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

References:

http://www.geocities.com/CapeCanaveral/1224/theory/theory.html

(and referenced links)

http://www.colorado.Edu/geography/gcraft/notes/datum/datum_f.html

Part I

SPATIAL REFRENCE SYSTEMS AND FRAMES

GS608

Now, where in the world am I?

Local and Global EllipsoidLocal and Global Ellipsoid

Maps are 2-dimensional abstractions of realityMaps 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

Reference systemsReference systems

Abstract (Cartesian)

Geographical (geometric)

Earth is projected into 2-dimensional space

Or, can be viewed in 3-dimensions

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

Sphere as an approximation of the Earth surface

Geographical coordinatesGeographical coordinates

Latitude, longitude and height above the sphere (measured along the normal to the sphere)

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

sin)(

sincos)(

coscos)(

hrz

hry

hrx

For points with elevation h above the reference sphere:

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.

Geographical coordinate systemsGeographical coordinate systems

Based on spherical shape of the Earth

Geodetic coordinate systemsGeodetic 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.

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

Terminology confusion with geographic coordinatesTerminology confusion with geographic coordinates

Spherical Coordinates Based on Ellipsoid of Spherical Coordinates Based on Ellipsoid of RevolutionRevolution

• 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

WGS84

Coordinate Frame GeometryCoordinate Frame Geometry

Coordinate Conversion: Cartesian Coordinate Conversion: Cartesian X,Y,Z to Geodetic Lat, Lon, hX,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)

Short Definitions 1/2Short 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).

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.

Short Definitions 2/2Short Definitions 2/2

Reference Systems: SUMMARY 1/2Reference 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)

• 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.

Reference Systems: SUMMARY 2/2Reference Systems: SUMMARY 2/2

• 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)

ITRF200 Reference Frame

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.

• 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 1997.0;

• Transformation parameters (at epoch 1997.0) and their rates from ITRF97 to each individual solution were also estimated.

• Transformation between ITRF at epoch 1997.0 and other frames:

• Ri represent rotations, D scale change and Ti stands for translation; i=1,2,3

ITRF97

TRANSFORMATION PARAMETERS AND THEIR RATES FROM ITRF94 TO OTHER FRAMES

----------------------------------------------------------------------------------------------

SOLUTION T1 T2 T3 D R1 R2 R3 EPOCH Ref.

cm cm cm 10-8 .001" .001" .001" IERS Tech.

. . . . . . . Note #, page

RATES T1 T2 T3 D R1 R2 R3

cm/y cm/y cm/y 10-8/y .001"/y .001"/y .001"/y

----------------------------------------------------------------------------------------------

ITRF93 0.6 -0.5 -1.5 0.04 -0.39 0.80 -0.96 88.0

RATES -0.29 0.04 0.08 0.00 -0.11 -0.19 0.05 18 82

ITRF92 0.8 0.2 -0.8 -0.08 0.0 0.0 0.0 88.0 18 80

ITRF91 2.0 1.6 -1.4 0.06 0.0 0.0 0.0 88.0 15 44

ITRF90 1.8 1.2 -3.0 0.09 0.0 0.0 0.0 88.0 12 32

ITRF89 2.3 3.6 -6.8 0.43 0.0 0.0 0.0 88.0 9 29

ITRF88 1.8 0.0 -9.2 0.74 0.1 0.0 0.0 88.0 6 34

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

Parameter Notation Magnitude

Semi-major Axis a 6378137.0 meters

Reciprocal of Flattening 1/f 298.257223563

Angular Velocity of the Earth 7292115.0 x 10 -11 rad sec -1

Earth’s GravitationalConstant GM 3986004.418 x 10 8 m 3 /s 2

(Mass of Earth’s Atmosphere Included)

WGS 84 Four Defining Parameters

a and 1/f are the same as in the original definition of 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 (s).

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.

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 1996. 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.

World Geodetic System 1984 (WGS 84)

Differences between WGS 84 (G873) Coordinates and WGS 84 (G730), compared at 1994.0

Station Location NIMA Station Number East (cm) North (cm) Ellipsoid Height (cm)

Air Force Stations

Colorado Springs 85128 0.1 1.3 3.3

Ascension 85129 2.0 4.0 -1.1

Diego Garcia(<2 Mar 97) 85130 -3.3 -8.5 5.2

Kwajalein 85131 4.7 0.3 4.1

Hawaii 85132 0.6 2.6 2.7

NIMA Stations

Australia 85402 -6.2 -2.7 7.5

Argentina 85403 -1.0 4.1 6.7

England 85404 8.8 7.1 1.1

Bahrain 85405 -4.3 -4.8 -8.1

Ecuador 85406 -2.0 2.5 10.7

US Naval Observatory 85407 39.1 7.8 -3.7

China 85409 31.0 -8.1 -1.5

*Coordinates are at the antenna electrical center.

• 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 1997.0) 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.

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.

World Geodetic System 1984 (WGS 84)World Geodetic System 1984 (WGS 84)

Other commonly used spatial reference systemsOther commonly used spatial reference systems

• North American Datum 1983 (NAD83)

• State Plane Coordinate System (SPCS) based on NAD83

• Universal Transverse Mercator (UTM)

North American Datum (NAD)North American Datum (NAD)

NAD27 established in 1927NAD27 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 In 1970’s and 1980’s NGS carried out massive readjustment of the horizontal datum, and redefined the ellipsoidof the horizontal datum, and redefined the ellipsoid

The results is NAD83 (1986)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)

Parameter Notation Magnitude

Semi-major Axis a 6378137.0 meters

Reciprocal of Flattening 1/f 298.2572221

Datum point – none

Longitude origin – Greenwich meridian

Azimuth orientation – from north

Best fitting – worldwide

NAD 83 Defining Parameters

X,Y,Z (Lat, Lon, h) based on the definition of GRS80 ellipsoid

State Plane Coordinate SystemState 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

Lambert projectionLambert projection

Lambert projectionLambert projection

Transverse Mercator ProjectionTransverse Mercator Projection

State Plane Coordinate SystemState 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!

