mathematical models for horizontal geodetic networks - UNB

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MATHEMATICAL MODELS FOR HORIZONTAL GEODETIC NETWORKS

E. J. KRAKIWSKY D. B. THOMSON

April 1978

TECHNICAL REPORT LECTURE NOTES NO. NO. 217 48

MATHEMATICAL MODELS FOR HORIZONTAL GEODETIC NETWORKS

E.J. Krakiwsky D.B. Thomson

Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton, N .B. Canada E3B 5A3

April1978 Latest Reprinting January 1997

PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text.

PREFACE

The purpose of these notes is to give the reader an appreciation for the mathematical aspects related to the establishment of horizontal geodetic networks by terrestrial methods.

By terrestrial methods we

mean utilizing terrestrial measurements (directions, azimuths, and distances).

vertical networks are not discussed and instrumentation

is described only indirectly via the accuracy estimates assigned to the observations. The approach presented utilizes the well known aspects of adjustment calculus which allow one to design and analyse geodetic networks.

These notes do not provide extensive derivations.

The relation-

ships between models for the ellipsoid and models for a conformal mapping plane are also given. These notes assume the reader to have a knowledge of differential and integral calculus, matrix algebra, and least squares adjustment calculus and statistics.

ii

?age ?HEFACE............................................................

ii

TABLE OF CONTENTS.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i ~;

LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i v

ACK.'lOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

1.

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2.

THE NEED FOR GEODETIC NETt'lORKS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3.

DISTANCE MATHEMATICAL MODEL

3.1

Ellipsoid

Differential Approach . . . . . . . . . . . . . . . . . . .

6

3.2

Spherical Differential Approach.....................

9

3.3

Relationship of the Ellipsoid Model with the Plane Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .

10

Illustrative Example................................

11

3.4 4.

5.

6.

AZIMUTH MATHEMATICAL MODEL

4.1

Development of the Mathematical Model...............

15

4.2

Relationship of Ellipsoid Model to Plane Model......

16

4.3

Illustrative Example................................

17

DIRECTION MATHEMATICAL MODEL

5.1

Development of Hathematical Model...................

19

5.2

Illustrative Example................................

21

TECHNIQUE OF PRE-ANALYSIS

6.1

Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

6.2

Mathematical Foundation.............................

25

6. 3

Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

iii

LIST Ot !='IGURES

Page 3-l

Differential Elements en the Ellipsoid . . . . . . . . . . . . . . . . . .

3-2

Quadrilateral with

5-l

Orientation Unknown . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . .

20

~irection

22

5-2

·

~NO

Fixed Points . . . . . . . . . . . . . . . . . . . . . .

and Distance Observations . . . . . . . . . . . . . . . . . • . . . .

iv

2.2

T!Je autnors express their sincere appreciation to John R. Adams and Bradford G. Nickerson for their assist.ance in preparing the final manuscript of these lecture notes. for her typing of these notes.

v

Theresa Pearce is acknowledged

l.

INTRODUCTION

l·le besin these notes by first ·2stablishing the :1eed for geodetic networks .

Secticn 2 also serves as motivation for

the mathematical developments in the remaining sections.

In Sections 3

to S, inclusive, the mathematical models for distance,

azimuth, and

direction observations on the ellipsoid are developed.

It should be

noted that the observations considered in these notes are assumed to be made on the ellipsoid; however, it is well known that terrain obsePJations must be properly reduced to the ellipsoid [Bamford,

197~].

These models give the functional relationships between the observables and the unknown coordinates to be estimated.

The models are also applied

to examples to show how solutions of networks can be made.

The

of pre-analysis of networks is described in the final Section.

techni~~e

This

procedure allows one to design a network before actually making the observations.

Traditionally, the emphasis placed on establishinlating distances and coordinates.

Coordinates are the geodetic latitude and

lonyitude referred to a reference ellipsoid.

Distances are the corresponding

geodesic lengths between points or1 the ellipsdid.

