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Lecture 8

- Introduction to Geodesy and Map Projections

Geodesy is a 21st Century science making use of

the most advanced space measurement and computer

technologies. A map projection is any

systematic arrangement of meridians and parallels

portraying the curved surface of the spheroid

upon a plane

- Definition of Geodesy
- Geodesy is the study of
- The size and shape of the earth
- The measurement of the position and motion

of points on - the earth's surface, and
- The configuration and area of large portions

of the earth's surface. - Geodesy serves as a foundation for the mapping

and referencing of all - geospatial data, it is a dynamic application of

scientific methods in support - of many professional, economic and scientific

activities and functions, - ranging from land titling to mineral exploration

from navigation, mapping - and surveying to the use of remote sensing data

for resource management - from the construction of dams and drains, to the

interpretation of - seismic disturbances.

Applications of Geodesy

Geodesists improve models to enable more precise

determination of satellite orbital positions.

They use radio astronomy to position the earth

and points on or above the earths surface in a

reference system based on quasars. Using

space-borne instruments, geodesists study

variations in mean sea level, mass transports

caused by atmosphere, ocean circulation, ground

water redistribution, and ice sheet

changes. Satellites are also used to mea-sure

earths gravity field and its temporal changes

and to study its role in climate change

phenomena and natural hazards. Geodesists use

new satellite navigation and positioning

capabilities to survey the land more accurately

and more economically to within a centimeter.

They use state-of-the-art navigation systems to

provide precise positions of science platforms

on board aircraft, ships, and satellites. Geodesy

plays an active role in the burgeoning geomatics

industry that includes the disciplines of land

surveying, photogrammetry, remote sensing,

hydrography, cartography, engineering surveying,

geographic information science, and geospatial

computing.

Ohio State University

Geodetic Surveying

- An accurate means of determining position and

height. - Geodetic survey is an effective means of

providing accurate position and - height on the earth's surface. Works that

require geodetic surveys for - the accurate positioning of control points

are - Large mapping projects
- Tunnelling and laying of pipelines
- Precise control positioning for large

survey works

Shape of the Earth

- Over limited area treat earth as a plane - simple
- Sometime as as a sphere - spherical trigonometry
- geoid
- Forces generated by the Earths rotation

flatten the Earth into an ellipse

.

Relationship Between Different Surfaces

Earth An irregularly shaped planet we have to

work on. Geoid An equipotential surface (a

fancy way of saying the pull of gravity is equal

everywhere along the surface) which influences

survey measurements and satellite orbits. A

plumb bob always points perpendicular to the

geoid, not to the center of the earth.

Ellipsoid An ellipse which has been rotated

about an axis. This provides a mathematical

surface on which we can perform our

calculations. The shape of the ellipsoid is

chosen to match the geoidal surface as closely

as possible.

Relationship Between Different Surfaces

- AHD coincident with geoid for most practical

purposes - geoid departs from a geocentric sphere by 22km

and from an oblate geocentric spheroid, flattened

at the poles by 100m - All conventional survey measurements are made

relative to the geoid, but geodesists choose a

spheroid to approximate the geoid for data

reduction and subsequent mapping

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Relationship between the Geoid and Ellipsoid

geoid - spheroid separation N h - H

H orthometric height approximated by AHD above

the geoid h spheroidal height above the

spheroid N geoid height above the spheroid

Deflection of the Vertical

- Instrument set up perpendicular to the geoid.

Because we work with the spheroid. Difference is

the deflection of the vertical. This can be

neglected if small.

Deflection of the Vertical

- deflection in the meridian is ve when deflected

to the north

The Best Fitting Spheroid

- the spheroid is deliberately chosen to be a best

fit to the geoid, so as to simplify survey data

reduction. This is achieved by minimizing x,h,N - 6.3m in geoid-spheroid separation 1ppm

horizontal scale error

Coordinate Systems

- Plane Coordinates
- local, possible arbitrary
- z axis vertical throughout
- right handed system
- bearings relative to y axis
- ignores shape of earth
- x,y,z

The most commonly used coordinate system today is

the latitude, longitude, and height system. The

Prime Meridian and the Equator are the reference

planes used to define latitude and longitude.

The geodetic latitude of a point is the angle

from the equatorial plane to the vertical

direction of a line normal to the reference

ellipsoid. The geodetic longitude of a point

is the angle between a reference plane and a

plane passing through the point, both planes

being perpendicular to the equatorial plane.

The geodetic height at a point is the distance

from the reference ellipsoid to the point in a

direction normal to the ellipsoid.

