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 non-conservative part of the disturbing p
THE METHOD OF LEAST SQUARES
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
The probability that x is less than or equal to x is the value of
(x ) and is represented by the total area under the curve from - 00
shown as the shaded area in Figure 2-2. Note
Frequency distributions have two important characteristics ca
?re-ana.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.
The inclusion or exclusion of certain elements in A and in
Direct.ion observations are relative to the "zero" of the
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.
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
Eliminating 81 from the above by (Figure 14-1)
where N can be approximated by the Gaussian mean radius at point 1.
The above equation is of sufficient accuracy for most work.
can be compared to the. following formula
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
7-44, the following expnssiondefines the constant K:
The above two expressions for the constants K and
the description of the direct problem - recall the complex mapping equation
M. T.l\1. ZONES
3' M.T.M. ZONES
FOr the remaining cases, the x-y equations will not be given.
The main point of interest lies in the shape of the isoscale
From 9-5, the longitude from the central meridian of all points
having a scale of
The higher derivatives have been worked out in Thomas
[1952, p. 101].
Substituting the derivatives in 6-57 and 6-58 along with the
above expressions results in the following expression for the longitude
A = sec
where the subscript 1 denotes tha
The second derivative is
= .9:_ cfw_df(q)
The higher derivatives, are derived in an c.nalogous manner,
fiii(. ) : - fiii (g)
= if'IV ( q)
( q) '
Substituting the evaluation of th
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.
restrict the surface to be mapped to be the ellipso
of the dependent complex variable z
+ i y.
If two arbitrary curves
in the domain D intersect at the point P(u , v ) at an angle
mapping is called isogonal if the corres.ponding curves intersect in the
domain D' also at the angle
dX . ax
= dsT =
dSr = a., . 7E .
After substitution of 3-18a and 3-18b into 3-18, we get
e = [ax
az az 1
and noting that the term in brackets is the Gaussian fundamental.quantity
= q_ = y = 0
Under thcse conditions our complex mapping function becomes
x + iy
= a( A +
'T'rk above mapping function is conformal since the Cauchy-Riemann
equations are satisfied, namely
3q - a '
= - lJ.x
Therefore, to force 4-38 to satisfy this condition, we choose e equal
to g in l1-38.
= _.;e:;._ = _
Now we can describe conformal projections in terms of Gaussian fundamental
The International Association of Geodesy recommended as early as
in 1950 to investigate its.effect on precise levelling networks.
4.4) Non-periodic Effects
The last group of effects are the effects that do not have a
periodic character, at leas
INSTRUMENTATION AND ANALYSIS OF OBSERVATIONS
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
represent theposition of the
=- 2 k
Substituting these results back in (22) we get finally
1 AB ,;, _ _.-._ k) aw _.1)
L-:_ - ( 1 + h l2_a_a
where 'dW/'da can be interpreted as the negatively taken gravimetric tide
for the rigid earth.
What change in the potential W would now an
surface of the earth observe?
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
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
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
phases of the individual components by several tens of degrees.
Bay of Fundy, Bristol Channel, Saint-Malo Bay (southwest part of the English
Channel) and the Liverpool Bay, to name just a few locations, the actually
observed magnitudes of the ti
Table 2 continued
(L, lunar; S, solar)
L declinational wave
major elliptic wave of 82
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.
trace the influence let us take two points - the center of gravit
This can be rewritten as
1/pl = 1/p y
may be recognized as the. generating function for
Legendre's polynomials (see
and we can write
(B.)m P (cos Z)
where P (cos Z) are the L
cycle per day and we can thus see that the sectorial
raise to mainly semidiurnal variations,
the tesseral to diurnal variations
and the zonal to long term variations.
Decompo.@ion of Tidal Potential into Freg:uencies