...
-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.