Summary of doctor thesis le van hung

MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF MINING AND GEOLOGY LE VAN HUNG STUDY OF ADJUSTING COMBINATION OF GPS AND TERRESTRIAL OBSERVATIONS IN THE LOCAL TOPOCENTRIC COOR

MINISTRY OF EDUCATION AND TRAINING

HANOI UNIVERSITY OF MINING AND GEOLOGY

LE VAN HUNG

STUDY OF ADJUSTING COMBINATION OF GPS AND

TERRESTRIAL OBSERVATIONS IN THE LOCAL

TOPOCENTRIC COORDINATE SYSTEM APPLIED FOR

ENGINEERING SURVEY NETWORK

Study field: Geodesy and mapping

Code: 62520503

SUMMARY OF DOCTORAL DISSERTATION

Ha Noi – 2014

The dissertation has been completed at the Department of Engineering surveying, of Surveying Faculty, Hanoi University of

Mining and Geology

Scientific Supervisors:

1 Assoc Prof Dr Dang Nam Chinh Hanoi university of Mining and Geology

2 Assoc Prof Dr Nguyen Quang Phuc Hanoi university of Mining and Geology

Examiner 1: Assoc Prof Dr Nguyen Quang Tac

Hanoi Architectural University

Examiner 2: Dr Dương Chi Cong

VietNam Institute of Geodesy and cartography

Examiner 3: Dr Nguyen Van Van

Military map service of general staff of Viet Nam army

The dissertation will be defended at the University examination Council at the Hanoi University of Mining and Geology , at… h, ……… 2014

This dissertation can be referenced at the National library

or at the library of the Hanoi University of Mining and Geology

1

INTRODUCTION

1. The importance of dissertation

For engineering survey (TðCT), we are likely to encounter

cases where control networks are built in region not convenient for

GPS observation due to limited satellite transmission signal; therefore,

replying on only GPS technology for establishment of control network

is not sufficient In such cases, it is essential for combination of GPS

and terrestrial observations in order to enhance feasibility and

effectiveness in establishment of engineering control network On the

basis of the abovementioned practical requirements, it is prerequisite

for studying means of adjusting combination of GPS and terrestrial

observations and selecting the coordinate system which can be used in

certain limitation as the basic coordinate system for engineering

survey Using the local topocentric coordinate system whose datum

point is established at the center of control network area can mitigate

both abovementioned issues

2 Objective, subjective and scope of the dissertation

- Studying methods for establishment of the local topocentric coordinate

system whose datum point is located at the center of the Works;

completing procedures for transformation and conversion between the

local topocentric coordinate system and other popular systems;

- Determining the appropriate regions for local coordinate system to be

used for civil and industrial engineering surveys;

- Completing the theoretical basis and proposal of procedures for

closely adjusting combination of GPS and terrestrial observations;

- Establishing the software for adjusting combination of GPS

observation and angular-side measurement, this has highly relevent

application in the context of Viet Nam;

- Evaluating the accuracy of software performance

3 Content of dissertation

- General study of GPS technology and total station (TððT) in order

to establish engineering control network;

- Study of the local topocentric coordinate system, conversion from the

local topocentric coordinate system to geodetic coordinate system and

to rectangular coordinate system in UTM projection;

2

- Determination of usage limit of the local topocentric coordinate system in engineering survey;

- Calculation of weight of relative measurement in adjustment of GPS network;

- Study of adjusting GPS network in geocentric coordinates system (X,Y,Z) and the local topocentric coordinate system;

- Study of adjusting combination of GPS observation and angular- side measurement in the local topocentric coordinate system;

- Study of establishing computer programs to adjust combination of GPS observation and angular- side measurement

4 Scientific and practical meaning of dissertation

- Findings of the study provide the basis for adjusting combination of GPS and terrestrial observations in the local topocentric coordinate system applying for engineering high precision network

