1. Introduction to GPS: Theory and Applications
GS 608
Introduction to GPS: Theory and Applications
Undergraduate and Graduate, 3 credit hours
AU 2001
Department of Civil and Environmental Engineering and Geodetic Science

2. SPATIAL REFRENCE SYSTEMS AND FRAMES
Part I
SPATIAL REFRENCE SYSTEMS AND FRAMES
GS608
References:
(and referenced links)

3. Now, where in
the world am I?

5. Local and Global Ellipsoid

7. Maps are 2-dimensional abstractions of reality
As such, they are not located in their “real” geographic setting
They are removed from real coordinates
Thus:
We must have a system for locating objects once they are depicted on a map, in geographic space
These systems are called Spatial Reference Systems

9. Reference systems
Abstract (Cartesian)
Geographical (geometric)
Earth is projected into 2-dimensional space
Or, can be viewed in 3-dimensions

10. Grid can vary from map to map and make comparison between maps difficult

15. Sphere as an approximation of the Earth surface
Geographical coordinates
Latitude, longitude and height above the sphere (measured along the normal to the sphere)

21. Point is located on the surface of the reference sphere
r is a radius of the reference sphere approximating the shape of the Earth
XYZ triad is placed at the center of the sphere
For points with elevation h above the reference sphere:

22. If the triad XYZ is ECEF (Earth-centered and Earth-fixed), centered at the center of mass of the reference sphere with radius r
And the plane XZ is located in the reference (zero) meridian plane
And the plane XY is located in the equatorial plane
The polar coordinates  and  can be referred to as geographical (spherical) coordinates, latitude and longitude.

24. Geographical coordinate systems
Based on spherical shape of the Earth
Geodetic coordinate systems
Based on ellipsoidal shape of the Earth
Varying systems use different reference ellipsoids (spheroids)
Ellipsoid is an approximation of the shape of the Earth: the Earth is an oblate ellipsoid, nearly spherical, but bulging at the equator.

27. Terminology confusion with geographic coordinates
3 types of co-ordinates define different perpendiculars:
astronomical coordinates
physically defined perpendicular, based on the gravity
geodetic coordinates
mathematically defined perpendicular, based on the reference surface, specifically ellipsoid, used for large and medium scale mapping, and in geodesy.
geographic coordinates
mathematically defined perpendicular, based on the reference surface, typically spheres, used for small scale mapping

30. Spherical Coordinates Based on Ellipsoid of Revolution
They are consistent from map to map, but making measurements necessary to use them may be difficult as
degrees of longitude vary in distance from about 69 miles at the equator to 0 mile at the poles
Geodetic coordinates are not directly measurable in the field, they can be observed by astronomical methods and reduced to the ellipsoid

31. WGS84

33. Coordinate Frame Geometry

36. Coordinate Conversion: Cartesian X,Y,Z to Geodetic Lat, Lon, h
Precise (accurate) conversion can be performed
Iteratively
direct computation of longitude
iterate for latitude and height
Closed formulas (Borkowski, 1989; see the handout, IERS Conventions, 1996, p.12)

37. Short Definitions 1/2
Map Projection
A Map Projection defines the mapping from geographic coordinates on a sphere or the geodetic coordinates on a spheroid to a plane.
Reference System
Shape, size, position and orientation of a (mathematical) Reference Surface, (e.g. Sphere or Spheroid). It normally is defined in a superior, geocentric three-dimensional coordinate system (for example WGS84, ITRF).

38. Short Definitions 2/2
Reference Surface
Mathematically (e.g. Sphere or Spheroid) or physically (Geoid) defined surface to approximate the shape of the earth for referencing the horizontal and/or vertical position.
Reference Frame
A set of control points to realize a Reference System.
Geodetic Datum
Traditional Term for Reference System.

