An Investigation of CORS Site Transformations

THE SCHOOL OF SURVEYING AND SPATIAL INFORMATION SYSTEMS

An Investigation of CORS Site Transformations GMAT4015 Thesis Part B

Author: Paul Wigmore

Supervisor: Dr Craig Roberts Submitted on 21 October, 2011

i

Abstract

New South Wales Land and Property Information (LPI) has established a network of

Continuously Operating Reference Stations (CORS), known as CORSnet-NSW, which

has greatly expanded the availability of high accuracy Global Navigation Satellite

System (GNSS) measurements across the state. However, coordinates obtained using

CORSnet-NSW are not consistent with the coordinates of survey marks contained in

the Survey Control and Information Management System (SCIMS). This is due to the

Geocentric Datum of Australia 1994 (GDA94), which is used to express SCIMS

coordinates, suffering from distortions that would degrade the performance of

CORSnet. As a stop-gap solution an ad-hoc realisation of GDA94, known as GDA(2010)

was developed to define the positions of CORSnet stations.

The difference between SCIMS and CORSnet derived coordinates creates a need for a

method of obtaining coordinates using GNSS that are consistent with SCIMS, as this

would deliver the high productivity of satellite positioning in tandem with the

reliability and legal standing of SCIMS. Site Transformations are the method

recommended by LPI to deliver SCIMS-consistent GNSS positioning for CORSnet-NSW

users.

This thesis examined the impact of changing the number and spatial distribution of the

transformation points used in the site transformation process. The results indicate

that when used as recommended site transformations are able to provide near

seamless positioning consistency for CORSnet users. When sub-optimal arrangements

of transformation points are used there is a corresponding fall in the performance of

the site transformation. The drop in performance is not reliably indicated by the errors

in position and height at the transformation points, which are commonly used when

assessing the quality of site transformations. This places the onus on surveyors to use

their judgement and knowledge of the importance of the spatial arrangement of

transformation points, rather than relying on numerical indicators, when assessing the

quality of a site transformation.

ii

Academic Honesty Statement

Table of Contents

Abstract .............................................................................................................................. i

Academic Honesty Statement .......................................................................................... ii

List of Figures ................................................................................................................... vi

List of Tables .................................................................................................................. viii

Acknowledgements ......................................................................................................... ix

1 Introduction ................................................................................................................... 1

2 Definition and Applications of Datums .......................................................................... 3

2.1 Defining a Datum .................................................................................................... 3

2.1.1 Reference Systems ........................................................................................... 3

2.1.2 Reference Frames ............................................................................................ 5

2.1.3 The International Terrestrial Reference Frame (ITRF) ..................................... 6

2.1.4 Coordinate Transformations Between Datums ............................................... 7

2.1.5 Coordinate Conversions from Cartesian to Geographical Coordinates .......... 9

2.2 Applying Map Projections to Datums ................................................................... 10

2.2.1 Coordinate Conversion from Geographical to Grid Coordinates .................. 10

2.2.2 The Map Grid of Australia 1994 (MGA94) ..................................................... 10

2.2.3 Distortions in Map Projections ...................................................................... 11

2.2.4 Scale Factors .................................................................................................. 12

3. High Accuracy GNSS Observation Methods ................................................................ 14

3.1 Carrier Phase Measurements (CPH) ..................................................................... 14

3.1.1 Real Time Kinematic (RTK) ............................................................................. 15

3.1.2 Continuously Operating Reference Stations (CORS) ..................................... 15

3.1.3 Network Real Time Kinematic (NRTK) ........................................................... 15

4 Geodetic Infrastructure in NSW ................................................................................... 17

4.1 Datums Used in Australia...................................................................................... 17

4.2 The NSW Statewide Control Network .................................................................. 18

4.2.1 The Survey Control Information Management System (SCIMS) ................... 18

4.3 CORSnet-NSW ....................................................................................................... 19

4.3.1 GDA94 and GDA94(2010) .............................................................................. 20

4.3.2 Site Transformations ...................................................................................... 21

5. Field Work ................................................................................................................... 24

5.1 Site Transformation Assessment .......................................................................... 24

5.1.1 Mark Selection ............................................................................................... 24

5.1.2 Instrument Configuration .............................................................................. 26

5.1.3 GNSS Constellation Visibility .......................................................................... 27

5.1.4 Data Collection ............................................................................................... 27

6. Data Processing ........................................................................................................... 28

6.1 Data Download and Formatting ........................................................................... 28

6.2 Precision Plots ....................................................................................................... 28

6.3 MATLAB Script Development................................................................................ 31

7. Results ......................................................................................................................... 36

7.1 Maximally Constrained Transformation ............................................................... 38

7.2 Regular Quadrilateral Transformation .................................................................. 39

7.3 Regular Triangle Transformation .......................................................................... 40

7.4 Dense Quadrilateral Transformation .................................................................... 41

7.5 Skewed Quadrilateral Transformation ................................................................. 42

7.6 Skewed East West Triangular Transformation ..................................................... 43

7.7 Skewed North South Triangular Transformation ................................................. 44

7.8 Maximum-Spaced Quadrilateral Transformation ................................................. 45

8. Analysis ....................................................................................................................... 46

8.1 Maximally Constrained Transformation ............................................................... 46

8.2 Regular Quadrilateral Transformation .................................................................. 47

8.3 Regular Triangular Transformation ....................................................................... 47

8.4 Dense Quadrilateral Transformation .................................................................... 48

8.5 Skewed Quadrilateral Transformation ................................................................. 48

8.6 Skewed East-West Triangular Transformation ..................................................... 49

8.7 Skewed North-South Triangular Transformation ................................................. 50

8.8 Maximum-Spaced Quadrilateral Transformation ................................................. 50

8.9 Summary ............................................................................................................... 51

9. Conclusion ................................................................................................................... 53

Appendix 1: SiteTran MATLAB Script .............................................................................. 54

Appendix 2: VBA Module from Redfearn’s Formulae Spreadsheet ............................... 62

Appendix 3: Trimble GNSS Planning Software GDOP Graphs ........................................ 63

Appendix 4: SCIMS Coordinate Information for Marks Included in Field Work ............. 64

Appendix 5: Precision Plots of GNSS Coordinate Data ................................................... 65

References ...................................................................................................................... 74

vi

List of Figures

Figure 1 - An oblate ellipsoid showing the semi-major axis (a) and the semi minor axis

(b). ..................................................................................................................................... 3

Figure 2 - The geoid with an exaggerated deformation scale to highlight global

variations (ESA, 2011). ...................................................................................................... 4

Figure 3 - The relationship between ellipsoidal heights (H), and geoid heights (h) and

the geoid separation (N) (Swisstopo, 2010). .................................................................... 4

Figure 4 - Visualisation of the orientation of a simplified global earth reference frame

(Slater, 2007). .................................................................................................................... 5

Figure 5 - The ITRF2008 station velocity vectors (Altamimi, 2011). ................................. 7

Figure 6 - The geometrical relationship between Datum A and Datum B as expressed

by the seven transformation parameters (Paul et al., 2004). .......................................... 9

Figure 7 - Diagram of the relationship between the reference ellipsoid, projection

plane and the central meridian used by the UTM projection system (Knippers, 2009).

........................................................................................................................................ 11

Figure 8 – A cross section of the UTM projection system (ICSM, 2009). ....................... 12

Figure 9 - MGA Zone Boundaries (DESWPC, 2005)........................................................ 13

Figure 10 - CPH measurement showing the ambiguity (N) and phase measurement

(Rizos, 2000). ................................................................................................................... 14

Figure 11 - The major system components for conducting Network RTK (Leica, 2010).

........................................................................................................................................ 16

Figure 12 - Tide gauges used to define AHD71 (GA, 2011). ........................................... 17

Figure 13 - The different heights used to compute AUSGeoid09 (GA, 2010). ............... 18

Figure 14 - CORSnet-NSW station locations and service coverage areas (LPMA, 2011).

........................................................................................................................................ 19

Figure 15 - Horizontal distortions in GDA94 at selected CORSnet sites across eastern

NSW (Janssen, 2011). ...................................................................................................... 21

Figure 16 - Mark selected for use in the site transformation assessment (Google Earth,

2011). .............................................................................................................................. 25

Figure 17 - Horizontal precision plot for PM50063. ....................................................... 29

Figure 18 - Vertical precision plot for PM50063. ............................................................ 30

vii

Figure 19 – Horizontal precision plots for SS35814 (left) and PM35827 (right). ........... 30

Figure 20 - Vertical precision plot for PM29680. ............................................................ 31

Figure 21 - File format for the GNSS data input file. ...................................................... 32

Figure 22 - File format for the SCIMS data input file. ..................................................... 32

Figure 23 - Flowchart of the calculation steps performed by the SiteTran MATLAB

script. Blue indicates raw field data, red indicates SCIMS data, and purple indicates

field data to which transformation parameters have been applied. ............................. 33

Figure 24 – Plot of the horizontal and vertical error vectors for the maximally

constrained transformation. ........................................................................................... 38

Figure 25 – Plot of the horizontal and vertical error vectors for the regular

quadrilateral transformation. ......................................................................................... 39

Figure 26 – Plot of the horizontal and vertical error vectors for the regular triangular

transformation. ............................................................................................................... 40

Figure 27 – Plot of the horizontal and vertical error vectors for the dense quadrilateral

transformation. ............................................................................................................... 41

Figure 28 - Plot of the horizontal and vertical error vectors for the skewed


Figure 29- Plot of the horizontal and vertical error vectors for the skewed east-west

triangular transformation. .............................................................................................. 43

Figure 30 - Plot of the horizontal and vertical error vectors for the skewed north-south

triangular transformation. .............................................................................................. 44

Figure 31 - Plot of the horizontal and vertical error vectors for the maximum-spaced


Figure 32 - Comparative histogram of linear errors produced by the MSQT and the

MCT. ................................................................................................................................ 50

viii

List of Tables

Table 1 - Type field string values and their corresponding functions. ........................... 32

Table 2 - Horizontal and vertical errors for the maximally constrained transformation.

........................................................................................................................................ 38

Table 3 - Horizontal and vertical errors for the regular quadrilateral transformation. . 39

Table 4 - Horizontal and vertical errors for the regular triangular transformation. ...... 40

Table 5 - Horizontal and vertical errors for the dense quadrilateral transformation. ... 41

Table 6 – Horizontal and vertical errors for the skewed quadrilateral transformation. 42

Table 7 - Horizontal and vertical errors for the skewed east-west triangular

transformation. ............................................................................................................... 43

Table 8 - Horizontal and vertical errors for the skewed north-south triangular

transformation. ............................................................................................................... 44

Table 9 - Horizontal and vertical errors for the the maximum-spaced quadrilateral

transformation. ............................................................................................................... 45

Table 10 - Linear error at TPs for the MCT and SNST. .................................................... 52

ix

Acknowledgements

I would like to thank my supervisor Dr Craig Roberts for his patience and valuable

assistance in the process of completing this thesis. Dr Bruce Harvey’s help and advice

on programming of least squares processing was also greatly appreciated.

I wish to extend my thanks to Doug Kinlyside, Robert Lock and Michael London at LPI

for their help in arranging for me to visit the Queen’s Square LPI office to download

SCIMS data.

The concept that became this thesis originated while I was completing a placement

with the LPI in their Bathurst office and I would like to thank all those I worked with for

deepening my knowledge of, and respect for, LPI’s operations and the dedication of its

staff. I am especially grateful to Simon McElroy, Joel Haasdyk and Volker Janssen for

their patient answers to my many questions, which formed the basis of my

understanding of many of the topics covered in this thesis.

I am very grateful for the financial support provided to me by the LPI over the last year

through the Surveyor General’s Undergraduate Scholarship for Surveying.

Finally, I wish to thank my family and friends who have supported me throughout my

time at UNSW.

1

1 Introduction

Since the establishment of the surveying profession the principle of operating ‘from the

whole to the part’ has been at the core of survey methods. In practice, this principle

requires surveyors to connect to survey marks, usually contained within a provincial or

national geodetic network, whose coordinates and accuracies were obtained through a

network adjustment. The ‘whole to the part’ principle was most applicable when surveyors

used what would now be considered low accuracy measurement methods. In this context,

having a strong connection to stable network marks gave surveyors the ability to assess the

quality of their data and improve error detection.

The accuracy and precision of the instruments used by surveyors has increased rapidly as

technologies such as laser electronic distance measurement (EDM) and GNSS have become

ubiquitous tools for capturing spatial data. Previously, with low accuracy methods, many

observations would be combined in network adjustments to give coordinates with greater

accuracy than could be achieved with a single measurement. As the accuracy of

measurement techniques has improved, there has been a corresponding increase in

sensitivity to errors or distortions in the geodetic networks that are used to constrain

observations in adjustments. This trend has progressed to point where the accuracy of a

single measurement using modern instruments may match, or even exceed, that of the

network.

GNSS is an example of a technology which is capable of detecting distortions in adjusted

geodetic networks. The strengths of GNSS include its high accuracy, versatility, and

commercial productivity. However, GNSS also has significant weaknesses including

sometimes unpredictable error behaviour influenced by atmospheric and environmental

factors. GNSS also lacks rigorous legal traceability, which is an essential consideration where

the legal standing of positioning information must withstand scrutiny, such as boundary

definition. It is, therefore, essential to have the capability to merge the productivity and

versatility of GNSS with the enormous reliability and legal force of long-standing networks of

survey marks, even if this is at the cost of degrading the accuracy of the measurement

method.