Cartesian Coordinate System

Map Grid Coordinates

- derived from geodetic coordinates
- transform 3D to 2D
- scale and line distortions present
- survey observables require corrections
- map projections

Geodetic Datums

- numerical or geometrical quantity or set of

quantities which serve as a reference or base for

other quantities - The adopted coordinates (after and adjustment of

measurements comprise the datum) - The spheroid is a simple geometrical reference

surface to which the coordinates are referred - horizontal or vertical, datums
- regional or global different best-fitting

reference spheroids have been defined in

different parts of the world because of the

undulating geoid. Eg. the Australian National

Spheroid.

NB Difference between spheroid and datum

Geodetic Datums

- consists of f, l or an initial origin the

azimuth for one line the parameters of the

reference ellipsoid and the geoid separation at

the origin. The deflection of the vertical and

geoid-spheroid separation are set to zero at an

origin point eg Johnson in Australia - geodetic latitudes and longitudes depend on both

the reference spheroid and coordinate datum - often the spheroid is implicitly linked to the

datum, so it has become common to use the datum

name to imply the spheroid and vice versa eg

WGS84 - the orientation and scale of the spheroid is

defined using further geodetic observations

Geodetic Datums

- Local/regional datum
- Approximates size and shape of the earth on a

local, regional scale - geometrical centre of the spheroid not

necessarily coincident with geocentre - well suited to surveying over the areas they were

defined for - inadequate for global satellite - surveying systems.

Regional Spheroids and Datums

- once the best-fitting spheroid is adopted, all

geodetic observations are reduced to this

spheroid, adjusted in a least squares sense,

which forms the geodetic datum.

eg the Australian Geodetic Datum (AGD) is based

on ANS

Global Spheroids and Datums

- satellite geodesy provides us with spheroids that

are geocentric, where their geometrical centre

corresponds with the Earths centre of mass since

the satellite orbits are close to the geocentre - orientation achieved by aligning its minor axis

with the Earths mean spin axis at a particular

epoch eg WGS84 - a modern global network of accurately coordinated

ground stations comprises a global datum called

the International Earth Rotation (IERS)

International Terrestrial Reference Frame (ITRF)

Global Spheroids and Datums

- ITRF is positioned relative to the geocentre

using a variety of space geodetic techniques,

such as Satellite Laser Ranging (SLR), Very Long

Baseline Interferometry (VLBI) and GPS. - The ITRF is considered to be a more reliable

datum than WGS84 and will form the backbone of

the GDA 10cm difference between them

Effect of Using Different Datums

- Datum and spheroid must be specified to define

horizontal position not just f, l - Without this information a single point can

refer to different positions

Map Projections

- Any systematic arrangement of meridians and

parallels portraying the curved surface of the

spheroid upon a plane

Map Projections

- The aim of a map projection is to represent the

Earths surface or mathematical representation of

the Earths surface on a flat piece of paper

with a minimum of distortion. - Recall
- Spheroidal Earth can be approximated to a plane

over small areas with minimal distortion - As the area of the spheroid becomes greater then

the distortion becomes greater

The Problem

P

Q

P

Y

?

?

Q

X

We can say that x f1(???) y f2(???)

Therefore, the coordinates on the plane have a

direct functional relationship with latitude and

longitude. It follows that should be a one to

one correspondence between the earth and the

map. However 1) some projections may not be able

to show the whole surface of the Earth. 2) some

points may be represented by lines instead of

points This is because the spheroid has a

continuous surface whereas a plane map must have

a boundary.

Projection surfaces

- developable surfaces
- geometric or mathematical
- gnomonic, stereographic, orthographic

Map projections are attempts to portray the

surface of the earth or a portion of the earth on

a flat surface. Some distortions of conformality,

distance, direction, scale, and area always

result from this process. Some projections

minimize distortions in some of these properties

at the expense of maximizing errors in others.

Some projection are attempts to only moderately

distort all of these properties.

Conformality When the scale of a

map at any point on the map is the same in any

direction, the projection is conformal.

Meridians (lines of longitude) and parallels

(lines of latitude) intersect at right angles.

Shape is preserved locally on conformal or

orthomorphic maps. Distance -

equidistant A map is equidistant

when it portrays distances from the center of the

projection to any other place on the map.

Direction - azimuthal A map preserves

direction when azimuths (angles from a point on a

line to another point) areportrayed correctly in

all directions. Scale

Scale is the relationship between a distance

portrayed on a map and the same distance on the

Earth. Area - equal-area When a map

portrays areas over the entire map so that all

mapped areas have the same proportional

relationship to the areas on the Earth that they

represent, the map is an equal-area map.