39. Reference Systems: SUMMARY 1/2
A coordinate system is most commonly referred to as three mutually perpendicular axes, scale and a specifically defined origin
An access to the coordinate system is provided by coordinates of a set of well defined reference points
Coordinate system and an ellipsoid create a datum; ellipsoid must be defined by two parameters (a and f or a and e); ellipsoid must be oriented in space
Modern systems, especially these derived from GPS observations are Earth-centered, Earth-fixed (ECEF)

41. Reference Systems: SUMMARY 2/2
Geodetic datum defines the size and shape of the earth and the origin and orientation of the coordinate systems used to map the earth.
Numerous different datums have been created and used so far, evolving from those describing a spherical earth to ellipsoidal models derived by modern techniques, such as satellite observations
Modern geodetic datums range from flat-earth models used for plane surveying to complex, global models, which completely describe the size, shape, orientation, gravity field, and angular velocity of the earth.
Potential problems:
Referencing geodetic coordinates to the wrong datum can result in significant position errors
The diversity of datums in use today and the technological advancements that have made possible global positioning measurements with sub-decimeter accuracies require careful datum selection and careful conversion between coordinates in different datums.

44. Datums are created by geodesists, while cartography, surveying, navigation, and astronomy are the end users
National Imagery and Mapping Agency (NIMA), former Defense Mapping Agency created WGS84 – World Geodetic Datum 84
National Geodetic Survey (NGS) created NAD83 – North American Datum 83
International Earth Rotation Service (IERS) created ITRFxx, where xx stands for the reference year at which the frame was (re)established or (re)computed
ITRF stands for International Terrestrial Reference Frame
ITRF coordinates can be expressed in WGS84 at 10 cm level
Newest ITRF refers to epoch 2000 (ITRF2000)

45. ITRF200 Reference Frame

46. What is ITRF ?
The International Earth Rotation Service (IERS) has been established in 1988 jointly by the International Astronomical Union (IAU) and the International Union of Geodesy and Geophysics (IUGG). The IERS mission is to provide to the worldwide scientific and technical community reference values for Earth orientation parameters and reference realizations of internationally accepted celestial and terrestrial reference systems
In the geodetic terminology, a reference frame is a set of points with their coordinates (in the broad sense) which realize an ideal reference system
The frames produced by IERS as realizations of ITRS are named International Terrestrial Reference Frames (ITRF).
Such frames are all (or a part of) the tracking stations and the related monuments which constitute the IERS Network, together with coordinates and their time variations.

47. ITRF97
The reference frame definition (origin, scale, orientation and time evolution) is achieved in such a way that ITRF97 is in the same system as the ITRF96
Station velocities are constrained to be the same for all points within each site;
ITRF97 positions were estimated at epoch ;
Transformation parameters (at epoch ) and their rates from ITRF97 to each individual solution were also estimated.
Transformation between ITRF at epoch and other frames:
Ri represent rotations, D scale change and Ti stands for translation; i=1,2,3

49. TRANSFORMATION PARAMETERS AND THEIR RATES FROM ITRF94 TO OTHER FRAMES
SOLUTION T1 T2 T D R R R3 EPOCH Ref.
cm cm cm " " " IERS Tech.
Note #, page
RATES T1 T2 T D R R R3
cm/y cm/y cm/y 10-8/y .001"/y .001"/y .001"/y
ITRF
RATES
ITRF
ITRF
ITRF
ITRF
ITRF

50. X,Y,Z (Lat, Lon, h) based on the definition of WGS84 ellipsoid

51. WGS 84 Four Defining Parameters
Parameter Notation Magnitude
Semi-major Axis a meters
Reciprocal of Flattening /f
Angular Velocity of the Earth w x rad sec -1
Earth’s GravitationalConstant GM x 10 8 m 3 /s 2
(Mass of Earth’s Atmosphere Included)
a and 1/f are the same as in the original definition of WGS 84

52. World Geodetic System 1984 (WGS 84)
The original WGS 84 reference frame established in 1987 was realized through a set of Navy Navigation Satellite System (NNSS) or TRANSIT (Doppler) station coordinates
Significant improvements in the realization of the WGS 84 reference frame have been achieved through the use of the NAVSTAR Global Positioning System (GPS).
Currently WGS 84 is realized by the coordinates assigned to the GPS tracking stations used in the calculation of precise GPS orbits at NIMA (former DMA).
NIMA currently utilizes the five globally dispersed Air Force operational GPS tracking stations augmented by seven tracking stations operated by NIMA. The coordinates of these tracking stations have been determined to an absolute accuracy of ±5 cm (1s).