2

The establishment of CORSnet-NSW has expanded the availability of high accuracy GNSS

positioning in terms of geographical coverage and the range of users. When CORSnet-NSW

users require consistency with SCIMS coordinates, site transformations are used to achieve

this outcome. Site transformations determine the relationship between the datum used for

SCIMS, and the WGS84 datum used by GPS, over a limited area. The relationship between

the two datums is determined based on a selection of transformation points (TPs) whose

coordinates are known in both datums and, when used correctly, should provide users with

near seamless consistency of coordinates.

The aim of this thesis is to examine the performance of site transformations in providing

consistency of coordinates as the number, geometry and distribution of the TPs is changed.

The effect of the change in the selection of TPs will be examined by applying the parameters

of each site transformation to GNSS data collected at survey marks to determine the level of

consistency between the transformed GNSS coordinates and the official coordinates in both

position and height.

2 Definition and Applications

2.1 Defining a Datum

2.1.1 Reference Systems

The first step in establishing a

purely mathematical model that approximates the shape of the earth

figure. The parameters of the geometrical figure are based on emp

sources such as Satellite Laser Ranging (SLR), Doppler Orbitography and Radio Posi

Satellite (DORIS), Very Long Baseline Interferometry (VLBI) and

Isaac Newton, the best approximation to the figure of the earth is an oblate ellipsoid of

revolution as shown in Figure

Figure 1 - An oblate ellipsoid showing the semi

The dimensions of an ellipsoid are defined by

in Equation 1 which are the lengt

flattening (f). Many different estimates of the best

been made and these are known as reference e

The shape a reference ellipsoid approximate

topographic, surface of the earth

equipotential surface of the earth

(MSL). Since gravity is an attraction between masses

3

Applications of Datums

establishing a datum is to define a reference system. A reference system

that approximates the shape of the earth as a

The parameters of the geometrical figure are based on empirical observations from

such as Satellite Laser Ranging (SLR), Doppler Orbitography and Radio Posi

Very Long Baseline Interferometry (VLBI) and GNSS. As first described by


Figure 1.

An oblate ellipsoid showing the semi-major axis (a) and the semi minor axis (b)

The dimensions of an ellipsoid are defined by fixing any two of the three parameters shown

the length of the semi-major axis (a) the semi-minor axis (b)

estimates of the best-fitting ellipsoid to the earth

known as reference ellipsoids.

� � � � ��

llipsoid approximates is not the physical, also known as

earth but the geoid (Janssen, 2009). The geoid is

earth’s gravity field that is approximated by mean sea l

Since gravity is an attraction between masses, and given the mass distribution of the

reference system is a

a simple geometric

irical observations from

such as Satellite Laser Ranging (SLR), Doppler Orbitography and Radio Positioning by

As first described by


(a) and the semi minor axis (b).

fixing any two of the three parameters shown

minor axis (b) and the

earth’s shape have

Equation 1

also known as the

eoid is an

that is approximated by mean sea level

mass distribution of the

4

earth is uneven, the geoid is an irregular shape. The shape of the geoid is sometimes

described as being ‘potato-like’ to indicate its irregularity as can be seen in Figure 2.

Figure 2 - The geoid with an exaggerated deformation scale to highlight global variations (ESA, 2011).

Since a reference ellipsoid can never be a perfect model of the irregular geoid surface it is

necessary to differentiate the forms of height that can be measured. Ellipsoidal heights (h)

are above and perpendicular to the surface of the reference ellipsoid. However, ellipsoid

heights cannot be used to determine the direction of water flow, which is essential for many

engineering applications. Determining water flow direction requires measurement of height

above the geoid (H). The two forms of height are related by the geoid separation (N) as

shown in Figure 3 and Equation 2.

� = ℎ + � Equation 2

Figure 3 - The relationship between ellipsoidal heights (H), and geoid heights (h) and the geoid separation (N)

(Swisstopo, 2010).

2.1.2 Reference Frames

The second step in defining a datum is to

the physical world to create a reference f

involves the pairing of points

directions in the physical world

three-dimensional Cartesian coordinate system

satellite-based positioning, a reference e

through the following steps as seen in

• The centre of the ellipsoid is defined to be the

• The Z-axis is defined as the

• Orientation is fixed by taking the X

to the intersection of the equatorial plane and Prime M

the line of longitude passing through the Greenwich Observatory in London

• The Y-axis is defined implicitly so as to

system.

Figure 4 - Visualisation of the orientation of a

The challenge in defining a reference system

world which makes creating and maintaining a global r

may first appear. Though the

suggest that points and directions defined by

the earth are fixed, they are in fact subject to constant variation. For example,

events are continually changing the distribution of mass within

the position of the geocentre.

5

g a datum is to ‘anchor’ the geometry of the reference s

to create a reference frame. Creating a reference frame

and orientations of the reference ellipsoid with points

in the physical world. As part of the definition of a reference frame a fundamental

coordinate system is also established. Since the introduction of

ased positioning, a reference ellipsoid is typically converted to a

steps as seen in Figure 4 (Slater, 2007).

The centre of the ellipsoid is defined to be the geocentre.

axis is defined as the earth’s rotational axis.

ation is fixed by taking the X-axis as the projection of a line from the geocentre

on of the equatorial plane and Prime Meridian which is defined as

passing through the Greenwich Observatory in London

implicitly so as to complete a right-handed Cartesian

Visualisation of the orientation of a simplified global earth reference frame

in defining a reference system arises in that there is variability

ating and maintaining a global reference frame more difficult than

Though the earth appears to be a stable system and intuition wou

suggest that points and directions defined by the mass, surface monuments and

they are in fact subject to constant variation. For example,

changing the distribution of mass within the earth leading to shifts in

the position of the geocentre. Continental drift causes entire land masses to move across

the geometry of the reference system to

rame for the earth

llipsoid with points and

rame a fundamental

Since the introduction of

cally converted to a reference frame

a line from the geocentre

which is defined as

passing through the Greenwich Observatory in London.

handed Cartesian coordinate

rame (Slater, 2007).

variability in the physical

erence frame more difficult than it

appears to be a stable system and intuition would

mass, surface monuments and rotation of

they are in fact subject to constant variation. For example, seismic

leading to shifts in

Continental drift causes entire land masses to move across

6

the face of the earth such that even the most stable landmarks, such as mountain peaks, are

in motion and their coordinates must be periodically updated to take account of this. Even

the movements of the earth exhibit variations such as the ‘wobbling’ of the earth about its

rotational axis.

To overcome the natural variation in the mass distribution and movement of the earth a

series of conventions are followed when defining global reference frames. The position of

the earth’s rotational axis is defined as the Conventional Terrestrial Pole (CTP) which is the

mean position of the rotational axis between 1900 and 1905 (Rizos, 2000). The orientation

of the X axis is defined as the International Reference Meridian (IRM) which is the position

of the Prime Meridian as it was defined on January 1, 1984 (NIMA, 2000). Using the CTP and

IRM a reference ellipsoid can be fixed to the earth at a particular point in time.

2.1.3 The International Terrestrial Reference Frame (ITRF)

The standard global reference frame is based on the CTP and IRM and is known as the

International Terrestrial Reference Frame (ITRF). Responsibility for maintaining and defining

the ITRF has been held by the International Earth Rotation Service (IERS) since it was

established in 1987 (IERS, 2010). The IERS fulfils these roles by using an extensive network of

ground stations that use a range of measurement techniques to monitor the variation in the

rotation, mass distribution and tectonic motion of the earth. Since the irregular movements

of the earth mean that seemingly fixed points on the planet’s surface are actually in motion,

each version of the ITRF is defined with reference to a particular moment in time known as

an epoch to which the global adjustment relates. The IERS periodically releases updated

realisations of the ITRF with the most recent being ITRF2008 (IERS, 2011). Each new ITRF is

made up of a list of the coordinates of the IERS ground stations at that particular epoch, as

well as a tectonic velocity field which states the direction and speed of the crustal motion at

each of the ground stations, which is shown in Figure 5.

7

Figure 5 - The ITRF2008 station velocity vectors (Altamimi, 2011).

2.1.4 Coordinate Transformations Between Datums

Each successive realisation of the ITRF as defined by the station coordinates and their

accompanying velocity field completely defines a unique datum. The relationship between

when a position is determined and which datum it is referenced to leads to a requirement

for a method that allows coordinates measured within datum A to be expressed in terms of

datum B. This process is known as a coordinate transformation and is familiar to surveyors

and geodesists who commonly need to work with multiple datums that may be related to a

particular country or province as well as a specific epoch.

One type of transformation that is used commonly in surveying is the affine transformation

in which a straight line will remain a straight line and parallel lines will remain parallel. A

similarity transformation is a particular type of affine transformation in which the applied

scale factor is the same in all directions, and is the most commonly used transformation

method in surveying. In order for a similarity to be applied all coordinates must be

expressed as Cartesian coordinates within their respective datums (Harvey, 2006).

Seven parameters are required to perform a similarity transformation. Three translations

along the Cartesian axes (TX, TY, TZ) are required to express the offset between the origin of

the two datums. These translation terms must express the position of the origin of the A

datum as a set of coordinates within the B datum. Furthermore, three rotations (ω, θ, κ)

around the x-, y- and z-axes respectively, as well a scale factor (s), are also necessary. The

8

rotation angles are used to calculate the rotation matrix (R), which must be in the rigorous

form shown in Equation 3 if the magnitude of the rotation terms is relatively large. Where

the rotation terms are approximately 3” or less, the full form of R is not required and can be

simplified to the form shown in Equation 4 (Harvey, 1985).

� = �� = cos sin 0

− sin cos 0

0 0 1

� cos� 0 − sin �0 1 0

sin � 0 cos � � 1 0 0

0 cos sin 0 − sin cos �

Equation 3

� = � 1 �� −��

−�� 1 �� 1

� Equation 4

The general equation to take a general point P, whose coordinates are known within datum

A (XA, YA , ZA ), and apply a similarity transformation to express those coordinates within

datum B (XB, YB , ZB ) is shown in Equation 5 (Harvey, 1985).

�� = ��,, �� + � � � �� Equation 5

The geometrical relationship between datum A and datum B as described by these seven

parameters is shown in Figure 6.

9

Figure 6 - The geometrical relationship between Datum A and Datum B as expressed by the seven transformation

parameters (Paul et al., 2004).

Transformations should be applied only when necessary as each transformation will

introduce errors in the resulting coordinates. These errors are an inevitable result of the

modelling of physical parameters that is required to relate the two datums.

2.1.5 Coordinate Conversions from Cartesian to Geographical Coordinates

Coordinate conversions are a method of representing a point within the same datum

through different coordinate systems. Beginning with Cartesian coordinates the first

conversion that is usually applied is to geographical coordinates of latitude (λ), longitude (φ)

and height above the reference ellipsoid (h). Equations 6, 7 and 8 are one method of

performing this conversion which makes use of the intermediate parameters given in

Equations 9, 10 and 11 (ICSM, 2009).

10

tan λ = ��

tan φ = ��1 − + �� 1 − �� − ��

ℎ = � cos φ + Z sin φ − a�1 − e� sin�φ

Equation 6

Equation 7

Equation 8

� = �� + ��

tan u = �� 1 − + ��

� = �� + ��

Equation 9

Equation 10

Equation 11

Coordinate conversions are an exact process which is perfectly reversible assuming no

round-off errors are present in the calculation.

2.2 Applying Map Projections to Datums

2.2.1 Coordinate Conversion from Geographical to Grid Coordinates

Geographical coordinates are an easily understood form of expressing position when

dealing with large portions of the earth’s surface in applications such as maritime navigation

and large geodetic networks. Geographical coordinates are not an intuitive system when

dealing with positioning applications over a small area where the assumption of a flat earth

is valid; especially since the angular units of latitude and longitude do not immediately

correspond to distance on the ground. Map projections are a mathematical tool to

overcome the weaknesses in geographical coordinates by converting angular geographical

coordinates to linear grid coordinates of easting and northing.

2.2.2 The Map Grid of Australia 1994 (MGA94)

The current grid coordinate system used in Australia is known as the Map Grid of Australia

1994 (MGA). MGA uses the Universal Transverse Mercator (UTM) projection system to

convert from geographical to grid coordinates. The UTM projection can be visualised by

11

placing the earth inside a cylinder that runs east-west so that the cylinder appears to ‘cut’

into the earth either side of a line of longitude known as the central meridian, as seen in

Figure 7. Points on the curved surface of the earth are then projected onto the flat surface

of the cylinder, which is known as the projection plane.

Figure 7 - Diagram of the relationship between the reference ellipsoid, projection plane and the central meridian used

by the UTM projection system (Knippers, 2009).

2.2.3 Distortions in Map Projections

Inherent in any projection system is some level of distortion in the representation of the

curved surface of the earth on a flat map. As can be seen in the common example of trying

to flatten an orange peel, some distortion is always required to wrestle a curved surface into

a plane. The UTM projection is known as a conformal projection which means that over

relatively small areas the shape of features is well preserved. The geometry of the UTM

projection also means that maximum possible distortion of distances is largely dependent

on the width of each projection zone as seen in Figure 9. The separation between the

projection plane (solid red line) and the ellipsoid surface (dashed red line) in Figure 8

represents a simplified relationship between the distance east or west of the central

meridian and the amount of distortion caused by projecting the curved ellipsoid surface

onto a plane.

