Satellite Geodesy Lecture Notes MSU 2015.pptx

MIDLANDS STATE
UNIVERSITY
FACULTY OF SCIENCE
DEPARTMENT OF SURVEYING
AND GEOMATICS
SVG302 GPS
LECTURE NOTES
LECTURER: D NJIKE

CHAPTER 1
INTRODUCTION TO SATELLITE GEODESY
Geodesy – is the science of the measurement and mapping of the earth’s
surface (Helmert, 1880) {classical definition}.
– includes the determination of the earth’s external gravity field as
well as the surface of the ocean floor (Torge, 1991)
Satellite – an artificial body placed in orbit round the earth or another planet in
order to collect information or for communication.
- a celestial body orbiting the earth or another planet.
Satellite Geodesy – comprises the observational and computational
techniques which allow the solution of geodetic problems
by the use of precise measurement to, from or between
artificial mostly near-earth satellites
- is the measurement of the form and dimensions of the Earth,
the location of objects on its surface and the figure of the
Earth's gravity field by means of artificial satellite techniques
- geodesy by means of artificial satellites.

Objectives of satellite geodesy
1. Determination of precise global, regional and local 3-D positions (e.g.
establishment of geodetic control
2. Determination of the earth’s gravity field and linear functions of this field
(e.g. precise geoid)
3. Measurement and modelling of geodynamical phenomena (e.g. polar
motion, earth rotation, crustal deformation)
Historical Developments of Satellite Geodesy
1957 – Launch of SPUTNIK 1
1958 - Earth Flattening from satellite data (f = 1/298.3)
1958 – LaUnch of EXPLORER IB

Cont’d
1959- Third Zonal Harmonic (Pear shape of the earth)
1959 – Theory of the Motion of Artificial satellites
1960 – Launch of TRANSIT-1B
1960 – Launch of ECHO-1
1960 –Theory of satellite orbits
1962 – Launch of ANNA-1B
1962 – Geodetic connection between France and Algeria
1964 – basic geodetic problems had been successfully tackled namely:
• Determination of a precise numerical value of the earth flattening
• Determination of the general shape of global geoid
• Determination of connections between most important geodetic datums
(to +50m)

Phases of development
1. 1958 – 1970
Development of basic methods for satellite observation, computational and analysis of
satellite orbits
2. 1970 – 1980
Scientific projects phase
New observation techniques were developed and refined – laser ranging to satellites and
to the moon and satellite altimetry
TRANSIT system was used for geodetic Doppler positioning
3. 1980 – onwards
Operational use of satellite techniques in geodesy, geodynamics and land surveying.
Two aspects:
a) Satellites methods are increasingly used by surveying community replacing the
conventional methods
b) Increased observation accuracy

Applications of Satellite Geodesy
Global Geodesy
• General shape of the earth’s figure and gravity field
• Dimensions of a mean earth ellipsoid
• Establishment of a global terrestrial reference frame
• Detailed geoid as a reference surface on land and at sea
• Connection between different geodetic datums
• Connection of national datums with a global datum
Geodetic Control
• Establishment of geodetic control for national networks
• Installation of 3-D homogeneous networks
• Analysis and improvements of existing terrestrial networks
• Establishment of geodetic connections between islands or with the mainland
• Densification of existing networks up to short interstation distances

Cont’d
Geodynamics
• Control points for crustal motion
• Polar motion, earth rotation
• Solid earth tides
Applied and Plane Geodesy
• Detailed plane surveying (cadastral, engineering, GIS, mapping etc,)
• Installation of special networks and control for engineering tasks
• Terrestrial control points in Photogrammetry and Remote Sensing
• Position and orientation of Photogrammetric cameras
• Control points for Cartography during expedition

Cont’d
Navigation and Marine Geodesy
• Precise navigation of land, sea and air vehicles
• Precise positioning for marine mapping exploration, hydrography,
oceanography, marine geology and geophysics
• Connections of tide gauges (unification of height systems)
Related Fields
• Position and velocity determination for geophysical observations
(gravimetric, magnetic, seismic survey) also at sea and in the air
• Determination of ice motion in glaciology

CHAPTER 2
FUNDAMENTALS OF COORDINATE SYSTEMS
Well defined and reproducible reference coordinate system are essentials for
description of satellite motion, modelling of observables, and the
representation and interpretation of results
• Reference coordinate systems in satellite geodesy are global and geocentric
by nature
• Terrestrial measurements are by nature local in character
• Relationship between both systems must be known with sufficient
accuracy
• Since relative position and orientation change with time, the recording
modelling of the observation time plays an important role
• The establishment of precise transformation formulas between systems is
one of the most important tasks in satellite geodesy

Cartesian Coordinate systems and Coordinate Transformations
In a Cartesian coordinate system with the axes x, y, z the position of a
point P is determined by its position vector:

Cartesian Coordinate systems and Coordinate Transformations
• The transformation to a second Cartesian coordinate system with
identical origin and the axes xI, yI, zI, which is generated from the first
one by a rotation around the z-axis by the angle y , can be realized
through the matrix operation

The representation is valid for a right-handed coordinate system.
When viewed towards the origin, a counter-clockwise rotation is
positive. Any coordinate transformation can be realized through
a combination of rotations. The complete transformation is

• The mathematical properties of rotation matrices are described using
linear algebra. The following rules are of importance:

• The relation between the position vectors in two arbitrarily rotated
coordinate systems is then
• In satellite geodesy the rotation angles are often very small, thus
allowing the use of the linearized form for R. With cos α ∼= 1 and
sin α ∼= α (in radians), neglecting higher order terms, it follows that

• To describe satellite motion, observables and models it is
necessary to have a well-defined and reproducible reference
coordinate system.
• Since the accuracy in satellite systems and the precision
requirements are tight, these reference systems have to be
accurate as well.
• It is important to note the difference between Reference
System and Reference Frame, two different concepts.
• The first one is understood as a theoretical definition,
including models and standards for its implementation. The
second one is its practical implementation through
observations and a set of reference coordinates, e.g. a set of
fundamental stars, for a Celestial Reference Frame, or
fiducial stations, for a Terrestrial Reference Frame.
Coordinate Systems

• The International Celestial Reference System (ICRS) was
proposed by the International Earth Rotation and Reference
Systems Service (IERS) and formally accepted by the
International Astronomical Union (IAU) in 1997.
• A realization of the ICRS is the International Celestial
Reference Frame (ICRF).
• On the other hand, IERS is in charge of defining, realizing
and promoting the International Terrestrial Reference
System (ITRS).
• Realizations of ITRS are the International Terrestrial
Reference Frames (ITRFs), being the ITRF2005 the current
reference realization of ITRS.
Coordinate Systems

• also known as Earth Centred Inertial (ECI) (Strictly
speaking this is a quasi-inertial system because of the
annual motion of the Earth around the Sun, and thus it
is subjected to a certain acceleration, but can be
thought of as inertial over short periods of time).
• mainly used for the description of satellite motion.
• the CRS has its origin in the Earth's centre of mass or
Geocentre,
• its fundamental plane is the mean Equator plane
(containing the Geocentre) of the epoch J2000.0,
• the principal axis x is pointing to the mean Vernal
equinox of epoch J2000.0.
Coordinate Systems
Conventional Celestial Reference System

Coordinate Systems

• The three axis defining this coordinate are shown in
Figure 1 above.
• xCRS axis: Its origin is the Geocentre, the Earth's centre of
mass, and its direction is towards the mean equinox at
J2000.0 (i.e., the intersection between the J2000 equatorial
plane and the ecliptic plane).
• zCRS axis: This axis is defined by the direction of the earth
mean rotation pole at J2000.0.
• yCRS axis: Is the orthogonal to the formers ones, so the
system is right handed.
Coordinate Systems

• This reference system is also known as Earth-Centred,
Earth-Fixed (ECEF), it is an earth-fixed, i.e. rotating (not
space-fixed as CRS reference system).
• Its origin is the Earth's centre of mass,
• the fundamental plane contains this origin and it is
perpendicular to the Earth's Conventional Terrestrial
Pole (CTP) (defined as an average of the poles from 1900
to 1905).
• Its principal axis is pointing to the intersection of the mean
Greenwich meridian and the equator.
• Since this coordinate system follows the diurnal rotation of
earth, this is not an inertial reference system.
Coordinate Systems
Conventional Terrestrial Reference System

Coordinate Systems

• The three axis that define this system are showed in
the figure above.
• zTRS: This axis is defined by the Conventional Terrestrial
Pole (CTP).
• xTRS: This axis is defined as the intersection between the
equatorial plane and the mean Greenwich meridian plane.
The equatorial plane is orthogonal to the CTP and in the
Mean Greenwich meridian direction. This meridian was
established by the Bureau International de l'Heure (BIH)
observatory.
• yTRS: It is orthogonal to the other axes so that the system is
right-handed.
Coordinate Systems

• The Conventional Terrestrial Pole is commonly referred to as the
Earth's North Pole. However it should be remembered that the Earth's
polar axis precesses and nutates,
• Thus the position of the "instantaneous" pole is given in seconds of arc
from the CTP,
• the International Earth Rotation Service (IERS) tracks the position of
the pole in relation to the CTP as a function of time,
Coordinate Systems

• An example of a CT system is the International
Terrestrial Reference Frame (ITRF) where stations are
located with reference to the GRS 80 ellipsoid using
VLBI and SLR techniques.
• This world-wide datum takes into account the temporal
effects such as plate tectonics and tidal effects. Thus it is
regularly updated and the date of the update is appended
to its name. For example, ITRF 00 is the datum as
defined in J2000.0. Previous versions were ITRF 97,
ITRF 96, and ITRF 94.
Coordinate Systems

• The datum known as WGS 84 (not to be confused
with the WGS 84 ellipsoid) is another example of a
TRF system of coordinates. Both of these systems of
points with coordinates are known as worldwide
datums.
• Since NAD 83 uses points only on the North American
continent, it is known as a local datum. NAD 83 is
also called a regional datum.
Coordinate Systems

