1 geodesy for neutrino physicists by wes smart, fermilab based on: “gps satellite surveying” by...

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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 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

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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

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Earth Modeled by Reference EllipseSpin 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)

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The Geodetic and Geocentric Cartesian Coordinate Systems

N

hP

xy

z

x

y

Surface Normal

Meridian

Looking from above Equator Looking from above North Pole

z is the spin axis

is latitude

is longitude

x=(N+h)coscos

y=(N+h)cossin

z=[(N(1-e2)+h]sin

P

N=a/(1-e2sin2)0.5

e2=1-(b/a)2

Greenwich+

-

North pole

+

-

North

South

East

West

+East

West

-(Not Origin)

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Local Geodetic Coordinates

up

P1(,h)

xy

z

Normal to

Ellipsoid

Looking from above Equator

z is the spin axis

is latitude

is longitude

A second point P2 relative to P1 is given by:

n=-(x2-x1)sincos-(y2-y1)sinsin+(z2-z1)cos

e=-(x2-x1)sin+(y2-y1)cos

u=(x2-x1)coscos+(y2-y1)cossin(z2-z1)sin

North polenorth

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

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Compare Coordinate SystemsSystem 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

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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

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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

ebBarcelona

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.

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Spreadsheet Results; Chicago to Barcelona

Part A Geodetic Coordinates Geocentric Cartesian Coordinates

A1 Ellipsoid height Latitude 0 Longitude 0 x y z

"Chicago" NAD 83 0.0000 42.0000000 -88.0000000 165667.8748 -4744107.2248 4245603.8360

A2

"Barcelona" NAD83 0.0000 42.0000000 2.0000000 4744107.2248 165667.8748 4245603.8360

Angles A1 to A2 d dx dy dz

azimuth vertical 6713270.3216 4578439.3500 4909775.0997 0.0000

0.98108986 -0.55328241 rad Local Geodetic Coordinates of A2; ref A1

56.21230819 -31.7007471 deg dn de du

3176362.2981 4746998.9683 -3527707.7199

Part B Geodetic Coordinates Geocentric Cartesian Coordinates

B1 Ellipsoid height Latitude 0 Longitude 0 x y z

"Barcelona" NAD83 0.0000 42.0000000 2.0000000 4744107.2248 165667.8748 4245603.8360

B2

"Chicago" NAD 83 0.0000 42.0000000 -88.0000000 165667.8748 -4744107.2248 4245603.8360

Angles B1 to B2 d dx dy dz

azimuth vertical 6713270.3216 -4578439.3500 -4909775.0997 0.0000

-0.98108986 -0.55328241 rad Local Geodetic Coordinates of B2; ref B1

303.7876918 -31.7007471 deg dn de du

All lengths in meters 3176362.2981 -4746998.9683 -3527707.7199

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Spreadsheet Results; NuMI Target to FarPart A Geodetic Coordinates Geocentric Cartesian Coordinates


ACTRN1 NAD 83 153.9683 41.8320235 -88.2655587 144058.5523 -4757396.9876 4231823.0745

A2

FARctr 2001 (NAD83) -248.3992 47.8202665 -92.2414120 -167796.9924 -4287098.7216 4703296.8722


azimuth vertical 735337.9371 -311855.5447 470298.2661 471473.7977

-0.41723537 -0.05829776 rad Local Geodetic Coordinates of A2; ref A1

336.0941743 -3.3402156 deg dn de du

671113.2727 -297478.1082 -42844.2768



FARctr 2001 (NAD83) -248.3992 47.8202665 -92.2414120 -167796.9924 -4287098.7216 4703296.8722

B2

ACTRN1 NAD 83 153.9683 41.8320235 -88.2655587 144058.5523 -4757396.9876 4231823.0745


azimuth vertical 735337.9371 311855.5447 -470298.2661 -471473.7977

2.67536239 -0.05718496 rad Local Geodetic Coordinates of B2; ref B1

153.2869734 -3.2764567 deg dn de du

-655781.0335 330010.3321 -42027.3543

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Spreadsheet Results; MINOS Near to FarPart A Geodetic Coordinates Geocentric Cartesian Coordinates