Map projections fall into four general classes.

Cylindrical projections result from projecting

a spherical surface onto a cylinder.

In the secant case, the cylinder touches the

sphere along two lines, both small circles.

When the cylinder is tangent to the sphere

contact is along a great circle

When the cylinder upon which the sphere is

projected is at right angles to the poles, the

cylinder and resulting projection are

transverse.

When the cylinder is at some other,

non-orthogonal, angle with respect to the

poles, the cylinder and resulting projection is

oblique.

Conic projections result from projecting a

spherical surface onto a cone.

When the cone is tangent to the sphere contact

is along a small circle.

In the secant case, the cone touches the sphere

along two lines, one a great circle, the other a

small circle.

Azimuthal projections result from projecting a

spherical surface onto a plane.

In the secant case, the plane touches the sphere

along a small circle if the plane does not pass

through the center of the earth, when it will

touch along a great circle.

When the plane is tangent to the sphere contact

is at a single point on the surface of the Earth.

The Universal Transverse Mercator UTM

The Universal Transverse Mercator projection is

actually a family of projections, each having in

common the fact that they are Transverse Mercator

projections produced by folding a horizontal

cylinder around the earth. The term transverse

arises from the fact that the axis of the

cylinder is perpendicular or transverse to the

axis of rotation of the earth. In the Universal

Transverse Mercator coordinate system, the earth

is divided into 60 zones, each 6 of longitude in

width, and the Transverse Mercator projection is

applied to each zone along its centerline, that

is, the cylinder touches the earth's surface

along the midline of each zone so that no point

in a given zone is more than 3 from the location

where earth distance is truly preserved.

- unit of length is the metre
- an ellipsoid is adopted as the shape and size of

the earth - coord obtained by a TM of f and l of points on

the ellipsoid - the true origin of coords is the intersection of

the equator and the central meridian of a zone - a central scale factor of 0.9996 is superimposed

on the central meridian - for points in the northern hemisphere, E and N

coords are related to a false origin 500,000m W

if the true origin and for points in the southern

hemisphere, E and N are related to a false origin

500,000m W and 10,000,000m S of the true origin - the projection has 60 zones, 6o wide in

longitude, beginning with zone 1 having a central

meridian of 177oW, numbered consecutively

eastwards, ending with zone 60 with a central

meridian of 177oE - the latitude extent of each zone is 80oS and 84oN

The Universal Transverse Mercator UTM

AMG and MGA

- The AMG and MGA are both systems of rectangular

coordinates based on TM projections of f and l

related to the AGD and GDA. - closely corresponds with the UTM grid used

globally - coordinates in metres
- zones are 6 wide (1/2 degree overlap)
- zones numbered from zone 49 with central

meridian 111E to zone 57 with central meridian

159E - central scale factor k0 0.9996
- origin of each zone is the intersection of

central meridian with the equator

- false origin S 10 000 000m, W 500 000
- coordinates described in Easting (E) and

Northing (N)

The TM graticule and the AMG/MGA

Projection Transverse Mercator Ellipsoid GRS

80 Central Meridian 141.00000 Reference

Latitude 0.00000 Scale Factor 0.99960 False

Easting 500000 False Northing 10 000 000.00000

These parameters mean that the Central Meridian

of Zone 55 is at 141E so that it covers from

138E to 144E the Reference Latitude is 0.0000

(the equator, which is 0N) the origin of the

coordinate system is at the intersection of

the Central Meridion with the Reference Latitude

and thus is at (0N,144E), where the coordinates

are (x, y) (500 000,10 000 000) m. The false

Easting of 500,000m is to ensure that all points

in the zone have positive x coordinates. The

y-coordinates are always positive in the Northern

hemisphere because 0 is at the equator. In the

Southern Hemisphere, a false Northing of

10,000,000m is applied to ensure that the

y-coordinate is always positive. The Scale

Factor of 0.9996 means that along the Central

Meridian, the true scale of 1.0 is reduced

slightly so that at locations off the true

meridian the scale factor will be more nearly 1.0

(the Transverse Mercator projection distorts

distance positively as you move away from the

Central Meridian).