This means ttat observed

distances must be reduced to the ellipsoid surface before they can be u'~

-ius ted. It is '"'orthwhile to note that, if the geodesic distance between

t•"'o points on the ellipsoid could be expressed in a closed form, as a function of the coordinates, then it ·.vould simply be a matter of linearization to obtain the linear form needed for the adjustment.

But since the inverse

pr.Jblem on the ellipsoid does not have a mathematically expedient closedfan~

solution, an alternate approach must be taken.

given below.

Two approaches are

The first is based on an ellipsoidal differential expression,

while the second begins with a spherical approximation.

The relationship

of t:1c ellipsoid model with the plane case model is also discussed, and an example using the ellipsoidal differential approach is worked out. 3.1

Ell:psoidal Differential Approach

The mathematical model for the geodesic distance, expressed as a function of two sets of coordinates, is symbolically written as F

ij

=S(.,A.)-S .. =O, l

l

J

J

(3-l)

lJ

where the first term is a non-linear function of the coordinates, while the second term is the value for the geodesic distance.

6

This non-linear

:no~: ·l

:..s :1?!-·rox::..I:'\ated by a linear Taylor se:-ies.

..-1t:'

·~

'-"'

j

(3-2)

..

:j .. 0

.\

s lJ ..

.)

J

:he

.,a:

.lt::o

s~~ , ~~0

T!'l.e resulting equation :o.s

fi:-s~

+ dS (-~~I l

...

? .-,.,, ~~~ -v

J, ~I l

J

J

+ •.• sij

o.

(3-3)

t..,·o ter:ns represent: the point. of expansion, namely t!:e

of the distance based on the approximate values of the coordinates r. •

....

I

~hird



"-

j

I

')

~.)

J

1

minus the observed value of the ellipsoid distanceS ... l.J

term is the differential.change in the distance due to differential

::;,a;:gcs in the coordinates and is described by the total differential [:-!·:L:-.ert, 1880; Tobey,l928]

(3-4)

as

0

-~;,~

. ...

= -Mi

0

cos Ctij

0

0

J

Jl

(3-5)

I

0

= N.sino. ..

(3-6)

cos ~j

0 0 3S =-H.coso. .. 36. J Jl

(3-7)

J

.,_

0

:-:::>

:'

,, = -N.J

0

0

sino.. cos 1>. .J Ji

(3-8) I

.J

•')

:1 ') Zt. i tC. N'being the radii of cur~ature of the ellipsoid in the meridian and 0

pri:;ie vertical planes 1 respectively, and o. is the geodetic azimuth. V

s ..

l.J

is the correction to the observed length.

Finally

,1.,..,I

observation equation, ..

v

S.

-, cos -ij

::..

=

,,

,_II

::.. j ·')

:--1.

si:1

N. sin

"

li

cos

~-

J

+

do.

--

·::L.

a..

J.j

d~'.'

~

+ 1-. '""·~j

l:l .

Ji

--

"J

·.)

~\.

d'" ''-

l

i

:...)

J

s ..

J

~J

+

(-as -

+

aq, .

( 3-13)

~

where 0

:ls Jrp.

0

R

-cos¢. [

~

.

s~n

...

-cos¢. cosd>. R o[

.0 COS
+':OS~.

1

(3-16)

·-'

sin

-·'

.\~·)

--'---1.-

.l.

'J

cos.:b·- cos'~~ sin 'i

R

J



s1.n

J

(3-17)

l.

;:,0 v

This is essentially the model used by Grant [1973].

Note that the units

of t.he t.erms in the linearized model (3 ·13) may be changed as ir. Again, the computation of the distance

fro~

done rigorously and not, for example by

approximate coordinates must be

(3-13).

Bessel's method is one of

the most reliable and accurate for this purposE. 3.~

as in Section

3.3

(3-10).

Again,_. for the same reasons

an iterative solution is required.

Relationship of the ElliPsoid Model with the Plane Model We know that the mathematical model for a distance·observation

on the plane is F .. =

(X.

l.J

J

and after linearization 0

F ..

l.J

::;

(S ..

l.J

0

-

X.)

2

l.

