1. Chapter 8: Three-Dimensional Coordinating Systems
Chapter7:3D Coordinating Systems
Chapter 8: Three-Dimensional Coordinating Systems
Dr. John Ogundare
BCIT, Burnaby

2. Chapter Objectives Be able to do the following:
Describe the commonly used three-dimensional coordinate reference systems
Discuss the needs and the common models for three-dimensional coordinating systems
Explain the concepts and principle of electronic coordinating system
Describe the features of three-dimensional coordination with Global Navigation Satellite System (GNSS)
Discuss the features and applications of three-dimensional coordination with electronic theodolites
Analyze the accuracy limitations of three-dimensional coordination with electronic theodolite, including three-dimensional traverse surveys
Describe the features and accuracy limitations of airborne laser scanning system as coordinating system
Describe the features and accuracy limitations of terrestrial laser scanning system as coordinating system

3. Coordinating System: Introduction (1/2)
Three-dimensional coordinating system is a system of hardware and software that allows three-dimensional (x, y, z) coordinates to be determined
Needed in industrial metrology
Measurement of antennas
Providing dimensional control on aircraft
Providing non-contact, real-time 3D coordinates of objects of different sizes
Needed in deformation monitoring applications
Three types of coordinating systems
(Global Navigation Satellite System (GNSS)
Electronic coordinating system
3D terrestrial laser scanning system

4. Coordinating System: Introduction (2/2)
Reference coordinate system is an integral part of coordinating system –needed to calculate positions and solve problems
Coordinate system is defined when origin, directions of axes and units (angular or linear) are defined
When origin and axes of the coordinate system are specified with regard to the earth, it forms a datum
A datum may be associated with a reference ellipsoid (for horizontal) or the geoid (for height system) on which measurements may be reduced
Three types of coordinate reference systems:
One-dimensional coordinate reference systems
Two-dimensional coordinate reference systems
Three-dimensional coordinate reference systems

5. One-Dimensional Coordinate System
One-dimensional coordinate system is about height determination for points on the earth surface or near the earth surface
It is a coordinate system with the origin on the geoid (or mean sea level) and z-axis along the direction of gravity
Heights are determined from measured elevation differences
The coordinate system is used to represent elevations or orthometric heights on maps

6. Two-Dimensional Coordinate System
Two-dimensional coordinate system can be divided into:
Coordinate reference systems on reference ellipsoid – locates positions in angular units as latitude and longitude on the surface of ellipsoid
Latitude and longitude coordinates are known as geodetic coordinates
Example of such coordinate system is North American Datum of (NAD83)
Coordinate reference systems on the plane –locates positions in linear units as easting and northing on the plane (as in plane surveying or map projections)
Example of such coordinate system is Universal Transverse Mercator projection (UTM)
Origin of coordinates system: center of map projection
X-axis direction in the East-West direction and y-axis direction in the North- South direction

7. Two-Dimensional Coordinate System: Advantages and Disadvantages
Some of the advantages of using 2D coordinate systems as computation model:
Separating horizontal (2D ) survey projects from leveling(1D) survey projects allow better positional accuracy to be achieved in each case
Orthometric heights (H) produced through leveling is practically meaningful in engineering (compared to 3D x, y, z coordinates)
Northing, easting and orthometric height coordinates are easy to manipulate in survey computations; some engineering projects are better done in map projection coordinate system
One disadvantage of using 2D coordinate system as computation model is that measurements must first be rigorously reduced to the reference ellipsoid, then to map projection plane be use in computations

8. Three-Dimensional Coordinate System
Three types of 3D coordinates systems
Conventional Terrestrial Reference System (CTRS) or International Terrestrial Reference System (ITRS)
Local geodetic (LG) system
Local astronomic (LA) system
The 3 systems locate positions in three linear dimensions (X, Y, Z) with respect to their origins
ITRS is the closest approximation of the geocentric natural coordinate system whose coordinate axes are defined by the directions of gravity and the spin axis of the earth
Natural coordinate system provides astronomic latitude and longitude and gravity potential as natural coordinates
LA system can also be considered as the natural coordinate system