12

Figure 8 – A cross section of the UTM projection system (ICSM, 2009).

2.2.4 Scale Factors

To account for the distortion of distances caused by projecting coordinates, scale factors are

used to convert between ground and grid distances. Taking Zone 55 in Figure 9 as an

example, and moving from west to east, the scale factor would be slightly greater than one

(1.006) at the zone 54/55 boundary where the projection plane lies above the ellipsoid. The

scale factor then falls as the projection plane and ellipsoid surface become closer and will

equal one where they intersect. Where the ellipsoid lies above the projection plane the

scale factor falls below one and will have a minimum value of 0.9996 at the central

meridian.

However, since the ellipsoid is merely an approximation to the earth’s shape it is also

necessary to consider the effect of the geoid separation and the topography. Elevation is

particularly important as it has a large impact on the magnitude of distortions and the scale

factors that compensate for them.

In order to limit the magnitude of the maximum distortion, the UTM system divides the

earth into zones that run north-south either side of their particular central meridian.

Typically each zone extends three degrees of longitude either side of the central meridian

with a half degree overlap, or buffer, where either of the neighbouring zones’ projections

can be used to express position. This means that the maximum magnitude of an ellipsoidal

scale factor is reasonable and only 60 UTM zones are required to cover the entire planet. It

13

is important to note that since the same range of values for easting and northing are used in

every zone a single UTM grid coordinate could refer to any one of 60 different points on the

earth’s surface. In the case of MGA coordinates any easting and northing pair could refer to

one of the eight zones that cover the Australian mainland as seen in Figure 9. To uniquely

define a position the grid coordinates must be combined with a zone number.

Figure 9 - MGA Zone Boundaries (DESWPC, 2005).

14

3. High Accuracy GNSS Observation Methods

High accuracy GNSS observations methods are generally classed as those methods capable

of achieving centimetre level accuracy or better, in either absolute or relative positioning.

Though the following description of those methods applies explicitly to GPS, the principles

remain the same when applied to other navigation satellite constellations.

3.1 Carrier Phase Measurements (CPH)

Carrier Phase Measurements (CPH) are based on exploiting the GPS signal structure to

achieve measurement precisions of around 1-2% of the wavelengths of the L1 and L2

frequency signals broadcast by Global Positioning System (GPS) satellites. The L1 and L2

wavelengths of 19cm and 24 cm respectively, mean that the signal processing resolution will

result in mm-level measurement precisions under ideal circumstances (Rizos, 2000). The

challenge in performing CPH is that phase observations are ambiguous due to their

sinusoidal signal structure which makes successive wavelengths indistinguishable from one

another. Therefore, CPH measurement requires two components: the integer number of

wavelengths in the signal path (N) as well as the phase of the wavelength fraction as shown

in Figure 10.

Figure 10 - CPH measurement showing the ambiguity (N) and phase measurement (Rizos, 2000).

The process of determining N is known as Ambiguity Resolution (AR) and is accomplished

through algorithmic processing. A full explanation of AR is beyond the scope of this thesis

though further detail can be found in (Xu, 2007).

15

3.1.1 Real Time Kinematic (RTK)

Real Time Kinematic (RTK) is a high productivity form of CPH. RTK requires two receivers

joined by a data link. One receiver is known as the base station, which remains stationary

over a mark with known coordinates. The second is known as the rover and is free to move.

By taking simultaneous measurements at the base and rover to a common constellation of

at least five navigation satellites, many systematic and random errors can be eliminated or

reduced. As a result, AR can be performed to provide coordinates with centimetre-level

accuracy at the rover. Increasing the baseline length from the base to rover will impact on

the accuracy of generated coordinates. The relationship between baseline length and error

magnitude will vary depending on atmospheric conditions and satellite geometry. A baseline

length of no more than 20 km should reliably produce accurate coordinates, though

baselines of up to 50km are possible (Gopi, 2006).

3.1.2 Continuously Operating Reference Stations (CORS)

Continuously Operating Reference Stations (CORS) are GNSS receivers placed at fixed

locations which can be used to calculate corrections to GNSS observations and model the

spatio-temporal behaviour of errors that impact on satellite positioning. Data from CORS

stations may be supplied for free or as part of a paid subscription and enables high accuracy

positioning to be performed by users. CORS are a relatively new form of positioning

infrastructure that are receiving heavy investment around the world due to the expected

economic returns of the readily-available, high-accuracy positioning that they enable. The

economic returns from CORS are expected to flow from industries such as agriculture and

mining as well as surveying, where the presence of an available nearby CORS will eliminate

the need for surveyors to set up a base station to conduct RTK surveys. Using a data link to

access the CORS in real-time, usually over mobile internet, will allow quasi-single receiver

positioning at centimetre level accuracy.

3.1.3 Network Real Time Kinematic (NRTK)

CORS that are linked to a central processing and distribution server are known as CORS

Networks. The typical arrangement and components of a CORS network is shown in Figure

11. By collecting data from each CORS, and limiting CORS spacing to around 70km, it is

possible to generate corrections to GNSS observations and determine atmospheric errors in

the network for supply to users. In particular, CORS networks enable Network RTK (NRTK)

16

CPH positioning which allows AR to be achieved at far greater separations between

receivers than is possible with single base RTK. There are several computational methods for

performing NRTK. The method used when conducting fieldwork for this thesis was the

Virtual Reference Station (VRS) method.

Figure 11 - The major system components for conducting Network RTK (Leica, 2010).

3.1.3.1 Virtual Reference Station (VRS)

VRS involves using the data obtained from CORS to interpolate the errors at a point close to

the rover in order to create a simulated, or virtual, base station at that location. By creating

a virtual base station, the length of the effective baseline to the rover can be minimised.

Since errors accumulate with base-rover separation this also reduces the errors that are

produced.

VRS involves the following steps being successfully completed by the network operator

(Janssen, 2009a):

1. Fixing the ambiguities in the network baselines and determining the atmospheric

and orbital errors across the network.

2. Displace the data of the CORS nearest the rover to simulate the VRS.

3. Using bi-linear or higher order interpolation to estimate errors where the VRS is

located.

4. Deliver the corrections to the user in real time.

17

4 Geodetic Infrastructure in NSW

4.1 Datums Used in Australia

The current positioning datum used in Australia is the Geocentric Datum of Australia 1994

(GDA94) which uses the Geodetic Reference System 1980 (GRS80) reference ellipsoid and is

a realisation of ITRF92 at epoch 1994 (ICSM, 2009). Height in Australia is expressed with

respect to the Australian Height Datum 1971 (AHD71). AHD71 is based upon tide gauge

measurements taken between 1966 and 1968 at 30 locations around the Australian

coastline as shown in Figure 12. The AHD71 datum, or zero height, is defined as the surface

that passes through MSL at each of the thirty tide gauges where the measurements were

taken (GA, 2011a).

Figure 12 - Tide gauges used to define AHD71 (GA, 2011).

AHD71 suffers from systematic distortions as a result of its definition. MSL defined by raw

tide gauge measurements is an approximation to, but is not coincident with, the geoid.

Ocean currents and atmospheric conditions perturb MSL so tide gauge observations need to

be corrected for these effects to correctly define the geoid. These corrections were not

applied when AHD71 was defined and the distortions were therefore carried through. The

main distortion present in AHD71 is a result of the change in the density of water between

the warmer northern seas and the colder southern seas which causes a one metre ‘tilt’ in

18

the AHD from north to south. Therefore, the AHD datum lies 0.5m above the geoid in the far

north and 0.5m below the geoid in far-south (GA, 2011).

Though AHD has errors as a result of its definition, the challenges in establishing a

replacement datum and migrating previous observations and networks to a new height

datum, have prevented the replacement of AHD. Instead, Geoscience Australia (GA) has

developed data products to convert ellipsoidal heights obtained by GNSS to AHD, which

allows GNSS observations to be made consistent with the current datum. The most recent

of these data products is AUSGeoid09. AUSGeoid09 is a combined geometric and

gravimetric model as it corrects for both the geoid undulation and the distortions in the

AHD, as shown in Figure 13. In effect, applying AUSGeoid09 to GNSS elevations is taking

results from a good measurement technique and distorting them to match the required

vertical datum (GA, 2010).

Figure 13 - The different heights used to compute AUSGeoid09 (GA, 2010).

4.2 The NSW Statewide Control Network

4.2.1 The Survey Control Information Management System (SCIMS)

The network of fixed survey marks across NSW is accessed through the Survey Control

Information Management System (SCIMS) which is maintained by LPI. The information

contained within SCIMS states the official coordinate of each survey mark in the GDA94

datum as well as an indication of the quality of those coordinates in terms of their class and

order.

The values contained within the SCIMS database are based on field work conducted by the

LPI as well as information taken from Deposited Plans (DPs) lodged by surveyors with the

LPI. The quality of coordinates in SCIMS is heavily dependent on the accuracy of the

19

methods used to gather observations. Errors in SCIMS data can also arise because of the

successive transformations and adjustments that are performed to update coordinates to

the new datums. Inevitably, this means that, in general, the accuracy of the values in SCIMS

will lag behind the capabilities of surveying methods as they increase in accuracy.

Periodically the SCIMS coordinates will improve their accuracy as values are updated

through network adjustments that include more accurate modern observations. However,

given the effort and expense required to gather and process new observations, as well as

the care that it is essential for the LPI to take to assure the reliability and integrity of the

information in SCIMS, updates are performed at a measured pace.

4.3 CORSnet-NSW

The state-wide network of CORS across NSW is known as CORSnet-NSW (hereafter

abbreviated to CORSnet). CORSnet is the greatly expanded successor to the SydNET CORS

network which covered the Sydney metropolitan area (White, et al, 2009). The current

expansion phase of CORSnet began in 2009 and has resulted in 61 operational CORS, not

including Lord Howe Island, as shown in Figure 14 (LPMA, 2011). LPI is committed to

providing the best possible positioning infrastructure and expanding CORSnet to cover the

entire population of NSW (White, et al, 2009).

Figure 14 - CORSnet-NSW station locations and service coverage areas (LPMA, 2011).

20

4.3.1 GDA94 and GDA94(2010)

In order to rapidly and correctly complete AR when using CORSnet to perform NRTK, the

relative coordinates of the CORS need to be known very accurately. When CORSnet was

being established it was found that GDA94 did not meet the accuracy standard required for

performing NRTK. GDA94 was adjusted in 1997 using a combination of terrestrial and

satellite observations including data that dated back to the definition of the Australian

Geodetic Datum 1996 (AGD66). Due to the poorer accuracy of some of the older

observations, as well as the layers of adjustments and transformations that the data went

through, GDA94 contains distortions. For many applications these distortions would not

have any impact. However, the distortions in GDA94 were significant enough to degrade

CORSnet’s NRTK performance because errors on the order of a few millimetres per

kilometre over the CORS spacing of 70-100km are enough to make NRTK unfeasible

(Janssen, 2011).

In order to overcome the distortions in GDA94 without having to define and roll-out a new

datum, a compromise was adopted by developing GDA(2010) as a 'stop gap' solution.

GDA2010 is an ad-hoc realisation of GDA94 which includes only GNSS observations of

baselines connected to stations included in the Australian Regional GNSS Network (ARGN)

that was calculated in 2010. This approach to defining a datum was able to control

distortions and generate coordinates accurate enough to allow CORSnet to operate at full

capacity. While GDA(2010) provides CORS coordinates of high accuracy, it also creates a

challenge in that it is not consistent with the SCIMS coordinates of fixed marks which are

expressed in GDA94. The magnitude of these distortions can be up to 30 cm in some areas

of NSW as shown in Figure 15.

21

Figure 15 - Horizontal distortions in GDA94 at selected CORSnet sites across eastern NSW (Janssen, 2011).

4.3.2 Site Transformations

Site transformations are the method recommended by LPI to allow high accuracy GNSS

measurements derived from CORSnet to be used consistently with SCIMS coordinates. The

site transformation acts as a map of the unique distortions that are present in the

coordinates of the marks that are used in the calculation process. Site transformations are

used by surveyors in applications where GNSS techniques, such as NRTK, are desirable but

need consistency with fixed marks. In order to perform a site transformation a set of TPs

that surround the survey site and have known coordinates obtained from SCIMS ,as well as

from high accuracy GNSS, is selected. These TPs are then used to determine the

transformation parameters between the coordinates of the selected points with respect to

the GNSS observations in WGS84, and the SCIMS values in GDA94. The area within the

boundary formed by the TPs for a given site transformation is known as the valid area.

Determining site transformation parameters is performed using a type of least squares

known as the combined method. A detailed description of least squares processing methods

is beyond the scope of this thesis but can be found in (Harvey, 2006) along with a detailed

description of the particular method presented here.

22

The aim of least squares is to take observations and their precisions to produce an estimate

of a related set of unknown values, also known as the parameters, and their precisions. The

essential components in performing least squares are the raw observations as well the

functional and stochastic models. The functional model expresses a known mathematical

relationship between the observations and the parameters. For combined least squares this

relationship must be in the form shown in Equation 13 in terms of the adjusted observations

(L) and the adjusted parameters (X).

