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Published on February 12, 2008

Author: Stella


Geodesy for Neutrino Physicists by Wes Smart, Fermilab :  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 Outline :  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 Earth Modeled by Reference Ellipse Spin Causes Larger Diameter at Equator than at Poles:  Earth Modeled by Reference Ellipse Spin Causes Larger Diameter at Equator than at Poles a=semi-major axis=6378137 m b=semi-minor axis=6356752.3141 f=flattening= 1/298.25722210 e=eccentricity=(0.00669438)0.5 f=(a-b)/a e2=2f-f2=1-(b/a)2 a-b= 21385 m b a GRS 80 (Geodetic Reference System) = Ellipse parameters in NAD 83 (North American Datum) The Geodetic and Geocentric Cartesian Coordinate Systems:  The Geodetic and Geocentric Cartesian Coordinate Systems j N h P xy z l x y Surface Normal Meridian Looking from above Equator Looking from above North Pole z is the spin axis j is latitude l is longitude x=(N+h)cosjcosl y=(N+h)cosjsinl z=[(N(1-e2)+h]sinj P N=a/(1-e2sin2j)0.5 e2=1-(b/a)2 Greenwich + - North pole + - North South East West + East West - (Not Origin) Local Geodetic Coordinates:  Local Geodetic Coordinates up P1(j,l,h) xy z Normal to Ellipsoid Looking from above Equator z is the spin axis j is latitude l is longitude A second point P2 relative to P1 is given by: n=-(x2-x1)sinjcosl-(y2-y1)sinjsinl+(z2-z1)cosj e=-(x2-x1)sinl+(y2-y1)cosl u=(x2-x1)cosjcosl+(y2-y1)cosjsinl+(z2-z1)sinj North pole north east Specified for a point P1, Cartesian up is along the normal to Ellipsoid north is the intersection of the plane perpendicular to the normal containing P1 and the plane containing the z (spin) axis and P1 east = the cross product: north x up Into screen h Compare Coordinate Systems:  Compare Coordinate Systems System Coordinates Range Cartesian/ Familiarity Easy Calcs ? . Geodetic Latitude global no medium Longitude Ellipsoidal ht. Geocentric x, y, z global yes low Cartesian Local north, local yes high Geodetic east, up Calculation Method:  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 Azimuth Example Chicago to Barcelona:  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 nc ec nb eb Barcelona Plane of right plot These 2 cities are both at 42o N Latitude and 90o apart in Longitude. Beam must leave Chicago north of east and would arrive in Barcelona from north of west. These directions are not 180o 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. Spreadsheet Results; Chicago to Barcelona:  Spreadsheet Results; Chicago to Barcelona Spreadsheet Results; NuMI Target to Far:  Spreadsheet Results; NuMI Target to Far Spreadsheet Results; MINOS Near to Far:  Spreadsheet Results; MINOS Near to Far Spreadsheet “Subroutines”:  Spreadsheet “Subroutines” Slide13:  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. Spreadsheet Results; Offaxis Detector:  Spreadsheet Results; Offaxis Detector Slide16:  Find latitude, longitude, and ellipsoidal height from geocentric Cartesian coordinates x,y,z First approximate solution for j tanj1=z/[(1-e2)(x2+y2)0.5] Then find j by iteration tanj=[z+ae2sinj/(1-e2sin2j)0.5]/(x2+y2)0.5 Finally tanl=y/x and h=[(x2+y2)0.5)/cosj]-N Inverse Transformation Heights:  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 Geoid Heights for North America:  Geoid Heights for North America Summary:  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

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