53. World Geodetic System 1984 (WGS 84)
Using GPS data from the Air Force and NIMA permanent GPS tracking stations along with data from a number of selected core stations from the International GPS Service for Geodynamics (IGS), NIMA estimated refined coordinates for the permanent Air Force and DMA stations. In this geodetic solution, a subset of selected IGS station coordinates was held fixed to their IERS Terrestrial Reference Frame (ITRF) coordinates.

54. World Geodetic System 1984 (WGS 84)
Within the past years, the coordinates for the NIMA GPS reference stations have been refined two times, once in 1994, and again in The two sets of self-consistent GPS-realized coordinates (Terrestrial Reference Frames) derived to date have been designated:
WGS 84 (G730 or 1994)
WGS 84 (G873 OR 1997) , where the ’G’ indicates these coordinates were obtained through GPS techniques and the number following the ’G’ indicates the GPS week number when these coordinates were implemented in the NIMA precise GPS ephemeris estimation process.
These reference frame enhancements are negligible (less than 30 centimeters) in the context of mapping, charting and enroute navigation. Therefore, users should consider the WGS 84 reference frame unchanged for applications involving mapping, charting and enroute navigation.

55. Differences between WGS 84 (G873) Coordinates and WGS 84 (G730), compared at 1994.0
Station Location NIMA Station Number D East (cm) D North (cm) D Ellipsoid Height (cm)
Air Force Stations
Colorado Springs
Ascension
Diego Garcia(<2 Mar 97)
Kwajalein
Hawaii
NIMA Stations
Australia
Argentina
England
Bahrain
Ecuador
US Naval Observatory
China
*Coordinates are at the antenna electrical center.

56. World Geodetic System 1984 (WGS 84)
The WGS 84 (G730) reference frame was shown to be in agreement, after the adjustment of a best fitting 7-parameter transformation, with the ITRF92 at a level approaching 10 cm.
While similar comparisons of WGS 84 (G873) and ITRF94 reveal systematic differences no larger than 2 cm (thus WGS 84 and ITRF94 (epoch ) practically coincide).
In summary, the refinements which have been made to WGS 84 have reduced the uncertainty in the coordinates of the reference frame, the uncertainty of the gravitational model and the uncertainty of the geoid undulations. They have not changed WGS 84. As a result, the refinements are most important to the users requiring increased accuracies over capabilities provided by the previous editions of WGS 84.

57. World Geodetic System 1984 (WGS 84)
The global geocentric reference frame and collection of models known as the World Geodetic System 1984 (WGS 84) has evolved significantly since its creation in the mid-1980s primarily due to use of GPS.
The WGS 84 continues to provide a single, common, accessible 3-dimensional coordinate system for geospatial data collected from a broad spectrum of sources.
Some of this geospatial data exhibits a high degree of ’metric’ fidelity and requires a global reference frame which is free of any significant distortions or biases. For this reason, a series of improvements to WGS 84 were developed in the past several years which served to refine the original version.

58. Other commonly used spatial reference systems
North American Datum 1983 (NAD83)
State Plane Coordinate System (SPCS) based on NAD83
Universal Transverse Mercator (UTM)

59. North American Datum (NAD)
NAD27 established in 1927
defined by ellipsoid that best fit the North American continent, fixed at Meades Ranch in Kansas
over the years errors and distortions reaching several meters were revealed
In 1970’s and 1980’s NGS carried out massive readjustment of the horizontal datum, and redefined the ellipsoid
The results is NAD83 (1986)
based on earth-centered ellipsoid that best fits the globe and is more compatible with GPS surveying
in 1990’s state-based networks readjustment and densification, accuracy improvement with GPS (HARN and CORS networks)

60. NAD 83 Defining Parameters
Parameter Notation Magnitude
Semi-major Axis a meters
Reciprocal of Flattening /f
Datum point – none
Longitude origin – Greenwich meridian
Azimuth orientation – from north
Best fitting – worldwide
X,Y,Z (Lat, Lon, h) based on the definition of GRS80 ellipsoid