- Proposal to use the local topocentric coordinate system as the basic coordinate system and replacement of rectangular coordinate system in UTM projection in civil and industrial engineering survey;

- Algorithms and procedures are converted into combined adjustment softwares that are applicable to practical engineering survey in Viet Nam

- Findings of the dissertation can be applicable in the field of education, teaching, scientific research and manufacturing

5 Rational arguments:

- First argument: It is significant for calculation of weight in GPS network and combination of GPS network and terrestrial observation Weight of GPS network should be standardized in accordance with two-step adjustment procedure

- Second argument: It is required to calculate angle distort for adjusting the measured angle before adjustment combination of GPS and terrestrial observations Factors (i.e dimensions) of the network after adjustment ensure the best suitability with field factors These factors are necessary for the design works, construction of civil and industrial works that requires high accuracy

- Third argument: The local topocentric coordinate system can be used

as basic coordinate system, which is the replacement of rectangular coordinate system in UTM projection in order to serve for surveying

3 stage (large scale ratio), design, and construction works Scope

(radius) of using the local topocentric coordinate system is estimated

according to the requirements of angle distor and length distort

6 New contributions

- Completing the formulation system in order to determine appropriate

region for the local topocentric coordinate system to be used in

engineering surveying;

- Introducing the formula for calculation of the horizontal angle distort

when adjusting combination of GPS and terrestrial observations in the

local topocentric coordinate system

- Establishing Data Processing Software called ADGT (ADjustment

GPS Topography) including transformation module, conversion

module, module for analyzing side processing data, module for

adjusting combination of GPS and terrestrial observations in the local

topocentric coordinate system applying for engineering surveying

7 Quantity and structure of dissertation

Content of dissertation is presented on 118 pages, 39 drawings

and diagrams, 21 tables

CHAPTER 1 OVERVIEW OF APPLICATION OF GPS TECHNOLOGY IN

ESTABLISHMENT OF ENGINEERING CONTROL NETWORK

AND ADJUSTMENT OF GPS NETWORK

1.1 METHODS FOR ESTABLISHMENT OF GEODETIC NETWORK

Two common methods for establishment of horizontal control

network as well as engineering surveying network are use of terrestrial

observation (traditional) and satellite positioning (GNSS) Each

method has its own advantages and disadvantages and requires

specific conditions for measurement and establishment of the network

1.1.1 Terrestrial observation method

In the past, in order to establish engineering control network, we

mainly use total stations However, this method is not fully efficient in

complicated topography and poor sky visibility In addition, one

disadvantage is that performance of terrestrial observation only can be

delivered in the appropriate weather conditions and appropriate times

4

in order to avoid the impact of weather conditions and meteorology on the measurement result

1.1.2 GPS observation method

Based on specific parameters a, b of GPS satellite and total station, draw the diagram (Figure 1.1) to compare visually the accuracy of side length measurement by GPS ( mD-GPS) and accuracy of side measurement by total station (mD-TD) According to the diagram, at the distance of D <2,3 km, side length error by GPS is larger than that by total station

Figure 1.1 - Comparison of side length error by GPS and by total station

1.1.3 Combination of GPS and terrestrial observations

The control network used in engineering surveying requires high accuracy, specific selection of appropriate location and construction time of the network If only using one method for establishment of network (either terrestrial observation or GPS observation), difficulties can arise due to the certain disadvantages of each method Therefore, combination of both methods for establishment of the control network will gain advantages, overcome drawbacks of each method, improve the efficiency and shorten construction duration of the network

1.2 OVERVIEW OF APPLICATION OF GPS TECHNOLOGY IN ESTABLISHMENT OF ENGINEERING CONTROL NETWORK

1.2.1 Application of GPS technology in establishment of engineering control network in foreign countries

In foreign countries, geodesists early made applications of GPS technology for engineering survey and results indicate that accuracy (plan, elevation) in short distance can be from 2mm to 5mm

