Making maps compatible with GPS - Ordnance Survey · PDF fileMaking maps compatible with GPS....

OrdnanceSurveyIreland

Making maps compatible with GPS

Colin Bray

Survey Ireland 99, Malahide, 25 May 1999


Transformations between The Irish Grid and the GPS Co-ordinate Reference Frame

WGS84 / ETRF89

Geodetic Reference Systems


• A point on the ground does not have a unique Latitude (φ ) and Longitude (λ ).

• The true co-ordinates of a ground point change with time.

• Therefore we need a transformation to relate one reference system to another.


• Cartesian– X, Y, Z

••• Geodetic Ellipsoidal (Geographic)Geodetic Ellipsoidal (Geographic)Geodetic Ellipsoidal (Geographic)––– Latitude, Longitude, Ht. Above Ref. EllipsoidLatitude, Longitude, Ht. Above Ref. EllipsoidLatitude, Longitude, Ht. Above Ref. Ellipsoid

••• Plane (Grid)Plane (Grid)Plane (Grid)––– Eastings, Northings, Elevation (above ref. Datum)Eastings, Northings, Elevation (above ref. Datum)Eastings, Northings, Elevation (above ref. Datum)

Pole

Z axis

Greenwich Meridian(0 degrees Longitude)

X axis

Y axis

Equator

Cartesian Reference System


••• CartesianCartesianCartesian––– X, Y, ZX, Y, ZX, Y, Z

• Geodetic Ellipsoidal (Geographic)– Latitude, Longitude, Ht. Above Ref. Ellipsoid

••• Plane (Grid)Plane (Grid)Plane (Grid)––– Eastings, Northings, Elevation (above ref. Datum)Eastings, Northings, Elevation (above ref. Datum)Eastings, Northings, Elevation (above ref. Datum)

Pole

Greenwhich Meridian(0 degrees Longitude)

Equator

LatLong

P

Ellipsoidal Reference System


••• CartesianCartesianCartesian––– X, Y, ZX, Y, ZX, Y, Z

••• Geodetic Ellipsoidal (Geographic)Geodetic Ellipsoidal (Geographic)Geodetic Ellipsoidal (Geographic)––– Latitude, Longitude, Ht. Above Ref. EllipsoidLatitude, Longitude, Ht. Above Ref. EllipsoidLatitude, Longitude, Ht. Above Ref. Ellipsoid

• Plane (Grid)– Eastings, Northings, Elevation (above ref. Datum)

Plane Co-ordinates

True Origin

False Origin

Eastings

Northings

Geodetic Co-ordinate Reference Systems

• Irish Grid Reference System– Ireland 1975 Mapping Adjustment

• GPS Reference systems– WGS84– ITRF96– ETRF89– IRENET95

IRENET95 Reference Frame

D106

D108 D109

D112D111D113

D115

D128

D101

D100Z009

D097 D098

D103

D110

D099

D104

D117

D118

D129

D119D122

D123

D121

D124

D126

D125D171D168D169

D166

D170

D093

D091 D092

D095D096

D089

D120

D159

D162

D164

D161

D084D087 D088

D131D157

D154D152 D153

D158

D074 D075

D079D080

D076

D081

D086

Z006

D082

D090D141

D173

D140

D142

Z007D148D151D147

D150

D067

D066

D072

Z005D068

D073D070

D071D135

D137

D062

D058

D063 D064D065

D048

D134D133

D055

D059 D060 D046

D030

D031

D044

D032

D047

D043D042

D041

D034

D052

D035

D040

D053D054

D038

D013D028

D014

D027D026

D016D017 D018

D023 D022

Z002

D021

D020

D037

D001

D002D003

D011

Z001

D004

D009D005

D008

D006

D107

D114

D116D105

D102

D127 D130Z010

D165

D172

Z011 D167

D094 D132

D160

D163

D083

Z008

D085 D155D156

D078

D077D143

D149

D144D145D146

D069D136

D138D139

D061 D050Z004D051

D029

D045Z003

D049

D033 D036D039

D015

D025D024

D019

D056

D012 D010D007

Relationship between Ireland 1975 and GPS Reference Systems

IRELAND

AIRY MODIFIED

GRS80

ELLIPSE

ELLIPSE

Transformations between Ireland 1975 Reference System & ETRF89 (GPS) Reference System

• Level 1 Transformation– Easting and Northing Shifts– Accuracy: Within 2 m 95% overall

• Level 2 Transformation– Helmert 7 Parameter– Accuracy: Within 1 m 95 % overall

Level 1 Transformation


• Easting and Northing Shift

dE +49.0 m to the EastingdN - 23.4 m to the Northing

ETRF89 Latitude & Longitude positions on the GRS80 reference ellipsoid are projected into Eastings & Northings using Transverse Mercator projection, with Irish Grid parameters to give GPS (Irish Grid) Co-ordinates. Easting and Northing shifts applied to give Irish Grid Co-ordinates.

