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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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