Converting spheroidal coordinates to grid (AMG)

coordinates

- E (K0nwCosf)1 term1 term2 term3
- Term1 (w2/6)Cos2f(y-t2)
- Term2 (w4/120)Cos4f4y3(1-6t2)y2(18t2)-y2t2t4

- Term3 (w6/5040)Cos6f(61-479t2179t4-t6)
- E E False Easting
- N K0m Term1 Term2 Term3 Term4
- Term1 (w2/2)nSinf Cosf
- Term2 (w4/24)nSinf Cos3f(4y2y-t2)
- Term3 (w6/720)nSinf Cos5f8y4(11-24t2)-28y3(1-6t

2)y2(1-32t2)-y(2t2)t4 - Term4 (w8/40320)nSinf Cos7f(1385-3111t2543t4-t6

) - N N False Northing

- Ko central scale factor 0.9996
- ? as defined in previous lecture
- ???????????
- ? geodetic latitude
- t tan?
- ??? geodetic longitude measured from central

meridian ?0, - positive eastwards ? - ?0
- m meridian distance
- To translate to false origin of AMG
- 10 000 000 m to northing
- 500 000 m to easting

Example from AGD to AMG

Data Station BUNINYONG Latitude f -37o 39

15.557 Longitude l c Zone 54 lo 141o

Computations w l - lo 143o 55 30.633 -

141o 2o 55 30.63

meridian distance

Example from AGD to AMG

m -4184650.83515514.577-8.259-.016

-4169144.533

Radii of curvature

Easting E .9996(258127.64828.736-.031

-.000036 258053.090 Northing N

.9996(-4169144.533-4025.327-2.435-.001.00000024

-4171503.027

E E False origin 500000.000 258053.090

758053.090 N N False origin

10000000.000 -4171503.027 5828496.973

Converting grid (AMG) coordinates to spheroidal

coordinates

- E E - False Easting x E/(K0n')
- f f - Term1 Term2 - Term3 Term4
- Term1 (t/K0r)(xE/2)
- Term2 (t/K0r)(Ex3/24)-4y29y(1-t2)12t2

- Term3 (t/(K0r)(Ex5)/720)8y4(11-24t2)-12y3

(21-71t2)15y2(15-98t215t4)180y(5t2-3t4)

360t4 - Term4 t/K0r)(Ex7/40320)(13853633t24095t4

1575t6) - w Term1 - Term2 Term3 - Term4
- Term1 x Secf Term2 (x3/6)Secf(y2t2)
- Term3 (x5/120)Secf-4y 3(1-6t2)y2(9-68t2)

72yt224t4 - Term4 (x7/5040)Secf(61662t21320t4720t6)
- l l0 w

Meridian Distance and Foot-point Latitude

m aA0f -A2Sin2fA4Sin4f -A6Sin6f where A0

1-(e2/4)-(3e4/64)-(5e6/256) A2

(3/8)(e2e4/415e6/128) A4 (15/256)(e43e6/4) A6

35e6/3072

The foot-point latitude (f) is the latitude for

which the meridian distance equals the true

northing divided by the central scale factor

(mN/k0). This value can be calculated directly,

once three other values are available. n

(a-b)/(ab) f/(2-f) G a(1-n)(1-n2)(1(9/4)n2(

225/64)n4)(p/180) s (mp)/(180G) The foot point

latitude (in radians) is then calculated by f

s((3n/2)-(27n3/32))Sin2s ((21n2/16)-(55n4/32))

Sin4s(151n3/96) Sin6s(1097n4/512)Sin8s

Example from AMG to AGD

Data Station BUNINYONG Zone 54 k0 0.9996 E

758053.090 N 5828496.973 False origin

-500000.000 False Origin -10000000.000

Computations E 258053.090 N

-4171503.027 m N/k0 -4173172.296 foot

point latitude

Example from AGD to AMG

f -37o 41 26.198 0o 2 10.7616 -

0.1206.0001-.00000015 -37o 39 15.557 l

141o 0o 55 36.9341 - 6.3081.0071-.0000099

143o 55 30.633

Redfearns formulae To transform to MGA use

parameters of the GRS80 ellipsoid

http//www.anzlic.org.au/icsm/gdatm/gdatm.htm

http//www.osg.vic.gov.au/mgadoc.htm

3 dimensional transformations

We have 2 problems

1. Transform GPS positions given in cartesian X,

Y, Z to geodetic ????, h (WGS84) to project onto

UTM

2. Convert GPS geodetic coordinates from WGS84

spheroid to AGD spheroid and hence to AMG.

With the advent of GDA2000 we are faced with the

further problem of converting AMG coordinates on

the AGD (ANS66) to MGA coordinates on the GDA

(GRS80).

