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CHAPTER 2 GPS GEODESY 2.1. INTRODUCTION...
Transcript of CHAPTER 2 GPS GEODESY 2.1. INTRODUCTION...
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CHAPTER 2
GPS GEODESY
2.1. INTRODUCTION
The science of geodesy is concerned with the earth by quantitatively
describing the coordinates of each point on the surface in a global or local
coordinate system. The real shape of the earth approximates that of an
oblate ellipsoid, which is a solid obtained by rotating an ellipse and is called
the Reference Ellipsoid or Spheroid (Leick, 1995). Geodetic investigations are
aimed at determining vectorially the coordinates of the real earth surface
locations with respect to the origin and surface of this Reference Ellipsoid.
Geodetic measurements are made with reference to two points which may be
a few kms apart on the earth's surface, coordinates of which have been
precisely determined. The line joining these points and forming a baseline is
used to constitute a triangle by including a third unknown point whose
azimuthal angles from the end points of the base line are measured by
theodolites. These two angular measurements, together with the base line
length, enable one to calculate the coordinates of the third unknown point
and the lengths of the other two sides. Using these other two sides as the
next bases for the new triangles, the process is then repeated to cover the
whole region with a network of triangles whose vertices can be accurately
located (Parkinson et al., 1996).
Modern geodetic investigations make use of the same principle, except that
the baseline is constituted by positions of a set of 24 specially configured
orbiting Global Positioning System (GPS) satellites up in the sky. Satellite
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positions, calculated by using Kepler's laws and periodically updated by the
actual tracking to correct any deviations caused by solar radiation pressure
are fed to the respective satellites which, in turn, beam that information to
the earth with transmission instant. Ground receivers are rigidly positioned at
selected locations for accurate coordinate determinations and convert this
into Satellite-Receiver Ranges Rjk (distances between the jth receiver and kth
satellite) by multiplying the travel time of radio signals (beamed
simultaneously at two frequencies of 1575.42 and 1227.6 MHz) with the
velocity of light (~ 3X 100 million m/sec) (Leick, 1995).
The heart of GPS Geodesy is the unprecedented accuracy of range and
thereby coordinates determinations are made possible by the fact that the
ruler used is the velocity of light, traveling at the rate of 300 billion mm/sec.
The lengths of a single wave at the transmitted frequencies are of the order
of 300 mm. Since each cycle consists of 360 degrees which can be accurately
made to within 1 degree using modern techniques, it enables one to obtain
an accuracy of ~ 3 mm in range determination. This potential accuracy of
3mm is, however, slightly degraded by the effect of the charged Ionosphere
and atmosphere in the troposphere, both of which introduce travel time
delays due to varying refractive index (Wells, 1989; Hoffmann et al., 1994;
Leick, 1995). Fortunately, the former effect can be calculated by exploiting
the dispersive (frequency dependent) effect of the ionosphere by making
measurements at two different frequencies (dual frequency GPS
transmission). Most of the tropospheric effect can be calculated from the
ground pressure and humidity by modeling the atmosphere as a
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hydrostatically stabilized medium. A source of inaccuracy stems from the
uncertainty in the knowledge of precise satellite coordinates at every instant.
However, the International Geodetic Service corrects this shortcoming by
calculating refined orbital of GPS satellites by jointly analyzing synchronous
data from several hundred permanent GPS stations around the world. This
exercise naturally takes time but refined orbital are available for geodetic
analysis with a delay of about a fortnight and disseminated for scientific use
worldwide through the internet. GPS is widely recognized for the precision
measurement of the baseline vector between pairs of receiver antennas. By
differencing the carrier phases simultaneously recorded by the receivers, the
coordinates of one end of the baseline (“remote” or “rover” site) can be
established with respect to the other end ("base" or "reference" site). This is
called as interferometry. The phase of fringe pattern is the difference of
phases of the interfering waves. In the case of GPS, the differences of the
carrier phases measured by two receivers are simply fringe phases (Wells,
1989; Parkinson et al., 1996). GPS carrier-phase measurements are used
mostly for positioning, with the coordinates of antenna determined either
from post processing of collected data or in real time with the aid of a
communication link. However, GPS interferometry can also provide
information on the orientation or direction of the baseline connecting the
antennas. If these antennas are rigidly mounted on a platform, one can
derive information on the platform's attitude.
2.2. ADVANTAGE OF GPS SURVEYS
1. It is three-dimensional, weather independent and site intervisibility is
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not needed.
2. The rapid data processing with quality control is possible with high
precision by using common reference system.