• Used to define coordinates of celestial bodies – stars,
• Established by first defining celestial sphere on which
stars are located,
• The celestial sphere is very large compared to the
earth such that it is considered as a point at the centre
of the sphere – dimensionless,
Coordinate Systems
Celestial Coordinate System

Coordinate Systems

• In the celestial coordinate system the North and South
Celestial Poles are determined by projecting the rotation
axis of the Earth to intersect the celestial sphere, which in
turn defines a Celestial Equator.
• The celestial equivalent of latitude is
called declination and is measured in degrees North
(positive numbers) or South (negative numbers) of the
Celestial Equator.
• The celestial equivalent of longitude is called right
ascension. Right ascension can be measured in degrees,
but for historical reasons it is more common to measure it
in time (hours, minutes, seconds): the sky turns 360
degrees in 24 hours and therefore it must turn 15 degrees
every hour; thus, 1 hour of right ascension is equivalent to
15 degrees of (apparent) sky rotation.
• The position of a star is given as (r,θ,λ)
Coordinate Systems

• In general there are the following celestial coordinate
systems:
• Ecliptic coordinate system
• commonly used for representing the positions
and orbits of Solar System objects.
• Because most planets (except Mercury), and many small solar
system bodies have orbits with small inclinations to
the ecliptic, it is convenient to use it as the fundamental plane.
• The system's origin can be either the center of the Sun or the
center of the Earth,
• its primary direction is towards the vernal equinox, and it has
a right-handed convention.
• It may be implemented in spherical or rectangular coordinates
Coordinate Systems

• The Ecliptic is the path that the Sun appears to follow across the sky
over the course of a year.
• It is also the projection of the Earth's orbital plane onto the Celestial
Sphere.
• The latitudinal angle is called the Ecliptic Latitude, and the
longitudinal angle is called the Ecliptic Longitude.
• Like Right Ascension in the Equatorial system, the zero point of the
Ecliptic Longitude is the Vernal Equinox.
Coordinate Systems

• Right Ascension Coordinate System
• In Figure 1, S is a celestial body on the celestial hemisphere whose
position is to be fixed by spherical coordinates. The earth is located
at the centre, O, of the sphere with its axis in the direction of OP,
• Hour circles on the celestial sphere compare with the meridian
circles or meridians of longitude of the earth. In the figure, PSU is
an hour circle arc.
• Parallels of declination of the celestial sphere compare with the
parallels of latitude of the earth.
• The equinoctial colure of the celestial sphere passes through the
vernal equinox, V, an imaginary point among the stars where the sun
apparently crosses the equator from south to north on March 21 of
each year. The E.C. compares with the prime meridian through
Greenwich.
• Right ascension of the sun or any star (comparable to the longitude
of a station on earth) is the angular distance, alpha, measured along
the celestial equator between the vernal equinox and the hour circle
through the body. Right ascensions are measured eastward from the
vernal equinox and may be expressed in degrees of arc (0� to
360�) or in hours of time (0h to 24h).
Coordinate Systems

• Right Ascension Coordinate System
• Declination of any celestial body is the angular distance, delta, of
the body above or below the celestial equator. It is comparable with
the latitude of the station on earth. If the body is above the equator
its declination is said to be north and is considered as positive; if it
is below the equator its declination is said to be south and is
considered negative. Declinations are expressed in degrees and
cannot exceed 90� in magnitude.
• Polar distance of any celestial body is = 90� - delta with due
regard to the sign of the declination.
• For the present purpose the vernal equinox is assumed to be a fixed
point on the celestial equator. However, the coordinates of celestial
bodies with respect to the celestial equator and the equinoctial
colure change slightly with the passage of time, due to:
• Precession and nutation,
• Proper motion
• Aberration
• parallax
Coordinate Systems

• The Hour Angle Coordinate System
• In Figure 1, let the plane of the hour circle MNPN'M' coincide at
the time of observation with the plane of the observer's meridian
circle, and let S be some heavenly body whose position with
respect to the observer's meridian and the equator MM'UV it is
desired to establish,
• The spherical coordinates of the star are given by (1) the angular
distance of the star above or below the equator, which in the
figure is given by the arc US, defined previously as the
declination, and (2) the angular distance measured along the
equator between the meridian and the hour circle through the star.
When this measurement is from east to west it is called an hour
angle. The hour angle of any celestial body may then be defined
as the angular distance measured westward along the equator
from the meridian of reference to the hour circle through the
body.
• Hour angles are expressed either in hours of time or in degrees of
arc.
Coordinate Systems

• The Horizontal Coordinate System
• uses the observer's local horizon as the Fundamental Plane. This conveniently
divides the sky into the upper hemisphere that you can see, and the lower
hemisphere that you can't (because the Earth is in the way).
• The pole of the upper hemisphere is called the Zenith. The pole of the lower
hemisphere is called the nadir.
• The angle of an object above or below the horizon is called the Altitude (Alt
for short). The angle of an object around the horizon (measured from the North
point, toward the East) is called the Azimuth.
• The Horizontal Coordinate System is sometimes also called the Alt/Az
Coordinate System.
• The Horizontal Coordinate System is fixed to the Earth, not the Stars.
Therefore, the Altitude and Azimuth of an object changes with time, as the
object appears to drift across the sky.
• because the Horizontal system is defined by your local horizon, the same
object viewed from different locations on Earth at the same time will have
different values of Altitude and Azimuth.
• Horizontal coordinates are very useful for determining the Rise and Set times
of an object in the sky. When an object has Altitude=0 degrees, it is either
Rising (if its Azimuth is < 180 degrees) or Setting (if its Azimuth is > 180
degrees).
Coordinate Systems

• zenith:the direction straight up, i.e., directly overhead.
• nadir:the direction diametrically opposite to the zenith.
• horizon:1. the great circle midway between zenith and nadir 2. the
great circle formed by the intersection of the celestial sphere with a
plane perpendicular to the line from an observer to the zenith.
• meridian:the great circle passing through the observer's zenith, and
north and south points on the horizon. It is both a vertical circle and
an hour circle. The observer's meridian is the most important of all
circles of reference.
• vertical circle:any great circle passing through both the observer's
zenith and nadir. Vertical circles receive their name from the fact that
they are perpendicular to the horizon.
• altitude:the angle from the horizon along the vertical circle to the
object.
• azimuth:the angle from the north point of the horizon clockwise to the
foot of the vertical circle through the object.
• transit:when a star crosses the observer's meridian; a.k.a.culminate
Coordinate Systems

• Surveyors generally use a three-dimensional Cartesian system called the
Local Astronomical (LA) coordinates to describe positions in reference to
their own location.
• The origin (0,0,0) corresponds with location of the instrument used to
make surveying measurements on the surface of the Earth: from now on
called the observer's station.
• The x axis (N) points from the origin towards the CTP (north) and is a
tangent with the curvature of the Earth.
• The z axis (U) points away from the surface of the Earth opposite the
direction of gravity towards the observer's zenith. Its negative axis points
in the direction of gravity and the observer's nadir.
• The y axis (E) creates a left-handed Cartesian coordinate system by being
perpendicular to both the x and z axes and pointing east from the
observer's station. This axis is tangent to the curvature of the Earth at the
observer's station.
• Note that unless the observer is at the North Pole, the direction of the U
axis (local astronomical z axis) will not align with the Z axis in the CT
coordinate system.
Coordinate Systems
Celestial Coordinate System: Local Astronomical System

Ellipsoidal and Cartesian Coordinates
Conversion
• The (x,y,z) ECEF cartesian coordinates can be expressed in the
ellipsoidal coordinates λ,φ,h, where λ and φ are, respectively, the
longitude and latitude from the ellipsoid, and h the height above it.
• Figure 1 illustrates the relation between Cartesian and ellipsoidal
coordinates.

Ellipsoidal and Cartesian Coordinates
Conversion

From Ellipsoidal to Cartesian coordinates

From Cartesian to Ellipsoidal coordinates

Reference Frames in GNSS
GPS reference frame WGS-84
• From 1987, GPS uses the World Geodetic System WGS-84,
developed by the US Department of Defense (DoD), which is a
unified terrestrial reference system for position and vector
referencing.
• The GPS broadcast ephemeris are linked to the position of the
satellite antenna phase centre in the WGS-84 reference frame,
thus, the user receiver coordinates will be expressed in the same
ECEF frame.
• The initial implementation of WGS-84 was realized from a set of
more than a thousand terrestrial sites, which coordinates were
derived from Transit observations.
• Successive refinements (which also lead to some adjustments of
the fundamental constants), using more accurate coordinates of
the monitor stations, approximate to some ITRS realizations.
• For instance, realizations WGS84(G730) and WGS84(G873)
correspond to ITRF92 and ITRF94, respectively. The refined
frame WGS84(G1150) was introduced in 2002, which agrees
with ITRF2000 at the centimetre level.

The parameters of the WGS-84 ellipsoid are given in the following table 1:
Table 1: Ellipsoidal parameters WGS-84 (revised in 1997).
GPS reference frame WGS-84

• The GLONASS broadcast ephemeris are given in the
Parametry Zemli 1990 (Parameters of the Earth 1990) (PZ-
90) reference frame.
• As the WGS-84, this is an ECEF frame with a set of
fundamental parameters associated (see table 2 from
[GLONASS ICD, 2008]).
• The determination of a set of parameters to transform the
PZ-90 coordinates to the ITRF97 was the target of the
International GLONASS Experiment (IGEX-98).
• [Boucher and Altamimi, 2001] presents a review of the
IGEX-98 experiment and, as a conclusion, they suggest
the following transformation from (x,y,z) in PZ-90
to (x',y',z') in WGS-84, with a meter level of accuracy.
GLONASS reference frame PZ-90

Following the notation of equation (3) in Transformation between Terrestrial Frames:
the previous transformation (1) is defined by the parameters table:

• According to the GLONASS modernisation plan, the
ephemeris information implementing the PZ-90.02
reference system was updated on all operational
GLONASS satellites from 12:00 to 17:00 UTC,
September 20th., 2007.
• From this time on, the satellites are broadcasting in the
PZ-90.02. This ECEF reference frame is an updated
version of PZ-90, closest to the ITRF2000.
• The transformation from PZ-90.02 to ITRF2000
contains only an origin shift vector, but no rotations
nor scale factor, as it is shown in equation (2)
[Revnivykh, 2007]

• The parameters associated to the PZ-90 and PZ-90.02 are
given in the next table 2 ([GLONASS ICD, 1998] and
[GLONASS ICD, 2008]):

TIME SYSTEMS
• to appreciate the role of time in GPS data analysis it is necessary to
review briefly the various time systems involved, and their
associated time scales.
• Some of th
• ese definitions are standard and inherent to all space positioning
technologies, while others are particular to the GPS system.
• In general there are three different time systems that are used in
space geodesy (KING et al, 1987; LANGLEY, 1991d; SEEBER, 1993)
based on various periodic processes as follows:

TIME SYSTEMS
• Dynamical time
• Atomic time
• Sidereal time
The major types of these systems are shown in Table 1
below.