9. Properties of Three Common 3D Coordinate Systems
ITRS System
LG System
LA System
Origin
Earth Center of mass
Instrument setup station projected to the reference ellipsoid.
Instrument setup point
Primary (or z-) axis
From the origin pointing to conventional terrestrial pole (CTP)
An orthogonal line passing through the origin on the reference ellipsoid
Along the direction of gravity or of plumb line when the survey instrument is level); referred to as Up direction.
Secondary (or x-) axis
A line from the origin passing through the intersection of the mean equator and the Greenwich mean meridian
A line tangent at the origin and aligned along the geodetic meridian, pointing towards the geodetic north
A line tangent at the origin and aligned along the astronomical meridian, pointing towards the true north; referred to as Northing direction.
Tertiary (or y-) axis
Along the line orthogonal to the z-x plane in a right-handed system.
Along the line orthogonal to the z-x plane in a left-handed system.
From the origin along the Easting direction.

10. Advantages of 3D Coordinate Systems
Important advantage of using 3D coordinate system as a computation model is that in using the measurements to compute the X, Y, Z coordinates of a point, one does not need to reduce the measurements to reference ellipsoid, but only needs to correct for atmospheric and instrumental errors
Commonly used in positioning nuclear accelerator and in alignment of radio telescope aerial arrays over a very long distance, relative to the center of mass of the earth
GNSS is an example of a system that provides coordinates in this model
Important disadvantage: ellipsoidal height (h) derived in this process is not practically useful as orthometric height, in engineering

11. Topographic Coordinate System
Assumes the earth is a plane (avoiding map projection)
Topographic surveying is a special type of 3D surveying for determining 3D (x, y, elevation) coordinates of features on the earth
Features include building corners, road centerlines, trees, ridges, valleys, etc.
Examples: aerial mapping and ground and underground surveys

12. Coordinate System for 3D Coordinating Systems
ITRS is commonly used in space or extraterrestrial techniques in 3D positioning involving large area
LG is commonly used in engineering projects, such as local deformation monitoring and alignment in industrial metrology where the earth’s curvature can be ignored
LG system is illustrated in Fig. 8.1
Right-handed LG (rLG) system in which x- and y-axes are switched in order for consistency with the convention used in North America
If (Northing, Easting and Up) terms are used instead of (x, y, z) rLG and LG systems will mean the same

13. Representation of Local Geodetic System
Fig. 8.1

14. 3D Coordination with GNSS
Positions by GPS are given as latitude (), longitude (λ) or as 3D Cartesian (X, Y, Z) coordinates based on WGS84 coordinate reference frame
WGS84 (G1150) is closely related to ITRF2000 (epoch ) so that
3D (X, Y, Z) coordinates by GPS is closely related to ITRS
It is earth-centered, earth-fixed datum
It produces 3D global coordinate system based on ITRF

15. 3D Coordination with Electronic Theodolites
Electronic coordinating system is for real-time calculations of 3D coordinates of target points
Electronic coordinating system consists
Two or more theodolites
Linked micro-computer
Applications: positioning of targets and deformation surveys over small area
Examples of coordinating system
Sokkia NET2100
Leica TMS (tunnel measurement system)

16. 3D Coordination with Electronic Theodolites: Operating Principle
Operation principle is based on 3D intersection for positioning based on free-stationing technique
Methodology:
Simultaneous use of 2 theodolites (T1 and T2) in Fig. 8.2
Or using one single theodolite in free-stationing technique
Observables (Fig. 8.2):
Horizontal angles 1 and 2
Zenith angles z1 and z2
Baseline measurement b
3D coordinates of target point P are determined from the measurements using trigonometric functions
Baseline b may not be measured directly since short baseline are difficult to measure precisely
Well-calibrated invar scaling bars of known lengths are located at suitable locations as part of the micro-network to be measured; markers fixed to the ends of the bars are used as targets and angles are measured to them in triangulation survey