1.2.2 Application of GPS technology in establishment of engineering control network in Viet Nam

5

In engineering survey, GPS technology has been researched to

be applicable for each stage of design, construction and operation

GPS technology was used for establishment of geodetic networks for

the following works: Dung Quat Industrial Zone, National

Convention Center, My Dinh National Stadium, But Son Cement

Plant, Thai Nguyen Cement Plant, Bai Chay Bridge, Thanh Tri Bridge,

Yen Phong Industrial Zone…

1.3 OVERVIEW OF PROCESSING DATA ON COMBINATION OF

GPS AND TERRESTRIAL OBSERVATIONS IN ENGINEERING

SURVEY

1.3.1 Processing data on combination of GPS and terrestrial

observations for enginneering survey in foreign countries

The Observations in the GPS network is correlative

measurement [54] and posterior covariance matrix after baselines

solutions will be apriori covariance matrix of the adjustment GPS

network [64] This is the difference between calculation of adjustment

of modern three-dimensional coordination network and traditional

two-dimensional rectangular coordination network Adjusting

combination of GPS and traditional terrestrial observation in

the local topocentric coordinate system was studied by Slawomir

Cellmer and Zofia Rzepecka since 2008 [65]; however, mentioned

terrestrial observation was only the length of sides measured by total

station, but not horizontal angle measurment

1.3.2 Processing data on combination of GPS and terrestrial

observations in engineering survey in Viet Nam

A number of studies related to this subject were conducted;

however, mostly in geodetic coordinate system and resolutions for

large network

1.4 ADJUSTMENT SOFTWARE FOR GEODETIC NETWORK IN

FOREIGN COUNTRIES

Besides a number of data processing softwares together with

GPS sattelites from well-known manufacturers such as Trimble, Leica,

there are some other softwares introduced in the websites of

Geoinformatics Software Development Companies:

- Software STAR*NET v.7.1 (2012) made in Canada

6

- Software MOVE3 v.4.0.2 made in Netherlands

- Software COLUMBUS v.3.8 made in United States etc

General comments:

Findings of above-mentioned foreign and domestic studies indicate that application of GPS technology for establishment of geodetic control network in general and engineering control networks

in particularly has become popular For engineering survey (TðCT),

we are likely to encounter cases where the Works are implemented in region not convenient for GPS observation due to limited satellite transmission signal Moreover, it is difficult to apply only GPS technology for establishment of network when the Works are built in adjacent to each other or in the area with barriers and signal jamming factors In such cases, GPS and terrestrial observations are combined

in order to improve the feasibility and effectiveness for establishment

of engineering control network

For GPS network, (3D) three-dimensional coordinate network, the network is adjusted in the geocentric coordinate system (X,Y,Z) or geodetic coordinate system (B,L,H) and then transformed to rectangular coordinate system in UTM projection The calculation procedure above is entirely reasonable when being applied for the national coordinate system, all over the country For engineering surveying network with small scope of control, using the local topocentric coordinate system to process GPS network are mentioned in some studies, including adjustment combination of GPS observation and distance measurement by total station In fact, there have not been any in-depth researches into above issue

Regarding to the local topocentric coordinate system, it is required to clarify some issues in order to apply the system appropriately and determine the largest area for using the the local topocentric coordinate system as basic coordinate system for engineering surveying during the process of surveying, design, construction and resolutions of relationship between the local topocentric coordinate system and national coordinate system A number of studies relating to adjustment of combination of GPS and