Time
Time scales - are based on the observation of uniform and repetitive
astronomical or physical phenomena
Time scale - time interval between two consecutive phenomena forms the scale
measure of a particular time scale
Time unit - a certain multiple or fraction of the scale measure
Second – basic time unit
The starting point or origin has to be fixed (eg astronomical event)
Datation – event of reading of the particular time scale
Epoch – datation in astronomy
Absolute time measurement – epoch determination
Relative time measurement – determination of time intervals between two epochs

Dynamical Time
• required to describe the motion of bodies in a particular
reference frame and according to a particular gravitational
theory.
• The most nearly inertial (non-accelerating) reference
frame to which we have access through gravitational
theory has its origin located at the centre-of-mass of the
solar system (the barycentre).
• Dynamical time measured in this system is
called Barycentric Dynamical Time (TDB -- the
abbreviation for this and most other time scales reflects
the French order of the words).
• A clock fixed on the earth will exhibit periodic variations as
large as 1.6 milliseconds with respect to TDB due to the
motion of the earth in the sun's gravitational field.

Dynamical Time
• However, in describing the orbital motion of near-earth
satellites we need not use TDB, nor account for these
relativistic variations, since both the satellite and the
earth itself are subject to essentially the same
perturbations.
• For satellite orbit computations it is common to
use Terrestrial Dynamical Time (TDT), which represents a
uniform time scale for motion within the earth's gravity
field and which has the same rate as that of an atomic
clock on the earth, and is in fact defined by that rate.
• In the terminology of General Relativity, TDB corresponds
to Coordinate Time, and TDT to Proper Time. The
predecessor of TDB was known as Ephemeris Time (ET).

Atomic Time
• The fundamental time scale for all the earth's time-
keeping is International Atomic Time (TAI). It results from
analyses by the Bureau International des Poids et Mesures
(BIPM) in Sèvres, France, of data from atomic frequency
standards (atomic "clocks") in many countries. (Prior to 1
January, 1988, this function was carried out by the Bureau
International de l'Heure (BIH).)
• TAI is a continuous time scale and serves as the practical
definition of TDT, being related to it by: TDT = TAI + 32.184
seconds
• The fundamental unit of TAI (and therefore TDT) is the SI
second, defined as "the duration of 9192631770 periods
of the radiation corresponding to the transition between
two hyperfine levels of the ground state of the cesium 133
atom". The SI day is defined as 86400 seconds and the
Julian Century as 36525 days.

• Because TAI is a continuous time scale, it has one
fundamental problem in practical use:
• the earth's rotation with respect to the sun is slowing down
by a variable amount which averages, at present, about 1
second per year. Thus TAI would eventually become
inconveniently out of synchronisation with the solar day.
• This problem has been overcome by
introducing Coordinated Universal Time (UTC), which
runs at the same rate as TAI, but is incremented by 1
second jumps ( so-called "leap seconds") when
necessary, normally at the end of June or December of
each year.
• During the period mid-1994 to the end of 1995, one
needed to add 29 seconds to UTC clock readings to
obtain time expressed in the TAI scale.
Atomic Time

• The time signals broadcast by the GPS satellites are
synchronised with atomic clocks at the GPS Master
Control Station, in Colorado Springs, Colorado.
• These clocks define GPS Time (GPST), and are in turn
periodically compared with UTC, as realised by the U.S.
Naval Observatory (USNO) in Washington D.C.
• GPST is a continuous measurement of time from an epoch
set to UTC at 0hr on 6 January, 1980 and is often stated in
a number of weeks and seconds from the GPS-Time
epoch. As a result there will be integer-second differences
between the two time scales.
• GPS-Time does not introduce leap seconds and so is
ahead of UTC by an integer number of seconds (10
seconds as of 1 July 1994, 11 seconds at 1 January 1996 ).
GPS Time is steered by the Master Control site to be
within one microsecond (less leap seconds) of UTC.
Atomic Time

• For example, in December 1994 clocks running on GPST
were offset from UTC by 10 seconds. There is therefore
a constant offset of 19 seconds between the GPST and TAI
time scales:
GPST + 19 seconds = TAI
• The GPS Navigation Message contains parameters that
allow the GPS user to compute an estimate of the current
GPS-UTC sub-microsecond difference as well as the number
of leap seconds introduced into UTC since the GPS epoch.
• GPS-Time is derived from the GPS Composite Clock (CC),
consisting of the atomic clocks at each Monitor Station and
all of the GPS SV frequency standards. Each of the current
(Block II) SVs contains two cesium and two rubidium clocks
(Langley 1991).
Atomic Time

• The U. S. Naval Observatory (USNO) monitors the GPS SV
signals. The USNO tracks the GPS SVs daily, gathering
timing data in 130 six-second blocks. These 780-second
data sets include a complete 12.5-minute Navigation
Message, containing a GPS-UTC correction and an
ionospheric model.
• Compared to the USNO Master Clock, a set of some sixty
cesium and from seven to ten hydrogen maser clocks,
these GPS SV data sets are used to provide time steering
data for introduction into the CC at a rate of 10-18
seconds per second squared.
• Each GPS SV signal is transmitted under control of the
atomic clocks in that SV. This space vehicle time (SV-
Time) is monitored and the difference between GPS-
Time and the SV-Time is uploaded into each satellite for
transmission to the user receiver as the SV Clock
Correction data.
Atomic Time

Universal Time and Sidereal Time
• A measure of earth rotation is the angle between a
particular reference meridian of longitude (preferably
the Greenwich meridian) and the meridian of a
celestial body.
• The most common form of solar time is Universal
Time (UT) (not to be confused with UTC, which is an
atomic time scale).
• UT is defined by the Greenwich hour angle (augmented
by 12 hours) of a fictitious sun uniformly orbiting in the
equatorial plane. However, the scale is not uniform
because of oscillations of the earth's rotational axis.
• UT corrected for polar motion is denoted by UT1, and
is otherwise known as Greenwich Mean Time (GMT).
The precise definition of UT1 is complicated because
of the motion both of the celestial equator and the
earth's orbital plane with respect to inertial space,
and the irregularity of the earth's polar motion.

• UT1 is corrected for:
• non-uniformities in the earth’s orbital speed,
• inclination of the earth’s equator with respect to its orbital plane,
• Polar motion
• Defines the actual orientation of the ECEF coordinate system
with respect to space and celestial objects,
• Is the basic time scale for navigation,
• Even with the corrections above, it remains a non-uniform
time scale due to variations in the Earth’s rotation,
• Drifts with respect to atomic time @ ̃several milliseconds
per day and can accumulate to 1 second per year,
• Civil and military time keeping applications require a time
scale with UT1 characteristics but with uniformity of an
atomic timescale – UTC has these characteristics.

• IERS determines when to add or subtract leap seconds to UTC so that the
difference between UTC and UT1 does not exceed 0.9 sec.
• UT1 is derived from the analysis of observations carried out by the IERS, and can
be reconstructed from published corrections (UT1) to UTC:
UT1 = UTC + UT1
• A measure of sidereal time is Greenwich Apparent Sidereal Time (GAST),
defined by the Greenwich hour angle of the intersection of the earth's equator
and the plane of its orbit on the Celestial Sphere (the vernal equinox). Taking
the mean equinox as the reference leads to Greenwich Mean Sidereal Time
(GMST).
• The conversion between mean solar time corrected for polar motion (UT1) and
GAST is through the following relation:
θg =1.0027379093.UT1 + θo + ∆Ψ.cos ε

• Where ∆Ψ is the nutation in longitude, ε is the obliquity of the ecliptic
and θo represents the sidereal time at Greenwich midnight (0hr UT). The
omission of the last term in the above equation permits the GMST to be
determined. θo is represented by a time series:
θo =24110.54841s + 8640184.812866s.To +0.093104s.To
2 6.2s.10-6.To
3
• where To represents the time span expressed in Julian centuries (of 36525
days of 86400 SI seconds) between the reference epoch J2000.0 and the
day of interest (at 0hr UT)

Relationship Between Time Scales

• The Figure above illustrates the relationship between
the various time scales discussed.
• The vertical axis indicates the relative offsets of the
origins of the time scales, and the slope of the lines
indicate their drift.
• Note that with the exception of UT1 (or GAST) all time
scales (nominally) have zero drift as defined by TAI.
Relationship Between Time Scales

TIME SYSTEMS - Summary
• TIME SYSTEMS
• The last concept essential in astronomical positioning is the concept of time. The
hour angle h of the star is the angle between the astronomical meridian of the
star and that of the observer. The local apparent sidereal time (LAST) is the hour
angle of the true vernal equinox. GAST (W) is the hour angle of the true vernal
equinox as seen at Greenwich.
• LAST and GAST can be linked together by the equation: LAST = GAST + LIT
• In practice, GAST is measured through universal time (UT) which differs from
every day standard time by an integral number of hours dependent on the hour
angle. Below are the different version of UT that are used.
• UT reflects the actual non-uniform rotation of the earth. It is affected by polar motion since
local astronomical meridians are slightly displaced.