17. Electronic Coordinating System: 3D Intersection Problem
Fig. 8.2
Triangulation network may be established instead with horizontal directions
and zenith angles as observables
Distances are not measured directly
Invar scale bars with known lengths between targets marked on the
bars are used as part of the network
Least squares adjustment is applied to all measurements to obtain
unknown coordinates of desired points

18. Electronic Coordinating System: Field Data Reductions (1/2)
Observables in electronic coordinating system:
Azimuth, horizontal directions (horizontal angles), zenith angles and slope distances
Observations must be corrected for
Instrumental errors
Meteorological effects –refraction effects are minimal because of short distances involved
Gravity effects (deflection of vertical) to reduce measurements to reference ellipsoid
The effects cause angular traverse loop misclosures as in instrument leveling errors
Short distances involved cannot be precisely measured; horizontal and zenith angles are typical observables instead of distances

19. Electronic Coordinating System: Field Data Reductions (2/2)
Slope distances are not affected by gravity and are not
corrected for gravity effects
Measured zenith angle (Z´ij) from i to j is corrected for gravity:
i, i are the North-South and East-West components of the
deflection of the vertical at the instrument station i to station j
ij is the geodetic azimuth of line i to j
Measured azimuth (or direction) Aij is corrected for gravity:
Correction for misalignment of LG and LA north directions is Laplace
correction:
Z is the corrected zenith angle and i is the geodetic latitude of point i.

20. Electronic Coordinating System: Angle Reduction (3/3)
Last term (Cij) in in corrected azimuth in Eqn. (8.6) is
similar to leveling error; the term is zero when z = 90º
Measured angle (´) will not be affected by Laplace correction,
and the effect of deflection of the vertical components is
minimal and will be zero if backsight and foresight zenith
angles are 90º in relatively flat terrains

21. Electronic Coordinating System: 3D Coordinate Determination
Every setup point has a separate LA coordinate system with
its origin at the instrument setup point
Directions of all z-axes of all the LA systems will not be parallel
because of the curvature of the earth (geometric cause)
Azimuth of the same line in each LA system will be different
because of convergence of meridian (geometric cause) and deflection
of the vertical if reduced to the ellipsoid (gravimetric cause)
Traditional approach to solve misalignment of LA systems:
Fix one of the LA systems at one of the setup points
Translate all the other systems to the fixed system
If deflection of the vertical components are the same in an area:
They will affect different directions of astronomic and geodetic
North differently when transforming from LA to LG at each station

22. Electronic Coordinating System: 3D Coordinate Transformation (1/6)
Transformation of coordinates (xj, yj, zj) of point j in LA system
to LG coordinate system with the origin at the instrument
setup station i can be given as:
(8.10)
If the coordinates of the setup station i are (0, 0, 0), coordinates
of point j in LAi system can be given:
Sij = measured slope distance
Aij = measured astronomic azimuth
Z´ij = measured zenith angle
(8.11)

23. Electronic Coordinating System: 3D Coordinate Transformation (2/6)
Effects of gravity are removed by Eqn. (8.10) – the LA system
at each station is transformed to corresponding LG system
Transformed LG coordinate systems of all theodolite stations are not
parallel to each other yet
Geometric (curvature) effects must be corrected for to align all the transformed LG systems
Rotate each LGi system to line up with the fixed reference LGk:
(8.12)
Δλ and Δ are small changes (in radians) in longitude and latitude of the
origins of the LGi and LGk systems; (x0, y0, z0) are coordinates of origin
of LGk coordinate system

24. Electronic Coordinating System: 3D Coordinate Transformation (3/6)
 is a small angle (radians) given as
(8.13)
d is the distance from the origin of the fixed LGk system to the
z-axis of the LGi system in the horizontal plane of LGk system,
r is radius of the earth, h is ellipsoidal height of the LGk system
origin

25. Electronic Coordinating System: 3D Coordinate Transformation (4/6)
In geodetic micro-networks transformation (8.13) is done
implicitly
Earth is assumed flat and datum is defined by fixing LGk system and (x0, y0, z0)
(Δλ, Δ, ) are assumed negligible
Assumption of flat earth is not usually acceptable for finding elevations as the geoid or reference ellipsoid may deviate from the tangent plane by several millimeters at 1 km from the point of contact
There is a limit on the length of sight beyond which the curvature of the earth must be considered in height system as shown in Fig. 8.3