Level 1 Comparison Irish Grid and GPS

49.0 m49.0 m

23.4 m

54.3 m

ETRF89 position

Ireland 1975 position

N

Level 1 (Forward Case)

Step 1:Step 1:Step 1:••• Irish Grid CoIrish Grid CoIrish Grid Co---ords ords ords --- GPS (Irish Grid) CoGPS (Irish Grid) CoGPS (Irish Grid) Co---ordsordsords

EEE111 ,N,N,N111 →→→ EEE222 , N, N, N222

Step 2:Step 2:Step 2:••• GPS (Irish Grid) GPS (Irish Grid) GPS (Irish Grid) --- ETRF89 Geodetic Ellipsoidal ETRF89 Geodetic Ellipsoidal ETRF89 Geodetic Ellipsoidal

CoCoCo---ordsordsordsEEE222 , N, N, N2 2 2 →→→ LatLatLat222 , Long, Long, Long222

Transformation from Irish Grid Co-ords to ETRF89 (GPS) Geodetic Ellipsoidal Co-ords


Step 1:• Irish Grid Co-ords - GPS (Irish Grid) Co-ords

E1 ,N1 → E2 , N2

Step 2:Step 2:Step 2:••• GPS (Irish Grid) GPS (Irish Grid) GPS (Irish Grid) --- ETRF89 Geodetic Ellipsoidal ETRF89 Geodetic Ellipsoidal ETRF89 Geodetic Ellipsoidal

CoCoCo---ordsordsordsEEE222 , N, N, N2 2 2 →→→ LatLatLat222 , Long, Long, Long222



Step 1:Step 1:Step 1:••• Irish Grid CoIrish Grid CoIrish Grid Co---ords ords ords --- GPS (Irish Grid) CoGPS (Irish Grid) CoGPS (Irish Grid) Co---ordsordsords

EEE111 ,N,N,N111 →→→ EEE222 , N, N, N222

Step 2:• GPS (Irish Grid) - ETRF89 Geodetic Ellipsoidal

Co-ordsE2 , N2 → Lat2 , Long2


Level 1 (Forward Case)Step 1: Irish Grid Co-ords - GPS (Irish Grid) Co-ords

Irish Grid (East1 , North1 ) 271,707.4 248,879.6Shifts - 49.0 + 23.4GPS (Irish Grid) 271,658.4 248,903.0

Step 2: GPS (Irish Grid) - ETRF89 (WGS84) Co-ords

Latitude (φ2 ) Longitude (λ2 )

53°

29’ 06”.96840 - 6°

55’ 13”.92478

Level 1 (Reverse Case)

Step 1:Step 1:Step 1:••• ETRF89 Geodetic Ellipsoidal ETRF89 Geodetic Ellipsoidal ETRF89 Geodetic Ellipsoidal --- GPS (Irish Grid) GPS (Irish Grid) GPS (Irish Grid)

CoCoCo---ordsordsordsLatLatLat222 , Long, Long, Long222 →→→ EEE222 , N, N, N222

Step 2:Step 2:Step 2:••• GPS (Irish Grid) CoGPS (Irish Grid) CoGPS (Irish Grid) Co---ords ords ords --- Irish Grid CoIrish Grid CoIrish Grid Co---ordsordsords

EEE222 , N, N, N2 2 2 →→→ EEE111 ,N,N,N111

Transformation from ETRF89 (GPS) Geodetic Ellipsoidal Co-ords to Irish Grid Co-ords


Step 1:• ETRF89 Geodetic Ellipsoidal - GPS (Irish Grid)

