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Week 08

Prepared by

Assist. Prof. Dr. Himmet KARAMAN

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Contents GPS Mathematical Models &

Observation Techniques

ISTANBUL TECHNICAL UNIVERSITY - DEPARTMENT OF GEOMATICS ENGINEERING

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Point positioning with code ranges Point positioning with carrier phases

Point positioning with doppler data

Precise point positioning

Differential positioning

Relative positioning

Phase differences

GPS surveying techniques Initialization methods

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Point Positioning with Code Ranges


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The code range at epoch tcan be modeled as

The desired coordinates of the receiver site are

implicit in the distance rs(t) and can be calculated

as; WhereXs(t), Ys(t), Zs(t) are the components of the

geocentric position vector of the satellite at epoch t

andXr, Yr, Zrare the three earth-centered, earth-

fixed (ECEF) coordinates of the observing receiversite.

)()()( tcttR srs

r

s

r

222

)()()()( rsrsrssr ZtZYtYXtXt

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Point Positioning with Carrier Phases


4

Pseudoranges can also be obtained from carrierphase measurements. The mathematical model for

these measurements is;

Where rs(t) is the measured carrier phaseexpressed in cycles.

)()(1

)()( tfNttft rss

r

s

rs

sss

r

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Point Positioning with Doppler Data


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The mathematical model for Doppler data is;

And can be denoted as the observed Doppler shift

scaled to range rate.

is the instantaneous radial velocity between thesatellite and the receiver.

is the time derivative of the combined clock

bias term.

)()()( tcttD srs

r

s

r

)(tsr

)(ts

r

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Precise Point Positioning


Desired ionosphere-free combinations of codepseudoranges and carrier phases for PPP are;

The unknown parameters to be determined are; The point position contained in

The receiver clock error contained in

The tropospheric delay Trop and Ambiguities

Based on this model, PPP may be applied either in staticor in kinematic mode.

2

2

2

1

2

222

2

2

2

1

2

111

2

2

2

1

2

222

2

2

2

1

2

111

2

2

2

1

2

22

2

2

2

1

2

11 ,

ff

fN

ff

fNcff

f

ff

f

cff

fR

ff

fR

Trop

Trop

6

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Differential Positioning


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Differential positioning with GPS abbreviated by DGPS, isreal-time positioning technique where two or more receiversare used.

One receiver, usually at rest, is located at the reference orbase station with known or assumed coordinates and

Remote receivers are fixed or roving and their coordinatesare to be determined.

The reference station commonly calculates pseudorangecorrections (PRC) and range rate corrections (RRC) whichare transmitted to the remote receiver in real time.

The remote receiver applies the corrections to themeasured pseudoranges and performs point positioningwith the corrected pseudoranges.

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Basic Concept of Differential Positioning


8

reference rover

corrections

j

k l

m

satellites

A B

j

A

j

B

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DGPS wit Code Ranges


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Code range at base stationA to satellite s measuredat epoch t0may be modeled as;

where, is geometric range an s are the biases.

The pseudorange correction for satellite s atreference epoch t0 is;

Adapting to the rover site B and epoch t, the code

pseudorange measured at the rover is modeled as;

)()()()()( 00000 tttttR Ass

A

s

A

s

A

)( 0ts

A

)()()()()()( 000000 ttttRttPRC Ass

A

s

A

s

A

s

)()()()()( tttttR Bss

B

s

B

s

B

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DGPS with Phase Ranges


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The phase pseudorange measured at the base stationA at

epoch t0can be modeled as;

The phase range correction at reference epoch t0 is;

The formulation of range rate corrections at the base station A

as well as the application of predicted range corrections to the

observed phase ranges at the rover site B is carried out in fullanalogy to the previously described code range procedure.

s

A

s

A

ss

A

s

A

s

A

s Nttttt )()()()()( 00000

s

A

s

A

ss

A

s

A

ss

A

s NttttttPRC )()()()()()( 000000

s

AB

s

AB

s

Bcorr

s

B

s Nttt )()()(

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Relative Positioning


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For determining the coordinates of an unknown pointwith respect to a known point which, for most

applications, is stationary.

Relative positioning aims at the determination of thevector between the two points, which is often called

the baseline vector or simply baseline.

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Basic Concept of Relative Positioning


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j

k l

m

satellites

A B

j

A

j

B

baseline

ABb

S C C S O G O CS G G

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The Concept


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LetA

denote the (known) reference point,B

theunknown point, and bAB the baseline vector.

The corresponding position vectors XA, XB;

The components of the baseline vector are;

ABABbXX

AB

AB

AB

AB

AB

AB

AB

Z

Y

X

ZZ

YY

XX

b

ISTANBUL TECHNICAL UNIVERSITY DEPARTMENT OF GEOMATICS ENGINEERING

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Phase Differences


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Single-differences; Two receivers and one satellite are involved.

Double-differences;

Assuming two points A,B, and the two satellites j,k, two

single differences acquired. Triple-differences;


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Phase Observables


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k

lm n

A

errorsotherttNtc

ft AkkAkAkA )()()()(

Where,

k

A : geometric range from A to kk

AN : initial unknown integer number of cycles between k & A

k

A: phase measured at A for k at time t

f : frequency of signal

c : speed of light

k : Satellite clock error

A : Receiver clock error

Other errors= Tropospheric refraction + ionospheric refraction

+ noise & biases + multipathing effects

+ antenna phase center offset & variation + etc..