3. This is cost effective method which can be operated day and night with
a very few skilled personnel.
4. All or any of the following values could be available directly in the field
or after post-processing of the GPS data; Latitude, longitude, geodetic
height and X, Y, Z Cartesian coordinates; State Plane or Project
coordinates; Forward and back geodetic azimuth of the baseline;
Geodetic distance or monument to monument slope distance of
baselines; Vertical angle from point to point and geodetic azimuth
directly between two points.
2.3. MODE OF GPS SURVEYING
Planning of GPS survey is most important, regardless of the technique used.
Presently the techniques (mode) being used are static, Fast static (rapid
static), Kinematic, Pseudo - kinematic (pseudo-static) and Real time
kinematic.
2.3.1. Static mode of GPS surveying
This method is used for surveying projects that require high accuracy. In this
each receiver logs data at each point continuously for a pre-planned length of
time and the duration of data collection depends upon required precision,
number of visible satellites, satellite geometry (DOP), type of the receivers
(single frequency or dual frequency) and distance between receivers.
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The duration of data collection, however, should be long enough for the post
processing software to resolve the integer ambiguity. The modest receivers
and processing software are capable of resolving the ambiguity with small
amount of data. However, a higher accuracy for the baseline components can
be achieved by collecting data for a longer period of time. Collection of data
using two or more receivers for a certain period of time is called a session.
The slope line between any two antennas is called a baseline vector or simply
baseline. If more than two receivers are used, multiple baseline vectors can
be determined simultaneously. Most GPS survey projects consist of multiple
baselines or networks, and the baselines can be measured individually using
only two receivers or several at a time. When the baseline between a known
point and a new point is measured, the new point can be used as a known
point for other baselines.
Unlike in conventional surveys, the accuracy obtainable from networks is
independent of the network geometry. Accuracy can be increased by
increasing the number of redundant measurements. Redundant
measurements are over and above the ones required to determine the
coordinates of unknown points. A redundant measurement should also be
independent, i.e., a measurement that is not related to or could not be
generated from other measurements. In a single session using more than
two receivers, there are both independent (non-trivial) and dependent
(trivial) baselines. Baselines measured in separate sessions are always
independent. In a network of GPS baselines, blunders can be detected by
checking the closure of loops formed by connecting independent baselines. If
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the loops are elongated in an east-west direction, a higher accuracy in the
position can be obtained (GPS measurements are stronger in north-south
direction) and this may taken care of in the beginning. The networks should
also have several control points, located at strategic locations in order to
strengthen the network. These control points should be preferably above or
at least equal to the order of accuracy. The number and locations of control
points depend on the size and shape of network (see geometric geodetic
accuracy standards and specifications for using GPS relative positioning
techniques, Federal Geodetic Control Sub-committee, 1988).
Figure 2.1. Glacier surface velocity measurements in fast static mode
2.3.2. Fast static mode of GPS surveys
Fast Static or Rapid Static was a method developed for dual frequency
receivers. The field requirements and procedure for the fast static are same
as those for static except for the short session lengths. However, fast static is
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only suitable for low order control surveys, e.g., ground control for
photogrammetric mapping and glacier surface velocity measurements (Figure
2.1).
2.3.3. Kinematic mode of GPS surveying
This is a mode of positioning from a moving platform (i.e., when the antenna
is in motion) and used in navigation where usually only a single receiver is
used. However, unlike in navigation, the kinematic method is a relative
positioning method in which one antenna with a receiver are stationary and
other antenna with a receiver moving. When the moving receiver is in
constant motion, it is called as ‘continuous’ kinematic. In most surveying
applications, a method called ‘stop-and-go’ kinematic is used. The stationary
receiver, called as the base receiver (Figure 2.2 a), is placed at a known point
while a second receiver called as "rover' (Figure 2.2 b)
Figure 2.2. (a) Base and (b) Rover during kinematic survey of Gangotri glacier
(Kumar et al., 2008)
visits all the unknown points. Rover occupies each unknown point for a very
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short time (less than two minutes), hence the term "Stop-and-Go" surveying
is used. It is also possible to combine both ‘continuous’ and ‘stop and go’
methods in the same survey to operate more than one ‘rover’ with the same
base station. The single most advantage of ‘stop and go’ surveying is its
speed. This method also has following limitations, such as an initialization
process to determine the integer biases of at least 4 satellites is needed at
the beginning. Secondly, the lock on the four or more satellites must be
maintained during the entire survey. For this reason, kinematic GPS
surveying is suitable for an area where there are no large over-hanging trees
and over-passes or such structures in rover’s route. If for any reason a cycle
slip occurs, the rover must return to any previous point which had been
determined without cycle slip.