Co-ordsLat2 , Long2 → E2 , N2

Step 2:Step 2:Step 2:••• GPS (Irish Grid) CoGPS (Irish Grid) CoGPS (Irish Grid) Co---ords ords ords --- Irish Grid CoIrish Grid CoIrish Grid Co---ordsordsords

EEE222 , N, N, N2 2 2 →→→ EEE111 ,N,N,N111



Step 1:Step 1:Step 1:••• ETRF89 Geodetic Ellipsoidal ETRF89 Geodetic Ellipsoidal ETRF89 Geodetic Ellipsoidal --- GPS (Irish Grid) GPS (Irish Grid) GPS (Irish Grid)

CoCoCo---ordsordsordsLatLatLat222 , Long, Long, Long222 →→→ EEE222 , N, N, N222

Step 2:• GPS (Irish Grid) Co-ords - Irish Grid Co-ords

E2 , N2 → E1 ,N1


Level 1 (Reverse Case)Step 1: ETRF89 (WGS84) - GPS (Irish Grid) Co-ords

ETRF89 (φ2 , λ2 ) 53°

29’ 06”.96840 - 6°

55’ 13”.92478GPS (Irish Grid) 271,658.4 248,903.0

Step 2: Irish Grid Co-ords - GPS (Irish Grid) Co-ords

GPS (Irish Grid) 271,658.4 248,903.0Shifts + 49.0 - 23.4Irish Grid (East1 , North1 ) 271,707.4 248,879.6

Level 2 Transformation• Helmert 7 parameter Transformation

– 3 Translations (ΔX , ΔY , ΔZ )– 3 Rotations (θ x , θ y , θ z )– 1 Scale change (ppm)

• Carried out between two earth centred Cartesian reference systems– Ireland 75 (X1 , Y1 , Z1 )– ETRF89 (X2 , Y2 , Z2 )

Helmert 7 Parameter Transformation

Translations Rotations

ΔX (m) +482.530 θx (”) +1.042

ΔY (m) -130.596 θy (”) +0.214

ΔZ (m) +564.557 θz (”) +0.631

Scale (ppm) +8.150


Step 1Step 1Step 1: Irish Grid : Irish Grid : Irish Grid --- Ireland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalStep 2Step 2Step 2: Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal --- Ireland 75 CartesianIreland 75 CartesianIreland 75 CartesianStep 3Step 3Step 3: Ireland 75 Cartesian : Ireland 75 Cartesian : Ireland 75 Cartesian --- ETRF89 (WGS84) CartesianETRF89 (WGS84) CartesianETRF89 (WGS84) CartesianStep 4Step 4Step 4: ETRF89 Cartesian : ETRF89 Cartesian : ETRF89 Cartesian --- ETRF89 (WGS84) Geodetic EllipsoidalETRF89 (WGS84) Geodetic EllipsoidalETRF89 (WGS84) Geodetic Ellipsoidal

EEE111 ,N,N,N1 1 1 →→→ LatLatLat111 , Long, Long, Long111 →→→ XXX111 ,Y,Y,Y111 ,Z,Z,Z111 →→→ TTT →→→ XXX222 ,Y,Y,Y222 ,Z,Z,Z222 →→→ LatLatLat222 , Long, Long, Long222

From Irish Grid Co-ords (Easting1 , Northing1 ) to ETRF89 (WGS84) Geodetic Ellipsoidal Co-ords (Latitude12 , Longitude2 ).


Step 1: Irish Grid - Ireland 75 Geodetic EllipsoidalStep 2Step 2Step 2: Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal --- Ireland 75 CartesianIreland 75 CartesianIreland 75 CartesianStep 3Step 3Step 3: Ireland 75 Cartesian : Ireland 75 Cartesian : Ireland 75 Cartesian --- ETRF89 (WGS84) CartesianETRF89 (WGS84) CartesianETRF89 (WGS84) CartesianStep 4Step 4Step 4: ETRF89 Cartesian : ETRF89 Cartesian : ETRF89 Cartesian --- ETRF89 (WGS84) Geodetic EllipsoidalETRF89 (WGS84) Geodetic EllipsoidalETRF89 (WGS84) Geodetic Ellipsoidal

E1 ,N1 → Lat1 , Long1 →→→ XXX111 ,Y,Y,Y111 ,Z,Z,Z111 →→→ TTT →→→ XXX222 ,Y,Y,Y222 ,Z,Z,Z222 →→→ LatLatLat222 , Long, Long, Long222



