Geometric Reduction of Distances. John Hamilton Terrasurv

Transcription

1 Geometric Reduction of istances John Hamilton Terrasurv

2 Outline Brief History Atmospheric Corrections Reduction to the Marks Reduction to the Ellipsoid Comparison with GPS Example Questions/iscussion

3 Caveats The concepts and formulas given assume short to medium distances Longer distances which require more refined methods to reduce are rarely measured anymore due to GPS Arc-to-chord differences ignored Second Velocity correction ignored (negligible)

4 The magical ppm part per million (ppm) mm in km ft in mile Many modern EM s are capable of measuring to ppm or better Why? eformation Surveys Precision Stakeout Calibration for less accurate EM s

5 History First EM: Russians in 936 First commercial EM: Geodimeter (Swedish) NASM- in 950 NGS used Model 4 s in many geodetic surveys Model 4: Range of about 60 km at night

6 BIG RE Geodimeter Model 4 Modified by NGS Longest line measured 50 km in Arizona Picture at right: 98 prisms used for Mconald Observatory Survey

7 EM Accuracy In the 80 s, typical accuracy was 5 mm ± 5 ppm 90 s: improved to 3 mm ± 3mm Modern Trimble instruments: S3, S6, S8: mm ± ppm S6, S8 High Precision model: mm ± ppm Leica TM30: 0.6 mm ± ppm Kern ME5000 (Mekometer): 0. mm ± 0. ppm

8 EM istances Measured from instrument to prism The raw slope distance will vary as the height of instrument (HI) and height of target (HT) vary, and with differences in elevation Horizontal distances will increase with elevation Curvature of earth Plumb lines are not parallel

9 Golden Gate Bridge Tower-to-tower distance: 80 meters (400 FT) Tower height: 7 meters (746 FT) ifference in distance at bottom and top: meters Sea Level istance is m shorter ELLIPSOIAL ISTANCE is m shorter IMPORTANT TO USE ELLIPSOI HEIGHTS FOR REUCTIONS!

10 EM istances Raw values must be corrected for atmospheric delay and prism/instrument offsets EM can have scale bias Manufacturer calibration Calibration baseline (CBL) EM may exhibit cyclical behavior over the unit wavelength (typically 0 meters) Minimized in modern EM s Can be accounted for in calibration

11 Atmospheric Corrections First correction to be applied is for atmospheric delay epends on temperature, pressure, and to a lesser degree on humidity Speed of light in a vacuum: C 0 =99, km/s At sea level, in atmosphere, C 99,700 km/s ifference of about 300 ppm

12 Atmospheric Corrections Refractive index n: n = C C Usually expressed as N=n * ( X 0-6 ) 300 N g is the Group Refractive Index, depends on wavelength (λ) High precision S6: λ S6HP =870 nm; N g =94.79 Regular S6: λ S6 =905 nm; N g =93.67 Manufacturers provide a formula for the particular EM

13 First Velocity Correction RAW J N p.7 v t 73.5 t 73.5 J and N are constants for a particular instrument Third term in [brackets] often ignored (humidity) 0 6 p=atmospheric pressure in millibars (mb) t=temperature of air in degrees celsius v=partial water vapor pressure in mb (measure of humidity)

14 J and N Constants Manufacturer of conventional instruments J constant N constant Trimble VX/S Series from instrument from instrument Trimble Trimble 3300/ Trimble TTS300/ Sokkia SET Topcon Geotronics 400/ Leica Zeiss Elta/Elta3/Elta Zeiss Elta C S6 High Precision: J=78.779; N= (from instrument) BUT, this N value is for P in mm Hg rather than millibars. Therefore N S6 = 0.75 * =

15 Humidity Humidity can often be ignored Error in ppm caused by ±0% Relative Humidity: Temp ⁰C 50% RH 00% RH +0 ±0.06 ppm ±0.06 ppm +0 ±0.0 ppm ±0.0 ppm +30 ±0.5 ppm ±0.5 ppm +40 ±0.30 ppm ±0.30 ppm ew Point from airport reports/relative Humidity from weather reports MUST be converted to partial water vapor pressure for use in formula