��, �� = 0

Equation 13

In the case of site transformations the functional model is obtained by rearranging Equation

1 into the form shown in Equation 14.

The stochastic model states the precisions and correlations of the observations in terms of a

covariance matrix (Q) which in the case of site transformations will be a function of the

precision of the coordinates of the control points as measured by GNSS and indicated by

their class and order in SCIMS.

The first step in this least squares method is to calculate the misclose vector (b) which is the

‘observed minus the corrected value’ as given by Equation 15, based on the vector of raw

observations ( �) and an initial approximation of the values of the parameters (��

).

� = −�(�, ��)

Equation 15

The two design matrices must then be formed by taking partial differentials of the

functional model with respect to the parameters (A) and the observations (B), as shown by

Equations 16 and 17, for each TP (i).

��,�,�� + � � � �� − �� = 0

Equation 14

23

�� =

��

��

��

��

��

��

�� !!!!!"

Equation 16

#� =

��$�

��%�

��&�

��$

��%

��& ��$�

��%�

��&�

��$

��%

��& ��$�

��%�

��&�

��$

��%

��& !!!!!" Equation 17

The corrections that must be applied to the initial estimates of the parameters (ΔX) can then

be estimated as shown in Equation 18.

∆� = '(��)*)��(+��(��)*)��

Equation 18

Least squares is an iterative process so, if the initial estimates of the parameters are very

different to their true values, ΔX may need to be calculated several times.

In a successful application of least squares the solution should converge, which means that

as more iterations are performed the magnitude of the corrections should tend to zero.

When a valid estimate of the parameters has been achieved their precisions (��

) can be

determined in the form of a covariance matrix as shown in Equation 19.

*� = '(��)*)��(+��

Equation 19

4.2.3.1 Factors Affecting the Accuracy of Site Transformations

The quality of a site transformation is dependent on several factors, especially the geometry

of the TPs as well as the quality of their coordinates. The ideal arrangement is for the TPs to

have coordinates of high and equal accuracy spaced in a regular pattern, approximating a

regular polygon, surrounding the intended survey area. Having a wide variation in accuracy,

too few a number, or an irregular geometry of TPs will bias the transformation.

24

5. Field Work

5.1 Site Transformation Assessment

5.1.1 Mark Selection

The process of selecting marks for the site transformation required considering factors such

as logistics, suitable network geometry and coordinate quality. The inital choice involved

selecting an approximate region for the field work. The test site needed to be close to

UNSW to allow short travel times as well as easy access to Univesity of NSW (UNSW)

resources and staff who could assist with any equpiment problems that were encountered.

The region extending south of the UNSW campus to the Maroubra-Pagewood area was

selected as the best region. The regions to north and west of UNSW were generally higher

road traffic areas with a more dense street pattern making them less desirable for

performing field work.

After selecting a region, the nest step was obtaining SCIMS data on the marks present in the

area. SCIMS data was obtained by visiting the LPI office at Queen’s Square to select and

export the relevant data in the appropriate ArcGIS and database file formats.

The desired geomtery for the site transformation assessment was to have seven marks of

B2/LBL2 horizontal and vertical class and order arranged in a regular pattern around the test

area to use as TPs. Within the test area several check marks would be required. If a check

mark has a class and order of B2/LBL2 it was used as a horizontal and vertical check (HV)

mark. If a check mark met the horizontal B2 classification but failed to meet the LBL2

requirement it is was used as a horizontal only (H) check mark. Two test lines extending

away from the area defined by the TPs were also needed to test whether and how quickly

the accuracy of the transformation degrades outside the valid area. The test lines were

designated the northern (N) and north-western (NW) test lines.

To assist in selecting marks the SCIMS data was filtered to remove marks that did not have a

horizontal class and order of B2 and converted to kml format for display on Google

Earth.The display of survey mark positions on Google Earth allowed an initial selection of

marks that would give the desired network geometry. Using the Google Earth imagery it was

possible to locate any large trees or structures that were likely to obstruct the sky view from

25

the survey marks. A list of marks was chosen based on the Google Earth display and then

field reconnaisance of each mark was performed to confirm suitability and check for any

sign of damage or disturbance. The majority of marks chosen were included in the field

work, apart from a small number that had new construction or tree growth that had

occured since the 2009 imagery date indicated by Google Earth. The final network design

included 26 survey marks and is shown in Figure 16. The intended purpose of each mark is

denoted by icons with red stars indicating the surrounding TPs, green squares identifying HV

marks, green diamonds showing H marks, yellow squares corresponding to N marks, and

blue squares being NW marks. The SCIMS coordinate information for each mark is included

in Appendix 4.

Figure 16 - Mark selected for use in the site transformation assessment (Google Earth, 2011).

26

5.1.2 Instrument Configuration

The Leica 1200 GNSS receivers owned by the UNSW School of Surveying and Spatial

Information Systems (SSIS) were used for all field work. Initial use of the equipment

revealed two main problems with the instrument configuration that required resolution.

The first problem was an inability to connect and initialise using CORS-net in any mode

other than single-base RTK. This was unacceptable as single-base RTK would be subject to

base-rover separation-dependent errors that could be avoided with NRTK. The second

problem was that the instrument would begin logging observations at the same time as it

began attempting to perform AR. This meant that the first 10-20 sec of observations were in

code-only or float solution mode and had relatively poor accuracy. The desired

configuration was to have the instrument connect and complete AR, and only after this was

achieved to begin recording data.

After consulting the technical manual supplied by Leica the appropriate settings to resolve

each issue were determined. The problem of initialising in NRTK was solved by changing the

solution type in the rover settings to ‘VRS’ and allowing more time for AR. Single base RTK

typically took 10-20 seconds to complete AR whereas NRTK took longer than this. The longer

delay in AR is most likely due to the greater time required for the CORSnet server to carry

out the steps required for VRS in comparison with simply streaming data from a single

CORS. The greater delay led to an impression that the NRTK initialisation had failed after the

20 second period that was required to initialise in single base mode, and therefore

attempting to re-start the AR process by disconnecting from CORSnet. The change of

settings combined with allowing greater time resolved this issue.

The problem of forcing AR to complete before logging data was also solved by a change of

instrument settings, in particular turning off the ‘AUTO CONEC’ function in the data link

configuration options. This gives manual control of when to connect and disconnect from

the CORS-net server to the user. This meant the instrument could be instructed to connect

and initialise, and only when this was completed, told to begin recording data. While looking

for a solution to this problem another useful setting was also activated that allowed setting

a threshold value for the three dimensional coordinate quality (3DCQ) of the recorded

observations. The meant that in the event that a loss-of-lock occurred and the accuracy

27

degraded past preset 3DCQ value no data would be recorded. For the duration of the field

work the threshold 3DCQ value was set to 0.05m.

In order to maximise the quality of the observations the instrument was set to observe both

GPS and GLONASS constellations.

5.1.3 GNSS Constellation Visibility

The number of visible satellites and the geometrical dilution of precision (GDOP) was

determined in advance of field work to assist in planning when observation sessions should

be conducted. Information regarding the number and arrangement of visible navigation

satellites was obtained using the Trimble GNSS planning software. The Trimble software

provides graphs of the number of visible satellites, as well GDOP values and sky plots to aid

in determining time periods that are suitable for the user’s particular application. The GDOP

values for the days intended for field work were for the most part favourable with almost all

of the daylight hours having a GDOP of less than three and a half, which was suitable for the

proposed field work. On the first and second days of field work there were some short spans

of less than one hour where the GDOP moved above the typical value of 3.5 so no

observations were conducted during these periods. The Trimble planning graphs are

included in Appendix 3.

5.1.4 Data Collection

At each mark in the network two sets of data were collected. Firstly, 1 Hz NRTK coordinate

observations were recorded for five minutes to give a total of 300 position solutions.

Following this the instrument was set to occupy the point for a further five minutes and

produce a single coordinate solution from the mean observation values over that period.

The two different data sets were intended to perform different functions. The 1 Hz data was

intended for use in plotting the individual point solution in order to give some indication of

the positioning precision. This would highlight any marks that should be removed from

further processing in the event that their precision was unacceptably poor. The five minute

occupation was intended as the data to be used in the further processing, after being

checking for consistency with the 1 Hz data.

28

6. Data Processing

Data processing was implemented using Leica Geo Office (LGO), Microsoft Excel, and

MATLAB. Sources of data included SCIMS, field work and N values obtained from

AUSGeoid09.

6.1 Data Download and Formatting

After the data had been collected in the field it was downloaded from the instrument and

imported into Leica Geo Office (LGO). In order to allow further processing, the data was

then exported from LGO in a comma separated values (CSV) file. The LGO Format Manager

was used to design a custom export format that included only the useful data fields and

arranged them in the desired layout to simplify further processing steps. After consulting

literature on how Leica GNSS instruments store data it was discovered that, regardless of

the datum in which a Leica instrument may display a measured GNSS coordinate, the raw

point data is always stored as WGS84 geographical coordinates (Leica Geosystems, 2004).

Therefore, to avoid the unanticipated effect of any transformations applied by LGO to the

coordinate data, all information was exported in the WGS84 datum.

It is important to note that though the instrument considers the exported coordinates to be

in the WGS84 datum they are somewhat biased by the GDA(2010) CORS coordinates used to

perform NRTK. The CORS coordinates are defined by the network operator, in this case

CORSnet, and their values propagate through to the results obtained by users. There will be

some disagreement between GDA(2010) and WGS84 due to the continental shift that

GDA(2010) has not been corrected for, but which has been accounted for in WGS84. The

magnitude of the shift would be on the order of one metre but does not affect the ability of

CORSnet to perform NRTK. For CORSnet the essential requirement is not absolute

consistency with WGS84, but insuring that that the relative CORS coordinates achieve high

accuracy. Due to the fact that Australia’s tectonic motion is uniform across the continent,

the crustal motion introduces no errors in relative CORS coordinates expressed in

GDA(2010) and therefore does not impede the performance of CORSnet.

6.2 Precision Plots

The GNSS data exported from LGO was imported into Microsoft Excel and sorted to

generate data sets for each occupied point. Using MATLAB the 1 Hz and five minute

29

occupation coordinate observations were then plotted to give an indication of their

precision, and are included in Appendix 5. On each horizontal precision plot the origin

corresponds to the average easting and northing value based on the 1 Hz data. The

surrounding blue points represent the offset from the mean of the individual epoch

solutions and the red star indicates the offset of the value obtained from the five minute

occupation. PM50063 has a typical horizontal precision plot with a tight symmetrical

distribution which is shown in Figure 17.

Figure 17 - Horizontal precision plot for PM50063.

Similarly, a vertical precision plot was also produced for each mark. On the vertical precision

plots the y-axis corresponds to the offset from the mean ellipsoidal height of the 1 Hz data.

The x-axis corresponds to the time elapsed from the first observation. The red star indicates

the offset from the mean of the ellipsoidal height given by the five minute occupation and is

plotted halfway along the x-axis on each plot. The vertical precision plot for PM50063 is

shown in Figure 18.

30

Figure 18 - Vertical precision plot for PM50063.

The expected precision of the NRTK data was ±20mm in each horizontal component and

±50mm in the vertical. The precision plots indicate that the majority of the points were

within these expectations. Two points, SS35814 and PM35827, exhibited an unusual

irregular scattering in their horizontal precision plots which are shown in Figure 19.

Figure 19 – Horizontal precision plots for SS35814 (left) and PM35827 (right).

Though the distribution of data for SS35814 and PM35827 was unusual in comparison with

the other marks, the agreement between the averaged single epoch and five minute

occupation solutions was within expected precision so both points were included in

subsequent processing. PM35827 had light foliage overhead which is the most likely cause

of its more scattered data. No such obstructions were present at SS35814 so the cause in

that case is unknown. PM29680 exhibited an unusually large difference of 78mm in

31

ellipsoidal height between the averaged 1 Hz data and the five minute occupation as shown

in Figure 20.

Figure 20 - Vertical precision plot for PM29680.

The large difference in height values at PM29680 was particularly unusual as the time series

of the single epoch solution was stable and within ±50mm precision. No obstructions to sky

view were present that may explain the offset. Therefore, PM29680 was included in

subsequent processing as unexpected but still valid data.

6.3 MATLAB Script Development

A MATLAB script named SiteTran was written to perform the least squares calculations and

display the results, and is included in Appendix 1. SiteTran requires two CSV input files with

one containing SCIMS data (scims_data.csv) and another containing GNSS field data

(gnss_data.csv). Since the aim of the site transformation assessment was to examine what

changes in results are caused by selecting different combinations of TPs, it was necessary to

have simple method of altering which marks were used in determining the transformation

parameters. This was achieved by including a field in the GNSS data file named ‘type’. The

type field contains a string that corresponds to one of the five different mark functions in

the network, as shown in Table 1.

32

Table 1 - Type field string values and their corresponding functions.

Type Field String Mark Function

tp Transformation point

hv Horizontal and vertical check mark

h Horizontal only check marks

n Northern check line

nw North western check line

In order to alter the function of a mark it would only be necessary to edit this field in the

gnss_dat.csv file to change how any mark was used in a particular transformation. The GNSS

data file included the point name, WGS84 Cartesian coordinates and the mark type. The

GNSS file format is shown in Figure 21.