• UT1, also depicts the non-uniform rotation of the earth, but does not
account for polar motion. UT1 corresponds to GAST and is needed for
transforming the true right ascension (TRA) system to the instantaneous
(IT) system.
• UTC (universal coordinated time) is the broadcast time that represents a
smooth rotation of the earth. (It does not account for propagation delays.)
UTC is kept to within ±0.7s of UT1 by the introduction of leap seconds.
• UT2 is the smoothest time, and has all corrections applied to it.
• International Atomic Time (IAT) is based on an atomic second. To keep IAT
and UT1 close, leap seconds are introduced.

• GPS time is also based on an atomic second. It coincided with UTC
time on January 6, 1980 at 0.0 hours. With the introduction of leap
seconds to IAT, there is now a constant offset of 19 seconds between
GPS time and IAT.
• Relationships in Time Standards
• IAT = GPS + 19.000
• ITS = UTC + 1.000 n where n was 32 in June of 2000.
• UTC = GPS + 13.000

references
• Time Scales in Satellite Geodesy,
http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap2/214time.ht
m#dynamical_time, accessed 31/10/2012, 1815 hrs
• Peter H. Dana, 1997, Global Positioning System (GPS) Time
Dissemination for Real-Time Applications,
http://pdana.com/PHDWWW_files/Rtgps.pdf

Signal propagation
• Signals, on their path between satellites and ground stations, propagate
through atmospheric regions of different nature and variable state
• Signals experience different kinds of influences.
• Perturbations may occur to the direction of propagation, to the velocity
of propagation and to the signal strength.
• The atmosphere introduces unwanted perturbations.
• The impacts on the observational results are, in many cases, much
larger than the accuracy required in satellite geodesy.
• Consequently, atmospheric influences have to be determined directly
by measurements and/or by modeling,

Some Fundamentals of Wave Propagation
Basic Relations and Definitions
• The relation between the wavelength, λ, the frequency, f , and the propagation velocity,v, is:
v = λ · f.
• The relation between frequency and period is:
f = 1/P
• The phase, Φ , of a periodic wave is the fractional part t/ T of the period, P , through which the
time t has advanced with respect to an arbitrary time origin t0
• Furthermore:
ω = 2πf the angular frequency
and
k = 2π/χ the phase constant or wave number

Cont’d
• It follows for the propagation velocity v, that
v = λ · f. = λ/P = ω/k
• A periodic wave which can be modeled by a sinusoidal function in space and time is a
sinusoidal wave. In what follows only waves that are periodic functions in time are
considered:
y = Asin 2π(t/P + Φ0)
where y is the magnitude of the disturbance at time t ; Φ0 is the phase of the wave at t = 0,
and A is the maximum magnitude or the amplitude of the wave. The phase at time t is then
Φ = t/P + Φ0
2πΦ is called the phase angle φ
It follows that
y = Asin(ωt + φ0)

Wave propagation
• the geometrical interpretation of equation

Cont’d
• The wavelengths of electromagnetic waves, and hence their
propagation velocity, depend on certain properties of the medium in
which the waves are propagating. In a vacuum the velocity is:
c = χ/p = fχ
• The value currently in use in satellite geodesy is (McCarthy, 2000)
c = 2.997 924 58 · 108
ms−1 .

Frequency domains
• The frequency spectrum of electromagnetic waves spans nearly 20
orders of magnitude
• In satellite geodesy only two rather small domains are used, namely
the visible light (0.4–0.8 ·1015 Hz) and microwave domains (107 –
1010 Hz).

Spectrum of electromagnetic waves

• Some prefixes and symbols which are commonly used for the
description of frequencies
prefix symbol value prefix symbol value
femto f 10-15 Peta P 1015
pico p 10-12 Tera T 1012
nano n 10-9 Giga G 109
micro μ 10-6 Mega M 106
milli m 10-3 Kilo K 103
centi c 10-2 Hecto H 102

Radar bands
• Different kinds of subdivisions and terminology are in use for
electromagnetic waves.
• In satellite geodesy the subdivision into radar bands is used
• The particular assignments to capital letters were generated in a
random way during World War II.

Radar bands
Denomination Frequency Mean wavelength
P-band 220–300 MHz 115 cm
L-band 1–2 GHz 20 cm
S-band 2–4 GHz 10 cm
C-band 4–8 GHz 5 cm
X-band 8–12.5 GHz 3 cm
Ku-band 12.5–18 GHz 2 cm
K-band 18–26.5 GHz 1.35 cm
Ka-band 26.5–40 GHz 1 cm

Structure and Subdivision of the Atmosphere
• The structure of the atmosphere can be described, as a set of concentric spherical
shells with different physical and chemical properties.
• Various subdivisions are possible,
• With respect to signal propagation a subdivision into troposphere and ionosphere
is advisable, because the particular propagation conditions are quite different.
The troposphere - is the lower part of Earth’s atmosphere which extends from the
surface to about 40 km.
• Signal propagation depends mainly on the water vapor content and on
temperature.
The ionosphere - is the upper part of Earth’s atmosphere between approximately 70
and 1000 km.
• Signal propagation is mainly affected by free charged particles.

Possible subdivision schemes of the earth’s atmosphere

TROPOSPHERE
• The gaseous atmosphere where the daily weather takes place.
• The temperature decreases with height by 6.50 C/km.
• Horizontal temperature gradients are only a few degrees/100 km.
• Charged particles are virtually absent.
• The uncharged atoms and molecules are well mixed, and thus the troposphere is
practically a neutral gas.
• The index of refraction is slightly greater than 1. It decreases with increasing
height and becomes nearly 1 at the upper limit of the troposphere
• Nearly 90% of the atmospheric mass is below 16 km altitude, and nearly 99% is
below 30 km (Lutgens, Tarbuck, 1998).
• The troposphere is not a dispersive medium.
• The index of refraction depends on air pressure, temperature, and water vapor
pressure.
• it is difficult to model the index of refraction.

THE IONOSPHERE
• That part of the high atmosphere where sufficient electrons and ions are
present to affect the propagation of radio waves (Davies, 1990; Langley,
1998b).
• The generation of ions and electrons is proportional to the radiation
intensity of the sun, and to the gas density.
• A diagram indicating the number of ions produced as a function of height
shows a maximum in ion production rate. Such a diagram is called the
Chapman-profile;
• the general behavior of this profile is illustrated below.
• The spatial distribution of electrons and ions is mainly determined by two
processes:

Cont’d
• photo-chemical processes that depend on the insolation of the sun, and govern
the production and de- composition rate of ionized particles, and
• transportation processes that cause a motion of the ionized layers.
• Both processes create different layers of ionized gas at different heights.
• The main layers are known as the D-, E-, F1 -, and F2 -layers. In particular, the F1
-layer, located directly below the F2 -layer, shows large variations that correlate
with the relative sun spot number.
• Geomagnetic influences also play an important role.
• Hence, signal propagation in the ionosphere is severely affected by solar activity,
near the geomagnetic equator, and at high latitudes
• The state of the ionosphere is described by the electron density ne with the unit
[number of electrons/m3 ] or [number of electrons/cm3 ].

Signal Propagation through the Ionosphere and the Troposphere
• Refractivity, N for the troposphere is positive, and independent of the
frequency used.
• For the ionosphere, N is negative, and depends on the frequency.
• The refractivity decreases with increasing frequency.
• One consequence is that higher accuracy can be obtained in
propagation modeling when higher frequencies are used
• Two considerations, however, limit the increase of the selected
frequencies:

Cont’d
− Higher frequencies are technically demanding. The frequency domain
above 10 GHz cannot easily be utilized with existing technology.
− With higher frequencies the atmospheric absorption in the troposphere
increases.
• Without rainfall, the absorption can be neglected for frequencies between 30
• MHz and 30 GHz.
• With precipitation, however, signals in the frequency domain > 1 GHz
experience considerable attenuation.

Effect of the ionospheric propagation delay on range measurements for single-
frequency observations, and residual errors for dual-frequency observations (Hieber,
1983
single-frequency 400 MHz 1600 MHz 2000 MHz 8000 MHz
average effect 50 m 3 m 2 m 0.12 m
for 90% <
maximum effect
250 m
500 m
15 m
30 m
10 m
20 m
0.6 m
1.2 m
dual-frequency 150/400 400/2000 1227/1572 2000/8000
MHz MHz MHz MHz
average effect 0.6 m 0.9 cm 0.3 cm 0.04 cm
for 90% <
maximum effect
10 m
36 m
6.6 cm
22 cm
1.7 cm
4.5 cm
0.21 cm
0.43 cm

Implications
• The selection of frequencies for a particular satellite system is always a
compro- mise.
• This was the case with the TRANSIT system [6] when 150/400 MHz were
selected reflecting the technological progress of the 1960’s.
• And this is true for the GPS system [7] with the selection of 1.2/1.6 GHz.
• Table above gives an impression of how the ionosphere affects the
propagation delay at different frequencies, and it indicates the residual
errors when measurements on two frequencies are available.
• It becomes clear that for the GPS system, operating with two frequencies,
the residual errors are mostly below 1cm.

CHAPTER 3
Satellite Orbital Motion

INTRODUCTION
• Precise time-dependent satellite positions in a suitable reference frame are required for
nearly all tasks in satellite geodesy.
• The computation and prediction of precise satellite orbits, together with appropriate
observations and adjustment techniques is, for example, essential for the determination of
− geocentric coordinates of observation stations,
− field parameters for the description of the terrestrial gravity field as well as for the
determination of a precise and high resolution geoid
− trajectories of land-, sea-, air-, and space-vehicles in real-time navigation
− Earth’s orientation parameters in space.
• Essentially, the accuracy of the final results depends on the accuracy of the available
satellite orbits. The requirement for 1 cm relative accuracy in coordinates implies the
requirement for the knowledge of satellite orbits on the few meter accuracy level or even
better.