2.3.4. Real time kinematic GPS survey
Real time kinematic (RTK) refers to a stop-and-go method where the
coordinates of the points are available in real time. Here, a radio
communication link is maintained between the base receiver and the rover.
The base receiver supplies pseudo-range and carrier phase measurements to
the rover which in turn computes its position and display the coordinates.
The rover keeps updating coordinates as it moves as long as the lock on
satellites is maintained. Kinematic GPS surveying is generally suitable for any
type of surveying or mapping. However, for stakeout surveys, RTK is
essential. Some RTK receivers have the capability of resolving integer
ambiguity and this can only be used with dual frequency receivers. This
means that there is no need to maintain the lock on satellites while the rover
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is in motion. With this technique, the integer ambiguity can be resolved while
the receiver is still in motion (Blewitt and Lichten, 1992).
2.3.5. Pseudo-kinematic (or pseudo-static) survey
This is a combination of both static and kinematic methods. It has the speed
of kinematic method but there is no need to maintain lock on 4 satellites.
However, newer receivers and algorithms can resolve the integer ambiguity
much faster. There is a reference (base) receiver and a roving receiver, the
former remains at the reference point during the entire survey while the later
visits the unknown points (Pant et al., 2008). There is no initialization as in
‘stop and go’ method. Each point is occupied for 5-10 minutes for baselines
of 10 km or less. Each point must be revisited multiple times. Multiple
observations at the same site at different times capture different epochs
along the satellite's orbit and allow the satellite configuration to change and
to resolve the integer ambiguity. This technique is suitable for areas where
there are obstructions to signal or the receivers are not equipped with the
kinematic software. Pseudo-kinematic is the least precise of all methods but
is more productive than static “Stop-and-Go” and suitable for lower order
control such as photogrammetric control etc.
2.4. GPS SEGMENTS
The Global Positioning System consists mainly of three segments, the space
segment, the control segment and user segment.
2.4.1. Space segment
At altitude of about 20,000 km, space segment contains 24 satellites, in 24
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hours near circular orbits with inclination of orbit 550 (Fig. 2.3). The
constellation ensures at least 4 satellites in view from any point on the earth
at any time for 3-D positioning and navigation on the world wide basis.
Three-axis controlled earth–pointing satellites continuously transmit
navigation and system data comprising predicted satellite ephemeris, clock
error etc., on the dual frequency L1 and L2 bands.
2.4.2. Control segment
Control segment consists of a master control station, monitor station and
upload stations. Major operational tasks of control segment are to estimate
satellite ephemeredes and atomic clock behavior to predict the position of
satellites, clock drifts and subsequently upload the information to satellites.
The monitor stations are transportable shelters with receivers and computers
which passively track satellites from the navigation signals. This data is
transferred to master control segment for computer processing to provide the
best estimates of satellite position, velocity and clock drift relative to the
system time.
Figure 2.3. Space segment containing GPS satellites
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The data processed there of generate refined information of gravity field
influencing the satellite motion, solar pressure parameters, position, clock
bias and electronic delay characteristics of ground stations and other
observable system influences. Future navigation messages, generated from
this are loaded into the satellite memory every day via upload station which
has a parabolic antenna, a transmitter and a computer.
At present, there are five monitor stations located at Hawaii, Colorado
Springs, Ascension Island (South Atlantic Ocean) Diego Garcia (Indian
Ocean) and Kwajalein (North Pacific Ocean) (Gouldman et al, 1989) (Figure
2.4). Each station equipped with a precise cesium time standard and
receivers that continuously measure pseudoranges to all satellites in view.
The pseudoranges are measured every 1.5 seconds and after using the data
of ionosphere, they produce 15 minute interval data which is finally sent to
the master control station (Hoffmann-Wollenhoff et al., 1992).
Figure 2.4.World-wide locations of control segment station (Gouldman et al, 1989)
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2.4.3. User segment
The GPS User Segment consists of the GPS receivers and user community.
The GPS receivers convert SV signals into position, velocity and time
estimates. Four satellites are required to compute the four dimensions of X,
Y, Z (position) and Time. The GPS receivers are used for navigation,
positioning, time dissemination and other researches. Navigation in three
dimensions is the primary function of the GPS. The navigation receivers are
made for aircraft, ships, ground vehicles and individuals. Precise positioning
is possible by using GPS receivers at reference locations providing corrections
and relative positioning data for remote receivers (Leick, 1995; Kaplan,
1996). Surveying, geodetic control and plate tectonic studies are the
examples.