7 terrestrial observations were conducted; however, mostly with

geodetic coordinate system and resolutions for large network Now,

usage of the rectangular coordinate system in UTM projection is not

conformed to the major Works that set high requirement for the level

of distort of the reference grid Therefore, research and selection of a

coordinate system that is conformed to the Works and completion of

procedures for processing measurement data in that coordinate system

is a practical requirement of engineering surveying in our country

Regarding to softwares, besides some data processing softwares

for GPS observation such as GPSurvey 2.35, TTC, TBC, etc , together

with adjustment module of GPS network, there will be adjustment

softwares for engineering surveying such as STAR*NET, MOVE3 and

COLUMBUS which are dedicated adjustment softwares for all geodetic

networks (1D, 2D, 3D) Similarities of these softwares are that results

are recorded in the form set by programmer and explained in English

Until now most manufacturers in our country has not used these

softwares, but applied the adjustment module of data processing

softwares of Global Navigation Satellite System (GNSS) Therefore, it

is crucial to establish computation programs for calculating adjustment

of combination of GPS and terrestrial observations in engineering

surveying for practical application in Viet Nam

CHAPTER 2 THE LOCAL TOPOCENTRIC COORDINATE SYSTEM AND

APPLICATION IN THE ENGINEERING SURVEY

2.1 REQUIREMENTS OF REFERENCE SYSTEM FOR GEODETIC

HORIZONTAL CONTROL NETWORK

When carrying out the establishment of engineering control

network, it is required that distance between points in the network

after adjustment shall be conformed to the actual dimension on the

field Because of the above requirements, selection of reference scale

(mo), central meridian (Lo) when using UTM projection is significant

to process the engineering control network

2.1.1 Correction horizontal projection

8 Correction to the measured edge length when calculated according to the UTM projection

' S R 2

y 1 m (

m

2 m 0

UTM = − +

∆ (2.1)

With zone 30 (m0=0,9999) and the relative distortion length limit is

1 / T = 1/200000, calculate ymax=92,3 km and ymin=87,8 km Such areas with distortion smaller than 1/200000 is only 4,5 km wide

With zone 60 (m0=0,9996) and the relative distortion length limit is

1 / T = 1/200000, calculate ymax=181,3 km and ymin=179,1 km Such areas with distortion smaller than1/200000 is only 2,25 km wide

2.1.2 Correction for elevation compared with the reference Ellipsoid

Correction for elevation calculated by the formula:

S R

H S

m

m

H=−

∆

(2.2)

In order to reduce this distortion, choosing one projection surface

elevation of approximately average height measured region

2.2 LOCAL TOPOCENTRIC COORDINATE SYSTEM 2.2.1 Establishment of the local topocentric coordinate system

The local topocentric coordinate system is used in the satelite geodesy, geodetic astronomy, for determining the instantaneous position of satelite or space objects (coordinates of satellites or space objects continously change in the topocentric coordinate system and therefore, it is necessary to take consideration

into the time factor) in the coordinate system establised at the observing position

on the earth surface The topocentric coordinate system is also used for transforming the equations of 3D coordinate system, which is mentioned in the dissertation [58],[62]

Figure 2.1- The local topocentric coordinate system

Z

Y

X

L G

BG O

H Gr

G

x

y z

9

2.2.2 Transformation and conversion

When establishing the rectangular coordinate system and using

this system in engineering surveying- map, it is the priority to build and

complete the equation for conversion of coordinates from this

coordinate system to different popular coordinate system The equation

for conversion of the coordinate (in the forward direction and reverse)

ensures accuracy, about 0,”00001 for geodetic coordinates B, L

For example: Conversion from geodetic coordinates to the local

topocentric coordinate system presented in this dissertation

Comment: After conversion from geographical ellipsoidal coordinates

(B, L, H) into the local topocentric coordinate system (x, y, z) and vice

versa, it is indicated that calculated error is only 0,00001”, which

satisfies the calculations requiring high accuracy Elliposidal height (H)

completely coincides with the initial value

2.2.3 Measurement of coordinate and elevation to the control

network

According to the requirements stated in current standards and

codes and national management works, it is necessary for measurement

of national coordinates to the control network in order to:

