29
Since the described coordinate system is not inertial with
respect to fixed stars, its acceleration (with respect to an inertial
system) can be observed as a change of geometry of the potential.
This appears as nonconservative part of the disturbing p
THE METHOD OF LEAST SQUARES
1.
INTRODUCTION
The method of least squares is the standard method used to obtain
unique values for physical parameters from redundant measurements of
those parameters, or parameters related to them by a known
model.
The first
15
The probability that x is less than or equal to x is the value of
,o
(x ) and is represented by the total area under the curve from  00
0
to x
0
shown as the shaded area in Figure 22. Note
Frequency distributions have two important characteristics ca
28
?reana.l:lsis is based cornplet.:ly on the above equation and is performed
by simply specifying the elements of the design matrix A and the variancecovariance matrix EL.
6.3
Procedure
The inclusion or exclusion of certain elements in A and in
E is
in
Direct.ion observations are relative to the "zero" of the
horizontal
The location of the zero relative to the north
dircsti.o:1 is c.n :1nknown "nuisance" parameter an: must be solved for
by the adjustment along with the unknown coordinates.
The directio
111
A, h).
Also, recall that computations on the surface of the ellipsoid
required that observations be reduced to the surface of the ellipsoid by
first taking into account the effects of the gravity field and the atmosphere, and then the effect of the he
133
Eliminating 81 from the above by (Figure 141)
s
1419
s
we get
1420
a=Tt=
where N can be approximated by the Gaussian mean radius at point 1.
1
The above equation is of sufficient accuracy for most work.
It
can be compared to the. following formula
122
13.
GENERAL FORMULAE FOR REDUCTION TO THE MAP PROJECTION PLANE
The general formulae for reduction of distances and directions
to the conformal map projection plane are discussed in this section.
These general formulae are valid for most conformal proj
From
744, the following expnssiondefines the constant K:
K
=
N1
N2
= =.:.
748
R.eR.q2
.
The above two expressions for the constants K and
complete
the description of the direct problem  recall the complex mapping equation
(735)
X +
7.5
iy
=K
ei.R,(
78
PROJECTED
M. T.l\1. ZONES
3" WIDE
\
\
I
I
\
4(1"
1
t
I
8
I
ll
10
79
ONTARIO
COORDINATE SYSTEM.
3' M.T.M. ZONES
&
CENTRAL MERIDIANS
0
co
I
2\
.
:31
t:l.
Approx.
20 Miles
1
I "30'
Approx.
56 Miles
Figure
u
z
<
a:; I
E
j_
I
I
<'
Zl
I
I
I
CROSS SECTION
A
100
X=
Y
8
.'
p
t
=
+ 2N
s
p
2
96
97
+
FOr the remaining cases, the xy equations will not be given.
The main point of interest lies in the shape of the isoscale
curves.
From 95, the longitude from the central meridian of all points
having a scale of
6(
The higher derivatives have been worked out in Thomas
[1952, p. 101].
Substituting the derivatives in 657 and 658 along with the
above expressions results in the following expression for the longitude
A:
A = sec
663
where the subscript 1 denotes tha
dz
617
i
618
619
The second derivative is
= .9:_ cfw_df(q)
dq
dq
dz
620
The higher derivatives, are derived in an c.nalogous manner,
that is
fiii(. ) :  fiii (g)
lCJ
rrv (iq)
rv (iq)
= if'IV ( q)
:
rv
'
621
'
( q) '
Substituting the evaluation of th
GENERAL
Conformal map projections are the class of projections in
which angles on the surface to be mapped are preserved, tnat is, corresponding angles on the map plane and the surface are equal.
We will
restrict the surface to be mapped to be the ellipso
12
of the dependent complex variable z
=x
+ i y.
If two arbitrary curves
in the domain D intersect at the point P(u , v ) at an angle
0
0
e,
then the
mapping is called isogonal if the corres.ponding curves intersect in the
e.
domain D' also at the angle
I
23
J2
dX . ax
= dsT =
1
7E '
318b
12 =
az.az
1
dSr = a., . 7E .