1. Cartesian to Geodetic Coordinates

Recall

3 dimensional cartesian coordinates

spheroidal geodetic coordinates

As we measure on surface of the Earth, each

spheroidal geodetic coordinate has an associated

ellipsoidal height, h

h is measured positive outwards from the surface

of the spheroid along the surface normal

Remember ellipsoidal height orthometric height

geoid height

Bowrings Forward Transformation

(x, y, z)

(????, h)

The forward transformation is simple because it

is a closed formula

For GPS applications, the inverse transformation

is more important

Bowrings formulae work for transformations on

any spheroid

Example Bowrings Forward Transformation

Given the following ANS geodetic coordinates,

compute ANS cartesian coordinates

ANS cartesian coordinates have their origin at

centre of origin of ANS66. This origin is NOT

geocentric

? -32.000 ? 11554? h 30.0m

ANS66 parameters a 6378 160m e2 0.006

694 542

radius of prime vertical ? 6384163.694m

X (6 384 163.694 30.0). cos(-32). cos(115.9)

-2 364 890.007m

Y (6 384 163.694 30.0). cos(-32). sin(115.9)

4 870 298.747m

Z (6 384 163.694(1- 0.006 694 542) 30.0).

sin(-32) -3 336 458.978m

Bowrings Reverse Transformation

(x, y, z)

(????, h)

The reverse formula is more complicated because

after we rearrange the previous equation set we

get

which has ? on each side of the equation (in???

equation has to be solved iteratively

Bowrings Inverse Transformation - Iterative

Solution

1. Compute longitude, ?. This can be done

directly

? tan-1(y/x)

2. Compute p (x2 y2)1/2

3. Compute an approximate value ?0 of the

latitude from

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Example - reverse of previous example

x -2 364 890.008 y 4 870 298.747 z -3 360

458.976

all on ANS66 but method holds for other spheroids

eg WGS84

1. computed longitude, ? tan-1(y/x)

tan-1 (4 870 298.747/-2 364 890.008) -6406?

or 11554?

Note ambiguity in tan-1 term. However, if you are

in Australia it is simple enough to work out

which is the correct longitude as one longitude

puts you on the other side of the world.

Example cont....

4. Compute an approximate value for the radius of

curvature in the prime vertical, ?0 from

6 384 163.694 (from program)

5. Compute the ellipsoidal height from

h p / cos?o - ?o

5414103.309/cos(-32.0000) - 6 384 163.694

30.00m

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2. Change of Spheroid

We now know how to transform cartesian (X, Y, Z)

coordinates to geodetic (????? h) coordinates

(and vice versa)

Because GPS coordinates are computed in WGS84 and

AGD is on the ANS66 spheroid we must be able to

transform from one spheroid to another

This can be done in a number of ways but the

easiest is using a 7-parameter transformation of

cartesian coordinates

A 7-parameter transformation is essentially a

helmert transformation in 3 dimensions

Z

z?

x?

O?

y?

Y

O

X

We wish to transform one cartesian coordinate

system to another

7 parameter transformation

where

X, Y, Z and x?, y?, z? are cartesian coordinates

centred on O and O? respectively

1 ds represents the scale factor to be applied

between the two coordinate systems

dx, dy, dz is the vector representing the origin

shift between O? and O

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The changes induced by the rotations can be

expressed in terms of the following rotation

matrices

When fully multiplied out the full rotation

matrix becomes

When the values of ?x, ?y, ?z are very small

(usually the case in change of spheroid

computations) the cosines can be assumed equal to

1 and the sines equal to the numerical values of

the angles themselves expressed in radians.

R becomes

where rx, ry, rz are the small rotations in

radians

The matrix R is known as a Rodrigues matrix

So the 7-parameter transformation can be written

as

To transform from one spheroid to another we need

to know the transformation parameters, dx, dy,

dz, ds, rx, ry, rz

Note for the reverse transformation we need only

change the signs of the seven transformation

parameters

Transformation Parameters in Australia

- A set of transformation parameters from

WGS84 to AGD84 have been computed by Higgins

(1987) - dx -116.00m
- dy -50.47m
- dz 141.69
- rx -0.23
- ry -0.39
- rz 0.344
- ds -0.0983ppm

- A set of transformation parameters from

AGD84 to GDA94 have been computed by Auslig - dx -117.763m
- dy -51.510m
- dz 139.061
- rx -0.292
- ry -.443
- rz -.277
- ds -.191ppm
- These supersede Higgins 1987 parameters

Example

convert WGS84 coordinates

to AGD84.

1. Form R matrix (convert angles to radians)

2. Apply matrix and origin shift

X -2 364 759.300 Y 4 870 348.935 Z

-3 360 610.238AGD

WGS84 cartesian to AMG

7 parameter

Bowring

Redfearn

Note height transformed by hWGS84 - N HAHD