Figure 21 - File format for the GNSS data input file.

The SCIMS data file included the mark name, MGA grid coordinates, AHD, GDA ellipsoidal

coordinates and the AUSGeoid09 N value for each mark. The N values were obtained from

the AUSgeoid online computation service (GA, 2011b). The SCIMS file format is shown in

Figure 22.

Figure 22 - File format for the SCIMS data input file.

Once both data files have been loaded the MATLAB script begins the process of

calculating and applying the transformation parameters, which is shown in flowchart form in

Figure 23. The first step is to pre-process the data by using the N values to convert the AHD

height of SCIMS points to ellipsoid heights. This is required due to the fact that similarity

33

Figure 23 - Flowchart of the calculation steps performed by the SiteTran MATLAB script. Blue indicates raw field data,

red indicates SCIMS data, and purple indicates field data to which transformation parameters have been applied.

34

transformations can only be applied to coordinates in Cartesian form, and the conversion

from ellipsoidal to Cartesian requires that the ellipsoidal height be known. The script then

sorts each mark in the input files by their type and uses those marked as TPs to perform

three iterations of the least squares solution for the transformation parameters. In all cases

three iterations were sufficient for the least squares solutions to converge.

Once determined, the transformation parameters are then applied to all the GNSS

observations to give transformed Cartesian coordinates. Following this, the transformed

Cartesian coordinates are converted to ellipsoidal coordinates on the GRS80 ellipsoid using

the ‘xyz2ell3’ MATLAB function obtained from (Craymer, 2011).

The rigorous method for converting from ellipsoidal coordinates to grid coordinates is

known as Redfearn’s Formulae. The formulae are somewhat complex due to the number of

intermediate terms that need to be calculated, which makes Redfearn’s Formulae time

consuming to program using software such as MATLAB. Therefore, ellipsoidal coordinates

were converted to grid form using the Redfearn’s Formulae spreadsheet taken from (GA,

2011). As provided by GA, the Redfearn’s Formulae spreadsheet is intended to be used to

manually enter ellipsoidal coordinates for a single point to obtain their grid form, or vice

versa. Since this would be too time consuming a Visual Basic for Applications (VBA) module

was added to the spreadsheet to automate the process, and is included in Appendix 2. The

ellipsoid heights of the transformed GNSS data were then converted back to AHD using

AUSGeoid09 N values.

At this point the error in easting, northing and AHD between the transformed GNSS data

and the SCIMS coordinates is calculated. After exploring different ways of displaying the

error information, the MATLAB plotting function named Quiver was selected. Quiver allows

the plotting of vectors at specified locations. In this case the position of the vectors is

determined by the SCIMS grid coordinates of each survey mark. The magnitude and bearing

of the vector plotted at each point is determined by the difference between the

transformed GNSS and SCIMS data for each mark. For each transformation two Quiver plots

were produced: one for the horizontal differences and another for the vertical differences.

The scale of the distortion vector was exaggerated in comparison with the main plot to aid

in analysing the results. In order to illustrate the relative scales an indicative vector is

35

displayed in the south-western corner of each plot showing the size of a 10mm and 20mm

difference on the horizontal and vertical plots respectively. An output text file is also

produced for each transformation which contains the differences in easting, northing and

AHD between the transformed GNSS and SCIMS coordinates for each point. The linear

difference, rather than the difference in easting and northing, as well as the bearing of the

distortion vector and the difference in AHD are also exported to a text file.

36

7. Results

The results for each transformation are presented in a graphical and numerical form. The

graphical display of results consists of the two Quiver plots showing the horizontal and

vertical error vectors. On each plot the marks and vectors are colour coded to indicate the

different function they were used for in each transformation. Red indicates marks that were

used as TPs. Blue indicates the marks that are used as H or HV check marks. The N points are

displayed in pink and the NW marks are displayed in fluorescent blue. On the plot of vertical

distortion vectors marks of type H are not displayed as they have a vertical class and order

of less than LBL2 and were therefore not appropriate for assessing accuracy in the height

component.

The numerical output is in a tabular form containing the linear horizontal error and the

vertical error at each survey mark. The numerical output is presented with the results sorted

into groups based on the function of each mark within the particular transformation. The

average error for each group is also included, with the vertical average using absolute error

values. To aid in highlighting any change in the error behaviour between different mark

functions, the field containing the function tag is colour coded consistently with the

graphical output using red for TPs, pink for N and fluorescent blue for NW. The H and HV

marks, which are both plotted in blue, are differentiated in the tabular form by lighter and

darker shades respectively. The vertical errors at the H marks are included in the numerical

output for the sake of completeness and give an indication of the impact of vertical class

and order on error behaviour.

The cells containing the errors are colour coded to indicate their relative magnitude. The

cells containing the horizontal errors are coded with a colour scale that moves from green to

red as the magnitude of the error increases. Since the vertical errors can be either positive

or negative values a colour scale was applied that changes from red for extreme negative

values, to yellow for mid-range values, and finally to blue for large positive values.

The nomenclature used in naming the transformations is intended to highlight what type of

effect each one is attempting to examine. The change from maximum constraint, using all

TPs, to quadrilateral and triangular transformations, is intended to examine the effect of

changing the number of TPs. The change from regular to skewed indicates a move from a

37

strong, regular geometry of TPs to an uneven spatial distribution. Dense transformations

examine the effect of restricting the separation of TPs and reducing the valid area.

As described in Section 4.3.2, part of the least squares output is the quality of the

parameters. Obtaining realistic values for the precision of the parameters from the least

squares process was not possible in this case. The reason for this was that, when

considering the two datums over the surface of the earth, the area where data was

obtained is extremely small. The tiny area of sampling in relation to the large distance from

the sampled points to the origin of the datums means that miniscule rotations can have a

large impact, and makes decorrelating rotations from translations very difficult. Since no

meaningful precisions could be obtained from the least squares calculations, and all marks

to be used as TPs had identical class and order, an equal arbitrary precision was applied to

all SCIMS and NRTK coordinates. Therefore, the quality of each site transformation will only

be assessed by the errors in position and height between transformed GNSS and SCIMS

coordinates.

38

7.1 Maximally Constrained Transformation

Figure 24 – Plot of the horizontal and vertical error vectors for the maximally constrained transformation.

Table 2 - Horizontal and vertical errors for the maximally constrained transformation.

Mark Name Usage Linear Error Group Average Vertical Error Group Average

(mm) (mm) (mm) (mm)

SS35582 h 17 9

SS35593 h 10 -4

SS35597 h 4 2

SS35609 h 11 10 22 19

SS35611 h 13 10

SS35814 h 3 20

SS35827 h 8 67

PM25477 hv 9 9

PM25494 hv 7 0

PM46887 hv 13 11 51 15

SS35825 hv 10 -7

SS54824 hv 14 -8

PM50063 n 10 -11

SS35648 n 19 19 -7 7

SS64963 n 8 -13

PM29680 nw 6 67

PM29694 nw 11 14 59 57

PM29700 nw 26 38

PM31543 nw 13 63

PM29568 tp 4 -7

PM29711 tp 2 10

PM83860 tp 10 -7

SS11257 tp 4 5 2 4

SS23291 tp 4 2

SS35530 tp 10 0

SS35829 tp 4 1

39

7.2 Regular Quadrilateral Transformation

Figure 25 – Plot of the horizontal and vertical error vectors for the regular quadrilateral transformation.

Table 3 - Horizontal and vertical errors for the regular quadrilateral transformation.


(mm) (mm) (mm) (mm)

SS35582 h 15 6

SS35593 h 12 -7

SS35597 h 1 -3

SS35609 h 10 10 16 17

SS35611 h 14 5

SS35814 h 6 17

SS35827 h 10 65

PM25477 hv 12 6

PM25494 hv 9 -4

PM29568 hv 5 -13

PM46887 hv 11 10 49 13

PM83860 hv 15 -13

SS35825 hv 10 -10

SS35829 hv 8 -1

SS54824 hv 12 -13

PM50063 n 16 -19

SS35648 n 24 18 -14 18

SS64963 n 12 -22

PM29680 nw 9 55

PM29694 nw 12 15 49 46

PM29700 nw 20 29

PM31543 nw 20 50

PM29711 tp 4 2

SS11257 tp 3 4 -2 2

SS23291 tp 3 2

SS35530 tp 8 -2

40

7.3 Regular Triangle Transformation

Figure 26 – Plot of the horizontal and vertical error vectors for the regular triangular transformation.

Table 4 - Horizontal and vertical errors for the regular triangular transformation.


(mm) (mm) (mm) (mm)

SS35582 h 16 11

SS35593 h 12 0

SS35597 h 8 5

SS35609 h 18 12 27 22

SS35611 h 17 14

SS35814 h 6 25

SS35827 h 8 69

PM25477 hv 9 3

PM25494 hv 11 -4

PM29568 hv 12 17

PM29711 hv 4 0

PM46887 hv 16 10 54 10

SS11257 hv 6 0

SS35825 hv 13 -3

SS35829 hv 2 3

SS54824 hv 20 -3

PM50063 n 7 1

SS35648 n 16 13 2 2

SS64963 n 17 2

PM29680 nw 16 81

PM29694 nw 15 19 72 70

PM29700 nw 37 49

PM31543 nw 7 79

PM83860 tp 10 7

SS23291 tp 5 8 0 6

SS35530 tp 9 11

41

7.4 Dense Quadrilateral Transformation

Figure 27 – Plot of the horizontal and vertical error vectors for the dense quadrilateral transformation.

Table 5 - Horizontal and vertical errors for the dense quadrilateral transformation.


(mm) (mm) (mm) (mm)

SS35582 h 20 18

SS35593 h 8 0

SS35597 h 7 6

SS35609 h 8 10 20 21

SS35611 h 11 8

SS35814 h 4 18

SS35827 h 9 72

PM25477 hv 7 14

PM25494 hv 4 6

PM46887 hv 14 54

PM83860 hv 12 10 -16 18

SS23291 hv 5 18

SS35530 hv 15 15

SS35825 hv 11 -5

SS54824 hv 13 -12

PM50063 n 17 -36

SS35648 n 23 18 -23 35

SS64963 n 13 -46

PM29680 nw 12 42

PM29694 nw 15 18 37 33

PM29700 nw 20 20

PM31543 nw 25 34

PM29568 tp 2 -3

PM29711 tp 4 4 3 3

SS11257 tp 3 -4

SS35829 tp 6 4

42

7.5 Skewed Quadrilateral Transformation

Figure 28 - Plot of the horizontal and vertical error vectors for the skewed quadrilateral transformation.

Table 6 – Horizontal and vertical errors for the skewed quadrilateral transformation.


(mm) (mm) (mm) (mm)

SS35582 h 20 1

SS35593 h 8 -9

SS35597 h 8 -8

SS35609 h 13 10 16 19

SS35611 h 12 7

SS35814 h 4 24

SS35827 h 8 67

PM25477 hv 6 4

PM25494 hv 4 -8

PM29568 hv 4 -21

PM46887 hv 16 10 53 16

SS11257 hv 6 9

SS35530 hv 14 -16

SS35825 hv 12 -6

SS54824 hv 16 -7

PM50063 n 11 5

SS35648 n 17 13 1 5

SS64963 n 11 10

PM29680 nw 11 68

PM29694 nw 15 17 60 56

PM29700 nw 29 38

PM31543 nw 15 60

PM29711 tp 4 2

PM83860 tp 7 5 -4 3

SS23291 tp 5 -2

SS35829 tp 4 4

43

7.6 Skewed East West Triangular Transformation

Figure 29- Plot of the horizontal and vertical error vectors for the skewed east-west triangular transformation.

Table 7 - Horizontal and vertical errors for the skewed east-west triangular transformation.


(mm) (mm) (mm) (mm)

SS35582 h 21 -11

SS35593 h 13 -25

SS35597 h 12 -30

SS35609 h 6 12 -8 23

SS35611 h 14 -12

SS35814 h 9 12

SS35827 h 8 60

PM25477 hv 12 -8

PM25494 hv 13 -26

PM29568 hv 14 -45

PM29711 hv 20 -31

PM46887 hv 9 14 45 26

PM83860 hv 26 -26

SS11257 hv 8 -1

SS35530 hv 21 -31

SS35825 hv 7 -17

SS54824 hv 8 -26

PM50063 n 34 -22

SS35648 n 38 34 -23 21

SS64963 n 31 -19

PM29680 nw 31 22

PM29694 nw 30 37 18 14

PM31543 nw 51 3

PM29700 tp 4 0

SS23291 tp 7 7 0 0

SS35829 tp 9 0

44

7.7 Skewed North South Triangular Transformation

Figure 30 - Plot of the horizontal and vertical error vectors for the skewed north-south triangular transformation.

Table 8 - Horizontal and vertical errors for the skewed north-south triangular transformation.


