1 Geodesy for Neutrino Physicists by Wes Smart, Fermilab Based on: “GPS Satellite Surveying” By Alfred Leick, Wiley (1990 ) Geodesy : a branch of applied mathematics that determines the exact positions of points and figures and areas of large portions of the earth’s surface, the shape and size of the earth, and the variations of terrestrial gravity and magnetism. Or: what’s needed beyond the Flat Earth Society

2 Outline Ellipsoid model of the earth Three geodetic coordinate systems and the.. transformations between them Method of calculation Excel spreadsheet to do these transformations Examples (in excel): Chicago – Barcelona, NuMI Height above sea level, geoid, geoid height Summary

3 Earth Modeled by Reference Ellipse Spin Causes Larger Diameter at Equator than at Poles a=semi-major axis= m b=semi-minor axis= f=flattening= 1/ e=eccentricity=( ) 0.5 f=(a-b)/a e 2 =2f-f 2 =1-(b/a) 2 a-b= m b a GRS 80 (Geodetic Reference System) = Ellipse parameters in NAD 83 (North American Datum)

4 The Geodetic and Geocentric Cartesian Coordinate Systems  N h P xy z x y Surface Normal Meridian Looking from above EquatorLooking from above North Pole z is the spin axis  is latitude is longitude x=(N+h)cos  cos y=(N+h)cos  sin z=[(N(1-e 2 )+h]sin  P N=a/(1-e 2 sin 2  ) 0.5 e 2 =1-(b/a) 2 Greenwich + - North pole + - North South East West + East West - (Not Origin)

5 Local Geodetic Coordinates up P 1 ( ,h) xy z Normal to Ellipsoid Looking from above Equator z is the spin axis  is latitude is longitude A second point P 2 relative to P 1 is given by: n=-(x 2 -x 1 )sin  cos -(y 2 -y 1 )sin  sin +(z 2 -z 1 )cos  e=-(x 2 -x 1 )sin +(y 2 -y 1 )cos u=(x 2 -x 1 )cos  cos +(y 2 -y 1 )cos  sin  (z 2 -z 1 )sin  North pole north east Specified for a point P 1, Cartesian up is along the normal to Ellipsoid north is the intersection of the plane perpendicular to the normal containing P 1 and the plane containing the z (spin) axis and P 1 east = the cross product: north x up Into screen h

6 Compare Coordinate Systems SystemCoordinatesRangeCartesian/Familiarity Easy Calcs ?. GeodeticLatitudeglobalnomedium Longitude Ellipsoidal ht. Geocentricx, y, zglobalyeslow Cartesian Localnorth,localyeshigh Geodeticeast, up

7 Calculation Method Get Geodetic coordinates of points: may need to find ellipsoidal heights from elevations Use Spreadsheet to find Geocentric Cartesian coordinates Do desired calculations in the Geocentric Cartesian coordinate system (which you already know how to do) If needed, use the inverse transformation to calculate Geodetic coordinates of results

8 Azimuth Example Chicago to Barcelona up xy z Normal to Ellipsoid Looking from above Equator North pole north east Into screen Looking from above North Pole Dashed lines are not in the plane y x Chicago ncnc ecec nbnb ebeb Barcelona Plane of right plot These 2 cities are both at 42 o N Latitude and 90 o apart in Longitude. Beam must leave Chicago north of east and would arrive in Barcelona from north of west. These directions are not 180 o apart because east is a different direction in each city. (This is also true for north and up.) This applies as well for an airplane on the great circle route between the two cities.

9 Spreadsheet Results; Chicago to Barcelona Part AGeodeticCoordinatesGeocentricCartesianCoordinates A1Ellipsoid heightLatitude 0 Longitude 0 xyz "Chicago"NAD A2 "Barcelona"NAD Angles A1 to A2ddxdydz azimuthvertical radLocal Geodetic Coordinates of A2; ref A degdndedu Part BGeodeticCoordinatesGeocentricCartesianCoordinates B1Ellipsoid heightLatitude 0 Longitude 0 xyz "Barcelona"NAD B2 "Chicago"NAD Angles B1 to B2ddxdydz azimuthvertical radLocal Geodetic Coordinates of B2; ref B degdndedu All lengths in meters