After substitution of 318a and 318b into 318, we get
cos
ax ax
e = [ax
+
ay aY
33:
+
az az 1
33:

319
lEG
and noting that the term in brackets is the Gaussian fundamental.quantity
F, t
= q_ = y = 0
53
.
Under thcse conditions our complex mapping function becomes
I
x + iy
= a( A +
iq) .
I
54
'T'rk above mapping function is conformal since the CauchyRiemann
equations are satisfied, namely
dX
"'3'I =
Ex _
3q  a '
'lilll
dX
3q
=  lJ.x
jll
Therefore, to force 438 to satisfy this condition, we choose e equal
to g in l138.
Thus
k2
= _.;e:;._ = _
2
2
N cos
_.g._
2
2
N cos
438a
Now we can describe conformal projections in terms of Gaussian fundamental
quantities, namely
f= 0
439
e=g
4
35
of arc.
The International Association of Geodesy recommended as early as
in 1950 to investigate its.effect on precise levelling networks.
4.4) Nonperiodic Effects
The last group of effects are the effects that do not have a
periodic character, at leas
25
3)
3.1)
INSTRUMENTATION AND ANALYSIS OF OBSERVATIONS
Tilt Measurements
To detect the tidal tilt one has to observe the variations of the
spatial angle between the local vertical (direction of gravity) and a line
thought to
represent theposition of the
21
( ).
d
u
'da
=
3
kW
.
3
= 2 k
aw
aa
Substituting these results back in (22) we get finally
1 AB ,;, _ _.._ k) aw _.1)
L:_  ( 1 + h l2_a_a
(22b)
where 'dW/'da can be interpreted as the negatively taken gravimetric tide
for the rigid earth.
Tidal
oth
16
What change in the potential W would now an
surface of the earth observe?
the
The potential W(A) obviously changes to W(B)
so that one would observe the difference W(B)  W(A).
be expressed as a summation of three terms: (i)
This difference can
W(, the
29
of. the individual components.
The amplitudes then can be compared to the
amplitudes of the theoretical tidal vawes and the values D and G computed.
Similarly, we can also compare the phases of the individual components to
get the phase lags of the the
18
Inversely, if the applied force is constant or secular, the earth responds
as an almost plastic body.
This is, of course, reflected in the fact
the the Love's numbers are also frequency dependent.
In the existing
literature this functional relationship
32
phases of the individual components by several tens of degrees.
In the
Bay of Fundy, Bristol Channel, SaintMalo Bay (southwest part of the English
Channel) and the Liverpool Bay, to name just a few locations, the actually
observed magnitudes of the ti
13
Table 2 continued
Symbol
Velocity
per hour
Amplitudelo5
Origin
(L, lunar; S, solar)
T2
29,958933
+ 2479
82
30,oooooo
+42286
30,041067

s
s
s
30,082137
+ 7858
L declinational wave
30,082137
+ 3648
s
R2
1
mK2
sK
2
354
major elliptic wave of 82
principal
1)
1.1)
TIDES OF THE RIGID EARTH
Tidal Force and Potential
To begin with, let us try to see how does the gravitation at
traction of a celestial body, say the moon (t), affect the earth.
To
trace the influence let us take two points  the center of gravit
3
This can be rewritten as
(4a)
and consequently
1/pl = 1/p y
Here y
1
1
(4b)
may be recognized as the. generating function for
Legendre's polynomials (see
y
1
1971],
2.22)
and we can write
00
=
2:
m=O
(B.)m P (cos Z)
P
m
(5)
where P (cos Z) are the L
10
cycle per day and we can thus see that the sectorial
raise to mainly semidiurnal variations,
constituent gives
the tesseral to diurnal variations
and the zonal to long term variations.
1. l1)
Decompo.@ion of Tidal Potential into Freg:uencies
_rhe class