Fundamentals of Celestial Mechanics, Two-Body Problem
• In celestial mechanics we are concerned with motions of celestial bodies under the
influence of mutual mass attraction.
• The simplest form is the motion of two bodies (two-body problem).
• For artificial satellites the mass of the smaller body (the satellite) usually can be
neglected compared with the mass of the central body (Earth).
• The two-body problem can be formulated in the following way:
Given at any time the positions and velocities of two particles of known mass
moving under their mutual gravitational force calculate their posi-tions and
velocities at any other time.
• Under the assumption that the bodies are homogeneous and thus generate the
gravitational field of a point mass the orbital motion in the two-body problem can
be described empirically by Kepler’s laws. It can also be derived analytically from
Newtonian mechanics.

Con’td
• To determine positions we need accurate information about
the position of satellites
• It is thus important to understand how GPS orbits are
characterised,
• All positioning of satellites today is based on the laws of
Johannes Kepler who lived in Germany from 1571 to 1630.
• Keplers work was based on observations carried out by the
Danish astronomer Tycho Brahe (1546-1601).
• Kepler developed a number of theorems and laws describing
the motion of the planets in their orbits around the sun.
• These laws do, in general, also describe the motion of a
satellite orbiting around the earth and the laws are therefore
repeated below.

KEPLERS 1st LAW
• The orbit of each planet is an ellipse with the sun in
one of the foci.
Effect on satellites:
• The orbit of a satellite is an ellipse with the gravitational
centre of the earth in one of the foci.
• Referring to Figure 1:
• F are the two foci of the ellipse
• P is perigee, the point on the orbit closest to the earth
• A is apogee, the point on the orbit farthest away from the earth
• a is the semi major axis of the ellipse
• b is the semi minor axis of the ellipse

KEPLERS 2nd LAW: LAW OF AREAS
• The planets revolve with constant area velocity, e.g. the radius vector
of the planet sweeps out equal areas in equal lengths of time,
independent of the location of the planet in the orbit.
Effect on satellites:
• Satellites revolve with a constant area velocity within the orbit. The speed of
the satellite is not constant, but varies with the location of the satellite in the
orbit, so the speed is higher when the satellite is close to the earth (see Figure
2).

KEPLERS 2. LAW
Figure 2. The satellite sweeps out equal areas in the ellipse in equal time intervals while
orbiting

KEPLERS 3rd LAW
• The relation between the square of the period, T, and the cube of the
semi major axis, a, is constant for all planets:
• Effect on satellites:
• Two satellite orbits with the same size of their semi major axes, will have the
same T even if the eccentricities of the orbital ellipses are different (see Figure
3).

KEPLERS 3rd LAW
Figure 3. Two orbits with same size of semi major axis and period, but with different
eccentricity.

KEPLERS 3rd LAW
• The value of the constant given in Equation (1) was determined
several years later by Isac Newton (1624 - 1727) based on his work on
gravity.
• Where GM is the earths gravitational constant of 3986004.418 x 108
m3/s2 (Misra and Enge, 2001)

KEPLERS 3rd LAW
• Keplers three laws would be true for satellites today if
the satellite and the earth were point masses (or
homogeneous bodies with a spherical mass
distribution), and if no other forces than earths gravity
were affecting the satellites.
• This is of course not the case, and the expressions of
satellite motions are therefore more complicated since
we have to account for the variations in the earths
gravity field, and several external forces e.g. lunar gravity
and solar radiation affecting the satellites.

ORBITAL COORDINATES SYSTEM
• In order to describe the motion of a satellite within its orbit, we
define an orbital coordinate system, called q.
• The axis of the coordinate system are defined so that:
• the origin is located in the mass center of the earth,
• the first axis, q1, is directed towards perigee,
• the second axis, q2, is located in the orbital plane, perpendicular to the first
axis in the direction of the satellite motion, and
• the third axis, q3, is perpendicular to both first and second axis to form a right
hand system.
• In Figure 4 the q3 axis is thus pointing out of the plot towards the
reader.

Figure 4. Elements of the orbital coordinate system, q.

• Further, in order to described the location of the satellite
within the orbital coordinate system we need to define a
number of parameters for the orbital ellipse (Figures 4
and 5):

Figure 5. Parameters for describing the location of a satellite in the orbital coordinate
system, q. Figure inspired by Kaula (1969).

CONTD
• The position of the satellite for a given epoch in time is given as:
• The q3 coordinate is zero, since the coordinate system is defined so
the q3-axis is perpendicular to the orbital plane. The satellite motion
is, according to the laws of Kepler, a 2D motion within the q
coordinate system.

CONTD
• Equation (3) can also be given as:
• where the satellite motion is described using the eccentric
anomaly as the angular variable.
• The eccentric anomaly, E and the true anomaly, are two
different angles, both indicating the satellite position in the
orbit as a function of time.
• Depending on the use of the expressions, and the variables
given, one expression is usually preferable to the other

The expressions given in equation (3) and (4) are solutions to the
basic equation of motion in a force field, Equation (10), which is a
second order non-linear differential equation.

CONVENTIONAL INERTIAL REFERENCE
SYSTEM (CIS)
• Having defined a coordinate system for describing the
motion of a satellite within its orbit, we now need a
relation between the orbital coordinate system and the
coordinate systems we use for referencing of the
positions on the surface of the earth (e.g., WGS84) in
order to use the satellites for positioning on the surface
of the earth.
• The Conventional Inertial System (CIS) is necessary as an
intermediate step in this conversion. The CIS is used for
positioning and orientation of the earth in space and is
defined by orienting the axes towards distant quasars.

CONVENTIONAL INERTIAL REFERENCE
SYSTEM (CIS)
• The Conventional Inertial System (CIS) is defined with:
• the origin coinciding with the center of mass of the earth.
• The third axis, Z, is defined to be coinciding with the
rotational axis of earth rotation,
• the first axis, X, is located in the equatorial plane towards the
vernal equinox, and finally
• the second axis, Y, is located in the equatorial plane to
complete a right handed cartesian coordinate system.

Figure 6. Coordinate axes of the inertial reference system.

• The vernal equinox is the point in space where the equatorial plane of
the earth intersects with the ecliptic (the plane of the earth and the
sun) in the spring time. I.e. the direction to the sun as seen from the
earth when the sun is moving from the southern to the northern
hemisphere. The point is also called the spring equinox.
• the CIS does not rotate with the earth, this property makes it
convenient for positioning of satellites.

• Since the mass distribution of the earth is not
homogenous, the rotational axis of the earth is time
variant, and the motion of the axis is composed of two
periodic movements called precession and nutation.
• Precession is caused by gravitational attraction of the
sun, the moon and other celestrial objects, and it
causes the spin axis to move in a slow circular motion
like a top.
• Nutation is a smaller movement with a shorter period
superimposed on the precession.
• The axis of the CIS are thus not constant in time, and
when converting positions from the inertial reference
system to an earth fixed system as for instance the
WGS84, this motion must be taken into consideration.

CONVERSION OF SATELLITES POSITIONS BETWEEN
ORBITAL SYSTEM AND CIS
• The CIS and the orbital coordinate system both have the center of
mass of the earth as origin. This means that conversion of coordinates
from one system to the other does not include translations, but only
rotations of the coordinate axes with respect to each other.
• The three rotation angles are given in the inertial reference system,
they are shown in Figure 7 and are denoted as:

• Ω - right ascension of the ascending node. The angle between the first axis
of the CIS, and the vector in the CIS pointing from origo to the point in the
Equatorial plane where the orbital plane intersects with the Equatorial
plane. This point is denoted the ascending node, and the right ascension of
the ascending node identifies the point where the satellite moves from the
southern hemisphere of the earth to the northern hemisphere.
• i - is called the inclination, and is the inclination angle of the orbital plane
with respect to the Equatorial plane.
• ω- is the argument of perigee. The angle between the position vector of
the ascending node and the position vector of the satellite at the current
epoch in time.

• Coordinates of the satellite position as given in the orbit coordinate
system can now be converted to coordinates in the inertial reference
system by rotating about the first and the third axis of the CIS, using
the three rotation angles; Ω, i, and ω , and corresponding rotation
matrices.

Figure 7. Rotation angles between orbital and inertial coordinate systems.

KEPLER ELEMENTS
• the parameters we need for describing the satellite
orbit and its relation to the inertial reference system
are the following six variables, which are normally
referred to as the Kepler Elements
• Satellite orbit size and shape:
• a – semi major axis
• e - eccentricity
• Location of orbit in the inertial reference system:
• i - inclination
• Ω– right ascension of the ascending node
• ω – argument of perigee
• Further, to describe the location of a satellite in its orbit, we
need:
• ν– true anomaly
• or
• E - eccentric anomaly

Perturbed Satellite Motion
• The satellite motion is affected by external forces dragging and pushing the
satellite from the theoretically smooth orbit
• The most important perturbing effect is, however, caused by variations in
earths gravity field.
• The earth is not a point mass and the mass is not homogeneously distributed
inside the earth.
• The deviation of the gravity field from a central sphere, and the variations in
the earth gravity field as a function of the distribution of masses inside the
earth are well modeled today, mainly because of many years of studies of
satellite orbit perturbations, but also because of a very dense network of
gravity reference stations on the surface of the earth, where gravity is
measured precisely at regular intervals.
• The models of the earths gravity field are therefore also used to model the
effect of the satellite orbits.

Perturbed Satellite Motion
• The non-spherical and non-central gravity field causes a rotation of the
orbital plane within the inertial coordinate system.
• The gravity field basically tries to drag the satellite orbit into the
equatorial plane.
• The effect on the Kepler elements, describing the size, shape and
location of the satellite orbit, is rather large, and must be considered
when dealing with real satellite positions.
• The effect is larger for satellites located in orbits close to the surface of
the earth, the so-called LEO satellites (low earth orbiters).

• Other forces affecting the satellite motion are:
• gravitational effects of the sun and the moon,
• solar radiation pressure,
• albedo (reflection of solar light from the surface of the earth back into space),
• effects of earth and ocean tides,
• radiation from space,
• atmospheric drag etc.