Step 1Step 1Step 1: Irish Grid : Irish Grid : Irish Grid --- Ireland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalStep 2: Ireland 75 Geodetic Ellipsoidal - Ireland 75 CartesianStep 3Step 3Step 3: Ireland 75 Cartesian : Ireland 75 Cartesian : Ireland 75 Cartesian --- ETRF89 (WGS84) CartesianETRF89 (WGS84) CartesianETRF89 (WGS84) CartesianStep 4Step 4Step 4: ETRF89 Cartesian : ETRF89 Cartesian : ETRF89 Cartesian --- ETRF89 (WGS84) Geodetic EllipsoidalETRF89 (WGS84) Geodetic EllipsoidalETRF89 (WGS84) Geodetic Ellipsoidal

EEE111 ,N,N,N1 1 1 →→→ Lat1 , Long1 → X1 ,Y1 ,Z1 →→→ TTT →→→ XXX222 ,Y,Y,Y222 ,Z,Z,Z222 →→→ LatLatLat222 , Long, Long, Long222



Step 1Step 1Step 1: Irish Grid : Irish Grid : Irish Grid --- Ireland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalStep 2Step 2Step 2: Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal --- Ireland 75 CartesianIreland 75 CartesianIreland 75 CartesianStep 3: Ireland 75 Cartesian - ETRF89 (WGS84) CartesianStep 4Step 4Step 4: ETRF89 Cartesian : ETRF89 Cartesian : ETRF89 Cartesian --- ETRF89 (WGS84) Geodetic EllipsoidalETRF89 (WGS84) Geodetic EllipsoidalETRF89 (WGS84) Geodetic Ellipsoidal

EEE111 ,N,N,N1 1 1 →→→ LatLatLat111 , Long, Long, Long111 →→→ X1 ,Y1 ,Z1 → T → X2 ,Y2 ,Z2 →→→ LatLatLat222 , Long, Long, Long222



Step 1Step 1Step 1: Irish Grid : Irish Grid : Irish Grid --- Ireland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalStep 2Step 2Step 2: Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal --- Ireland 75 CartesianIreland 75 CartesianIreland 75 CartesianStep 3Step 3Step 3: Ireland 75 Cartesian : Ireland 75 Cartesian : Ireland 75 Cartesian --- ETRF89 (WGS84) CartesianETRF89 (WGS84) CartesianETRF89 (WGS84) CartesianStep 4: ETRF89 Cartesian - ETRF89 (WGS84) Geodetic Ellipsoidal

EEE111 ,N,N,N1 1 1 →→→ LatLatLat111 , Long, Long, Long111 →→→ XXX111 ,Y,Y,Y111 ,Z,Z,Z111 →→→ TTT →→→ X2 ,Y2 ,Z2 → Lat2 , Long2



Step 1: Irish Grid - Ireland 75 Geodetic Ellipsoidal

Easting1 = 271,707.427 mNorthing1 = 248,879.641 m

φ1 = 53º 29’ 06”.17996λ1 = - 6º 55’ 10”.77000

Level 2 (Forward Case)Step 2: Irl 75 Geodetic Ellipsoidal - Irl 75 Cartesian

2/11

22 )sin1( φeav

−=

111 coscos)( λφhvX +=

111 sincos)( λφhvY +=

12

1 sin))1(( φhevZ +−=

= 6,391,123.911 m

= 3,775,226.258 m

= - 458,166.888 m

= 5,102,293.084 m

h = 0.000 m

Level 2 (Forward Case)Step 3: Ireland 75 Cartesian - ETRF89 Cartesian

• Helmert 7 parameter transformation

θ x = 1”.042 = 0.0000050518 Radiansθ y = 0”.214 = 0.0000010375 Radiansθ z = 0”.631 = 0.0000030592 Radians

XYZ

XYZ

XYZ

z y

z x

y x

2

2

2

1

1

1

11

1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

+ −− +

− +

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

μ θ θθ μ θθ θ μ

ΔΔΔ


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

557.564596.130

530.482

084465.5102293888768.458166258140.3775226

0000081500.10000050518.00000010375.00000050518.00000081500.10000030592.00000010375.00000030592.00000081500.1