10 Spreadsheet Results; NuMI Target to Far Part AGeodeticCoordinatesGeocentricCartesianCoordinates A1Ellipsoid heightLatitude 0 Longitude 0 xyz ACTRN1NAD A2 FARctr 2001(NAD83) Angles A1 to A2ddxdydz azimuthvertical radLocal Geodetic Coordinates of A2; ref A degdndedu Part BGeodeticCoordinatesGeocentricCartesianCoordinates B1Ellipsoid heightLatitude 0 Longitude 0 xyz FARctr 2001(NAD83) B2 ACTRN1NAD Angles B1 to B2ddxdydz azimuthvertical radLocal Geodetic Coordinates of B2; ref B degdndedu

11 Spreadsheet Results; MINOS Near to Far Part AGeodeticCoordinatesGeocentricCartesianCoordinates A1Ellipsoid heightLatitude 0 Longitude 0 xyz NearNAD A2 FARctr 2001(NAD83) Angles A1 to A2ddxdydz azimuthvertical radLocal Geodetic Coordinates of A2; ref A degdndedu Part BGeodeticCoordinatesGeocentricCartesianCoordinates B1Ellipsoid heightLatitude 0 Longitude 0 xyz FARctr 2001(NAD83) B2 NearNAD Angles B1 to B2ddxdydz azimuthvertical radLocal Geodetic Coordinates of B2; ref B degdndedu All lengths in meters

12 Spreadsheet “Subroutines” Convert degrees, minutes, seconds to degreesConvert degrees to degrees, minutes, seconds 0 '" 000 '" Enter absolute value input data only in cells with black borders using paste special (value) For negative input data, enter absolute value and append minus sign to result. Find ellipsoid height, h, in meters Linear Interpolation from elevation, H, in feet. h=H+N ParameterResult H (feet)N (m)h (m) Find h, Latitude, Longitude from x,y,z Only change cells with black borders.  Lat k-1 0 h ( m) Longitude 0 x inputy inputz input Lat k  Lat 1  Iterate Latitude equation by copying red border cell to 1. Enter x,y,z using paste special (value) black border cell, using paste special (value), 2. Copy "Lat 1" or red border cell to black until the red and black bordered cells agree. border Lat k-1cell using paste special 4. Copy h, Lat, Long to desired locations, using (value) paste special (value),

13 Linear Interpolation Use to find the speadsheet input parameter which gives the desired result for an output value. All data input should be by typing or paste special value. Input only into cells marked for input. Select the input parameter and output result you wish to use, put desired value of result into the answer line of the “subroutine” Guess a value for the parameter, put in spreadsheet, copy parameter and result into line 1 of the “subroutine” Repeat for line 2 Put answer parameter value in spreadsheet, copy it and result into line 1 or 2 (pick the line which has its result further from the desired value). Repeat last step until the speadsheet result has the desired value. Linear Interpolation ParameterResult answer7-70

14 Spreadsheet Results; Offaxis Detector GeodeticCoordinatesGeocentricCartesianCoordinates A1Ellipsoid heightLatitude 0 Longitude 0 xyz ACTRN1NAD A2 10 mradNAD Angles A1 to A2ddxdydz azimuthvertical rad degLocal Geodetic Coordinates of A2; ref A1 dndedu along y' (m)total transverseangle (rad)angle (deg) A2 Relative to NuMI Beam 10 mrad Longitude 0 Angle to B (rad) ParameterResult answer All lengths in meters

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16 Find latitude, longitude, and ellipsoidal height from geocentric Cartesian coordinates x,y,z First approximate solution for  tan  1 =z/[(1-e 2 )(x 2 +y 2 ) 0.5 ] Then find  by iteration tan  =[z+ae 2 sin  /(1-e 2 sin 2  ) 0.5 ]/(x 2 +y 2 ) 0.5 Finally tan =y/x and h=[(x 2 +y 2 ) 0.5 )/cos  ]-N Inverse Transformation

17 Heights Geoid Ellipsoid H N h P H, Orthometric height, is above “sea level”, ie elevation h is the ellipsoidal height, GPS measures in h directly N, the geoid height, is about -32 m at Soudan and Fermilab To calculate N: Geoid is the equipotential surface with gravity potential chosen such that on average it coincides with the global ocean surface. N accounts for the difference between the real earth and the ideal reference ellipsoid used for calculation. N varies with latitude and longitude. h=H+ N

18 Geoid Heights for North America

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20 Summary Earth is modeled well by ellipsoid 3 geodetic coordinate systems  Geodetic: Latitude, Longitude, Ellipsoidal height  Geocentric Cartesian: x, y, z  Local Geodetic: north, east, up Transformations between them with Excel Transform points to Geocentric Cartesian where calculations are easy and familiar If desired, transform answers back to Geodetic Coordinates