Perturbing forces
• Perturbing forces are in particular responsible for:
1. Accelerations due to the non-spherically and inhomogeneous mass distribution within
Earth (central body), r¨ E .
2. Accelerations due to other celestial bodies (Sun, Moon and planets), mainly r¨ S , r¨ M .
3. Accelerations due to Earth and oceanic tides, r¨ e , r¨ o .
4. Accelerations due to atmospheric drag, r¨ D .
5. Accelerations due to direct and Earth-reflected solar radiation pressure, r¨ SP , r¨ A .
• The perturbing forces causing 1 to 3 are gravitational in nature; the remaining forces are
non-gravitational. The total is:
ks = rË + r¨S + r¨M + rë + rö + r¨D + r¨SP + r A .

Perturbing forces acting on a satellite
sun
moon
O
orbit rA
Earth
satellite
rE, rO
rM
rSP
rD
rS

Implications of perturbations on GPS satellite orbit. From Seeber (2003)
Perturbation Effect on satellite
acceleration m /
S2
Deviation of earth gravity field
from a sphere
Variations in earth gravity field
Solar and lunar gravitation
Earth and ocean tides
Solar radiation pressure
Albedo
5 · 10-5
3 · 10-7
5 · 10-6
1 · 10-9 each
1 · 10-7
1 · 10-9

CHAPTER 4
Basic Observation Concepts and Satellites
Used in Geodesy

Satellite Geodesy as a Parameter Estimation Problem
• The fundamental equation of satellite geodesy can be formulated as:
r S (t ) = r B (t ) + ρ(t )
or
rj (t ) = r i (t ) + ,r ij (t ).

Basic relations for satellite observations
ri
rij
Bi
Sj
Y
X
Z
rj

Observation Concepts
• For a solution to equation above we have to establish a relation
between the observations, characterized by:
• the vector, r ij (t ),
• the parameters which describe the satellite position rj (t ),
• the location of the observation station r i (t ).
• In the estimation process either all parameters can be treated as
unknowns, or some of the parameters are considered to be known, in
order to stabilize and to simplify the solution.

Classification of parameters
• The parameters in the equation can be subdivided into different groups, for
instance into:
(1) Parameters describing the geocentric motion of the observation station rB(t ).
• The first of these are the geocentric station coordinates.
• Then there are geo-dynamic parameters, describing the relation between the Earth-fixed
terrestrial reference system and the space-fixed inertial reference system, namely the polar
motion and Earth rotation parameters.
• Also belonging to this group are the parameters used for the modeling of solid Earth tides
and tectonic crustal deformations.
• Finally, the transformation parameters between geocentric and particular geodetic or
topocentric reference frames may be considered.

Classification of parameters
(2) Parameters describing the satellite motion r s (t ).
• The satellite coordinates,
• the harmonic coefficients of Earth’s gravity field,
• parameters describing other gravitational or non-gravitational perturbations,
like the solar radiation pressure.
(3) Parameters influencing directly the observations ρ(t ).
• atmospheric parameters,
• clock parameters,
• signal propagation delays.

Observables and Basic Concepts
• The observation techniques used in satellite geodesy can be
subdivided in different ways. One possibility has been already
introduced, namely a classification determined by the location of the
observation platform
− Earth based techniques (ground station → satellite),
− satellite based techniques (satellite → ground station),
− inter-satellite techniques (satellite → satellite).
• Another classification follows from the observables in question.
• A graphical overview is given below.

Overview of observation techniques in Satellite Geodesy

Determination of Directions
• Photographical methods are almost exclusively used for the determination of
directions.
• An artificial satellite which is illuminated by sunlight, by laser pulses, or by some
internal flashing device, is photographed from the ground, together with the
background stars.
• The observation station must be located in sufficient darkness on the night side of
Earth.
• The stars and the satellite trajectory form images on a photographic plate or film
in a suitable tracking camera, or on a CCD sensor.
• The photogram provides rectangular coordinates of stars and satellite positions in
the image plane, which can be transformed into topocentric directions between the
observation station and the satellite, expressed in the reference system of the star
catalog (equatorial system, CIS).

Determination of Directions
• Two directions, measured at the same epoch from the endpoints of a
given baseline between observing stations, define a plane in space
whose orientation can be determined from the direction cosines of the
rays.
• This plane contains the two ground- stations and the simultaneously
observed satellite position.
• The intersection of two or more such planes, defined by different
satellite positions, yields the inter-station vector between the two
participating ground stations.

The use of directions with satellite cameras

Determination of Ranges
• For the determination of distances in satellite geodesy the propagation
time of an electromagnetic signal between a ground station and a
satellite is measured.
• According to the specific portion of the electromagnetic spectrum we
distinguish between optical systems and radar systems
• Optical systems are weather-dependent. Laser light is used
exclusively, in order to achieve the required signal strength and
quality.
• Radar systems are weather- independent; wavelengths of the
centimeter and decimeter domain are used. The propagation behavior,
however, is significantly affected by atmospheric refraction.

Determination of ranges
• We distinguish the one-way mode and the two-way mode.
• In the two-way mode the signal propagation time is measured by the
observer’s clock.
• The transmitter at the observation station emits an impulse at epoch tj . The
impulse is reflected by the satellite at epoch tj +Δ tj, and returns to the
observation station where it is received at epoch tj + Δtj
• The basic observable is the total signal propagation time ,tj .
• In the one-way mode we assume that either the clocks in the satellite and in
the ground receiver are synchronized with each other, or that a remaining
synchronization error can be determined through the observation technique.
This is, for instance, the case with the Global Positioning System (GPS).
j

• Further we distinguish between either impulse or phase comparison methods.
• When a clear impulse can be identified, as is the case in satellite laser ranging,
the distance is calculated from the signal propagation time
• phase comparison method, the phase of the carrier wave is used as the
observable.
• In the two-way mode the phase of the outgoing wave is compared with the phase
of the incoming wave.
• In the one-way mode the phase of the incoming wave is compared with the
phase of a reference signal generated within the receiver.
• In both cases the observed phase difference, corresponds to the residual portion,
,λ, of a complete wavelength.
• The total number, N , of complete waves between the observer and the satellite
is at first unknown. This is the ambiguity problem.

• Different methods are used for the solution of the ambiguity term N ,
for example:
− measurements with different frequencies (e.g. SECOR),
− determination of approximate ranges with an accuracy better
than λ/2 (e.g. GPS with code and carrier phases),
− use of the changing satellite geometry with time (e.g. GPS
carrier phase observations),
− ambiguity search functions (e.g. GPS).

Determination of Range Differences (Doppler method)
• The range differences are derived from the measurement of the
frequency shift caused by the change of range between the observer
and the satellite during a given satellite pass.
• The satellite transmits a signal of known frequency fs which is
tracked by a ground receiver. The relative motion d s/d t between the
receiver and the transmitter causes the received frequency fr (t ) to
vary with time
• This is the well-known Doppler effect.
• The frequency shift in a given time interval tj , tk is observed, and is
scaled into a range difference ,rij k

Doppler effect
• The observation of the Doppler effect is frequently used in satellite geodesy.
• The technique is always applicable when a satellite, or a ground-beacon,
transmits on a stable frequency.
• The orbital elements of the very first satellites were determined by observing the
Doppler-shift of the satellite signals.
• The most important application of the Doppler method in geodesy has been with
the Navy Navigation Satellite System (TRANSIT).
• A current space system based on the Doppler technique is DORIS
• The Doppler effect can also be used for the high precision determination of range
rates |,r˙jk | between satellites.
• This method is named Satellite-to-Satellite Tracking (SST), and it can be applied to
the mapping of a high resolution Earth gravity field.

Satellite Altimetry
• Altimetry is a technique for measuring height,
• Satellite altimetry was the first operational satellite-borne observation
technique in satellite geodesy.
• Satellite radar altimetry measures the time taken by a radar pulse to
travel from the satellite antenna to the surface and back to the
satellite receiver,
• The altimeter emits a radar wave and analyses the return signal that
bounces off the surface
• Surface height is the difference between the satellite’s position on
orbit w.r.t an arbitrary reference surface (the Earth’s center or the
Earth’s ellipsoid)
• We can also measure wave height and wind speed over the oceans,
backscatter coefficient and surface roughness for most surfaces off
which the signal is reflected, by looking at the return signal’s amplitude
and waveform

Satellite Altimetry
• Altimetry satellites are able to measure the distance between
the satellite and the surface of the Earth.
• This distance is called range.
• Altimetry satellites transmit a radar signal to the Earth.
• This signal is reflected by the Earth's surface and the satellite
receives the reflected signal.
• The time elapsed between transmission and reception of the
radar signal is the key parameter in calculating the distance
between the satellite and the ground surface.

Satellite Altimetry
• Precise orbit altitude is needed to calculate the range.
• The SENTINEL-3 instruments, GNSS and DORIS, retrieve the orbit
altitude.
• The orbit altitude is the distance between the satellite and an
arbitrary reference surface (the reference ellipsoid or the geoid).
• The scientific community is usually interested in the surface
height in relation to this reference surface (the reference ellipsoid
or the geoid) instead of being referenced to the position of the
satellite.

Satellite Altimetry
• The surface height can be approximately derived from range and
altitude using the following equation:
Surface Height = Altitude - Range
• The complete calculation of surface height should also include all
corrections due to environmental conditions.
• Examples of these corrections are atmospheric propagation
corrections (ionosphere and troposphere) and geophysical
corrections (tides and atmospheric pressure loading).
• satellite altimetry can be used to determine the geoid over the
oceans.

Satellite altimetry
MSL (geoid)
M
A
MSL (geoid)
H

Interferometric Measurements
• The basic principle of interferometric observations is shown in Fig.
below.
• A1 and A2 are antennas for the signal reception.
• When the distance to the satellite S is very large compared with the
baseline length b, the directions to S from A1 and A2 can be
considered to be parallel.
• From geometric relations we obtain
d = b.cosθ

Interferometric measurements
P
A2
A1
d
b
χ
S
S
θ

• If λ is the wavelength of a continuous signal from the satellite, then the
phase difference Φ, caused by the range difference d , can be observed
at both antennas.
• The observed phase difference is uniquely determined only as a frac-
tion of one wavelength; a certain multiple, N , of whole wavelengths
has to be added in order to transform the observed phase difference
into the range difference d .
• The basic interferometric observation equation is hence
d = b · cos θ = 1/2π λ + Nλ.