26. Electronic Coordinating System: 3D Coordinate Transformation (5/6)
Curve k-i is a level surface through setup points
By leveling to point i (distance d from k) the horizontal plane deviates from level surface by Δh given as
(8.16)
Δh is the error in height if the curvature of the earth is ignored over a distance d from a tangent plane
For d = 100 m, Δh is less than 0.8 m
Fig. 8.3

27. Electronic Coordinating System: 3D Coordinate Transformation (6/6)
In practice, the corresponding axes of all the LG systems constituting a micro-network will be parallel:
When the projections of all the setup stations are on the same plane surface (assuming the reference ellipsoid is a plane with no curvature)
This can be assumed in industrial metrology applications where distances are rarely greater than 100 m (allowing plane surface to be assumed for the region)

28. Factors Influencing Accuracy of Electronic Coordinating Systems
Some of the factors affecting accuracy of electronic coordinating systems are:
Equipment and target design
Geometry of measurement scheme
Influence of environment, such as vibration, wind, temperature fluctuations, refraction, varying lighting conditions

29. Accuracy of Electronic Coordinating Systems: Equipment and Target Design
Instrument precision -systematic errors eliminated, optical quality (pointing), leveling and centering errors minimized
Design of targets – allowing precise centering of cross-hairs over angular range of (60-120)
Other considerations: scale and centering of instrument
Scale: short distances cannot be measured accurately enough (0.05 mm) to satisfy scale requirement – calibrated invar scale bars of known lengths are used and distances are not measured
Two targets defining the scale bar length are tied to the network through triangulation with known distance (to accuracy of 0.01 mm) between them used as distance observable in adjustment

30. Accuracy of Electronic Coordinating Systems: Equipment Centering
Centering error of coordinating system: 0.1 mm centering accuracy requires that permanent pillars be used, which is impossible because of restricted space
Location of instrument whenever it is setup can be determined by resection by observing to distant control points using free-stationing method – if there is no requirement to physically set up on a point
Wall targets established from previous surveys can be used as control points
With this instrument can be located anywhere in the project area
Resected coordinates of instrument locations are used to obtain intersected coordinates of targets
Any error in the resected coordinates of instrument stations will be reproduced in the object point coordinates.

31. Accuracy of Electronic Coordinating Systems: Effects of Geometry & Environment
Effect of Geometry of measurement scheme:
Length of baseline to be 5 – 10 m
Intersection angle range is (78-142)
Angle of intersection at targets on the scale bar is close to 90
Effect of the Environment:
Important stations in metrology networks are wall targets (serving as control points) and object points on aligned structures
Horizontal and vertical refractions must be considered, which can be minimized by
keeping sight distances short
Keep line of sight away from heat radiating sources
Keep temperature distribution constant in the work area

32. Analysis of 3D Traverse surveys
(x, y) and elevation coordinates are determined in 3D traverses
Total station or EDM/theodolite combination is used in trigonometric/two-dimensional traverse approaches
QA/QC measures before 3D traverse surveys include:
Determining acceptable discrepancies between sets of angles, directions, zenith angles, and distances
Specifying reductions needed before ending a station occupation, such as mark-to-mark for comparison of sets

33. 3D Traverse Survey: Steps
Reconnaissance survey to choose locations of control points and other traverse points
Design and simulation to identify equipment, techniques, specifications for QA/QC; predicting precision and accuracy of results, etc.
Equipment testing, including optical plummets, additive constants of total station, collimation errors, rod constants, etc.
Field observation to collect data with appropriate field notes done with care
Data processing, including post-analysis, reduction of data, estimation of coordinates and elevations, statistical assessment of results, etc.
Reporting and presenting the deliverables