2

2

2

ZYX

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

456504.5102905992351.458286860986.3775732

2

2

2

ZYX

X2 = 3,775,732.860986 mY2 = - 458,286.992351 mZ2 = 5,102,905.456504 m

Level 2 (Forward Case)Step 4: ETRF89 Cartesian - ETRF89 Geodetic Ellipsoidal

2

212 tan

XY−=λ

2/12

22 )sin1( φeav

−=

2/122

22

22

212 )(

sintan

YXveZ

+

+= − φ

φ

= - 6º 55’ 13”.92595

No.1 ν

= 6,391,971.872757 m No.3 ν

= 6,391,971.950556 mφ2 = 53º.4852663624 φ2 = 53º.4852668787

No.2 ν

= 6,391,971.950371 m No.4 ν

= 6,391,971.950557 mφ2 = 53º.4852668774 φ2 = 53º.4852668787

φ2 = 53º 29’ 06”.96076


Step 1Step 1Step 1: ETRF89 (WGS84) Geodetic Ellipsoidal : ETRF89 (WGS84) Geodetic Ellipsoidal : ETRF89 (WGS84) Geodetic Ellipsoidal --- ETRF89 CartesianETRF89 CartesianETRF89 CartesianStep 2Step 2Step 2: ETRF89 (WGS84) Cartesian : ETRF89 (WGS84) Cartesian : ETRF89 (WGS84) Cartesian --- Ireland 75 CartesianIreland 75 CartesianIreland 75 CartesianStep 3Step 3Step 3: Ireland 75 Cartesian : Ireland 75 Cartesian : Ireland 75 Cartesian --- Ireland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalStep 4Step 4Step 4: Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal --- Irish GridIrish GridIrish Grid

LatLatLat222 , Long, Long, Long222 →→→ XXX222 ,Y,Y,Y222 ,Z,Z,Z222 →→→ TTT →→→ XXX111 ,Y,Y,Y111 ,Z,Z,Z111 →→→ LatLatLat111 , Long, Long, Long111 →→→ EEE111 ,N,N,N111

From ETRF89 (WGS84) Geodetic Ellipsoidal Co-ords (Latitude12 , Longitude2 ) to Irish Grid Co-ords (Easting1 , Northing1 ).


Step 1: ETRF89 (WGS84) Geodetic Ellipsoidal - ETRF89 CartesianStep 2Step 2Step 2: ETRF89 (WGS84) Cartesian : ETRF89 (WGS84) Cartesian : ETRF89 (WGS84) Cartesian --- Ireland 75 CartesianIreland 75 CartesianIreland 75 CartesianStep 3Step 3Step 3: Ireland 75 Cartesian : Ireland 75 Cartesian : Ireland 75 Cartesian --- Ireland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalStep 4Step 4Step 4: Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal --- Irish GridIrish GridIrish Grid

Lat2 , Long2 → X2 ,Y2 ,Z2 →→→ TTT →→→ XXX111 ,Y,Y,Y111 ,Z,Z,Z111 →→→ LatLatLat111 , Long, Long, Long111 →→→ EEE111 ,N,N,N111



Step 1Step 1Step 1: ETRF89 (WGS84) Geodetic Ellipsoidal : ETRF89 (WGS84) Geodetic Ellipsoidal : ETRF89 (WGS84) Geodetic Ellipsoidal --- ETRF89 CartesianETRF89 CartesianETRF89 CartesianStep 2: ETRF89 (WGS84) Cartesian - Ireland 75 CartesianStep 3Step 3Step 3: Ireland 75 Cartesian : Ireland 75 Cartesian : Ireland 75 Cartesian --- Ireland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalStep 4Step 4Step 4: Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal --- Irish GridIrish GridIrish Grid

LatLatLat222 , Long, Long, Long222 →→→ X2 ,Y2 ,Z2 → T → X1 ,Y1 ,Z1 →→→ LatLatLat111 , Long, Long, Long111 →→→ EEE111 ,N,N,N111



Step 1Step 1Step 1: ETRF89 (WGS84) Geodetic Ellipsoidal : ETRF89 (WGS84) Geodetic Ellipsoidal : ETRF89 (WGS84) Geodetic Ellipsoidal --- ETRF89 CartesianETRF89 CartesianETRF89 CartesianStep 2Step 2Step 2: ETRF89 (WGS84) Cartesian : ETRF89 (WGS84) Cartesian : ETRF89 (WGS84) Cartesian --- Ireland 75 CartesianIreland 75 CartesianIreland 75 CartesianStep 3: Ireland 75 Cartesian - Ireland 75 Geodetic EllipsoidalStep 4Step 4Step 4: Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal : Ireland 75 Geodetic Ellipsoidal --- Irish GridIrish GridIrish Grid