0

i;.

l.J

J

0

1

l/2

-

S .. ::; Q 1

l.J

(3-18)

0

-

dy. J

- vs .. l.J

l.

l.J

0

2

l.

(y.-y.) J l.

dx.

s ~.

0

Y.)

0

(x. -x.) s .. ) - J l.

-

(y. -

sol.J ..

dy. l.

0

(x. -x.) (y. -y.) 1 ~ l. J + dx. + s :-'. s?. J l.J l.J

::;

0,

(3-19)

or

vs ..

0

-

-

sina .. dx. l. l.J

-

cosa .. dy. + Jl. J

l.J

0

0 . 0.. dx. cosa .. d y. -sl.na l. J l. l.J J

0

s ..

l.J

s .. l.J

(3-20)

l

•::l, ·"·: n·-.::: .:m

1

::::1e el:!.ipsoid and differential elements on the :_Jlane: d·;.

= I~~' ~" - l

d,+, '.'

T1

-':]

M. l

dyi =

d\f>'.'

o"

(3-21)

I

1

0

:-1 . _J_

dy. =

p"

J

(3-22)

dcjl ·:

J

= p"

d\'.'

1

N·~ cos ~ 1

dx.

1

p"

1

(3-23)

d;\.". 1.

0

0

N. cos if>.

=

dx. J

J

J

p"

d>..':

(3-24)

J

SurYtituting (3-21) to (3-24) into (3-20) yields an expression similar to (3-9) except for d\ '.' term.

It

1

a~proximately

is

has been shown by Tobey [1928]

that N. coS. sina. .. l.

l

lJ

equal to -N.cos¢.sina. .. , allowing us to deduce the ellipsoid J

J

J

1

model from the plane model.

3.4

Illustrative Example ~~e

bet:-.;een

no•.v extend the linearized mathematical model (3-9) to many distances

:nany stations.

The observation equation in matrix form is p

+ w = v n,u u,l n,l n,l A X

n,n

2

= a on

2:

-1

(3-25)

n

where n i3 the total number of distances observed and u the total number of coordinates of unknown stations. L:

a

2 0

is the a priori variance factor and

tbe variance-covariance matrix for the observed distances.

12

Figure 3-1

j~scJ~ces

~bserved,

:Fi~ure

0

0

'" 14

C-14

dl3

' J3

-, ~

::.: )"'

c23

l

:

c

0

' d.

b

34

~-

-3'

~

34

d

d

d.\

24

s

observatio~

1

. l

oa ..

3a.

oa ..

da .. = _2:.2

df. + 1.

_2:.2

a:\.

d:\.

~

~

1 c:;

+

~~

aq,j

d/

s~

:-1. sin

...,

,·iSl.D ''i~

dQ~ +

.

J

d~~ J

Jl.

s:;, .

1.

l.J

.

1

1

A., 1

orientation uni... "l.

1

2,2

2,2

l::. J, i

I:

2,2

2,2

(6-16)

·~. ,\.

J J

J

':~e variance-covariance information from I:-X

=

(ATI:~u 1 .Z\.) -l

To increase the probability level of the standard two-dimensional confidence region {about 38%) to a higher level, say 95%, we revert to (Wells and Krakiwsky 1971, pp.

the basics of multivariate statistics 126-130].

The quadratic form for the parameters in the case of a

2 0

known

is

T -1 d 2 (X-X) l:v (x-x) + X u, 1-a

(6-17)

.(1,

-

where X is the least squares estimate,

x

is the true value of the

parameters, u is the dimensionality of the problem, a is the desired 2 confidence level, and xu,l-a is a random variable with a chi-square

distribution and degrees of freedom u.

In other words, we can establish

a confidence region for the deviations from the least squares estimate

X,

of some other set X, based on the statistics of

x .

Returning to equation (6-1,7.) we see that

(X -

X) T

I:: l

~X

- X)

X

is an equation of a hyperellipsoid that defines the limits of the associated confidence region.