(mm) (mm) (mm) (mm)

SS35582 h 20 7

SS35593 h 5 -3

SS35597 h 5 7

SS35609 h 12 10 29 20

SS35611 h 5 13

SS35814 h 12 18

SS35827 h 7 61

PM25477 hv 10 7

PM25494 hv 6 3

PM29711 hv 10 23

PM46887 hv 15 45

PM83860 hv 12 13 -2 12

SS11257 hv 20 -2

SS23291 hv 21 -12

SS35825 hv 9 -10

SS35829 hv 15 -8

SS54824 hv 15 -5

PM50063 n 27 33 -2 2

SS64963 n 38 -2

PM29680 nw 28 92

PM29694 nw 28 30 81 81

PM29700 nw 44 56

PM31543 nw 20 97

PM29568 tp 4 0

SS35530 tp 6 5 0 0

SS35648 tp 3 0

45

7.8 Maximum-Spaced Quadrilateral Transformation

Figure 31 - Plot of the horizontal and vertical error vectors for the maximum-spaced quadrilateral transformation.

Table 9 - Horizontal and vertical errors for the the maximum-spaced quadrilateral transformation.


(mm) (mm) (mm) (mm)

SS35582 h 13 2

SS35593 h 15 -11

SS35597 h 7 -18

SS35609 h 17 13 4 20

SS35611 h 19 1

SS35814 h 7 26

SS35827 h 11 75

PM25477 hv 13 5

PM25494 hv 12 -13

PM29568 hv 11 -33

PM29711 hv 9 -20

PM46887 hv 14 11 60 19

PM83860 hv 11 -14

SS11257 hv 5 13

SS35825 hv 13 -3

SS35829 hv 7 15

SS54824 hv 18 -13

PM50063 n 5 13 -10 11

SS35648 n 21 -11

PM29680 nw 7 30

PM29694 nw 7 15 27 22

PM29700 nw 30 10

PM31543 tp 6 9

SS23291 tp 3 5 16 13

SS35530 tp 4 -17

SS64963 tp 6 -8

46

8. Analysis

In the analysis of results the typical precision of NRTK was taken to be ±20mm in easting and

northing, and ±50mm in height. The positional component precisions correspond to a

precision in linear error of 28mm. For simplicity, the term ‘noise level’ will be used to

describe to the amount of variation expected due solely to the precision level of NRTK

positioning. Since the heights on the H marks are not LBL2 only the position component of

these marks is considered in the analysis unless explicitly stated otherwise.

8.1 Maximally Constrained Transformation

The maximally constrained transformation (MCT) was used as the baseline result against

which other transformations were assessed, as it combined having the highest number of

TPs with a strong geometrical arrangement. The results of the MCT were consistent with

expectations. The TPs exhibited the smallest group average error in both position and height

of 5mm and 13 mm respectively. As expected, the average linear error among the HV and H

marks was very similar at 11mm and 10mm respectively. The average vertical errors among

the H and HV were 15mm and 19mm, confirming the more accurate heights on the HV

marks. However, both the HV and H groups had a single large vertical error which, if

removed would have reduced the group errors to 6mm and 11mm. Larger linear errors were

found in the N and NW marks of 19mm and 14mm, most likely caused by them being

outside the valid area of the MCT. The vertical errors at the N and NW marks did not behave

as expected with the N group having an average error of only 7mm while the NW average

was 57mm. This means that the N group has a smaller average vertical error than the HV

group despite being outside the valid area. The cause of the large consistent vertical error in

the NW group is unknown. One possible cause is that those marks were part of a network

adjustment that did not include the remainder of the occupied marks.

With the single exception of the height component at PM46887, all the HV and H marks

within the valid area exhibited errors that were less than the noise level expected with

NRTK. The orientation of the error vectors for the HV and H marks was well distributed,

showing a slight trend to the east with some north-south variation.

47

8.2 Regular Quadrilateral Transformation

The regular quadrilateral transformation (RQT) and the MCT had near identical average

group linear errors, with differences of at most 2mm. The vertical errors were also largely

consistent with the MCT, though there was a change in the average vertical errors for the N

marks which increased from 7mm to 18mm. Within the valid area the linear and vertical

errors were within the noise level, with the exception of the vertical component of

PM46887. The orientation of the positional error vectors in the valid area, as well as the H

and HV marks, also remained consistent with the MCT. The large vertical error at each NW

marks was also repeated.

The increase in vertical error among the N marks is most likely due to removal of PM83860,

which was the northernmost TP in the MCT. As a result, in comparison to the MCT, the RQT

would have a very slight bias to the south which appears to have caused a corresponding

increase in errors among points located north of the valid area. The height error at

PM46887 remained very large in comparison with other marks in the valid area, though the

error reduced fractionally from 51mm to 49mm.

The RQT was able to able to produce the same level of error as the MCT which indicates that

reducing the number of TPs from the relatively high number of seven to four, while

maintaining a strong geometry, will not degrade the consistency of the resulting site

transformation.

8.3 Regular Triangular Transformation

All H and HV marks in the regular triangular transformation (RTT), with the exception of

height on PM46887, were within in the NRTK noise level. In comparison with the MQT the

RTT produced different group error behaviour. The MQT showed the linear error of 5mm

increasing by a factor of two when compared with the average error at the HV and H marks

of 11mm and 10mm respectively. The RTT did not have the same relationship, instead it

produced a smaller range between the linear error at the TP, HV and H marks with average

values of 8mm, 10mm and 12mm respectively. These values are larger than those produced

by the MCT and the smaller spread of average errors between groups indicates the RTT has

performed marginally poorer than the MCT in achieving consistency between the two

datums. An interesting feature of the vertical error plot for the RTT is the near zero vertical

48

errors produced by marks who lie on a line that runs through PM83860, the northernmost

TP, and SS35530, the south-west TP. It appears that along the alignment between these two

TPs a very accurate match in height was achieved, perhaps as a result of the triangular

geometry.

8.4 Dense Quadrilateral Transformation

The much tighter arrangement of control marks in the dense transformations was intended

to examine if reducing the size of the valid area would lead to a change in the consistency

between the transformed GNSS data and the ground coordinates. The four TPs in the dense

quadrilateral transformation (DQT) were chosen because they would enclose the majority of

the check marks and give a good indication of any change in the performance of the site

transformation.

Comparing the plot of horizontal error vectors of marks that lie in the valid area of the DQT

with the errors of only those points in MCT indicates no change in consistency. Both the size

and orientation of the horizontal error vectors within the valid area of the DQT exhibit only

fractional changes. There are also small changes in the vertical errors within the valid area

between the MCT and the DQT but, like the horizontal errors, the magnitude of the changes

is too small to indicate a systematic effect due to the change in TPs. Only the height

component of the N and NW marks show a significant change with the average error for the

NW marks improving 57mm to 33mm while the N mark average error increased from 7mm

to 35mm.

8.5 Skewed Quadrilateral Transformation

The skewed quadrilateral transformation (SQT) exhibits the same group error behaviour as

the MCT with a doubling in the average linear error of 5mm at the TPs in comparison with

10mm at the HV and H check marks. The average error values for the vertical errors are also

consistent between the MCT and the SQT to within 2mm.

The very high level of consistency with the MCT indicated by the numerical values, as well as

the orientation of the horizontal error vectors, indicates that the SQT was able to achieve

the same level of consistency between transformed GNSS data and ground coordinates as

the MCT. This was despite the SQT having three fewer TPs and a less regular geometry. The

49

SQT results indicate that a fairly slight degradation in the geometry of the TPs in this case,

had no detectable impact on the quality of the transformation.

8.6 Skewed East-West Triangular Transformation

The final three transformations presented here depart from the intended functions that

were assigned to the marks when the test network was planned. The change from the

intended function of the marks in the final three transformations is due to the robustness of

the transformations to the changes in geometry and number of TPs that had been applied in

the previous cases. More challenging examples of poor geometry with fewer TPs, as well as

wide separation were included to determine their impact of the performance of site

transformations. The final three transformations should never be used in practical

applications, and would only be considered where survey marks were very sparse and no

other option was available.

The skewed east-west triangular transformation (SEWT) used three nearly collinear TPs that

approximate a line running north-west to south-east. Using collinear TPs is an extremely

poor geometry as it heavily biases the transformation toward modelling errors that

dominate along the line of best fit through the location of the TPs.

The SEWT produced error vectors with significantly different magnitudes and orientations in

comparison with the previous transformations. The eastward trend in orientation seen in

the first six transformations was reversed in the SEWT to become a westward trend. In all

function groups, except for the TPs, the average errors were larger than those for the MCT,

confirming the detrimental impact of the poor TP geometry.

The TP marks had average horizontal and vertical errors of 7mm and 0mm respectively. The

perfect height consistency is most likely a result of using only three TPs with a collinear

geometry which means that the height component is effectively being modelled as line in

three dimensions rather than as a plane. This means that the least squares process is able to

generate parameters that produce a perfect match in height at the TPs but is unreliable

when the distance from the TPs increases. Also, changing the function of PM29700 from NW

to TP biased the transformation so as to greatly reduce the large vertical errors that were

observed at NW points in the first five transformations. It is also interesting to note that,

50

within the strict bounds of the valid area, the SEWT was able to keep the linear and vertical

errors within the noise level of NRTK despite its very poor geometry.

8.7 Skewed North-South Triangular Transformation

The skewed north-south triangle transformation (SNST) has nearly collinear TPs arranged in

an approximately north south line. There was a significant change in error behaviour in

comparison with the SEWT. The most striking feature of the error plots for the SNST is the

north-south line of check marks which all have vertical errors very close to zero. As

discussed in Section 8.6 this is most likely due to the three collinear TPs leading to the

height component being modelled as a line rather than a surface. Similarly to the SEWT, the

horizontal error at marks within the valid area of the SNST was within noise level of NRTK.

The magnitude of horizontal errors increases rapidly away from the valid area, especially at

the N and NW marks.

8.8 Maximum-Spaced Quadrilateral Transformation

The maximum-spaced quadrilateral transformation (MSQT) was included to give an

indication of whether dispersing the TPs while maintaining a regular geometry would have a

detectable change on the consistency between the transformed GNSS and ground

coordinate values. The MSQT resulted in a broad increase in error at many points, in

contrast to the MCT where there is a clearer delineation between the magnitude of errors

within and outside the valid area. The change in the distribution of errors is shown as a

comparative histogram in Figure 32. The comparative histogram shows that in the MSQT

fewer marks have very small or very large errors, with a large proportion of marks sharing a

similar magnitude of error.

Figure 32 - Comparative histogram of linear errors produced by the MSQT and the MCT.

10 5 0 5 10

5

15

25

More

Frequency

Linear Error

(mm) MSQT

MCT

51

The MSQT results demonstrate the degradation in the quality of a site transformation

caused by having too few TP’s placed at large separations. This approach forces the least

squares process to attempt to model a larger area, which will usually mean an increased

level of network distortions, based on a sparse sample which will inevitably degrade the

resulting transformation.

8.9 Summary

Certain marks in the test network showed consistent errors across all transformations.

PM46887 showed a significant positive vertical error in all transformations. The significant

horizontal error produced by PM29700 initially seemed to be due to lying outside the valid

area for most of the transformations. However, the large error at PM29700 remained in the

MSQT, which produced a centralised error distribution. The error behaviour of both

PM46887 and PM29700 suggests that an error was introduced in those marks from GNSS or

SCIMS coordinates, but determining which is not possible using the available data.

Viewing the results as a whole indicates the flexibility of similarity transformations in

achieving positioning consistency for users, even under challenging circumstances. With the

exception of the MSQT, and consistently erroneous marks, every transformation produced

errors within the noise level of NRTK for those marks within the valid area.

Making general comments about the errors produced at marks outside the valid area is

difficult due to the range of behaviours that were observed. The N and NW marks were

intended to contribute to this issue by giving an indication of how fast distortions

accumulate with distance from the valid area. The results that were achieved show some

common error behaviours within the separate N and NW groups, but no strong

commonality in terms of direction and magnitude of error, or increasing distance from the

valid area. This may indicate that the N and NW marks should have been extended further

to obtain a clearer indication of the relationship between distance from the valid area and

the loss of consistency with the ground coordinates.

One interesting aspect of the results is the very poor ability of errors at the TPs to indicate

the quality of a transformation when viewed in isolation. The residuals at the TPs are relied

upon as a quality indicator by surveyors when they perform site transformations using on-

52

board software in the field. Among the site transformations presented here, residuals alone

are a very poor indicator of the quality of the results. To take the most extreme example,

when viewed in isolation the linear error for the MCT and SQT, which have maximum values

of 10mm and 6mm, give an inverted indication of their true relative quality, as shown in

Table 10.

Table 10 - Linear error at TPs for the MCT and SNST.

Max. Constraint Skewed N-S Triangular

(mm) (mm)

Linear Error at TPs

4 4

2 6

10 3

4

4

10

4

The poor performance of linear error in indicating the quality of a transformation places the

onus on surveyors to have an understanding that a certain level of error is expected in site

transformations. In instances where drastically smaller than expected errors are produced,

this may indicate an unreliable result and require further examination.

53

9. Conclusion

The results presented here confirm the reliability of site transformations in achieving

consistency between GNSS and ground coordinates. Where a standard, or even slightly

compromised, geometry of marks was used in the site transformation the level of

agreement between the transformed GNSS data and SCIMS coordinates in the valid area

was, in the vast majority of cases, within the noise level of NRTK and therefore provided

seamless positioning from the user perspective. When the selection of TPs deviated severely

from the optimal configuration, marks within the valid area had errors within the NRTK

noise limit, though any move beyond the valid areas resulted in rapid accumulation of

errors.