34. 3D Traverse Surveys: Observables (1/2)
Heights of instrument (HI) and
reflectors/targets (HR or HT)
Horizontal angles when 2 rays
are measured at a station
Horizontal directions when
more than 2 rays are measured
at a station
Zenith angles to targets
Slope distances with reductions
Meteorological data
Fig. 8.4
Observations from B to A and A to B are done
Directions and zenith angles are observed at least in two sets
Because of possible changes in HI, HR and meteorological conditions observations
B to A may not be the same as those from A to B and would need to be treated as
separate observables or use the average of mark-to-mark values

35. 3D Traverse Surveys: Observables (2/2)
For traverse legs longer than 200 m, the effect of refraction on zenith angles will have to be corrected for
For n sets of direction (or angle) measurements, the standard deviation of the average is reduced by a factor of square root of n
The limit on the discrepancy between any two sets will be the standard deviation of one set times square root of 2.
Standard deviation of distance measurement is not reduced by repetition and averaging except there is a significant difference in atmospheric conditions between sets
Only mark-to-mark zenith and distance values forward should be compared with mark-to-mark values in the reverse for accuracy checks
Mark-to-mark values are used in map projections to obtain the Easting and Northing coordinates of the horizontal traverse

36. 3D Traverse Survey: Data Processing
Over-constrained least squares adjustment of 3D traverse is misleading:
more errors show up in observations or residuals (control points considered errorless are not) – making outlier detection more difficult and making measurements appear less precise
There will be more false outliers and higher variance factor
Calculated coordinates become more precise
In minimal constrained adjustment:
Errors in measurements are unbiased
Coordinates become less accurate due to errors in fixed point and uncontrolled errors in measurements
Measurement outliers are appropriate and are easier to detect

37. Check of Traverse Closure
Given the known coordinates of the last point k of a traverse as (xk, yk)
and the calculated coordinates based on unadjusted measurements
as (xk´, yk´) with propagated standard deviations as (xk´, yk´), the
traverse can be evaluated at (1-)% confidence level using:
(8.19)
(8.20)
If Eqns. (8.19) and (8.20) are satisfied, the linear misclosures of the
traverse are not significant at (1-)% confidence level.
The tests apply to loop traverse
Misclosure is higher when closed on new set of control points
compared to closing back on the same point (loop traverse) due
to relative error of positioning control points

38. Effect of Correlation on Traverse Closure
If correlated set of coordinates are to be checked for closure
(at (1-)% confidence level) using the coordinates of the last
traverse point k, the following test can be done:
Accept if: or
is the vector of computed correlated coordinates of point k and x is the
corresponding known vector of coordinates of point k
is the vector of covariance matrix of computed correlated coordinates of point k
u is the number of unknown parameters being tested; df2 = degrees of freedom

39. 3D Coordination with Laser Systems
Two types of laser (3D coordinating) systems:
Airborne laser scanning system (also known as LiDAR system)
Terrestrial laser scanning system
Coordination with airborne laser scanning system:
Operational principle is similar to that of laser profile that measures distance from airborne platform to the ground along a line
It is an upgrade of laser profiler with scanning mechanism added for mapping topographic features of an area instead of just measuring elevation values along a line
Lasers are mounted beneath airplane or helicopter to produce LiDAR 3D point cloud
(x, y, z) coordinates of scanned objects are obtained using
Aircraft position determined with GNSS measurements and
the distance measurements from aircraft to the ground

40. Airborne Scanning System: Main Components
Components of Airborne scanning system are:
Airborne platform
Laser unit to provide distance measurements to targets
Position and orientation system (POS), which consists of
GNSS system to provide positional information
Inertial measurement unit (IMU) for attitude determination
Ground segment consisting of GPS reference stations, processing software, etc.
Observables of scanning system:
Laser pulse travel to and from targets and the angle of line from the nadir, producing line-of-sight slant ranges
GPS data including carrier phase information stored in POS
IMU attitude data (roll, pitch, yaw) of aircraft
Each calculated slant distances are corrected for atmospheric conditions and for roll, pitch and yaw