- Carry out construction works in accordacne with the general planing;

inspect the red line boundary and construction boundary line

- Uniform the elevation system of the control network with national

elavation system, meeting the requirements of space planning, water

supply and drainage works for the works

- Manage the Works by national geographic database

2.3.DETERMINATION OF THE LIMITATION WHEN USING OF

LOCAL GEODETIC COORDINATE SYSTEM FOR ENGINEERING

SURVEY

2.3.1.Theoretical basis

Basic plane playing an important role in the local topocentric

coordinate system is horizontal plane (horizon plane) which is

perpendicular to the normal direction of Ellipsoid at the reference

point On that horizontal plane, a rectangular coordinate system x, y

(or N, E) is established and can be used as ground coordinates of the

building Using this method, we can set a cartesian coordinate system

10

in which the basic plane is close to normal horizontal plane of the building This is necessary for small area and high-altitude building in mountainous areas such as hydroelectric projects, industrial zones, etc

Figure 2.2- Options for establishment of topocentric coordinate system For three cases shown in the Figure 2.2, the length L in the horizon plane will be compared with the length of S geodetic line on the Ellipsoid which replaced by the length of large semi- circle with the radius Rm+HG (Figure 2.2.b) In the third case (Figure 2.2.c), the position of point on relief plan is projected toward normal direction at

G on the horizontal plane without using practical ellipsoid

2.3.2.Determination of radius of area to be used for the local topocentric coordinate system

1.Calculation of scope of measurED area acoording to length distortion limit

- For the case that the distortion of the length L is based on the

datum: L ≤ 15,6 km;

- For the case that the distortion of the length L is not based on the

datum: L ≤ 20,1km;

- For general case (estimated according to the requirement on

distort): L≤ 2,45.R 1 /T

2 The equation for calculating the horizontal angle distort

Adjustment of the horizontal angle distort due to elevation differences between points is the difference:

∆ = − = ∆ P P ΦP−∆T T ΦT

T P

d L z d

L z R

sin sin

"

ρ δ δ β ββ

Figure 2.3 - Adjustment of the horizontal angle distort

11

2.3.3.Surveying horizontal angle distort

It can be realized that in the case, measured angle is far from the

reference point of the topocentric coordinate system with a distance of L=

1km and elevation difference of 55m (200m long side), horizonal angle

distort due to elevation difference of points is 13” If the length of side is

greater, the horizontal angle distort will be smaller in accordance with the

inverse proportion In general, this quite big distort should be taken

consideration into when adjusting combination of horizontal angle

measurement and GPS observation in the topocentric coordinate system

Table 2.1- Deformation values of horizontal angle δ and correction of β

deformation∆β

No Angle symbol

(T – M – P)

Angle on Ellipsoid (β)

Angle on plane (β') δβ (“) ∆β(“)

1 2 - 1 - 3 90 00 00.00 90 00 08.89 8.89 8.90

2 3 - 1 - 4 60 00 00.00 60 00 06.65 6.65 6.66

3 4 - 1 - 5 30 00 00.00 30 00 01.43 1.43 1.43

4 5 - 1 - 6 59 59 59.92 59 59 56.08 -3.84 -3.85

5 6 - 1 - 2 120 00 00.08 119 59 46.95 -13.13 -13.15

6 2 - 6 - 1 30 00 00.00 30 00 04.43 4.43 4.43

7 1 - 2 - 6 29 59 59.92 30 00 08.63 8.71 8.72

Sum of three numbers of angle distort adjustment ∆β in the

triangle 1-2-6 are checked in the three last rows of table 2.1 and have

the value of 0, which is completely matched with spherical excess that

is appropriately 0 in this case

2.3.4 Calculation of scope of measured area according to

horizontal angle distort limit

Table 2.2- Deformation value after using topocentric coordinate system

Option L (km) Case A : δβ Case B: δ(βH)

