MGA Concepts andGrid Calculations

Geodetic Surveying B

Objectives

Apply fundamental knowledge of MGA to grid calculations

Calculate and apply grid convergence.

Determine grid coordinates of a point given known coordinates of a start point and grid bearing and spheroidal distance from that start point.

Determine grid bearing and spheroidal distance between known points

Overview of Coordinates

There are three aspects to Understanding and Using Coordinates Datum Projections Observations

Datum, Projections and Observations

A “datum” is the underlying basis for coordinate systems

Positions on the datum can be “projected” to create grid coordinates

“Observations” (bearings and distances) in the real world need to be corrected to conform to the datum and projection

Why Coordinates?

The use of a uniform system of coordinates allows spatial information from various sources to be integrated

Increasing requirement for coordination in all types of surveys

At the heart of Australian Spatial Data Infrastructure (ASDI), GIS and GPS

Required in International Standards

Approximation - an Important Underlying Concept

“All exact science is dominated by the idea of approximation” Bertrand Russel

Coordinates are simply a way to approximate the “real world” using a mathematical model

Some models are better approximations than others

Understanding and Using Datums

The Geoid(Mean Sea Level)

Local DatumAGD84

(best fits Australia)

Geocentric Datum(best fit globally)

Ellipsoids and Geoids

AGD - The Old Datum

Terrestrial Observations

Systematic Errors Constrained by

Doppler (transformed)

Distribution Homogeneity Location of Marks

GDA - International Basis

International Terrestrial Reference Frame (ITRF) is a particular “realization” of an idealized reference system...

observation at certain sites and with certain factors in the processing produces...

set of positions and velocities of those sites at a certain time.

reference ellipsoid - GRS80

Link to ITRF by GPS observationsat IGS sites and the Australian National Network (500km). GDA’s link to ITRF makes it compatible with WGS84

GDA and the ITRF

Queensland GDA94 Data Set

QUT1

SUGA

TEXA

MULA

NORM

BREA

BRDV

WILF

WOLL

MUCK

BANZ

BARC

PI EB

HOWI

GREN

OLVE

BASSEMUU

TOWA

Qld 100kmNetwork

Magnitude of Shift

All coordinatesapparently shift inexcess of 200m.

Distortions between Transformed AGD84 and GDA94

Western Qld Central Coast

Types of Coordinates Systems

X

Y

Z

- X+ Y- Z

Cartesian

hGeodetic

Semi-major axis (a)

Semi-minor axis (b)

N

EProjection

Projection Coordinates onGDA and AGD

NMGA

EMGA

Map Grid Australiaon GDA

NAMG

EAMG

Australian Map Gridon AGD

Terminology

GDA94GDA94AGD84AGD84 Latitudes & Longitudes

AMG84AMG84 MGA94MGA94Eastings, Northings& Zone

Universal Universal Transverse Mercator MercatorStd. 6 Degree Zones, with

the same Central Meridians etc.

Understanding and Using Projections

UTM Projection

6 Degree zonesLongitude of Zone

1 : 3 east longitude0.9996 Scale Factor

on Central Meridian

500 000 m false easting10 000 000 m false northing1/2 degree overlap

Ref: Chapter 1. GDA Technical Manual: ICSM Web Site

Zone

54

Cen

tral

Mer

idia

n

Scal

e Fa

ctor

1.0

Scal

e Fa

ctor

0.9

996

AMG/MGA - UTM Projection

Terrain

Surface

Geoid

NEllipsoid

Sca

le F

acto

r 1.

0

Scal

e Fa

ctor

1.0

Sca

le F

acto

r 0.

9996

Sca

le F

acto

r 1.

0006

Scal

e Fa

ctor

1.0

006

Zon

e B

ound

ary

Zon

e B

ound

ary

Cen

tral

Mer

idia

n

Zone 55

Projection Plane

AMG/MGA - UTM Projection

AMG - Redfearn’s Approx (See Study Book)

ER, NR = Rectangular CoordsNote meridian distance (m) = NR

ET, NT = Transverse Mercator CoordsE’, N’ = AMG Coords without false originE, N = AMG Coordinates

GDA94 to MGA94(Redfearn’s Formulae)

Datum Parameters Semi-Major Axis (a) Inverse Flattening (f)

Projection Parameters Longitude of Central

Meridian (Zone) Scale Factor on Central

Meridian False Easting, False

Northing

Input Data Latitude, Longitude & Height

Computed Parameters Radius of Curvature: Meridian Distance: Foot-Point Latitude :

Function (semi-major axis, inverse flattening and latitude)

Output Easting, Northing, Zone, Grid

Conv. , Point Scale Factor

Ref: Chapter 5. GDA Technical Manual: ICSM Web Site

GDA94 - MGA94 (Example)

Easting NorthingMeridian Dist -4,168,963.528

1st term 258,248.359 1st term -4,028.8902nd term 28.781 2nd term -2.4393rd term -0.031 3rd term -0.0014th term 0.000 4th term 0.000Sum 258,277.108 Sum -4,172,994.858Sum*K0 258,173.797 Sum*K0 -4,171,325.660False Origin 500,000.000 False Origin 10,000,000.000Easting 758,173.797 Northing 5,828,674.340Grid Convergence Point Scale1st term 1° 47' 15.806" 1st term 1.0008211112442nd term 0° 00' 03.553" 2nd term 0.0000002905073rd term 0° 00' 00.002" 3rd term -0.0000000001304th term 0° 00' 00.000" Sum 1.000821401621Convergence 1° 47' 19.360" Point Scale 1.000421073060

Ref: Redfearn.xls : GDA Technical Manual : ICSM Web Site

Geographic Coordinates Converted in Overlapping Zones.