Translating the origin of the coordinate system to

and assuming the dimensionality , u, to be 2, we get the equation of an ellipse

x

31

-1

X

1

-

2 X = X~.

"'-

X

0

2 xl

c

(6-18)

, -l - ( l -1

xlx2

xl.

l

=

[ x1 x2 } 0

0

2

xlx2

x2

x2

2 x2,1-a

(6-19)

where x 1 and x 2 represents the coordinates ( 4,. , A.) or the coordinate l l c1 i

Eferences

(.~¢

.. , fo), . . ) • lJ l.J

An equation without cross product terms can be obtained via the eigenvalue problem, namely

-1 (J

2

0

max

l [yl v -2 0

v..rhere y 1 and y 2 are the rot~tcJ

0

2

.

m:Ln

[~:l

=

2

x2,l-a

(6-20)

transformed x 1 x 2 coordinates v.•i th respect to the

coordinate axes resulting from the eigenvalue problem. To obtain the equation of the ellipse from the above, we write

y (J

.

2 ma::s::

;. y 2 2 + 2 2 c x2,l-a: min

2 1 2 x2,1-a

=

l

1

(6-21)

which says that the semimajor (a) and semiminor (b) axes of the ellipse are ...,

a

=

b

~ x;,l-o:

I X2•,_., 2



...

-

0

max

a .

nu.r.

Thus, to obtain a confidence region \''i th a certain probability level, the axes of the £actor as shov,'Tl above. [i

= • 05

,

/

)'2 '2,1-.05

st~~dard

conEidence region must be multiplied by a

Note for any 2 dimensional adjustment ·with 2.45 .

32

For the case where __

2

lS

0

unknown, a similar develo?ment :or

ti:~eodasie.

Leipzig.

Jordan, W. an~ ?·mEggert ~1962). Handbuch der Vernessengskunde, Bd. III. Engl~sn 4ranslat~on, Army Map Service, Washington. Konecny, G. (1969}. Control Surveys as a Foun d a t·~on f or an Integrated Data System. The Canadian Surveyor, XXII, 1.

34

Krakiwsky, E.J. (1975). A Synthesis of Recent Advances in the Hethod of Least Squares. Department of Surveying Engineering, Lecture Notes #42, University of New Brunswick, Fredericton. Krakiwsky, E.J. and P. vanicek (1974). "Geodetic Research Needed for the Redifinition of the Size and Shape of Canada". A paper presented to the Geodesy for Canada Conference, Ottawa, Proceedings. Federal D.E.M.R., Ottawa. Linkwitz, K. (1970). The Use of Control for Engineering Surveys. of t~e 1970 P~nua1 Meeting, Halifax, Canadian Institute ~f Surveying, Ottawa. ~cLellan,

C.D. (1972). Survey Control for Urban Areas. Surveyor, Vo. 26, No. 5.

Mikhail, E.M. ~Civil

(1976). Observations and Least Squares. Engineering, New York.

Papers

The Canadian

IEP Series in

NAS (1969). Useful Applications of Earth-Oriented Satellites. Academy of Sciences, Washington, D.C., U.S.A. NASA (1972). Earth and Ocean Physics Application Program. Washington, D.C., U.S.A.

National

NASA,

Roberts, W.F. (1960). The Need for a Coordinate System of Survey Control and Title Registration in New Brunswick. The Canadian Survevor, XV, 5. Roberts, W.F. (1966). Integrated Surveys in New Brunswick. Surveyor, XX, 2.

The Canadian

Sebert, L.M. (1970). The History of the 1:250,000 Map of Canada. Survevs and Mapping Branch Publication No. 31, Department of Energy, Mines and Resources, Ottawa. Toby, W.M. (1928). Ottawa.

Geodesy.

Geodetic Survey of Canada, Publication #11,

Van Everdingen, R.O. and R.A. Freeze (1971). Subsurface Disposal of Waste in Canada. Technical Publication No. 49, Inland Waters Branch, Department of the Environment, Ottawa. Vanicek, P. and E.J. Krakiwsky (in prep).Concepts of Geodesy. Wells, D.E., Krakiwsky, E.J. The Method of Least Squares. Department of Surveying Engineering, Lecture Notes #18, University of New Brunswick, Fredericton.