12 According to the result of table 2.2, it can be seen that, so that angle distort (or error after adjustment) is no more than 0”,2 equivalent

to 20% of accurate horizontal angle measurement error (taking of 1”), radius (L) in the topocentric coordinate system can be 13 km if measured area is plain For the of difference in height, scope of usage

is smaller, only using in the radius of 9 km and calculating adjustment

of horizontal angle distort according to the equation (2.1)

2.3.5 Comments:

1 The local topocentric coordinate system is established on the principle of orthogonal projection representing the ground surface onto the horizontal plane (Local topocentric coordinate system) at the reference point (the datum) The length of side will not change when changing the elevation of local topocentric coordinate system according to normal at initial point

2 In case that elevation of points is considerably different from each other, it is necessary to take consideration into the angle distort for adjustment into measured angle before adjustment combination of GPS and terrestrial observations

3 Equations for conversion from geodetic coordinate system to topocentric coordinate system, as well as local topocentric coordinate system are exactly determined in the aspect of mathematics, which is convenient for programming on the computer in order for automation of conversion and transformation between local coordinate system and national coordinate system

4 Topocentric coordinate system is also a means of representing the positions in national coordinate system, similar to rectangular coordinate system in UTM projection in which central meridian LO is not defined according the general regulations [41]

5 Scope (radius) of using the local topocentric coordinate system in engineering survey is estimated according to requirements on distortion

by the equation: L≤ 2,45.R. 1/T

2.4 USING THE LOCAL TOPOCENTRIC COORDINATE SYSTEM

AS REPLACEMENT OF UTM PROJECT IN CIVIL AND INDUSTRY ENGINEERING SURVEYING

2.4.1 Comparison between UTM horizontal network and the local topocentric coordinate system

13

For example: Thanh Hoa Province According to above calculation,

only a narrow 4.5km-wide strip, which is in symmetric position and 90

km far from the prime meridian, varies in length with less than

1/200000 (Figure 2.5) All zones beyond two above narrow strips bear

the variation in length with over 1/200000 due to UTM projection,

such variation scale is too

great in engineering surveying

and exceed the requirement in

the Standard 309:2004 [3]

Obviously, according to the

regulation, cadastral map

separated into plot plans is not

suitable for large-scale

topographic map in design and

construction of works

Therefore, a separated prime

meridian is used so as to gurantee that length distort varies slightly

Using a separated prime meridian by UTM projection, it also means

that a topocentric coordinate system is set up

For above reasons, technically speaking, using the topocentric

coordinates is as meaningful as using UTM projection with a separated

prime meridian (as appropriate) and the advantage is that while the

width of an area is greater, length distort is still ensured to be

1/1000000 and height distort of measured area (if considering into

mountainous areas) is corrected

2.4.2 Advantages and disadvantages of using local topocentric

coordinate system

1.Processing GPS and terrestrial observations is convenient in a

appropriate region

2 Direction of the x-axis of the topocentric coordinate system is

northern direction and coincides with the meridian passing the datum of

the topocentric coordinate system However, direction of the x-axis in

UTM projection is the direction of the meridian of zone

3.The local topocentric coordinate system is 3D Coordinates system,

with the third axis z (or U) containing information about the elevation,

Figure 2.5- Thanh Hoa Map and Position of

prime meridian (Lo=105o)

14 which is more preeminent than a grid azimuth, 2D coordinate system

4 By measurement of national coordinates to the datum point of topocentric coordinate, conversion from topocentric coordinate system

to geodetic coordinate system and national coordinate system is accurately defined by rigorous formulas; moreover, meeting the requirement of managing positions in national coordinate system

2.4.3 Remarks of using the local topocentric coordinate system

1 Objects of usage: For civil and industry egineering surveying (in the area of about 250 km2), the local topocentric whose datum located adjacent to the center of the Works can be used as the basic coordinate system for engineering survey serving for design and construction