Universal Transverse Mercator, UTMUniversal 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

Universal Transverse MercatorUniversal 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

UTM ZonesUTM 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

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

Universal Transverse Mercator, UTMUniversal Transverse Mercator, UTM

Vertical Datum Definition 1/2Vertical Datum Definition 1/2

Horizontal control networksHorizontal control networks provide positional information (latitude and longitude) with reference to a mathematical surface called sphere or spheroid (ellipsoid)

By contrast, vertical control networksvertical 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 orthometric height (height above the geoid or mean sea level(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 heightellipsoidal height. This kind of height information is provided by GPS

Height Systems Used in the USAHeight 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

Vertical datumVertical datum is defined by the surface of reference – geoid or ellipsoid

An access to the vertical datum is provided by a vertical control vertical control networknetwork (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

Vertical Datum Definition 2/2Vertical Datum Definition 2/2

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

- 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

Orthometric vs Ellipsoidal HeightOrthometric vs Ellipsoidal Height

(Orthometric height)(computed from a geoid model)

So, how do we determine orthometric height?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? 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?Why?

Level Surfaces and Plumb Lines 1/2Level Surfaces and Plumb Lines 1/2

Equipotential surfaces are not parallel to each other

The level surfaces are, so to speak, horizontal everywherehorizontal 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

OrthometricOrthometric heights are measured along the curved plumb linesheights are measured along the curved plumb lines

Equipotential surfaces are rather complicated mathematically and they are not parallel to each othernot parallel to each other

Consequently:

Orthometric heights are not constant on the equipotential Orthometric heights are not constant on the equipotential surface !surface !

Thus, points on the same level surface would have different Thus, points on the same level surface would have different orthometric height !orthometric height !

Level Surfaces and Plumb Lines 2/2Level Surfaces and Plumb Lines 2/2

Spirit levelingSpirit leveling

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

Orthometric Height vs. Spirit LevelingOrthometric Height vs. Spirit Leveling

dh1

dh3

dh2

dh4

C1

C3

C2

C4

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 !

dhi H

Because equipotential surfaces are not parallel to each othernot parallel to each other

Geopotential Numbers 1/3Geopotential Numbers 1/3

The difference in heightdifference in height, dh, measured during each set up of leveling can be converted to a difference in potentialconverted 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 CGeopotential 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!

P

P

WWCgdh 0

0

Geopotential Numbers 2/3Geopotential Numbers 2/3

Since the computation of C is not path-dependentcomputation 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) surfaceGeopotential number is constant for the geopotential (level) surface

Consequently, geopotential numbers can be used to define height Consequently, geopotential numbers can be used to define height and are considered a natural measure for heightand are considered a natural measure for height

REMEMBER: Orthometric heights are not constant on the equipotential REMEMBER: Orthometric heights are not constant on the equipotential surface !surface !

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

H1

Orthometric height measured along the plumb line direction

S1

S2

Reference surface (geoid)H2

dhdown

dhup

H = H1-H2 dhup + dhdown 0

P2P1

No direct geometrical relation between the results of leveling and orthometric heights

S3

What then, if not orthometric height, is directly obtained What then, if not orthometric height, is directly obtained by leveling?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 Consequently, orthometric height are considered as quantities derived from potential differencesquantities derived from potential differences

Thus, leveling without gravity measurements introduces error (for short lines might be neglected) to orthometric height

Geopotential Numbers 3/3Geopotential Numbers 3/3

Let’s summarize: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/gravityHeight = C/gravity

Height Systems 1/5Height Systems 1/5 In order to convert the results of leveling to orthometric heights we need In order to convert the results of leveling to orthometric heights we need gravity inside the earth (along the plumb line) 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/gH = C/gmm) or

The reduction formula used to compute mean gravity, based on normal The reduction formula used to compute mean gravity, based on normal gravity at point P on the Earth surface leads togravity at point P on the Earth surface leads to:

Normal (also called normal orthometric) height, (H* = C/ H* = C/ m 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”

Height Systems 2/5Height Systems 2/5

We can also define dynamic heights

use normal gravity, 45, defined on the ellipsoid at 45 degree latitude, (HHDD = C/ = C/ 4545))

Note:Note: term “normal gravity” always refers to the gravity defined for the reference ellipsoid, while “gravity” relates to geoid or Earth itself

Height Systems 3/5Height Systems 3/5

Sometimes, instead of formulas provided above (involving C), it is Sometimes, instead of formulas provided above (involving C), it is convenient to use correction terms and apply them to the sum of convenient to use correction terms and apply them to the sum of leveled height differences: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!)

Height Systems 4/5Height 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

Height Systems 5/5Height 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

Vertical Datums: NGVD 29 and NAVD 88 Vertical Datums: NGVD 29 and NAVD 88

NGVD 29 – National Geodetic Vertical Datum of 1929NGVD 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 1988NAVD 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

Vertical Datums: NGVD 29 and NAVD 88Vertical Datums: NGVD 29 and NAVD 88

Difference between NGVD 29 and NAVD 88Difference 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 (www.ngs.noaa.gov)

International Great Lake Datum (IGLD) International Great Lake Datum (IGLD) 19851985

IGLD 85IGLD 85

• 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!

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

www.ngs.noaa.gov

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

Vertical DatumsVertical Datums