• The interferometric principle can be realized through observation techniques in
very different ways.
− the baseline length b between the two antennas,
− the residual distance d between the antenna and the satellite, and
− the angle θ between the antenna baseline and the satellite.
• In each case it is necessary to know, or to determine, the integer ambiguity term
N .
• The determination is possible through a particular configuration of the ground
antennas, through observations at different frequencies, or through well defined
observation strategies.
• With increasing baseline lengths the antennas cannot be connected directly with
cables.
• The phase comparison between the antennas must then be supported by the use
of very precise oscillators (atomic frequency standards).
• This is, for instance, the case with the Very Long Baseline Interferometry (VLBI)
concept.

• When artificial Earth satellites are used in the VLBI technique, it cannot be
assumed that the directions from the antennas to the satellites are parallel.
• Instead, the real geometry has to be introduced by geometric corrections;
• The interferometric principle has been widely used in the geodetic application of
the GPS signals.
(a) The signals from the GPS satellites can be recorded at both antenna sites
without any a priori knowledge of the signal structure, and later correlated for
the determination of the time delay τ .It is used to some extent in modern GPS
receiver technology, in order to access the full wavelength of L2 under “Anti-
Spoofing” (A-S) conditions.
(b) The phase of the carrier signal at both antenna sites can be compared, and the
difference formed.
• These so-called single phase differences can be treated as the primary
observables.
• The method is now widely used for processing GPS observations

Very Long Baseline Interferometry
• the most accurate of all the extraterrestrial positioning
techniques.
• initially developed by astronomers as a tool to improve the
resolution of radio telescopes, but even before the first
successful tests of the concept in 1967, it was realized that it
would be an ideal geodetic instrument.
• uses the principle of wave interference.
• Signals from a radio source, usually the random noise signals
of a quasar or other compact extragalactic object, are
received at the antennas of two or more radio telescopes.

Very Long Baseline Interferometry
• These signals are amplified and translated to a lower
frequency band under control of a hydrogen maser frequency
standard.
• The translated signals are digitized, time-tagged, and recorded
on wide bandwidth magnetic tape. Subsequently the tape
recordings are played back at a central processing site.
• The processor is a computer-controlled cross-correlator which
delays and multiplies the signals from the tapes recorded at a
pair of radio telescopes.
• The output of the processor is a sampled cross-correlation
function equivalent to the fringes of Young's experiment.

Very long baseline interferometry
• The primary observable of geodetic VLBI is the group delay,
the difference in arrival times of the quasar signal wave
fronts at the radio telescopes.
• In principle, the delay can be measured in the correlation
process by noting the time offset between a pair of tape
recordings required to achieve maximum correlation.
• The phase (delay) of the correlation function and its time
rate of change, the delay rate, are also measured.
• In practice, the group delay is obtained from measurements
of the phase delay at different frequencies.

• The primary component of the measured group delays is the
geometric delay,
• where B is the baseline vector connecting two radio telescopes, S
is the unit vector in the direction of the radio source, and c is the
speed of light.
• From observations of a dozen or more radio sources during a
nominal 24-hour session, the three components of the baseline
vector can be retrieved.
• A number of biases in the data must be carefully measured or
modelled.

Basic principle of VLBI Geometrical relationship
for VLBI

Applications of VLBI
• The accuracy of the celestial reference system, for modern needs
was achieved with the astrometric satellite mission HIPPARCOS
(Kovalevsky et al., 1997), and Very Long Baseline Interferometry
(VLBI).
• On January 1, 1988 the International Earth Rotation Service
(IERS) took over the task of determining Earth rotation
parameters. The principle observation techniques used are laser
ranging to satellites and to the Moon and Very Long Baseline
Interferometry.

Applications of VLBI
• the main contributions of VLBI to space geodesy are:
• to establish and maintain the International Celestial Reference Frame (ICRF),
• to establish and maintain the International Terrestrial Reference Frame (ITRF),
• to establish and maintain the time dependent Earth Orientation Parameters (EOP)
that relate the ITRF to the ICRF.
• VLBI is unique in that it is the only technique for establishing and
maintaining the ICRF, and the relationship between the ITRF and the
ICRF, by directly monitoring the nutation parameters and UT1.
• As well as this, it is the only geodetic space technique that contributes
to all three of the above mentioned items. Other advantages, when
compared with satellite techniques, come from the fact that VLBI is
independent of the gravity field. As a consequence (Drewes, 2000):
• VLBI is not affected by satellite orbit errors caused by gravity field mismodeling,
• VLBI is not influenced by variations of the geocenter, and
• VLBI is independent of the uncertainty of the GM value and hence of the related
scale problems.
• Compared with satellite laser ranging, VLBI has the advantage of being
weather independent.

Disadvantages of VLBI
• VLBI is a rather expensive technology, hence only a limited number
of telescopes is available,
• instrumental errors, like telescope deformation, are difficult to
handle,
• results are not yet available in real-time.
• VLBI also does not provide absolute coordinates with respect to
the geocenter, but baselines between stations or relative
coordinates with respect to some arbitrarily selected origin.
• Due to the high efficiency of modern satellite techniques like GPS,
the VLBI technology is not used for operational positioning in
geodesy and geodynamics.
• VLBI, due to its unique capacities, will however remain the primary
geodetic technique for maintaining the fundamental reference
frames and their inter-relationship.

Images of associated telescopes
Transportable 6-mVLBI telescope
20-m VLBI telescope

Satellites Used in Geodesy
• Most of the satellites which have been used, and still are used, in
satellite geodesy were not dedicated to the solution of geodetic
problems; their primary goals are various.
• Typical examples of this group are the navigation satellites of the
TRANSIT and of the GPS systems, and remote sensing (Earth
observation) satellites carrying a radar altimeter.
• Examples of satellites which were exclusively, or primarily, launched
for geodetic and/or geodynamic purposes are:

• PAGEOS (PAssive GEOdetic Satellite) USA 1966,
• STARLETTE, STELLA France 1975, 1993,
• GEOS-1 to 3 (GEOdetic Satellite 1 to 3) USA 1965, 1968, 1975,
• LAGEOS-1, 2(LAser GEOdynamic Satellite) USA 1976, 1992,
• AJISAI (EGS, Experimental Geodetic Satellite) Japan 1986,
• GFZ-1 (GeoForschungs Zentrum) Germany 1986,
• CHAMP (CHAllenging Mini Satellite Payload) Germany 2000.

• A frequently used distinction for the purposes of subdivision is passive
and active satellites.
• Passive satellites are exclusively used as targets. They have no
“active” electronic elements, and are independent of any power supply.
Their lifetime is usually extremely long.
• Active satellites in most cases carry various subsystems like sensors,
transmitters, receivers, computers and have a rather limited lifetime.
Table below gives an overview of the most important satellites that are
in use, or have been used, in satellite geodesy.

PASSIVE SATELLITES ACTIVE SATELLITES
ECHO-1 ETALON-1 ANNA-1B ERS-2
ECHO-2 ETALON-2 GEOS-3 TOPEX/POSEIDON
PAGEOS GFZ-1 SEASAT-1 GFO (Geosat Follow On)
STARLETTE NNSS satellites CHAMP
STELLA NAVSTAR satellites JASON
LAGEOS-1 GLONASS satellites ENVISAT
LAGEOS-2 GEOSAT GRACE
EGS (AJISAI) ERS-1

• Another possible subdivision is into:
− Geodetic Satellites,
− Earth Sensing Satellites,
− Positioning Satellites, and
− Experimental Satellites.
Geodetic satellites are mainly high targets like LAGEOS, STARLETTE, STELLA,
ETALON, ASIJAI, and GFZ which carry laser retro-reflectors.
They are massive spheres designed solely to reflect laser light back to the
ranging system. The orbits can be computed very accurately, because the
non-gravitational forces are minimized.

• Earth sensing satellites like ERS, GFO, TOPEX, JASON, ENVISAT carry
instruments designed to sense Earth, in particular to monitor
environmental changes. Many of these satellites carry altimeters. The
satellites are rather large with irregular shape, hence drag and solar
radiation forces are also large and difficult to model
• Positioning satellites are equipped with navigation payload. To this class
belong the former TRANSIT, GPS, GLONASS, and future GALILEO satellites.
Some of the spacecraft carry laser reflectors (e.g. GPS-35, -36, and all
GLONASS satellites).
• Experimental satellites support missions with experimental character. They
are used in the development of various other kinds of satellites, to test
their performance in real space operations.

The Doppler effect
• discovered by Christian Doppler a nineteenth century
Austrian physicist,
• is familiar to anyone who has waited patiently at a railway level
crossing for a train to pass. The pitch of the train's horm or whistle
changes as the train passes. It starts out high, changing
imperceptibly as the train approaches, then drops noticeably as the
train goes through the crossing, and maintains a lower pitch as the
train recedes in the distance.
• This same phenomenon which is so readily apparent at audio
frequencies also affects electromagnetic waves.
• The frequencies of both radio and light waves are shifted if
the source (transmitter) and the observer (receiver) are in
relative motion.

The Doppler effect
• The classical explanation of the effect is that the observer receives
more wave crests per second, i.e., the frequency is increased if the
source and the observer are moving closer together, whereas
fewer wave crests per second are received, i.e., the frequency is
decreased, if the source and the observer are moving farther
apart.
• If the relative speed of the source and observer is much less than
the speed of light, then the received frequency is given
approximately as
•
• where fs is the frequency at the source, c is the speed of light, and
S the distance or range between the source and the observer;
dS/dt is the range rate.

