Geodesy for Neutrino Physicists by Wes Smart, Fermilab

1
Geodesy for Neutrino Physicistsby 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 earths surface, the shape
and size of the earth, and the variations of
terrestrial gravity and magnetism. Or whats
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
  http//home.fnal.gov/smart/geodesy/calcs.xls
• Examples (in excel) Chicago Barcelona, NuMI
• Height above sea level, geoid, geoid height
• Summary

3
Earth Modeled by Reference EllipseSpin Causes
Larger Diameter at Equator than at Poles

• asemi-major axis6378137 m
• bsemi-minor axis6356752.3141
• fflattening 1/298.25722210
• eeccentricity(0.00669438)0.5
• f(a-b)/a
• e22f-f21-(b/a)2
• a-b 21385 m

b
a
GRS 80 (Geodetic Reference System) Ellipse
parameters in NAD 83 (North American Datum)
4
The Geodetic and Geocentric Cartesian Coordinate
Systems
Looking from above Equator
Looking from above North Pole
z
y
P
North pole
h
Surface Normal
P
N



j
l
Greenwich
North
East
East
x
South
West
West
xy
Meridian
-
-
-
(Not Origin)
z is the spin axis j is latitude l is longitude
x(Nh)cosjcosl y(Nh)cosjsinl z(N(1-e2)hsinj
Na/(1-e2sin2j)0.5 e21-(b/a)2
5
Local Geodetic Coordinates
Looking from above Equator
up
z
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
north
P1(j,l,h)
North pole
h
east
Into screen
Normal to Ellipsoid
xy
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
z is the spin axis j is latitude l is longitude
6
Compare Coordinate Systems
System Coordinates Range Cartesian/ Familiarity
Easy Calcs ?
. Geodetic Latitude global no medium Longitu
de Ellipsoidal ht. Geocentric x, y,
z global yes low Cartesian Local north, loc
al yes high Geodetic east, 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
Looking from above Equator
Looking from above North Pole Dashed lines are
not in the plane
z
Normal to Ellipsoid
y
north
up
North pole
Plane of right plot
east
eb
Into screen
x
Barcelona
nb
xy
nc
Chicago
ec
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.
9
Spreadsheet Results Chicago to Barcelona
10
Spreadsheet Results NuMI Target to Far
11
Spreadsheet Results MINOS Near to Far
12
Spreadsheet Subroutines
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.

14
Spreadsheet Results Offaxis Detector
15
(No Transcript)
16

Inverse Transformation

• Find latitude, longitude, and ellipsoidal height
• from geocentric Cartesian coordinates x,y,z
• First approximate solution for j
  tanj1z/(1-e2)(x2y2)0.5
• Then find j by iteration
  tanjzae2sinj/(1-e2sin2j)0.5/(x2y2)0.5
• Finally tanly/x and
  h(x2y2)0.5)/cosj-N

17
Heights
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.
P
H
h
Geoid
N
Ellipsoid
hHN
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 http//www.ngs.noaa.gov/GEOID/GEOID03/download
.html
18
Geoid Heights for North America
19
(No Transcript)
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