∆XAP2+ ∆YAP2 AzAB = atan(∆XAB / ∆YAB ) + C LAB =  ∆XAB2+ ∆YAB2 AzAP C a AzAB A(XA,YA) a =AzAB – AzAP LPC = LAB sin LAC = LAB cos a b

  30. THE INTERSECTION Y X 0 0 Given A(XA,YA), B(XB,YB), AzAP and AzBP compute P(XP,YP) β = AzBA – AzBP a = AzAB – AzAP AzAB = atan(∆XAB / ∆YAB ) + C LAB =  ∆XAB2+ ∆YAB2 γ = 180° – a– β LAP= LAB (sin rule) sin β sin γ LAP = LAB (sin β/ sin γ ) XP = XA + LAP sin AzAP YP = YA + LAP cos AzAP C=0° for∆XAB >0 and ∆YAB >0 C=180° for∆YAB <0 and C=360° for∆XAB <0 and ∆YAB >0 B(XB,YB) AzBP β AzAP AzBA Similarly (as a check on the calculations): LBP= LAB (sin rule) sin a sin γ LBP = LAB (sin β/ sin γ ) XP = XA + LBP sin AzBP YP = YA + LBP cos AzBP Outside Orientation AzAB a γ A(XA,YA) P(XP,YP) WARNING: USE A THIRD KNOWN POINT TO CHECK ORIENTATIONS

  31. Y X 0 0 INTERSECTION OF A LINE WITH A CIRCLE Given A,B and C and radius R, compute P1 and P2 Note a = AzAC – AzAB and LBP1 = LBP2 = R Apply the cos rule to triangle ABP1: LBP12 = LAB2+ LAP12-2(LAB)(LAP1)cos a or LAP12 –(2LABcos a)LAP1+ (LAB2 - LBP12) = 0 which is a quadratic equation with LAP1 as unknown C(XC,YC) B(XB,YB) R Note that since LBP1 = LBP2 = R the above equation also applies to triangle ABP2. Hence the two solutions of the quadratic equation are AP1 and AP2. P2(XP2,YP2) a P1(XP1,YP1) AP=2LABcos a ± (2LABcos a)2-4(LAB2 - LBP12) 2 A(XA,YA) Now use AzAC and the solutions of AP to compute P1(XP1,YP1) and P2(XP2,YP2)

  32. INTERSECTION OF TWO CIRCLES Y X 0 0 Given A,B and C and radius R, compute P1 and P2 Compute AzAB and LAB from given coordinates of A and B Note LAP1 = LAP2 = RA and LBP1 = LBP2 = RB Apply the cos rule to triangle ABP1: a = acos ((LAB2+ LAP12-LBP12)/(2LABLAP1)) = acos ((LAB2+ RA2-RB2)/(2LABRA)) B(XB,YB) AzAP1 = AzAB – a and AzAP2 = AzAB + a P1(XP1,YP1) RB P2(XP2,YP2) a XP1 = XA + RAsin (AzAP1) YP1 = YA + RAcos(AzAP1) and XP2 = XA + RAsin (AzAP2) YP2 = YA + RAcos(AzAP2) a RA A(XA,YA) WARNING Trilatertion of a point with only two distances yields two positions!!!!!

  33. HOMEWORK: LECTURE 13 (CHAPTER 11 SEC 1-6) 11.1, 11.9, 11.13, 11.15, 11.17

  34. RESECTION

  35. THE RESECTION Determine Coordinates of P from observations to known points A, B and C. Note: Since P is un-known, the raw observations at P need to be oriented. Use the Q-Point Method to orient the observations at P and then intersect P with the oriented forward directions from A and B. • Compute α and β from observations • a = 180° - α , b = 180° - β • Compute AzAB from XA, YA, XB,YB • AzAQ = AzAB – b, AzBQ = AzBA + a • Compute XQ and YQ with AzAQ and AzBQ • from A and B (forward intersection) • Compute AzQC (= AzPC) from XQ, YQ, XC,YC • AzAP = AzPC + α ± 180°, AzBP = AzPC - β ± 180° • 8. Compute XP and YP with AzAQ and AzBQ • from A and B (forward intersection) Q A b a α P b a Note:The solution is ambiguous when A,B,C and P all lie on a circle (the danger circle) β B C For best orientation results use the furthest point for Orientation.

  36. THE INACCESSIBLE POINT AP = AB sin (B) = AB sin (B) sin(180-(A+B) sin (A+B) BP = AB sin (A) sin (A+B) ∆hA = AP tan (vA) ∆hB = BP tan (vB) Elev.PA = Elev.A + ∆hA + hiA Elev.PB = Elev.B + ∆hB + hiB Elev P = Elev.PA + Elev.B 2 Can be applied in reverse to determine height of A (or B) from known height of P. Provided sufficient outside orientation is given, this model is also used to close traverses on inaccessible points with known coords. (e.g. church spire) ∆hA ∆hB vA B hiA p Outside Orientation A Caution: Steep sights – observe on both faces and level instrument carefully.

