MGA Concepts and Grid 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

Ellipsoids and Geoids Geocentric Datum (best fit globally) Local Datum The Geoid (Mean Sea Level) Geocentric Datum (best fit globally) Local Datum AGD84 (best fits Australia)

AGD - The Old Datum Terrestrial Observations Systematic Errors Constrained by Doppler (transformed) Distribution Homogeneity Location of Marks 8

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

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

Queensland GDA94 Data Set QUT1 SUGA TEXA MULA NORM BREA BRDV WILF WOLL MUCK BANZ BARC PI EB HOWI GREN OLVE BASS EMUU TOWA Qld 100km Network

Magnitude of Shift All coordinates apparently shift in excess of 200m. 4

Distortions between Transformed AGD84 and GDA94 Western Qld Central Coast

Types of Coordinates Systems X Y Z Semi-minor axis (b) Semi-major axis (a) N E Projection l f h Geodetic - X + Y - Z Cartesian

Projection Coordinates on GDA and AGD Map Grid Australia on GDA NMGA NAMG EAMG Australian Map Grid on AGD EMGA

the same Central Meridians etc. Terminology GDA94 AGD84 Latitudes & Longitudes Universal Transverse Mercator Std. 6 Degree Zones, with the same Central Meridians etc. AMG84 MGA94 Eastings, Northings & Zone

Understanding and Using Projections

UTM Projection 6 Degree zones Longitude of Zone 1 : 3 east longitude 0.9996 Scale Factor on Central Meridian 500 000 m false easting 10 000 000 m false northing 1/2 degree overlap Ref: Chapter 1. GDA Technical Manual: ICSM Web Site

AMG/MGA - UTM Projection Zone Boundary 150E Zone Boundary 144E Central Meridian 147E Zone 55 Projection Plane Zone 54 Central Meridian 141E Scale Factor 1.0 Scale Factor 0.9996 Terrain Surface Geoid N Ellipsoid Scale Factor 1.0 Scale Factor 0.9996 Scale Factor 1.0006

AMG/MGA - UTM Projection

AMG - Redfearn’s Approx (See Study Book) ER, NR = Rectangular Coords Note meridian distance (m) = NR ET, NT = Transverse Mercator Coords E’, N’ = AMG Coords without false origin E, 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) Ref: Redfearn.xls : GDA Technical Manual : ICSM Web Site

Geographic Coordinates Converted in Overlapping Zones.

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) 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 (q) + Arc-to-Chord Correction () = Azimuth (a) + 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) North L = 54992.279 S  L A s = 54972.271 K = 1.000 363 97 Bearing (AB ) Grid Plane Bearing () Plane Distance (L) AB = 1251721.18 A = -20.67 AB = 1251741.86 B = 19.18 BA = 3065205.37 B Grid Distance (S) Grid Bearing (BA ) Ref: Test Data. GDA Technical Manual : ICSM Web Site

Grid Calculations in Overlapping Zones Plane Distance Ellipsoid Distance Line Scale Factor Arc-to-Chord (A) Arc-to-Chord (B) Plane Bearing Grid Bearing (AB) Grid Bearing (BA) Grid Convergence 55003.307 54972.271 1.00042107 +23.94 -25.19 128 58 08.37 128 57 44.44 308  58 33.56 +1  47 19.36 54992.279 54972.271 1.00036397 -20.67 +19.47 125 17 21.18 125  17 41.86 305  17 01.72 -1  52 43.22 Ref: GridCalc.xls GDA Technical Manual : ICSM Web Site

Plane Coordinates Zone 55 Error 0.4 300 km Central Meridian Ellipsoid Zone Boundary Central Meridian Projection Plane Zone 55 300 km Error 0.4

Plane Coordinates Zone 55 /2 Error 0.06 100 km Central Meridian Ellipsoid Zone Boundary Central Meridian Projection Plane Zone 55 /2 100 km Error 0.06

Plane Coordinates E,N E,N X,Y X,Y Plane Bearing Grid Bearing Grid Distance E,N E,N X,Y Grid Bearing Grid Distance X,Y Plane Bearing Plane Distance

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