Near NAD 83 93.4971 41.8405633 -88.2706209 143617.7785 -4756732.2723 4232489.4513

A2

FARctr 2001 (NAD83) -248.3992 47.8202665 -92.2414120 -167796.9924 -4287098.7216 4703296.8722



-0.41729429 -0.05813482 rad Local Geodetic Coordinates of A2; ref A1

336.0907986 -3.3308801 deg dn de du

670153.5882 -297099.9605 -42664.2796



FARctr 2001 (NAD83) -248.3992 47.8202665 -92.2414120 -167796.9924 -4287098.7216 4703296.8722

B2

Near NAD 83 93.4971 41.8405633 -88.2706209 143617.7785 -4756732.2723 4232489.4513


azimuth vertical 734298.6171 311414.7709 -469633.5508 -470807.4209

2.67536239 -0.05718496 rad Local Geodetic Coordinates of B2; ref B1

153.2869734 -3.2764567 deg dn de du

All lengths in meters -654854.1585 329543.8985 -41967.9533

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Spreadsheet “Subroutines”Convert degrees, minutes, seconds to degrees Convert degrees to degrees, minutes, seconds

0 ' " 0 0 0 ' "

41 49 55.2846 41.83202351 41.83202351 41 49 55.284625

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 Parameter Result

H (feet) N (m) h (m) 1 0 0

1210 -32.0 336.8080 2 -10 100

7 -70

Find h, Latitude, Longitude from x,y,z Only change cells with black borders.

Lat k-1 0

h ( m) 41.83202351 Longitude 0 x input y input z input

153.9683 41.83202351 -88.26555873 144058.5523 -4757396.98764231823.074

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Lat k 41.83202351 0

Lat 1 41.83202813 0 3. 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),

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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

Parameter Result

1 0 0

2 -10 100

answer 7 -70

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Spreadsheet Results; Offaxis Detector

Geodetic Coordinates Geocentric Cartesian Coordinates


ACTRN1 NAD 83 153.9683 41.8320235 -88.2655587 144058.5523 -4757396.9876 4231823.0745

A2

10 mrad NAD83 305.0000 47.1500000 -91.8152727 -137659.0444 -4343503.4468 4653345.3173



-0.42425381 -0.05116410 rad

335.6920471 -2.9314871 deg Local Geodetic Coordinates of A2; ref A1

dn de du

595683.9779 -269061.1748 -33471.6487

along y' (m) total transverse angle (rad) angle (deg)

A2 Relative to NuMI Beam

10 mrad 654454.6086 6544.7643 0.010000 500635.6119 -6054.0536 2486.4382 0.5730

Longitude 0 Angle to B (rad)

-91.8152727 0.010000

Parameter Result

1 -91.81527 0.010000

2 -91.81500 0.009981

answer -91.81527266 0.010000 All lengths in meters

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Lines of Constant Angle from the NuMI Beam (at 1106' elevation)

46

47

48

49

50

-97 -96 -95 -94 -93 -92 -91 -90 -89

(W) Longitude (deg)

N L

ati

tud

e (

de

g)

15 mrad

10 mrad

Beam Axis

Far Detector

At Surface

100 km

100 km

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Find latitude, longitude, and ellipsoidal heightfrom geocentric Cartesian coordinates x,y,z

First approximate solution for tan1=z/[(1-e2)(x2+y2)0.5]

Then find by iteration tan=[z+ae2sin/(1-e2sin2)0.5]/(x2+y2)0.5

Finally tan=y/x and h=[(x2+y2)0.5)/cos]-N

Inverse Transformation

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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: http://www.ngs.noaa.gov/GEOID/GEOID03/download.html

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

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Geoid Heights for North America

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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

1 geodesy for neutrino physicists by wes smart, fermilab based on: “gps satellite surveying” by...

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