16 What does it matter? To affect the distance by ppm: Temperature: C (.8 F) Pressure: 3 mb (0. in Hg) Humidity: 5 mb partial pressure.5 C in wet bulb temperature Ignoring humidity altogether causes the following errors, in ppm: Temp C RH=50% RH=00%

17 Temperature most critical component Simple glass thermometer gives reasonable accuracy Must be kept out of direct sunlight Changes by 6.4 C per 000 m (3.5 F per 000 ) Varies irregularly by area, proximity to ground C = 5 ( F 3) 9

18 Pressure Less critical ΔP=-0. Hg per 00 feet increase in elevation S6 has on board pressure sensor mb=0.75 mm Hg= inches Hg If using Sea Level Pressure (SLP), must correct for altitude to station pressure oes not vary over small areas, except when front passes through

19 3 mb= PPM BAROMETRIC PRESSURE

20

21 Sea Level Pressure isseminated in weather reports NOAA weather radio AWOS/ASOS at airports Usually accessible by phone Must convert to station pressure (at station height above sea level): P stn P SL H 88 m 5.56 H m is height above sea level, in meters Approximately - Hg per 000 ft increase in elevation/-0. per 00 ft

22 Humidity AWOS/ASOS reports at airports give dew point temperature (T P ) RH Weather reports typically give relative humidity (RH) Saturated Vapor Pressure: 00 5( T TP ) v S 6.e 7.67T T 43.5 Partial Water Vapor Pressure: v v RH 00 S Sling psychrometer gives wet bulb temperature (T W ) W v W 6.e 7.67T W T 43.5 v W sta T T T v p 005 W W

23 Weather Calculator El Paso, TX National Weather Service Weather Calculator (

24 Portable Instrumentation Brunton AC Pro ± C ±.5 mb pressure ±3-5% RH Will log data in memory for later download to PC Current street price about $30-$50

25 Instruments and ata Collectors If T and P are entered, most instruments display CORRECTE values Raw data output to data collector is UNCORRECTE Values stored in.dc file are UNCORRECTE Some instruments may store corrected values (Zeiss S0 for example) TGO/TBC displays CORRECTE values

26 Geometric Corrections =ray path arc =ray path chord G =mark-to-mark 3 =ellipsoidal chord 4 =ellipsoidal arc 5 =horizontal distance H =horizontal distance at height of H =horizontal distance at height of HI=Height of instrument HR=Height of reflector

27 Radius of Curvature a semi-major axis=6,378,37 m b semi-minor axis=6,356,75.34 m f flattening /f= e eccentricity e a b f a e = Φ Latitude of station a a b

28 Radius of Curvature R distance from center of ellipse Radius varies by azimuth M=radius in N-S direction (meridian) N=radius in E-W direction (prime vertical)

29 Radius of Curvature Meridian Radius of Curvature Prime Vertical Radius of Curvature M N a e 3 e sin a e sin Average Radius of Curvature R M N M Radius of Curvature along a line of azimuth R N M N cos M sin

30 Radius of Curvature At equator M 0 b a 6,335,439 m N a 0 6,378,37 m At poles M=N=R=6,399,594 m

31 Arc to chord ( ) Usually can be neglected except for long lines Less than mm for lines under 38 km under normal conditions k 3 4 R k is coefficient of refraction

32 Coefficient of refraction k Ratio of curvature of ray path (r) to curvature of ellipsoid (R) Average value k=0.3 (i.e. r L =8R) Varies significantly depending on time of day, height above ground, etc Affects vertical angles much more than distances k Refraction angle R

33 Chord-to-chord ( 3 ) Often (incorrectly) called sea level correction More correct term is reduction to ellipsoid 3 H R H = HAE +HI = Elev +HI+N N=geoid height, distance from geoid to ellipsoid H = HAE +HR = Elev +HR+N H H H R

34 Chord to Arc ( 3 4 ) R Ellipsoidal chord to ellipsoidal arc m at 5 km (0.0 ppm) 0.00 m at 0 km (0. ppm) 0.8 m at 50 km (.5 ppm) Can be ignored for lines under 0 km

35 Arc-to-arc ( 4 ) 4 4 sin sin H R H R k H H k R k R R

36 How accurate? istance values in meters elta Height Mean Elevation esired Accuracy Accuracy Required elta Height refraction mm k (no units)