2 The local topocentric coordinate system shall be established and used at the stage of drawing the large-scale topographic map serving for deign works, construction works until the as-built drawing of completed Works Coordinate transformation and conversion from the local coordinate system to UTM rectangular coordinate systems and vice versa will be calculated via geographical ellipsoidal coordinates(B,L,H) by close formulas In the case of transformation from the topocentric coordinate system into local coordinate system, a number of common points will be used for determining the transformation parameters on the basis of some algorithms like Helmert, Affine transformation vv

3.In order to ensure angle and length distort, for the flat terrain, the measured radius can be up to 13km For the unflat terrain (slope of 0,275), the measured radius can be 9 km (diameter of 18 km) and before adjustment combination of horizontal angle measurement and GPS observation in the local topocentric coordinate system, it is essential to calculate the adjusment of horizontal angle distort due to elevation difference in the value of measured angle

4 When using the topocentric coordinate system for layout of the Works, it should be avoided to use point of sight with high elevation difference in order to prevent angle distort adjusment in layout angle

5 In order to calculate the elevation of adjusted points GPS network, it

is necessary to transform the local topocentric coordinate system to geographical ellipsoidal coordinates (B, L, H) and determine differential leveling on the basis of Geoid model or using common points - GPS leveling to interpolate anomalous elevation

15

CHAPTER 3 ADJUSTMENT COMBINATION OF GPS AND TERRESTRIAL

OBSERVATIONS IN THE LOCAL TOPOCENTRIC

COORDINATE SYSTEM

3.1 MEASUREMENT IN GPS NETWORK

3.1.1 GPS Baselines vector and effects of covariance matrix in

adjustment of GPS network

1 When carrying out close adjustment of GPS network (option

1) , in which the weight of baselines vector is the inverse of covariance

matrix Cxyz , mean square error of unit weight µ after adjustment will

be many times greater than 1 and verification Chi-squared test is

failed This indicates that covariance matrix Cxyz of baselines vector

only shows the correlation relationship (dependent) of elements of

baselines vector, but not accurately shows the value of actual

adjustment and covariance matrix This remark should be taken

consideration into when weight calculation for adjustment

combination of GPS and terrestrial observations

2 When using simple weight without the correlation component

(option 2) and considered equal weight (option 3), not only result of

accuracy assessment, but also coordinates, elevation, side length and

azimuthal side after adjustment can be changed

3 To get close and highly reliable adjustment results, when

adjusting the network, it is required to take full consideration into

covariance matrix Cxyz of measurement Covariance elements in

covariance matrix Cxyz can be omitted or weight matrix can be used in

the case of approximate adjustment of GPS network

3.1.2 Checking loop of closure in GPS Network

Checking measurement value before adjustment by loop of

closure limit is to discover and detect gross error This is a significant

work to be implemented before adjustment of the network so that

result adjustment will be accurate and reliable

COORDINATES SYSTEM

Indirect adjustment of GPS network in topocentric coordinate

system will be carried out in the following steps:

16 Step 1: Process baselines

Step 2: Check loop of closure before adjustment Step 3: Establish equation of correction

Step 4: Calculate the weight matrix P of the system of equations Step 5: Establish the standard equation

Step 6: Solve the system of standard equations Step 7: Calculate correction for measurement Step 8: Evaluate accuracy after adjustment

TERRESTRIAL OBSERVATIONS IN THE LOCAL TOPOCENTRIC COORDINATE SYSTEM.