The results also reinforce the importance of surveyors’ understanding of the spatial

behaviour, and expected magnitude, of differences between transformed GNSS and SCIMS

coordinates. This understanding is essential is to ensuring that when site transformation are

performed numerical errors are not accepted at face value, but are weighed against the

surveyor’s judgement and knowledge of the accuracy that can be expected from the

technology and methods they are using.

54

Appendix 1: SiteTran MATLAB Script

%Site Transformation Script using Condition Method Least Squares %Based on method shown in (Harvey, 2006) clc format compact format short g %______________________load coordinate observation to input files START %Need scims input as csv as follows: %MARK_ID,mgaeasting,mganorthin,ahdheight,latitude_d ecdeg, %longitude_decdeg,Ausgeoid09 %import scims data of all accupied marks into array fid = fopen('scims_data.csv'); scims_array = textscan(fid,'%s %f %f %f %f %f % f %s', 'delimiter',','... ,'headerlines',1); fclose(fid); %convert array into matrices column by column %NOTE scims_pid remains an array as a matrix cannot contain string values scims_pid = scims_array{:,1,:}; scims_e = scims_array{:,2,:}; scims_n = scims_array{:,3,:}; scims_en = [scims_e scims_n]; ahd = scims_array{:,4,:}; scims_lat = degtorad(scims_array{:,5,:}); scims_lon = degtorad(scims_array{:,6,:}); ausgeoid = scims_array{:,7,:}; type = scims_array{:,8,:}; gda_ellht = ahd + ausgeoid; scims_llh = [scims_lat scims_lon gda_ellht]; clear scims_array %build single matrix with all scims data scims_data = [scims_e scims_n scims_lat scims_lon a hd ausgeoid gda_ellht]; %Need GNSS input as csv as follows: %Mark_ID,GPS_x,GPS_y,GPS_z,type %import GNSS field data of all accupied marks into array fid = fopen('gnss_data.csv'); gps_array = textscan(fid,'%s %f %f %f %s','del imiter', ',','headerlines',1); fclose(fid); %extract gps_array to matrices gps_pid = gps_array{1,1}; gps_x = gps_array{1,2}; gps_y = gps_array{1,3}; gps_z = gps_array{1,4}; gps_tag = gps_array{1,5}; gps_xyz = [gps_x gps_y gps_z]; [num_gps,~] = size(gps_pid); clear gps_array %__________________________load coordinate observat ion from input files END %_______________________________sort GNSS data into variable by type START gps_tp = []; gps_tp_id = {}; gps_hv = []; gps_hv_id = {}; gps_h = []; gps_h_id = {}; gps_n = []; gps_n_id = {}; gps_nw = []; gps_nw_id = {};

55

for row = 1:num_gps if strcmp(gps_tag{row},'tp') gps_tp = [gps_tp; gps_xyz(row,:)]; gps_tp_id = [gps_tp_id; gps_pid{row}]; end if strcmp(gps_tag{row},'hv') gps_hv = [gps_hv; gps_xyz(row,:)]; gps_hv_id = [gps_hv_id; gps_pid{row}]; end if strcmp(gps_tag{row},'h') gps_h = [gps_h; gps_xyz(row,:)]; gps_h_id = [gps_h_id; gps_pid{row}]; end if strcmp(gps_tag{row},'n') gps_n = [gps_n; gps_xyz(row,:)]; gps_n_id = [gps_n_id; gps_pid{row}]; end if strcmp(gps_tag{row},'nw') gps_nw = [gps_nw; gps_xyz(row,:)]; gps_nw_id = [gps_nw_id; gps_pid{row}]; end end %store variable with number of each type of mark [num_tp,~] = size(gps_tp); [num_hv,~] = size(gps_hv); [num_h,~] = size(gps_h); [num_n,~] = size(gps_n); [num_nw,~] = size(gps_nw); %_______________________________sort GNSS data into variable by type END %_______________________________sort SCIMS data int o variable by type START [num_scims,~] = size(scims_data); [scims_x scims_y scims_z] = ell2xyz(scims_llh(:,1), scims_llh(:,2),... scims_llh(:,3)); scims_xyz = [scims_x scims_y scims_z]; scims_tp = []; scims_tp_id = {}; scims_tp_en = []; tp_ahd = []; tp_ausg = []; % scims_hv = []; scims_hv_id = {}; scims_hv_en = []; hv_ahd = []; hv_ausg = []; % scims_h = []; scims_h_id = {}; scims_h_en = []; h_ahd = []; h_ausg = []; % scims_n = []; scims_n_id = {}; scims_n_en = []; n_ahd = []; n_ausg = []; % scims_nw = []; scims_nw_id = {}; scims_nw_en = []; nw_ahd = []; nw_ausg = [];

56

%seperate scims data into coord types for row = 1:num_scims; %extract transf pts for row_a = 1:num_tp if strcmp(gps_tp_id{row_a},scims_pid{row}); scims_tp = [scims_tp;scims_xyz(row,:)]; scims_tp_id = [scims_tp_id; scims_pid{ro w}]; scims_tp_en = [scims_tp_en; scims_en(row ,:)]; tp_ahd = [tp_ahd; ahd(row)]; tp_ausg = [tp_ausg; ausgeoid(row)]; end end %extract hv pts for row_a = 1:num_hv if strcmp(gps_hv_id{row_a},scims_pid{row}); scims_hv = [scims_hv;scims_xyz(row,:)]; scims_hv_id = [scims_hv_id; scims_pid{ro w}]; scims_hv_en = [scims_hv_en; scims_en(row ,:)]; hv_ahd = [hv_ahd; ahd(row)]; hv_ausg = [hv_ausg; ausgeoid(row)]; end end %extract h pts for row_a = 1:num_h; if strcmp(gps_h_id{row_a},scims_pid{row}); scims_h = [scims_h;scims_xyz(row,:)]; scims_h_id = [scims_h_id; scims_pid{row} ]; scims_h_en = [scims_h_en; scims_en(row,: )]; h_ahd = [h_ahd; ahd(row)]; h_ausg = [h_ausg; ausgeoid(row)]; end end %extract n for row_a = 1:num_n if strcmp(gps_n_id{row_a},scims_pid{row}); scims_n = [scims_n;scims_xyz(row,:)]; scims_n_id = [scims_n_id; scims_pid{row} ]; scims_n_en = [scims_n_en; scims_en(row,: )]; n_ahd = [n_ahd; ahd(row)]; n_ausg = [n_ausg; ausgeoid(row)]; end end %extract nw for row_a = 1:num_nw if strcmp(gps_nw_id{row_a},scims_pid{row}); scims_nw = [scims_nw;scims_xyz(row,:)]; scims_nw_id = [scims_nw_id; scims_pid{ro w}]; scims_nw_en = [scims_nw_en; scims_en(row ,:)]; nw_ahd = [nw_ahd; ahd(row)]; nw_ausg = [nw_ausg; ausgeoid(row)]; end end end %_______________________________sort SCIMS data int o variable by type END %load vector of initial parameter values from file load 'init.txt'; %_________________________________________________C onstruct Q matrix START Q = 0.05^2*diag(ones(6*num_tp,1)); %_________________________________________________C onstruct Q matrix END %==========================Begin Iterative Solution ======================== %set max number of iterations

57

max_iter = 3; %set counter iter = 0; while iter <= max_iter %____________________________________Set Initia l Parameter Values START %name initial value variables: s is scale (NOT IN PPM), o is omega, %t is theta, k is kappa, input as radians, t is axis translat s = init(1); o = init(2); t = init(3); k = init(4); tX = init(5); tY = init(6); tZ = init(7); %______________________________________Set Init ial Parameter Values END %____Construct Rotation/Translation using inita l estimates matrix START Rx = [[1 0 0];[0 cos(o) sin(o)];[0 -sin(o) cos( o)]]; Ry = [[cos(t) 0 -sin(t)];[0 1 0];[sin(t) 0 cos( t)]]; Rz = [[cos(k) sin(k) 0];[-sin(k) cos(k) 0];[0 0 1]]; R = Rz*Ry*Rx; %construct full R matrix T = [tX;tY;tZ]; %______Construct Rotation/Translation using ini tal estimates matrix END %______________________________________________ Construct A matrix START A = []; for row = 1:num_tp; A_temp = []; Xa = gps_tp(row,1); Ya = gps_tp(row,2); Za = gps_tp(row,3); %scale A_temp(1,1)= R(1,1)*Xa+R(1,2)*Ya+R(1,3)*Za; A_temp(2,1)= R(2,1)*Xa+R(2,2)*Ya+R(2,3)*Za; A_temp(3,1)= R(3,1)*Xa+R(3,2)*Ya+R(3,3)*Za; %omega A_temp(1,2)= s*(-R(1,3)*Ya+R(1,2)*Za); A_temp(2,2)= s*(-R(2,3)*Ya+R(2,2)*Za); A_temp(3,2)= s*(-R(3,3)*Ya+R(3,2)*Za); %theta A_temp(1,3)= -s*cos(k)*(R(3,1)*Xa+R(3,2)*Ya +R(3,3)*Za); A_temp(2,3)= s*sin(k)*(R(3,1)*Xa+R(3,2)*Ya+ R(3,3)*Za); B1 = cos(k)*cos(k)*cos(t)+sin(k)*sin(k)*cos (t); B2 = cos(k)*R(1,2)-sin(k)*R(2,2); B3 = cos(k)*R(1,3)-sin(k)*R(2,3); A_temp(3,3)= s*(B1*Xa+B2*Ya+B3*Za); %kappa A_temp(1,4)= s*(R(2,1)*Xa+R(2,2)*Ya+R(2,3)* Za); A_temp(2,4)= -s*(R(1,1)*Xa+R(1,2)*Ya+R(1,3) *Za); A_temp(3,4)= 0; A_temp = [A_temp eye(3,3)]; A = [A;A_temp]; end %______________________________________________ _Construct A matrix END %______________________________________________ Construct b matrix START b = [];

58

for p=1:num_tp; PtsA = gps_tp(p,:)'; %extract one set of GN SS coords row-wise PtsB = scims_tp(p,:)'; %extract one set of SCIMS coords row-wise temp = -(s*R*PtsA + T - PtsB); b = [b;temp]; end %______________________________________________ __Construct b matrix END %______________________________________________ Construct B matrix START B_lefthand=[]; for p=1:num_tp; B_lefthand = blkdiag(B_lefthand,s*R); %const r blkdiag left half of B end B = [B_lefthand -eye(3*num_tp)]; %append negati ve ident matrix on RHS %______________________________________________ _Construct B matrix END %_____________________________calculate correct ions and adjusted values %delX = inv(A'*inv(B*Q*B')*A)*A'*inv(B*Q*B')*b %use backslash operator not inv function for be tter accuracy left_hand = ((A'/(B*Q*B'))*A); right_hand = ((A'/(B*Q*B'))*b); delX = left_hand\right_hand; adj_param = init + delX; init = adj_param; iter = iter + 1; end %========================End of Iterative Solution= ======================== %__________________________________________________ ____calculate precisions para_prec = inv(A'*((B*Q*B')\A)); %__________________________________________________ ____calculate residuals v = (-Q*B')/(B*Q*B')*(A*delX-b); %__________________________________________________ ____calculate VF VF = (v'*(Q\v))/((num_tp*3)-7); if max(v)>0.005 fid = fopen('trans_out.txt','w'); fprintf(fid,'WARNING: Residuals larger than 5mm .\n'); fprintf(fid,'\n'); fclose(fid); end %____________________________convert rotations to d egrees for display adj_param(2:4) = rad2deg(adj_param(2:4)); %____________________________export transformation param to text file fid = fopen('trans_out.txt','a'); fprintf(fid, 'adjusted parameters\n'); fprintf(fid, '%.6f\n', adj_param'); fprintf(fid, '\n'); fprintf(fid, 'VF\n'); fprintf(fid, '%.6f\n', VF'); fprintf(fid, '\n'); fprintf(fid, 'parameter precisions\n'); fprintf(fid, '%.6f %.6f %.6f %.6f %.6f %.6f %.6f\n' ,para_prec); fprintf(fid, '\n'); fprintf(fid, 'residuals\n'); fprintf(fid, '%.6f\n', v'); fclose(fid); %____________________________convert rotations to r adians