37 Using Zenith Angles: 4 ε is deflection of the vertical in azimuth of line Observed zenith angle Z Z and ε in radians ε usually small, often neglected EFLEC09 software from NGS ouble precision variables critical! R k Z H R R k Z R cos sin arctan 4

38 Horizontal istance 5 =horizontal distance at height H 5 sin Z sin Effect on 5 of ignoring second term: k 4 R Z istance Z =90 Z =80 Z =70 00 m 0 mm -0.3 mm -0.5 mm 300 m 0 mm -.3 mm -4. mm 500 m 0 mm -6.3 mm -.8 mm 000 m 0 mm -5. mm -47. mm 000 m 0 mm mm mm

39 Horizontal to ellipsoid 5 3 ppm error per 6 m uncertainty in H Using orthometric h instead of ellipsoidal H (N=-30 m): 5 ppm error Accuracy of R H 3 5 R Height Meters Sigma R Meters Error in PPM

40 Radius of Curvature Function of latitude and azimuth 640 R a d i u s o f c u r v a t u r e, K M Azimuth in degrees 0⁰ 0⁰ 40⁰ 60⁰ 80⁰ 90⁰ 6,37,000

41 Radius of curvature Mean R= M N R a d i u s o f C u r v a t u r e, K M PV Meridian Mean Radius 6,37, Latitude in egrees

42 Reduction to marks ( G ) G =Mark-to-mark slope distance Unambiguous, does not depend on choice of ellipsoid or height system Comparable to GPS vectors Archiving of observations Least Squares Adjustments Mark-to-mark distance G Mark-to-mark zenith angle Z G

43 Reduction to marks

44 Zenith Angle Reduction Z Z G 3 3 Z HT HI HT HI Z sin Z sin 3 G 6 G =angle between ellipsoidal normals HT=height of target (modern equipment HT=HR) If >6 meters, and HT-HI<0.5 meters, then Z sin Z R H cos Z HT HI G sin Z Z G Z Z

45 istance Reduction G G H H H H edm R G edm G R cos Z G G is on both sides, iteration required st iteration, G = and Z G =Z Then compute Z G using G Recompute G using updated values

46 istance Reduction using Heights If heights of markers are known: H and H are heights above ellipsoid of low and high marker, respectively h and h are heights of instrument/reflector at low and high marker, respectively Second term <0.00 m for <450 meters R h h h h H H G

47 Trigonometric Elevations One way: H H cos Z k sin Z R if using G and Z G then HI and HR=0 second term is curvature and refraction correction k is usually unknown, average value=+0.3 often used HI HR

48 Trig Heights-Reciprocal Obs Z G is mark-to-mark Z from to Z G is mark-to-mark Z from to H G H cos ZG cos G If using Z and Z then HI, HT, HI, HI must be applied istance Simultaneous Non- Simultaneous 00 m < mm < mm 300 m ± mm ± mm 500 m ±3 mm ±5 mm 000 m ± mm ±0 mm Z

49 ECEF LOCAL HORIZON ECEF=Earth Centered Earth Fixed Origin at center of mass of earth X-Y plane=equator X axis at 0⁰ Longitude Y axis at 90⁰ East Longitude Z axis perpendicular to equator (spin axis) GPS Vectors in ECEF ΔX, ΔY, ΔZ Independent of ellipsoid

50 Local Horizon System North-East-Up istance (Horizontal): = E + N E Azimuth: α = tan N H = UP ,000,000 Assumes standard refraction, k=0.3 Note: Up ΔH due to curvature and refraction

51 GPS Vector ΔX ΔY ΔZ E = sin λ ΔX + cos λ ΔY N = sin φ cos λ ΔX sin φ sin λ ΔY + cos φ ΔZ ΔU = cos φ cos λ ΔX + cos φ sin λ ΔY + sin φ ΔZ X = sin λ ΔE sin φ cos λ ΔN + cos φ cos λ ΔU Y = cos λ ΔE sin φ sin λ ΔN + cos φ sin λ ΔU ΔZ = cos φ ΔN + sin φ ΔU Mark-to-mark distance G = ΔX + ΔY + ΔZ G = ΔE + ΔN + ΔU