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Single Differences


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2 receivers & 1 satellite (substitute 2 phaseobservable)

AB

k


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Single Difference Equation


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errorsotherttNtc

ft A

kk

A

k

A

k

A )()()()(

Phase equation for stationA and satellite k

errorsotherttNtc

ft B

kk

B

k

B

k

B )()()()(

Phase equation for station B and satellite k

(1)

(2)

Substituting (1) in (2)

errorsothertNtc

ft AB

k

AB

k

AB

k

AB )()()(


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Double Differences


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2 receivers & 2 satellites (substitute 2 singledifferences)

AB

k m


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Double Difference Equation


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Single Difference for satellite kerrorsothertNt

c

ft AB

k

AB

k

AB

k

AB )()()(

Single Difference for satellite m

errorsothertNtc

ft AB

m

AB

m

AB

m

AB )()()(

(3)

(4)

Substituting (3) in (4)

errorsotherNtc

ft kmAB

km

AB

km

AB )()(


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Generalized Mathematical Model for Double

Differencing


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Acronyms


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m

A

k

A

m

B

k

B

km

AB

Phase difference

Rate of change on ranges

ABt Arithmetic mean of the receiver clock errors at A & B

ABt Difference between the two receiver clock errors

m

A

k

A

m

B

k

B

km

ABNNNNN Total integer ambiguity


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Triple Differences


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So far only one epoch t has been considered. Toeliminate the time-independent ambiguities,

differencing double-differences between two epochs

were suggested.

The Triple-difference equation is;

jk

AB

jk

AB

jk

AB

jk

AB

jk

AB

jk

AB

Ntt

Ntt

)(1

)(

)(1

)(

22

11

)(1

)(

form;simplifiedin

)()(

1

)()(

1212

1212

tt

tttt

jk

AB

jk

AB

jk

AB

jk

AB

jk

AB

jk

AB


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Static Relative Positioning


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In a static survey of a single baseline vector betweenpoints A and B, the two receivers must stay

stationary during the entire observation session.

In the following, the single, double, and triple

differencing are investigated with respect to thenumber of observation equations and unknowns.

It is assumed that the two sites A and B are able to

observe the same satellites at the same epochs.

The practical problem of satellite blockage is notconsidered here.


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Kinematic Relative Positioning


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In kinematic relative positioning, the receiver on theknown point A of the baseline vector remains fixed.

The second receiver moves, and its positions is to

be determined for arbitrary epochs.

The models for single, double and triple differenceimplicitly contain the motion in geometric distance.


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Pseudokinematic Relative Positioning


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The pseudokinematic method can be identified asstatic surveying with large data gaps.

The mathematical model for double differences

corresponds where generally two sets of phase

ambiguities must be resolved since the point isoccupied at different times.

The time span between two occupations is an

important factor affecting accuracy.

The minimum time span should be one hour.


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Virtual Relative Stations


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When processing a baseline, the effects of orbiterrors, ionospheric and tropospheric refraction are

reduced by forming differences of the observables.

These effects grow with increasing baseline length.

Therefore, it is good practice to use short baselinesrequiring a reference station close to the rover.

Real-Time Kinematic Survey is a good example for

this case.


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Static survey

Rapid static survey

Stop and go survey

Continuous kinematics survey

Real-time kinematic (RTK) survey

Surveying Techniques

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Stable platforms or pillars Long distances (10 km to thousands of kilometres)

Long occupation time (hours to days)

Control surveys

Simultaneous recording at several stations

Observation rates varying from 5 to 30 seconds

Reducing multipath effects

Post-processing required

Static Survey

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S U C C U S O G O CS G G


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Shorter distances (up to 10 km)

Shorter occupation time (10 minutes)

Densification of control networks

Observation rates varying from a

second to a few seconds

Post-processing required

2 reference receivers required

Reference

receiver 1

Reference

receiver 2

1

2

3

4

Rapid Static Survey

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Distances less than 1 km

1 minute occupation time

Observation rates of seconds

Initialisation required

Repeat initialisation when less

than 4 satellites are being

trackedReference receiverinitialisation

Stop-and-Go Survey

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Initialization required

Non-stop occupation

Observation rates of 1 second

Continuous Kinematic Survey

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receiverreceiver

radio

radio

antennaantenna

Real-Time Kinematic Survey

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Initialization Methods

Static survey

static survey between any two points (usually short baseline) is performed with

sufficient measurements. Specific details are in equipment documentation.

A B

Known baseline

survey is performed between any two

points whose coordinates are

previously determined. Usually one

epoch is sufficient. Only ambiguitiesare estimated with constraining the

position vector.

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Antenna swapStep 1: Reference & rover receivers are located over well defined marks,

collecting simultaneous observations for a period of 1 minute (A)

Step 2: Reference & rover receivers are swapped without changing the

tripods, collecting observations for a period of 1 minute (B)

Step 3: Reference & rover receivers are swapped again to return back totheir original locations, for a period of 1 minute (C)

In general, the first two steps are sufficient to resolve the integer ambiguities.

However, the third step is recommended for a further check.

Reference Rover Rover Reference Reference Rover

A B C34


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On the fly

the first three methods require the receivers to be stationary

there are restrictions in some applications, such as aerial

photogrammetry where camera positions are determined with

GPS. It is not possible to stop the aircraft to perform the above

initialisation techniques.

The on the fly method resolves the integer ambiguities while the

receiver is moving.

5 satellites with good geometry are required, 6 or more are preferred.

Dual frequency receivers are required.

Ambiguity resolution in 5 minutes, 2 minutes with 6 or 7 satellites.

Specific details given in the equipment documentation.


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Documents

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