3.3.1 GPS network adjustment in the local topocentric coordinate system

1 Calculation of the approximate coordinates of GPS points

2 Equation for measurement correction in the local topocentric coordinate system

V = AX + L (3.1)

3 Estabishment and solution of standard equation

Based on correction equation (3.1) and covariance matrix, the standard equation is established as follow:

ATPA.X + ATPL =0 (3.2) With P is diagonal block matrix, Pi is the inverse matrix of covariance matrix

Mxyz(3x3): 1

i

i M

P = − (3.3)

In which: M=RTCxyzR, R is rotation matrix

With the equation (3.3), actually take constant c=1 for calculating the weight in network adjustment

Standard equation (3.2) will be solved in the normal method:

X=−(ATPA)−1ATPL (3.4)

4 Calculation of measurement after adjustment and assessment of accuracy

Assessing the accuracy of the network after adjustment includes:

- Mean square error of unit weight µ; error of position

- Mean square error of weight function side and weight function azimuthal side

17

3.3.2.Adjustment combination of GPS and terrestrial observations

in the local topocentric coordinate system

Terrestrial observation values are adjusted in combination with

GPS observation includes: Value of horizontal angle measurement,

value of side length measurement

3.3.2.1 Horizontal angle measurement

Equation of linear angular correction as follow:

β

β=(a −a )dx +(b −b )dy −a dx−b dy +a dx+b dy+l

v m,p mt m m,p mt m m,p p m,p p mt t mt t (3.5)

It should be noticed that horizontal angle will be deformed due

to orthogonal projection on the plane when points are different in

elevation For this case, it is essential to calculate correction of

deformation∆βin measured angle β in the following formula (2.1):

t

p−δ δ

=

∆β (3.6)

3.3.2.2 Side length measurement

Equation of horizontal angle correction as follow:

i k i o i k

o o i i o i k

o o i k o i k

o o i k o

i

k

o

o

i

i

D ) y y ( dx D ) x x ( dy D ) y y ( dx D

)

x

x

(

The length of incline side also can be used for establishment of

correction equation

3.3.2.3 Establishment and solution of common standard equation

0 PL A X PA

= + (3.8) Standard equation of coefficient matrix is established in the

following formula:

T MD MD

MD GPS GPS T GPS T

A P A A P A PA

A = + (3.9) and: T MD MD

MD GPS GPS T GPS T

L P A L P A PL

A = + (3.10) Solving the standard equation (3.8) will reveal the unknown

quanity of combination adjusment calculation

When only adjusting GPS network, mean square error of unit

weight µGPS is many times bigger than 1; this indicates that covariance

matrix CXYZ of sides has not truely reflects GPS observation error

Chi-squared test are usually not passed

For processing weight in adjustment combination, steps are performed

as follow:

18

Step 1: Using the equation of correction GPS Observations, the priori

covariance matrix of weight Mxyz and coordinate of datum point C to adjust only GPS in topocentric coordinate system in order to get first adjustment coordinate of points and mean square error of unit weight µ GPS

Step 2: Adjustment of combination of GPS observation and angular-

side measurement The weight of the GPS basseline in this step must

be calculated by the following the formula:

( ) 1

xyz 2 GPS

1 xyz 2 GPS

µ

= µ

= (3.11) Values of

GPS

µ is the basis for estimating the priori covariance matrix of weight, when weight of GPS observation is compatible with angular- side measurement

3.3.2.4 Assessment of accuracy

Assessment of accuracy of adjustment result includes:

1 Calculation of unit reference standard deviation

t n n n PV V

2 1

T

− + +

=

µ (3.12)

2 Assessment of accuracy of point location and mutual accuracy

Mean square error of position:

yy xx 2

2

m = + = µ + (3.13) Mean square error of side:mS= µ QS (3.14) Standard deviation of the bearings mα=µ Qα (3.15)

3.3.3 Application of Kalman filter in combination adjustment (divided into several stages)

The multi-stage consecutive adjustment step using Kalman filtering solution as follows:

Step 1: Use the measurement step (i) and adjusted unknowns in step

(i-1) to calculate the free class L(i) Gain matrix K(i) is based on co-weight matrix

) 1 i X

Q − of previous stages (i-1), weight matrix additional measurement stages (i) P(i) and matrix system A number of (i) the correction equation measurement stages (i)

1 T ) i ( X ) i ( 1 ) i ( T ) i ( X

i Q A (P A Q A )

K = − + − (3.16)