52 Example (Real ata) GPS observations independent lines EM Zeiss S0 / mm ± ppm

53 Profile

54 GPS Observations uration X Y Z Length Ratio Variance RMS 3 minutes minutes MEAN ifference

55 EM Observations Shot from high station at 0:45 AM local Actual temperature and pressure: Low station: 65⁰ F/9.3 High station: 67⁰ F/8.7 Relative Humidity: 93% Zeiss S0 Onboard sensors 67⁰ F/8.7 (assumes RH=60%) PPM correction=+8.6 (computed by S0) Zenith: 93⁰ (mean +R) Slope: m (mean +R) HI:.435 m/ht:.00 m

56 Computed PPM corrections Station st term (J) nd term Humidity Term (93% RH) Total PPM High Low MEAN +6.0 Instrument 60% RH +8.6

57 Computed Values EM: m (corrected by S0) RAW: m (uncorrected) EM : m (using mean PPM)

58 Mark-to-Mark Zenith istance Z HT HI G sin Z Z sin 933'57.9" Δ Z = rad = Z G =

59 Mark-to-mark istance G G G EM G Z HT HI HT HI cos ) ( ) ( 34'09.8" cos( ) ( ) ( G G

60 ifferences istance Length GPS EM High only ( m) EM Low only ( m) EM Mean PPM MARK-TO-MARK values in meters

61 Reduction to ellipsoid Using mean R=6,37,000 m: m Using R AZ =? R H R H H H G 3 ) (

62 Radius of Curvature Meridian Radius of Curvature Prime Vertical Radius of Curvature M N a e 3 e sin a e sin Average Radius of Curvature R M N M Radius of Curvature along a line of azimuth R N M N cos M sin

63 Reduction to ellipsoid 3 G ( H H) H H R R Using mean R=6,37,000 m: m Using R AZ =6,385,604. m: m IFFERENCE OF 0.05 PPM Using Ortho H instead of Ellip H: m IFFERENCE OF 6.7 PPM! Inverse Adjusted GPS coords: m MUST USE ELLIP H FOR REUCTION TO ELLIPSOI!

64 GRI Grid Factor (GF): Ellipsoid to Grid Elevation Factor (EF): Horizontal (?) to Ellipsoid Combined Factor (CF): Horizontal to Grid /(CF)=Grid to Horizontal (GROUN) State Plane/UTM: completely different GF, same EF

65 State Plane (PA SOUTH) Low Point: GF= (40 PPM) EF= (9 PPM) CF= (69 PPM) High Point: GF= (40½ PPM) EF= (48 PPM) CF= (88 PPM) OK to use average GF, caution using average CF SPC Grid istance: m

66 UTM ZONE 7 High GF: (34 PPM) Low GF: (37 PPM) Average: UTM7 Grid istance: m NOTE: UTM grid distance is 0.53 m smaller than PA S SPC distance (.7 FT)

67 SUMMARY Atmospheric Corrections Temperature & Pressure (& Humidity??) Reduce to Mark-to-Mark HI, HT, Zenith distance or ΔHeight Reduce to Ellipsoid Height above Ellipsoid, Earth Radius EllipsoidGrid epends on grid

68 Questions/iscussion Contact: John Hamilton Terrasurv; Pittsburgh, PA

69 References Laurila, Simo H. (983). Electronic Surveying in Practice, John Wiley & Sons Inc. Robertson, Kenneth. (979). The use and calibration of distance measuring equipment for precise mensuration of dams (Revised), U.S. Army Engineer Topographic Laboratories, Fort Belvoir, VA Rueger, J. M. (989). Electronic istance Measurement, Springer-Verlag. Rueger, J. M. (98). Practical Results of EM-Height Traversing, The Australian Surveyor, Vol. 30, No. 6. Rueger, J. M. (98). EM-Height Traversing versus Geodetic Leveling, The Canadian Surveyor, Vol. 36, No.. Krakiwsky, Edward and Vanicek, Petr (986). Geodesy: The Concepts, North- Holland Torge, Wolfgang (980). Geodesy, Walter de Gruyter Vicenty, T. (986). Geometric Reduction of Measured Lines, Surveying and Mapping, Vol. 46, No. 3 Burkholder, Earl F. (008). The 3- Global Spatial ata Model, CRC Press