LatLatLat222 , Long, Long, Long222 →→→ XXX222 ,Y,Y,Y222 ,Z,Z,Z222 →→→ TTT →→→ X1 ,Y1 ,Z1 → Lat1 , Long1 →→→ EEE111 ,N,N,N111



Step 1Step 1Step 1: ETRF89 (WGS84) Geodetic Ellipsoidal : ETRF89 (WGS84) Geodetic Ellipsoidal : ETRF89 (WGS84) Geodetic Ellipsoidal --- ETRF89 CartesianETRF89 CartesianETRF89 CartesianStep 2Step 2Step 2: ETRF89 (WGS84) Cartesian : ETRF89 (WGS84) Cartesian : ETRF89 (WGS84) Cartesian --- Ireland 75 CartesianIreland 75 CartesianIreland 75 CartesianStep 3Step 3Step 3: Ireland 75 Cartesian : Ireland 75 Cartesian : Ireland 75 Cartesian --- Ireland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalIreland 75 Geodetic EllipsoidalStep 4: Ireland 75 Geodetic Ellipsoidal - Irish Grid

LatLatLat222 , Long, Long, Long222 →→→ XXX222 ,Y,Y,Y222 ,Z,Z,Z222 →→→ TTT →→→ XXX111 ,Y,Y,Y111 ,Z,Z,Z111 →→→ Lat1 , Long1 → E1 ,N1


Level 2 (Reverse Case)Step 1: ETRF89 Geodetic Ellipsoidal - ETRF89 Cartesian

φ2 = 53º 29’ 06”.96076 h2 = 125.355 mλ2 = -6º 55’ 13”.92595

2/12

22 )sin1( φeav

−=

222 coscos)( λφhvX +=

222 sincos)( λφhvY +=

22

2 sin))1(( φhevZ +−=

= 6,391,971.950 m

3,775,774.923 m

= - 458,292.097 m

= 5,102,962.686 m

Level 2 (Reverse Case)Step 2: ETRF89 Cartesian - Ireland 75 Cartesian

• Helmert 7 parameter transformation

θ x = 1”.042 = 0.0000050518 Radiansθ y = 0”.214 = 0.0000010375 Radiansθ z = 0”.631 = 0.0000030592 Radians

XYZ

XYZ

XYZ

z y

z x

y x

1

1

1

2

2

2

11

1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

− −− −

− −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

μ θ θθ μ θθ θ μ

ΔΔΔ


X1 = 3,775,268.317 mY1 = - 458,171.994 mZ1 = 5,102,350.308 m

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

557.564596.130

530.482

686942.5102962097739.458292923481.3775774

9999918500.00000050518.00000010375.00000050518.09999918500.00000030592.0

0000010375.00000030592.09999918500.0

1

1

1

ZYX

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

308243.5102350994840.458171317241.3775268

1

1

1

ZYX


Step 4: Ireland 75 Geodetic Ellipsoidal - Irish Grid

φ1 = 53º 29’ 06”.17992λ1 = -6º 55’ 10”.77009

Easting1 = 271,707.425 mNorthing1 = 248,879.640 m

Transformation Reversibility

• To maintain accuracy– The same level of transformation that was

originally used to transform a position forward should be used to transform that position back again.

The Next Step

GPS Mapping

• Reference Ellipsoid– GRS80

• Geodetic Co-ordinate Reference Frame– ETRF89 (IRENET95)

• Projection to be associated with GPS mapping system– Various projections under consideration

Projections for GPS Mapping

• Universal Transverse Mercator (UTM)– Scale factor: 0.999 600

• Modified Transverse Mercator– Scale factor: 0.999 820

Height

• GPS height measurements are related to the GRS80 Reference Ellipsoid.

• Knowledge of the Geoid is required to convert GPS ellipsoidal heights into orthometric heights.

• Work has started into the development of a Geoid Model for Ireland.

OrdnanceSurveyIreland


C D Bray

Survey Ireland 99, Malahide, 25 May 1999

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