41. Airborne Scanning System: Accuracy Analysis (1/2)
DGPS and inertial data are used to determine position of laser scanner to centimeter or decimeter accuracy and its orientation to better than 40"
Scanner and POS data sets are synchronized and transformed into an earth- fixed coordinate system, producing geocoded laser data
Registered laser scanner data with accuracy better than 10 cm in 3D space is possible
This accuracy is primarily due to the accuracy of POS
Derived footprint from laser scanning system are not based on redundant measurements, making LiDAR data and products less reliable
Calibration process of LiDAR system is still not transparent
Typical horizontal accuracy of the system is quoted as 1/2000th of the flying height and vertical accuracy is between 15 and 35 cm depending on flying height
Operating altitudes of LiDAR projects are m or up to 3000 m
Laser spots at nadir are more accurate than those at the outside edge of the swath

42. Airborne Scanning System: Accuracy Analysis (2/2)
Considerations in determining the accuracy of airborne LiDAR data are
Contributing error budgets from the system components such as errors related to laser ranger, GPS, IMU, atmospheric conditions, etc.
Wrong interpretation of what is meant by accuracy of LiDAR data
Details on how planimetric accuracy (x, y) is verified, which are not clear
Accuracy is also affected by a variety of conditions such as steep slope
Geoid height model errors will directly impact final accuracy of LiDAR product
Skill of personnel in project planning and execution will impact data accuracy and quality

43. Coordination with Terrestrial Laser Scanning System (1/2)
Terrestrial laser scanners are neither automated total stations nor digital cameras
Some scanners are equipped with leveling, centering and orienting devices
Scanners do not use crosshairs in measuring specific points; they automatically measure several nonspecific points in a short time
They provide permanent historical record of raw data
Observables measured by scanners:
Slant range
Horizontal and vertical angles passing through the center of the instrument
Intensity of the reflected laser beam at each object point

44. Coordination with Terrestrial Laser Scanning System (2/2)
Measured observables are used to calculate positions of each returned laser signal in the scanner’s internally defined coordinate system
Scanner coordinate system is defined:
Origin: electro-optical center of the scanner
z-axis: from the origin along the instrument vertical (rotation) axis
x-axis: from origin along instrument optical axis
y-axis: orthogonal to x-z plane in a right-handed system
The coordinate system and the relationship among the observables are shown in Fig. 8.5

45. Coordinate system of a terrestrial laser scanner
The range (s), horizontal direction() and vertical angle (v) are related as
(8.27)
(x, y, z) output coordinates of
several points constitute point
cloud (or scan)
Fig. 8.5
(x, y, z) output coordinates are
treated as observables in
scanners

46. Georeferencing Problem with Scanners
Point cloud (scan) referenced to instrument’s internally defined coordinates (x, y, z) are transformed to ground coordinates (X, Y, Z) through georeferencing
Two methods of georeferencing scan data:
Direct method – a method more familiar to surveyors
Indirect method
Direct georeferencing method
Scanner is used like total station: height of instrument measured and instrument is leveled, centered and telescope is oriented towards the target
Similar to reflectorless total station in operation
Not possible to optically orient telescopes of scanners toward target points –centers of targets are estimated

47. Scanner Space Coordinates and Ground Coordinates
Typical relationship between a vector of directly
georeferenced ground coordinates of a point P and vector
of its scanner space coordinates of point P:
Object coordinates
of setup station O
Object coordinates
of point P
scanner coordinates
of point P
k is the derived azimuth from the setup station to the backsight station

48. Accuracy Analysis of Terrestrial Laser Scanning System
Random error budget for points in a scanned point cloud:
Internal sources due to noise in observations and beam width uncertainty
External sources due to instrument setup errors and errors due to survey points used in georeferencing
Error budgeting for direct georeferencing:
Random errors in coordinates of the electro-optical center of scanner and of the coordinates of the center of the backsight target
Random error in derived azimuth from scanner to backsight target
Errors in horizontal direction measurement from the scanner setup point to backsight target due to leveling and centering of scanner and targets; pointing error to backsight target
Errors in vertical angle measurement from scanner setup point to backsight target due to leveling errors of scanner and target
Errors in scanner measurement of range, horizontal direction and vertical angle