The Doppler effect
• Returning to the train at the level crossing, the closer you are to
the track, the faster the change in pitch of the horn. And even if
you could not see or feel the train, you can tell when it passes the
crossing (the point of closest approach) by noting the instant when
the pitch of the horn is mid-way between the high and low
extremes (fs).
• Therefore by monitoring the frequency of the received sound as
the train passes and knowing its assumed constant speed, you can
establish your position in a two-dimensional coordinate system
where the x-axis, say, runs along the track and they-axis runs
perpendicular to it.
• The origin may be assigned arbitrarily. This is the principle of
Doppler positioning.

The Doppler effect
• In the case of a Transit satellite (or any other satellite for that
matter), the position of a receiver can be established by
continuously recording the Doppler shift of the received signals (or
the number of cycles of the Doppler frequency which is a more
precisely obtained observable).
• Subsequently these data are combined with accurate coordinates of
the satellite to determine the position of the receiver.
• As with the passage of a train, a single satellite pass can provide at
most only two coordinates of the receiver's position.
• Whereas this may be satisfactory for navigation at sea where the
height above the reference ellipsoid is approximately known, three-
dimensional positioning requires observing multiple satellite
passes.

The Doppler effect
• The approximate frequency of a received satellite radio signal (ignoring
relativistic effects) is given by
• fr ≈ fs (1 - 1/c dS/dt) ,
• where fs is the frequency of the signal measured at the satellite, cis the
speed of light, and dS/dt is the range rate.
• The Doppler shift frequency, fr - fs, is approximately proportional to the
range rate, the component of the satellite‘s velocity vector along the line of
sight from the receiver.
• The maximum range rate of a Transit satellite is about 7.4 km/s implying a
maximum Doppler shift when the satellite rises or sets of 25 ppm of the
transmitted frequency.
• This corresponds to 8.4 kHz at a frequency of 400 MHz.

The Doppler effect
• The Doppler shifts may be measured by differencing the received
frequencies from constant reference frequencies in the receiver.
• For most Transit receivers, these frequencies are 400 MHz and 150 MHz
precisely. The satellite transmitter frequencies are approximately 80 ppm
lower than the receiver reference frequencies in order that the Doppler
shift does not go through zero.
• If the transmitter frequencies were not offset, the receiver would have
difficulty distinguishing between positive and negative Doppler shifts.
• A record of the Doppler shift of a Transit signal during a typical pass is
shown in the upper part of this figure.
• The point of closest approach of the satellite, when the Doppler shift is
zero, occurred 6 minutes after the receiver locked onto the signal.

The Doppler effect
• Most Transit Doppler receivers count the number of accumulated
cycles of the Doppler frequency (actually, f0 - fr) rather than measure
the instantaneous Doppler frequency itself, since counting cycles can
be carried out more precisely than measuring the instantaneous
frequency. The counter is read out at intervals and the data stored.
The counter is reset either after each two minute paragraph or at the
end of the pass. Sequential differences in counter readings actually
constitute a series of biased range differences.
• The curves in this figure are based on actual data collected from
Oscar 19 by a Canadian Marconi CMA-722B receiver near Ottawa,
Canada, on 30 July 1983. ·

CHAPTER 6
THE GLOBAL POSITIONING SYSTEM
(GPS)

What is GPS?
• Official name of GPS is NAVigational Satellite Timing And Ranging Global
Positioning System (NAVSTAR GPS)
• Global Positioning Systems (GPS) is a form of Global Navigation Satellite
System (GNSS):
• GPS - USA
• GLONASS – Russian
• GALELIO – European Union
• BeiDou/CAMPSS – Chinese
• QZSS - Japanese
• Developed by the United States of
America Department of Defense (USA DoD)

What is GPS?
• The Global Positioning System (GPS) was designed for military applications.
• Its primary purpose was to allow soldiers to keep track of their position and
to assist in guiding weapons to their targets.
• The satellites were built by Rockwell International and were launched by
the U.S. Air Force.
• The entire system is funded by the U.S. government and controlled by the
U.S. Department of Defense.
• The total cost for implementing the system was over $12 billion
• It costs about $750 million to manage and maintain the system per year

History of GPS
• Initiated by U.S. Department of Defense
• Military planners wanted a technology where a position could be
obtained without the use of radio transmissions
• Feasibility studies begun in 1960’s.
• Pentagon appropriates funding in 1973.
• First satellite launched in 1978.
• System declared fully operational in April, 1995.
• Open to the public, 2000.

How does GPS work?
 Stations on earth, and a GPS
receiver, the distances between
each of these points can be
calculated.
 The distance is calculated based
on the amount of time it takes for
a radio signal to travel between
these points.
 Using satellites in the sky,
ground allows the GPS receiver
to know where you are, in terms
of latitude and longitude, on the
earth.
 The more satellites the GPSr can “see”, the more accurate your
reading.
 The GPSr must “see” the satellites, so it does not work well in dense
forests, inside caves, underwater, or inside buildings.

GPS SEGMENTS
GPS is made up of 3 segments
• Space Segment (SS)
• Control Segment (CS)
• User Segment (US)

Control Segment
Space Segment
User Segment
Three Segments of the GPS
Monitor Stations
Ground
Antennas
Master Station

Space Segment
• Satellite constellation consist of 24 satellites
• 21 satellite vehicles
• 3 spare satellite
• GPS satellites fly in circular orbits at an altitude of 20,200 km
• Orbital period of 11 hrs. 55 mins.
• Powered by solar cells, the satellites continuously orient
themselves to point their solar panels toward the sun and their
antenna toward the earth.
• Orbital planes are centered on the Earth
• Each planes has about 55° tilt relative to Earth's equator in order
to cover the polar regions.

Space Segment (Continued)
• Each satellite makes two complete orbits each
sidereal day.
• Sidereal - Time it takes for the Earth to turn 360 degrees in
its rotation
• It passes over the same location on Earth once each
day.
• Orbits are designed so that at the very least, six
satellites are always within line of sight from any
location on the planet.

Space Segment (Continued)
• Redundancy is used by the additional satellites to
improve the precision of GPS receiver calculations.
• A non-uniform arrangement improves the reliability
and availability of the system over that of a uniform
system, when multiple satellites fail
• This is possible due to the number of satellites in the
air today

GPS Satellite Vehicle
• Four atomic clocks
• Three nickel-cadmium batteries
• Two solar panels
• Battery charging
• Power generation
• 1136 watts
• S band antenna—satellite control
• 12 element L band antenna—user
communication
Block IIF satellite vehicle (fourth
generation)

GPS Satellite Vehicle
• Weight
• 2370 pounds
• Height
• 16.25 feet
• Width
• 38.025 feet including
wing span
• Design life—10 years
Block IIR satellite vehicle
assembly at Lockheed
Martin, Valley Forge, PA

Control Segment
• The CS consists of 3 entities:
• Master Control System
• Monitor Stations
• Ground Antennas

Kwajalein Atoll
US Space Command
Control Segment
Hawaii
Ascension
Is.
Diego Garcia
Cape Canaveral
Ground Antenna
Master Control Station Monitor Station

Master Control Station
• The master control station, located at Falcon Air Force
Base in Colorado Springs, Colorado, is responsible for
overall management of the remote monitoring and
transmission sites.
• GPS ephemeris is the tabulation of computed
positions, velocities and derived right ascension and
declination of GPS satellites at specific times for
eventual upload to GPS satellites.

Monitor Stations
• Six monitor stations are located at Falcon Air Force
Base in Colorado, Cape Canaveral, Florida, Hawaii,
Ascension Island in the Atlantic Ocean, Diego Garcia
Atoll in the Indian Ocean, and Kwajalein Island in the
South Pacific Ocean.
• Each of the monitor stations checks the exact altitude,
position, speed, and overall health of the orbiting
satellites.

Monitor Stations (continued)
• The control segment uses measurements collected by
the monitor stations to predict the behavior of each
satellite's orbit and clock.
• The prediction data is up-linked, or transmitted, to
the satellites for transmission back to the users.
• The control segment also ensures that the GPS
satellite orbits and clocks remain within acceptable
limits. A station can track up to 11 satellites at a time.

Monitor Stations (continued)
• This "check-up" is performed twice a day, by each
station, as the satellites complete their journeys
around the earth.
• Variations such as those caused by the gravity of the
moon, sun and the pressure of solar radiation, are
passed along to the master control station.

Ground Antennas
• Ground antennas monitor and track the satellites
from horizon to horizon.
• They also transmit correction information to
individual satellites.

User Segment
• The user's GPS receiver is the US of the GPS system.
• GPS receivers are generally composed of an antenna,
tuned to the frequencies transmitted by the satellites,
receiver-processors, and a highly-stable clock,
commonly a crystal oscillator).
• They can also include a display for showing location
and speed information to the user.
• A receiver is often described by its number of
channels this signifies how many satellites it can
monitor simultaneously. As of recent, receivers
usually have between twelve and twenty channels.

User Segment (continued)
• Using the RTCM SC-104 format, GPS receivers may
include an input for differential corrections.
• This is typically in the form of a RS-232 port at 4,800 bps
speed. Data is actually sent at a much lower rate, which
limits the accuracy of the signal sent using RTCM.
• Receivers with internal DGPS receivers are able to
outclass those using external RTCM data.

Trilateration
• GPS can be compared to trilateration.
• Both techniques rely exclusively on the measurement of distances to fix positions.
• One of the differences between them, however, is that the distances, called ranges in GPS, are not measured
to control points on the surface of the earth.
• Instead they are measured to satellites orbiting in nearly circular orbits at a nominal altitude of about 20,183
km above the earth.
• Trilateration is based upon distances rather than the intersection of lines based on angles.
• Now, in a terrestrial survey as indicated in this image here, there would probably be a minimum of three
control stations and from them would emanate three intersecting distances, i.e. L1, L2, and L3.
• This is very similar to what's done with GPS except instead of the control points being on the surface of the
Earth, they are orbiting the Earth. The GPS satellites are the control points orbiting about 20,000 kilometers
above the Earth.
• There's another difference, instead of there being three lines intersecting at the unknown point, there are
four.
• Four are needed because there are four unknown - X, Y, Z, and time - that need to be resolved.

Satellite Geodesy Lecture Notes MSU 2015.pptx

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