2.5. POSITIONING CONCEPTS
Various methods of determining the unknown geographic coordinates of the
receiver can be used depending upon the information collected by the
receiver. Two common methods of position determination are known as
pseudo-range positioning and differential carrier phase tracking. These
methods can be used with a combination of various mathematical positioning
models to determine the unknown geographic coordinates of the receiver
(Leick, 1995; Parkinson et al., 1996).
2.5.1. Deferential carrier phase tracking
Carrier Phase Tracking is accomplished by tracking the fractional phase of the
L1 or L2 carrier signals as they arrive at two or more GPS receivers at the
same time. The fractional phase of the L1 or L2 carrier signals arriving from
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multiple satellites is tracked over time and is used to infer the distance to
each satellite. As the GPS satellites are at far distances, the signals at two
receiver locations have essentially the same errors, induced from signal
propagation through the ionosphere and troposphere (Figure 2.5). By using
differences in the observations of multiple receivers, several errors are
removed. This procedure can be done using a single frequency or both the L1
or L2 frequencies (dual frequency). Dual frequency differential carrier phase
tracking yields accurate geographic position on the millimeter scale if
properly processed (Wells, 1989; Leick, 1995).
Figure 2.5. Differential GPS requires that the satellites are observed by two or more receivers at the same time
High quality survey grade GPS equipment and advanced processing software
are required for differential carrier phase positioning. Since the GPS receivers
are at two different locations, it is possible that all the satellites are not
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simultaneously visible to both receiving sites. The equations of mathematical
positioning for this method are outside the scope of this thesis, however the
topic is thoroughly described in Leick (1995) and Hoffmann-Wellenhof et al.
(1997).
Figure 2.6. Pseudo-range positioning (p1, p2, p3 and p4) relies on the estimate of the geometric distances between the satellite and receiver
2.5.2. Pseudo-range positioning
Pseudo-range positioning relies on determining the amount of time it takes
for the signal to propagate from the satellite to the receiver. This
transmission time is then used to determine the geometrical distance from
the receiver to the satellite as depicted in Figure 2.6.
Each GPS satellite transmits an unique pseudo-random signal modulated
onto the L1 carrier frequency, known as the coarse acquisition (C/A) code.
Each GPS receiver contains a copy of the C/A code for each satellite. By
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correlating the signal received from the satellite with one stored in the
receiver, the transmission time can be estimated. Once a propagation time is
estimated, the geometric distance between GPS receiver antenna and
transmitting satellite can also be estimated. The pseudo-range is the
apparent propagation time multiplied by the speed of light in a vacuum.
Since the satellite and receiver clocks are not synchronized to the same time
frame, there is an unknown timing error known as the clock bias. The
pseudo-range differs from the actual geometrical distance by the clock bias,
propagation delays and other errors including relativistic and doppler effects
(Wells, 1989; Hoffmann, et al., 1994; Leick, 1995; Parkinson, et al., 1996,).
The pseudo-range for the jth satellites can be expressed as Pj = pj + c ∆t +
Ttrop + Tion + Trel + Є.
Where Pj is the measured pseudo-range, pj is precise geometric distance
between the receiver and the jth satellite, c is the speed of light in a vacuum,
∆t is the unknown clock bias, Ttrop is the signal path delay due to the
troposphere, Tion is the signal path delay due to the ionosphere, Trel is the
signal delay due to relativistic errors caused by high satellite velocity and Є is
an estimate of the noise. A non-linear equation relates to geometric distance
between the jth satellite, and unknown positions of receiver (pj = ρρρρ(Xj – X)2 +
(Yj – Y)2 + (Zj – Z)2, Where (X, Y, Z) are three unknown coordinates of the
GPS receiver and (Xj, Yj, Zj) are known coordinates of GPS satellite as
transmitted in the ephemeris. A minimum of four satellites must be observed
to solve 3 unknown receiver coordinates and receiver clock bias (Wells,
1989; Hoffmann, et al., 1994; Leick, 1995; Parkinson, et al., 1996).
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2.6. ERROR BUDGET IN GPS POSITIONING
There are three main error sources in GPS positioning, the satellite, signal
propagation and the receiver. Table 2.1 summarizes the error sources and
their effects. These errors affecting the resulting position to varying degrees
are summarized in Table 2.2.