61. State Plane Coordinate System
Based on Lambert and Transverse Mercator projections
Developed in 1930’s and redefined in 1980’s and 90’s
NAD ellipsoid was projected to the conical (Lambert) and cylindrical (Transverse Mercator) flat surfaces
Allowed the entire USA to be mapped on a set of flat surfaces with no more than one foot distortion in every 10,000 feet (maximum scale distortion 1 in 10,000)
Coordinates used are called easting and northing; derived from NAD latitude, longitude and ellipsoidal parameters

62. Lambert projection

63. Lambert projection

64. Transverse Mercator Projection

65. State Plane Coordinate System
The scale of the Lambert projection varies from north to south, thus, it is used in areas mostly extended in the east-west direction
Conversely, the Transverse Mercator projection varies in scale in the east-west direction, making it most suitable for areas extending north and south
Both projections retain the shape of the mapped surface
Each state is usually covered by more than one zone, which have their own origins – thus, passing the zone boundary would cause the coordinate jump!

67. Universal Transverse Mercator, UTM
Developed by the Department of Defense for military purpose
It is a global coordinate system
Has 60 north-south zones numbered from west to east beginning at the 180th meridian
The coordinate origin for each zone is at its central meridian and the equator

68. Universal Transverse Mercator
UTM zone numbers designate 6-degree longitudinal strips extending from 80 degrees south latitude to 84 degrees north latitude
UTM zone characters designate 8-degree zones extending north and south from the equator
There are special UTM zones between 0 degrees and 36 degrees longitude above 72 degrees latitude, and a special zone 32 between 56 degrees and 64 degrees north latitude

69. UTM Zones
Each zone has a central meridia. Zone 14, for example, has a central meridial of 99 degrees west longitude. The zone extends from 96 to 102 degrees west longitude
Easting are measured from the central meridian, with a 500 km false easting to insure positive coordinates
Northing are measured from the equator, with a 10,000 km false northing for positions south of the equator

70. Ohio State Plane (Lambert projection, two zones) and UTM Coordinate Zone

71. Universal Transverse Mercator, UTM

72. Vertical Datum Definition 1/2
Horizontal control networks provide positional information (latitude and longitude) with reference to a mathematical surface called sphere or spheroid (ellipsoid)
By contrast, vertical control networks provide elevation with reference to a surface of constant gravitational potential, called geoid (approximately mean see level)
this type of elevation information is called orthometric height (height above the geoid or mean sea level) determined by spirit leveling (including gravity measurements and reduction formulas).
Height information referenced to the ellipsoidal surface is called ellipsoidal height. This kind of height information is provided by GPS

73. Height Systems Used in the USA
Orthometric
Normal (orthometric normal)
Dynamic
Ellipsoidal
Variety of height systems (datums) used requires careful definition of differences and transformation among the systems

74. Vertical Datum Definition 2/2
Vertical datum is defined by the surface of reference – geoid or ellipsoid
An access to the vertical datum is provided by a vertical control network (similar to the network of reference points furnishing the access to the horizontal datums)
Vertical control network is defined as an interconnected system of bench marks
Why do we need vertical control network?
to reduce amount of leveling required for surveying job
to provide backup for destroyed bench marks
to assist in monitoring local changes
to provide a common framework

75. The height reference that is mostly used in surveying job is orthometric
Orthometric height is also commonly provided on topographic maps
Thus, even though ellipsoidal heights are much simpler to determine (eg. GPS) we still need to determine orthometric heights

76. P terrain geoid ellipsoid
- angle between the normal to the ellipsoid and the vertical direction (normal
to the geoid), so-called deflection of the vertical
H – orthometric height
h – ellipsoidal height h = H + N
N – geoid undulation (computed from geoid model provided by NGS)
terrain
geoid
ellipsoid P 
Normal to the geoid (plumb line or vertical)
Normal to the ellipsoid H N h

77. Orthometric vs Ellipsoidal Height
(Orthometric height)
(computed from a geoid model)

78. So, how do we determine orthometric height?
By spirit leveling
And gravity observations along the leveling path, or
Recently -- GPS combined with geoid models (easy!!!) but not as accurate as spirit leveling + gravity observations
H = h-N
But why do we need gravity observations with spirit leveling?
Because the sum of the measured height differences along the leveling path between points A and B is not equal to the difference in orthometric height between points A and B
Why?