Zone 54 Zone 55

Easting 228 854.052 758 173.797

Northing 5 828 259.038 5 828 674.340

Grid Convergence -1 52 43.22 1 47 19.36

Point Scale Factor 1.00050567 1.00042107

MGA94 to GDA94(Redfearn’s Formulae)

Datum Parameters Semi-Major Axis (a) Inverse Flattening (f)

Projection Parameters Longitude of Central

Meridian (Zone) Scale Factor on Central

Meridian False Easting, False

Northing

Input Data Easting, Northing, Zone &

Height

Computed Parameters Foot-Point Latitude : Radius of Curvature: Meridian Distance:

Function (semi-major axis, inverse flattening and latitude)

Output Lat, Long, Grid Conv, Point SF

Ref: Chapter 5. GDA Technical Manual: ICSM Web Site

MGA94 - GDA94 (Example)

Latitude Deg Min Secs Longitude Deg Min SecsFoot point latitude -37° -41' -26.198 23" Central meridian 141° 00' 00.000 00"1st term 00° 02' 10.761 58" 1st term 02° 55' 36.934 06"2nd term 00° 00' -00.120 59" 2nd term 00° 00' -06.308 14"3rd term 00° 00' 00.000 13" 3rd term 00° 00' 00.007 11"4th term 00° 00' 00.000 00" 4th term 00° 00' -00.000 01"Latitude -37° -39' -15.557 12" Longitude 143° 55' 30.633 02"

Grid Convergence Deg Min Secs Point Scale1st term 01° 47' 22.253 77" 1st term2nd term 00° 00' -05.589 02" 2nd term3rd term 00° 00' 00.006 96" 3rd term4th term 00° 00' -00.000 01" sumGrid Convergence 01° 47' 16.671 70" Point Scale 1.000 420 2992

Ref: Redfearn.xls : GDA Technical Manual : ICSM Web Site

Scale & Convergence

Line Scale Factor (K)= L/s (plane / ellipsoidal) S/s (grid / ellipsoidal)

Grid Bearing () = Plane Bearing () + Arc-to-Chord Correction ()= Azimuth () + Grid Convergence ()

Ref: Glossary of Terms. GDA Technical Manual : ICSM Web Site

Grid Bearing & Ellipsoidal Dist from MGA94 Coordinates

Grid Bearing: function (plane bearing & arc-to-chord correction ) Arc-to-chord correction: function ( eastings,

northings and approx mean latitude)

Ellipsoidal Distance: function (plane distance & line scale factor ) Line Scale Factor: function ( CM scale factor,

eastings & approx. mean latitude)

Ref: Chapter 6. GDA Technical Manual : ICSM Web Site

Grid Bearing & Ellipsoidal Dist from MGA94 Coordinates (Example)

GridNorth

Plane Bearing ()

Plane Distance (L)

A

B

Grid

Bearing (

AB

)

Grid Bearing (BA )

Grid

Distance (S)

s = 54972.271K = 1.000 363 97

A = -20.67 AB = 1251741.86

B = 19.18

BA = 3065205.37 Ref: Test Data. GDA Technical Manual : ICSM Web Site

AB = 1251721.18

L = 54992.279

S L

Grid Calculations inOverlapping Zones

Plane DistanceEllipsoid DistanceLine Scale FactorArc-to-Chord (A)Arc-to-Chord (B)

Plane BearingGrid Bearing (AB)Grid Bearing (BA)Grid Convergence

54992.27954972.271

1.00036397-20.67+19.47

125 17 21.18125 17 41.86305 17 01.72-1 52 43.22

55003.30754972.271

1.00042107+23.94-25.19

128 58 08.37128 57 44.44308 58 33.56+1 47 19.36

Ref: GridCalc.xls GDA Technical Manual : ICSM Web Site

ZONE 54 ZONE 55

Plane Coordinates

Ellipsoid

Zon

e B

ound

ary

Zon

e B

ound

ary

Cen

tral

Mer

idia

n

Projection Plane

Zone 55

300 kmError 0.4

Zone 55 /2

100 kmError 0.06

Plane Coordinates

Ellipsoid

Zon

e B

ound

ary

Zon

e B

ound

ary

Cen

tral

Mer

idia

n

Projection Plane

X,Y

X,Y

X,YX,Y

X,Y

X,Y

X,Y

Plane BearingPlane Distance

E,N

X,Y

X,YX,Y

X,Y

X,Y

E,N

Grid BearingGrid Distance

Grid BearingGrid Distance

E,N

E,N

E,NE,N

E,N

E,N

E,N

Plane Coordinates

Summary

We investigated methods to:

Calculate and apply grid convergence.

Determine grid coordinates of a point given known coordinates of a start point and grid bearing and spheroidal distance from that start point.

Determine grid bearing and spheroidal distance between known points

Self Study

Read Module 6 (first part)

Review Questions