59

adj_param(2:4) = deg2rad(adj_param(2:4)); %create matrix with GNSS XYA coords sorted by type gps_sorted = [gps_tp;gps_hv;gps_h;gps_n;gps_nw]; %Apply tranformation parameters to all GNSS coordin ates and convert %to lat lon and ell height %xyz2ell function taken from: %http://www.craymer.com/software/matlab/geodetic/ gps_tran_llh = []; for row = 1:num_gps a = gps_sorted(row,:)'; b = s*R*a+T; [lat lon h] = xyz2ell3(b(1),b(2),b(3)); temp1 = degrees2dms(radtodeg(lat)); temp2 = degrees2dms(radtodeg(lon)); temp3 = [temp1 temp2 h]; gps_tran_llh = [gps_tran_llh;temp3]; end %Break point required here to convert lat/lon to gr id in excel spreadsheet %and add data to current folder in file: gps_tran_e n.mat load gps_tran_en.mat %seperate gps_tran_en.mat into variables based on t ype gps_tp_en = gps_tran_en(1:num_tp,:); gps_tp_eht = gps_tran_llh(1:num_tp,7); row = num_tp; gps_hv_en = gps_tran_en(row+1:row+num_hv,:); gps_hv_eht = gps_tran_llh(row+1:row+num_hv,7); row = row+num_hv; gps_h_en = gps_tran_en(row+1:row+num_h,:); gps_h_eht = gps_tran_llh(row+1:row+num_h,7); row = row+num_h; gps_n_en = gps_tran_en(row+1:row+num_n,:); gps_n_eht = gps_tran_llh(row+1:row+num_n,7); row = row+num_n; gps_nw_en = gps_tran_en(row+1:row+num_nw,:); gps_nw_eht = gps_tran_llh(row+1:row+num_nw,7); row = row+num_nw; %calculate east/north difference between grid coord s from SCIMS %and those from transformed GNSS data tp_en_err = scims_tp_en - gps_tp_en; hv_en_err = scims_hv_en - gps_hv_en; h_en_err = scims_h_en - gps_h_en; n_en_err = scims_n_en - gps_n_en; nw_en_err = scims_nw_en - gps_nw_en; %calculate AHD difference between SCIMS and transfo rmed GNSS data tp_ahd_err = tp_ahd - (gps_tp_eht - tp_ausg); hv_ahd_err = hv_ahd - (gps_hv_eht - hv_ausg); h_ahd_err = h_ahd - (gps_h_eht - h_ausg); n_ahd_err = n_ahd - (gps_n_eht - n_ausg); nw_ahd_err = nw_ahd - (gps_nw_eht - nw_ausg); %Plot horzontal distortion vector %set scale (exaggeration) factor for distortion vec tor display s = 30000; subplot(1,2,1) %plot indicate distrion vector in lower left of gra ph quiver(334500,6241000,s*0.010,0,'k','AutoScale','of f'); hold on %plot distortion vectors quiver(scims_tp_en(:,1),scims_tp_en(:,2),s*tp_en_er r(:,1),s*tp_en_err(:,2),...

60

'r','AutoScale','off','LineWidth',2); hold on quiver(scims_hv_en(:,1),scims_hv_en(:,2),s*hv_en_er r(:,1),s*hv_en_err(:,2)... ,'b','AutoScale','off'); hold on quiver(scims_h_en(:,1),scims_h_en(:,2),s*h_en_err(: ,1),s*h_en_err(:,2)... ,'b','AutoScale','off'); hold on quiver(scims_n_en(:,1),scims_n_en(:,2),s*n_en_err(: ,1),s*n_en_err(:,2),... 'm','AutoScale','off'); hold on quiver(scims_nw_en(:,1),scims_nw_en(:,2),s*nw_en_er r(:,1),s*nw_en_err(:,2)... ,'c','AutoScale','off'); hold on %plot distinctive markers to differentiate marks by type quiver(scims_tp_en(:,1),scims_tp_en(:,2),zeros(num_ tp,1),zeros(num_tp,1),'rp'); hold on quiver(scims_hv_en(:,1),scims_hv_en(:,2),zeros(num_ hv,1),zeros(num_hv,1),'bd'); hold on quiver(scims_h_en(:,1),scims_h_en(:,2),zeros(num_h, 1),zeros(num_h,1),'bd'); hold on quiver(scims_n_en(:,1),scims_n_en(:,2),zeros(num_n, 1),zeros(num_n,1),'md'); hold on quiver(scims_nw_en(:,1),scims_nw_en(:,2),zeros(num_ nw,1),zeros(num_nw,1),'cd'); axis equal xlabel({'Easting','Distortion vector of 10mm in sou th-western corner of plot'}... ,'FontSize',12,'FontWeight','bold') ylabel('Northing','FontSize',12,'FontWeight','bold' ) xlim([333500 339000]) ylim([6240000 6246500]) set(gca,'XTick',334000:2000:338000) set(gca,'XTickLabel',{'334000','336000','336000'}) set(gca,'YTick',6240000:2000:6246000) set(gca,'YTickLabel',{'6240000','6242000','6244000' ,'6246000'}) title('Horizontal Errors','FontWeight','bold') %Plot vertical distortion %vector%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %set scale (exaggeration) factor for distortion vec tor display s = 14000; subplot(1,2,2) quiver(334500,6241000,0,s*0.05,'k','AutoScale','off '); hold on quiver(scims_tp_en(:,1),scims_tp_en(:,2),zeros(num_ tp,1),s*tp_ahd_err,'r'... ,'AutoScale','off'); hold on quiver(scims_hv_en(:,1),scims_hv_en(:,2),zeros(num_ hv,1),s*hv_ahd_err,'b'... ,'AutoScale','off'); hold on quiver(scims_n_en(:,1),scims_n_en(:,2),zeros(num_n, 1),s*n_ahd_err,'m',... 'AutoScale','off'); hold on quiver(scims_nw_en(:,1),scims_nw_en(:,2),zeros(num_ nw,1),s*nw_ahd_err,'c'... ,'AutoScale','off'); hold on quiver(scims_tp_en(:,1),scims_tp_en(:,2),zeros(num_ tp,1),zeros(num_tp,1),'rp'); hold on quiver(scims_hv_en(:,1),scims_hv_en(:,2),zeros(num_ hv,1),zeros(num_hv,1),'bd'); hold on quiver(scims_n_en(:,1),scims_n_en(:,2),zeros(num_n, 1),zeros(num_n,1),'md'); hold on quiver(scims_nw_en(:,1),scims_nw_en(:,2),zeros(num_ nw,1),zeros(num_nw,1),'cd'); axis equal xlabel({'Easting','Distortion vector of 50mm in sou th-western corner of plot'}... ,'FontSize',12,'FontWeight','bold') ylabel('Northing','FontSize',12,'FontWeight','bold' ) xlim([333500 339000]) ylim([6240000 6246500])

61

set(gca,'XTick',334000:2000:338000) set(gca,'XTickLabel',{'334000','336000','336000'}) set(gca,'YTick',6240000:2000:6246000) set(gca,'YTickLabel',{'6240000','6242000','6244000' ,'6246000'}) title('Vertical Errors','FontWeight','bold'); %following code displays title over two subplots ob tained from: %http://www.mathworks.com/matlabcentral/newsreader/ view_thread/171543 ha = axes('Position',[0 0 1 1],'Xlim',[0 1],'Ylim', [0 1],'Box','off',... 'Visible','off','Units','normalized', 'clipping ' , 'off'); text(0.5, 1,'\bf INSERTNAME Transformation','Horizo ntalAlignment',... 'center','VerticalAlignment','top') %output of variable named excel_out for import and display with excel point_sequence = [gps_tp_id gps_hv_id gps_h_id gps_n_id gps_nw_id]; en_err = [tp_en_err hv_en_err h_en_err n_en_err nw_en_err]; [bearing lin_err] = cart2pol(en_err(:,1),en_err(:,2 )); ahd_err = [tp_ahd_err hv_ahd_err h_ahd_err n_ahd_err nw_ahd_err]; excel_out = [lin_err bearing ahd_err]; %export to text file for use in excel fid = fopen('excel_out.txt','w'); fprintf(fid, 'excel out\n'); fprintf(fid, '%.6f %.6f %.6f\n', excel_out'); fclose(fid);

62

Appendix 2: VBA Module from Redfearn’s Formulae Spreadsheet

Option Explicit

Sub ll2en()

Dim col_lat_d, col_lat_m, col_lat_s, col_lon_d, col_lon_m, col_lon_s, col_h As

Integer

Dim lat_d, lat_m, lon_d, lon_m As Integer

Dim lat_s, lon_s As Double

Dim active_row, input_row As Integer

Dim E, N As Double

col_lat_d = 2

col_lat_m = 3

col_lat_s = 4

col_lon_d = 5

col_lon_m = 6

col_lon_s = 7

col_h = 4

active_row = 4

input_row = 4

Sheets("Thesis").Select

'Debug.Print col_lat; ActiveSheet.Cells(active_row, col_lat)

While ActiveSheet.Cells(active_row, col_lat_d).Value <> ""

lat_d = ActiveSheet.Cells(active_row, col_lat_d)

lat_m = ActiveSheet.Cells(active_row, col_lat_m)

lat_s = ActiveSheet.Cells(active_row, col_lat_s)

lon_d = ActiveSheet.Cells(active_row, col_lon_d)

lon_m = ActiveSheet.Cells(active_row, col_lon_m)

lon_s = ActiveSheet.Cells(active_row, col_lon_s)

'Debug.Print lat_d; lat_m; lat_s; lon_d; lon_m; lon_s

Sheets("Latitude & Longitude to E,N,Zne").Select

ActiveSheet.Cells(input_row, 3) = lat_d

ActiveSheet.Cells(input_row, 4) = lat_m

ActiveSheet.Cells(input_row, 5) = lat_s

ActiveSheet.Cells(input_row, 9) = lon_d

ActiveSheet.Cells(input_row, 10) = lon_m

ActiveSheet.Cells(input_row, 11) = lon_s

E = ActiveSheet.Cells(40, 5)

N = ActiveSheet.Cells(40, 11)

Sheets("Thesis").Select

ActiveSheet.Cells(active_row, 9) = E

ActiveSheet.Cells(active_row, 10) = N

active_row = active_row + 1

Wend

End Sub

63

Appendix 3: Trimble GNSS Planning Software GDOP Graphs

64

Appendix 4: SCIMS Coordinate Information for Marks Included in Field Work

Mark

Name

MGA Easting MGA Northing MGA Zone GDA Class GDA Order AHD AHD Class AHD Order GDA Latitude GDA Longitude

(m) (m) (m) (dec. degrees) (dec. degrees)

1 PM25477 337418.843 6242451.456 56 B 2 24.943 LB L2 -33.9458723889 151.2406897750

2 PM25494 336884.563 6242342.625 56 B 2 24.660 LB L2 -33.9467707194 151.2348904056

3 PM29568 336228.277 6242260.474 56 B 2 20.548 LB L2 -33.9474092722 151.2277762083

4 PM29680 335348.770 6244776.014 56 B 2 22.133 LB L2 -33.9245952222 151.2187352139

5 PM29694 335610.439 6244562.018 56 B 2 19.970 LB L2 -33.9265652361 151.2215247667

6 PM29700 335876.544 6244221.048 56 B 2 19.740 LB L2 -33.9296805111 151.2243386500

7 PM29711 335917.498 6243220.696 56 B 2 20.224 LB L2 -33.9387047472 151.2245943500

8 PM31543 334412.104 6244927.635 56 B 2 15.898 LB L2 -33.9230814444 151.2086349333

9 PM46887 338007.845 6242799.731 56 B 2 48.887 LB L2 -33.9428236806 151.2471251917

10 PM50063 337342.188 6245163.076 56 B 2 47.718 LB L2 -33.9214159528 151.2403636278

11 PM83860 337118.004 6243693.544 56 B 2 28.388 LB L2 -33.9346287417 151.2376665083

12 SS11257 338061.541 6243541.116 56 B 2 38.618 LB L2 -33.9361485250 151.2478429417

13 SS23291 338444.935 6241697.098 56 B 2 33.800 LB L2 -33.9528309111 151.2516495972

14 SS35530 336763.006 6241428.628 56 B 2 28.791 LB L2 -33.9549912639 151.2334052611

15 SS35582 337262.490 6242130.424 56 B 2 29.445 D 4 -33.9487422139 151.2389389361

16 SS35593 337168.004 6242585.904 56 B 2 22.638 D 4 -33.9446215583 151.2380015167

17 SS35597 336540.572 6242372.895 56 B 2 18.906 D 4 -33.9464444250 151.2311752000

18 SS35609 336578.548 6242915.916 56 B 2 22.988 D 4 -33.9415551583 151.2316872306

19 SS35611 337064.095 6242965.105 56 B 2 24.366 D 4 -33.9411870722 151.2369480833

20 SS35648 337189.688 6244342.021 56 B 2 31.112 LB L2 -33.9287939889 151.2385621778

21 SS35814 337806.179 6243179.399 56 B 2 29.066 D 4 -33.9393699722 151.2450141639

22 SS35825 337742.287 6242863.595 56 B 2 44.562 LB L2 -33.9422070139 151.2442646583

23 SS35827 338005.295 6242573.292 56 B 2 41.333 D 4 -33.9448645750 151.2470557500

24 SS35829 338326.114 6242835.254 56 B 2 43.985 LB L2 -33.9425524194 151.2505742361

25 SS54824 337238.039 6243275.118 56 B 2 29.670 LB L2 -33.9384193222 151.2388870306

26 SS64963 337450.961 6245944.893 56 B 2 70.507 LB L2 -33.9143848556 151.2416846694

65

Appendix 5: Precision Plots of GNSS Coordinate Data

PM25477

PM25494

PM29568

66

PM29680

Note: The range on vertical precision plot for this point has been increased.

PM29694

PM29700

67

PM29711

PM31543

PM46887

68

PM50063

PM83860

SS11257

69

SS23291

SS35530

SS35582

70

SS35593

SS35597

SS35609

71

SS35611

SS35648

SS35814

72

SS35825

SS35827

SS35829

73

SS54824

SS64963

74

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An Investigation of CORS Site Transformations

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