79. Level Surfaces and Plumb Lines 1/2
Equipotential surfaces are not parallel to each other

80. Level Surfaces and Plumb Lines 2/2
The level surfaces are, so to speak, horizontal everywhere, they share the geodetic importance of the plumb line, because they are normal to it
Plumb lines (line of forces, vertical lines) are curved
Orthometric heights are measured along the curved plumb lines
Equipotential surfaces are rather complicated mathematically and they are not parallel to each other
Consequently:
Orthometric heights are not constant on the equipotential surface !
Thus, points on the same level surface would have different orthometric height !

81. Spirit leveling
Height differences between the consecutive locations of backward and forward rods
correspond to the local separation between the level surfaces through the bottom of the
rods, measured along the plumb line direction

82. Orthometric Height vs. Spirit Leveling
dh4 C3
dh3 C2
dh2
dh1 C1
dhi  H
C1, C2, C3, C4 – geopotential numbers corresponding to level (equipotential) surfaces
dh1, dh2, dh3, dh4 – height difference between the level surfaces (determined by spirit leveling, path-dependent); their sum is not equal to H !
Because equipotential surfaces are not parallel to each other

83. Geopotential Numbers 1/3
The difference in height, dh, measured during each set up of leveling can be converted to a difference in potential by multiplying dh by the mean value of gravity, gm, for the set up (along dh).
geopotential difference = gm*dh
Geopotential number C, or potential difference between the geoid level W0 and the geopotential surface WP through point P on the Earth surface (see Figure 2-8), is defined as
Where g is the gravity value along the leveling path. This formula is used to compute C when g is measured, and is independent on the path of integration!

84. Geopotential Numbers 2/3
Since the computation of C is not path-dependent, the geopotential number can be also expressed as
C = gm*H,
where H is the height above the geoid (mean sea level) and gm represents the mean value of gravity along H (along the plumb line at point P on Figure 2-8; see “orthometric height vs. spirit leveling)
the last relationship justifies the units for C being kgal*meter; it is not used to determine C!
Finally:
Geopotential number is constant for the geopotential (level) surface
Consequently, geopotential numbers can be used to define height and are considered a natural measure for height
REMEMBER: Orthometric heights are not constant on the equipotential surface !

85. Observed difference in height depends on leveling route
Points on the same level surface have different orthometric heights
Local normal (plumb line direction) to equipotential (level) surfaces
dhup
dhdown P1 P2 S3 H1 H2
Reference surface (geoid) S2
Orthometric height measured along the plumb line direction S1
No direct geometrical relation between the results of leveling and orthometric heights
H = H1-H2  dhup + dhdown  0

86. What then, if not orthometric height, is directly obtained by leveling?
If gravity is also measured, then geopotential numbers, C (defined by the integral formula shown earlier), result from leveling
Thus, leveling combined with gravity measurements furnishes potential difference, that is, physical quantities
Consequently, orthometric height are considered as quantities derived from potential differences
Thus, leveling without gravity measurements introduces error (for short lines might be neglected) to orthometric height

87. Geopotential Numbers 3/3
Let’s summarize:
The sum of leveled height differences between two pints, A and B, on the Earth surface will not equal to the difference in the orthometric heights HA and HB
The difference in height, dh, measured during each set up of leveling depends on the route taken, as level (equipotential) surfaces are not parallel to each other
Consequently, based on the leveling and gravity measurements
the geopotential numbers are initially estimated (using the integral formula introduced earlier), based on the leveling and gravity measurements along the leveling path
geopotential numbers can then be converted to heights (orthometric, normal or dynamic – see definitions below) if gravity value along the plumb line through surface point P is known
Height = C/gravity