Error Source Effect
Satellite
Signal Propagation
Receiver Variation
Clock bias
Orbital errors
Ionospheric refraction
Tropospheric refraction
Antenna phase center
Clock bias
Multipath
Table 2.1. Sources of errors in GPS surveying (Hoffman-Wellenhof, 1997)
Error Source RMS Error (m)
Ionosphere
Troposphere
Clock & Ephemeris
Receiver Noise
Multipath
7.0
0.7
3.6
1.5
1.2
Table 2.2. Error sources and their RMS effect on the determined receiver coordinates
(Langely, 1997)
2.6.1. Satellite errors
The satellite coordinates used in determining the geographic coordinates of
the receiver are transmitted on the L1 and L2 frequencies along with
parameters describing the satellites orbit and time. The orbit along which the
satellite travels must be known ahead of time. External effects on the
satellite, such as, solar radiation pressure, can shift it out of predicted orbit
by as much as 20 m with RMS (root-mean-square) errors of 5m (Langely,
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2000). Each GPS satellite contains four atomic clocks to ensure that a stable
timing system is maintained. Although these clocks are extremely accurate,
yet they can drift slightly resulting in each satellite’s clock not synchronized
to each other. These errors in the satellites orbital position and clocks can
result in the range of 1–5m in the final geographic position. The International
GPS Service (IGS) uses data collected by these sites to determine the true
orbital path and the clock drift for each satellite. These are offered at no cost
to the GPS community on a variety of time scales. The IGS final orbits are
available for download over the internet two weeks after the observation
date. Currently, the IGS final orbits have an accuracy of 3–5 cm in the orbital
position of satellite and an accuracy of 0.1–0.2 nanoseconds on the satellite
clock drift (Heroux et al., 2001). A substantial improvement in the accuracy
of the geographic receiver coordinates can be made by re-calculating the
receiver coordinates with the new satellite orbit and clock drift.
2.6.2. Propagation errors
GPS satellite signals experience various propagation delays as they travel
through the Earth’s atmosphere. These errors are mainly due to the
ionosphere and troposphere (Langley, 1997). The ionosphere is located
approximately 50 km–1000km above the surface, while the troposphere
begins at the surface of the Earth and extends up to an altitude of 14 km.
the satellites having low elevations with respect to the horizon have higher
ionospheric and tropospheric noise components because of the greater
amount of time spent in travelling through these two layers. The ionosphere
is most active in a region extending approximately 20± on either side of the
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magnetic equator, with high frequency scintillations experienced both in this
region and over the poles (Janssen, 2002). Dual-frequency GPS receivers are
able to remove the ionospheric effect by using a linear combination of
measurements on both frequencies (Janssen et al., 2002).
2.6.3. Receiver errors
The distance measured by the GPS receiver is the distance between physical
phase centers of the receiver and the satellite. However, phase center of the
GPS receiver is unstable and gets changes, with the changing direction of the
satellite signal (Mader, 2002). The phase center variations can be accounted
for by modelling the response of the satellite antenna. The effect of phase
center variation is quite small and is not taken into account for our GPS
prototypes. A significant amount of receiver error can be generated through
a process, known as multipath. This is where the GPS signals are reflected
from surface (such as ground surface or buildings) and directed towards the
antenna. Because the signal has traveled along a longer path, it appears that
the satellite is further away than it actually does.
The GPS receivers contain inexpensive quartz oscillators controlling the
clocks. By using a relatively inaccurate time keeping method, there is an
inherent inaccuracy of the receiver clock resulting in positioning errors.
Although an unknown clock drift is accounted and later solved by iterative
solution method, it can still incorporate large errors in the resulting position.
There are some common errors present in GPS observations which can be
eliminated by taking precautions at the time of GPS data generation.
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2.7. GPS SURVEYING CONSIDERATION
2.7.1. Visible satellites
In order to solve the positioning equations, four or more GPS satellites must
be visible to the GPS receiver. For higher latitudes, the geometry of satellite
constellation can create difficulty in having enough satellites in view of long
periods of time, as the satellites appear low on the horizon. This also makes
the satellite orbits susceptible to being blocked by high topography (Figure
2.7), which can be problematic if the GPS receivers are situated in the
valleys.
Figure 2.7.GPS satellite orbits. The orbital planes of the satellites do not pass directly over the poles
2.7.2. Elevation cutoff mask
To prevent large errors from the ionospheric and tropospheric delays,
satellites below a certain cutoff elevation are usually excluded from being
used in the positioning solution. In this work, the satellites at angle of less
than 150 with respect to the horizon are not incorporated into the solution.
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