  37. THE ‘DOG’S LEG’ CHECK When fixing or placing a point from only one known station by means of a ‘single polar’, the ‘dog’s leg’ is often applied to obtain an independent check. O U • From known point B observe two orientation points to ensure correct orientation. • Measure direction and distance to unknown point U • Place auxiliary point A in such a way that at least one known point in addition to B is visible and that the angle at U is greater than 30°. • Measure direction and distance to A • Move instrument to A and observe outside orientation point O and B and measure direction and distance to U. • Compute Coordinates of A and U to obtain independent coordinates for U A B To strengthen the check A is sometimes placed on line from B to an orientation point.

  38. THE 2D CONFORMAL COORDINATE TRANSFORMATION 2. Compute the Scale  ∆EAB2+ ∆NAB2 Scale s =  ∆XAB2+ ∆YAB2 N Given A(E,N) and B(E,N) and A(X,Y) and B(X,Y) compute C(E,N) from C(X,Y) 1. Compute the Swing Az(XY)AB = atan (∆XAB/∆YAB) Az(EN)AB = acot (∆NAB/∆EAB) ‘Swing’θ = Az(XY)AB - Az(EN)AB Az(EN)AB Az(XY)AB C B θ ∆NAB X D Y A XAB YAB ∆XAB ∆YAB ∆EAB E 0 0

  39. THE 2D CONFORMAL COORDINATE TRANSFORMATION 2. Compute the Scale  ∆EAB2+ ∆NAB2 Scale s =  ∆XAB2+ ∆YAB2 N Given A(E,N) and B(E,N) and A(X,Y) and B(X,Y) compute C(E,N) from C(X,Y) 1. Compute the Swing Az(XY)AB = atan (∆XAB/∆YAB) Az(EN)AB = acot (∆NAB/∆EAB) ‘Swing’θ = Az(XY)AB - Az(EN)AB 3. Compute the Translations TX and TY a = sYAsin θ, b=sXAcos θ Hence X’A = s(XAcos θ - YAsin θ) c = sXAsin θ, d=sYAcos θ Hence Y’A = s(XAsin θ + YAcos θ) EA = X’A + TX hence TX = EA – X’Aand NA = Y’A + TY hence TY = NA – Y’A C NC Y’ B D Y X A A YA d θ Y’A XA 4. Apply the transformation parameters EC = s(XCcos θ - YCsin θ) + TX NC = s(XCsin θ + YCcos θ) + TY c θ θ TY X’ X’A a b E 0 Tx EC EA 0

  40. SOME ‘GOLDEN RULES’ A B Orientation • Unless traversing, always observe two orientation rays to control correct orientation of instrument. • Long observation rays give better orientation than short rays. Orientation rays should always be longer than fixing rays. • The formulae derived for trilateration, intersection and resection use the minimum number of observations to determine one unchecked set of coordinates for unknown points. Additional independent observations should be made to provide sufficient redundancy and control of the quality of coordinates. • Pay attention to the strength in the geometry – keep intersection angles larger than 30°. • Be aware of situations that yield multiple solutions. (Intersection/Trilateration). • Add value to your survey by connecting it to the most widely used coordinate system in the area. If possible, avoid local coordinate systems. • Interpolate – do NOT extrapolate! • Connect to nearest available known points Mirror image due to orientation mistake of 180° Geometry Ø ✔ Interpolation Ø ✔ ✔ Neighborhood Principle Ø

  41. GEODESY

  42. Geodesy, Map Projections and Coordinate Systems • Geodesy - the shape of the earth and definition of earth datums • Map Projection - the transformation of a curved earth to a flat map • Coordinate systems - (x,y) coordinate systems for map data

  43. Types of Coordinate Systems • (1) Global Cartesian coordinates (x,y,z) for the whole earth • (2) Geographic coordinates (f, l, z) • (3) Projected coordinates (x, y, z) on a local area of the earth’s surface • The z-coordinate in (1) and (3) is defined geometrically; in (2) the z-coordinate is defined gravitationally

  44. Z Greenwich Meridian O • Y X Equator Global Cartesian Coordinates (x,y,z)

  45. Global Positioning System (GPS) • 24 satellites in orbit around the earth • Each satellite is continuously radiating a signal at speed of light, c • GPS receiver measures time lapse, Dt, since signal left the satellite, Dr = cDt • Position obtained by intersection of radial distances, Dr, from each satellite • Differential correction improves accuracy

  46. Global Positioning using Satellites Dr2 Dr3 Number of Satellites 1 2 3 4 Object Defined Sphere Circle Two Points Single Point Dr4 Dr1

  47. Geographic Coordinates (f, l, z) • Latitude (f) and Longitude (l) defined using an ellipsoid, an ellipse rotated about an axis • Elevation (z) defined using geoid, a surface of constant gravitational potential • Earth datums define standard values of the ellipsoid and geoid

  48. Shape of the Earth It is actually a spheroid, slightly larger in radius at the equator than at the poles We think of the earth as a sphere

  49. Ellipse An ellipse is defined by: Focal length =  Distance (F1, P, F2) is constant for all points on ellipse When  = 0, ellipse = circle Z b O a X   F1 F2 For the earth: Major axis, a = 6378 km Minor axis, b = 6357 km Flattening ratio, f = (a-b)/a ~ 1/300 P

  50. Ellipsoid or SpheroidRotate an ellipse around an axis Z b a O Y a X Rotational axis

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