88. Height Systems 1/5
In order to convert the results of leveling to orthometric heights we need gravity inside the earth (along the plumb line)
since we cannot measure it directly, as the reference surface lies within the Earth, beneath the point, we use special formulas to compute the mean value of gravity, along the plumb line, based on the surface gravity measured at point P
reduction formulas used to compute the mean gravity, gm, based on gravity measured at point P on the Earth surface lead to:
Orthometric height, (H = C/gm) or
The reduction formula used to compute mean gravity, based on normal gravity at point P on the Earth surface leads to:
Normal (also called normal orthometric) height, (H* = C/ m )
Where  is so-called normal gravity (model) corresponding to the gravity field of an ellipsoid of reference (Earth best fitting ellipsoid), and subscript “m” stands for “mean”

89. Height Systems 2/5 We can also define dynamic heights
use normal gravity, 45, defined on the ellipsoid at 45 degree latitude, (HD = C/ 45)
Note: term “normal gravity” always refers to the gravity defined for the reference ellipsoid, while “gravity” relates to geoid or Earth itself

90. Height Systems 3/5
Sometimes, instead of formulas provided above (involving C), it is convenient to use correction terms and apply them to the sum of leveled height differences:
Consequently, the measured elevation difference has to be corrected using so-called orthometric correction to obtain orthometric height (height above the geoid)
Max orthometric correction is about 15 cm per 1 km of measured height difference
Or, the measured elevation difference has to be corrected using so-called dynamic correction to obtain dynamic height (no geometric meaning and factual reference surface; defined mathematically)
Or, normal correction is used to derive normal heights
All corrections need gravity information along the leveling path (equivalent to computation of C based on gravity observations!)

91. Height Systems 4/5
Dynamic heights are constant for the level surface, and have no geometric meaning
Orthometric height
differs for points on the same level surface because the level surfaces are not parallel. This gives rise to the well-known paradoxes of “water flowing uphill”
measured along the curved plumb line with respect to geoid level
Normal height of point P on earth surface is a geometric height above the reference ellipsoid of the point Q on the plumb line of P such as normal gravity potential and Q is the same as actual gravity potential at P.
measured along the normal plumb line (“normal” refers to the line of force direction in the gravity field of the reference ellipsoid (model))
All above types of heights are derived from geopotential numbers

92. Height Systems 5/5
A disadvantage of orthometric and normal heights is that neither indicates the direction of flow of water. Only dynamic heights possess this property.
That is, two points with identical dynamic heights are on the same equipotential surface of the actual gravity field, and water will not flow from one to the other point.
Two points with identical orthometric heights lie on different equipotential surfaces and water will flow from one point to the other, even though they have the same orthometric height
The last statement holds for normal heights, although due to the smoothness of the normal gravity field, the effect is not as severe

93. Vertical Datums: NGVD 29 and NAVD 88
NGVD 29 – National Geodetic Vertical Datum of 1929
defined by heights of 26 tidal stations in US and Canada
uses normal orthometric height (based on normal gravity formula)
NAVD 88 – North American Vertical Datum of 1988
defined by one height (Father Point/Rimouski, Quebec, Canada)
585,000 permanent bench marks
uses Helmert orthometric height (based on Helmert gravity formula)
removed systematic errors and blunders present in the earlier datum
orthometric height compatible with GPS-derived height using geoid model
improved set of heights on single vertical datum for North America

94. Vertical Datums: NGVD 29 and NAVD 88
Difference between NGVD 29 and NAVD 88
ranges between – 40 cm to 150 cm
in Alaska between 94 and 240 cm
in most stable areas the difference stays around 1 cm
accuracy of datum conversion is 1-2 cm, may exceed 2.5 cm
transformation procedures and software provided by NGS (

95. International Great Lake Datum (IGLD) 1985
replaced earlier IGLD 1955
defined by one height (Father Point/Rimouski, Quebec, Canada)
uses dynamic height (based on normal gravity at 45 degrees latitude)
virtually identical to NAVD 88 but published in dynamic heights!

96. Vertical Datums
Use of proper vertical datum (reference surface) is very important
Never mix vertical datums as ellipsoid – geoid separation can reach 100 m!
Geoid undulation, N, is provided by models (high accuracy, few centimeters in the most recent model) developed by the National Geodetic Survey (NGS) and published on their web page
So, in order to derive the height above the see level (H) with GPS observations – determine the ellipsoidal height (h) with GPS and apply the geoid undulation (N) according to the formula H = h - N