R + (2h' -h)
+ 2Rh) - (R
- 2Rh') = 2R (h - h') .
Obviously even the term 2h'-h in the first equation can be neglected and we end up with
after having denoted
/cos S' by q,.
Now the spatial distance of P and
Discussion of the Molodenskij deflections
The Molodenskij deflections are indeed different from both kinds of
deflections used in classical geodesy.
6.3) we use:
In classical geodesy (see Physical Geodesy
The gravimetric deflections (on the
gravity vector Yp' both at the surface.
few hundreds of second of arc.
These two definitions vary by at most a
They differ by the term due to the curvature of the
normal plumb-1 ine between the telluroid and the surface.
It can be shown that the compon
Hence in the
approximation the two last
terms of the kernel may be neglected.
, which can be done for a
flat terrain around the point of interest we can write the zero approximation as
for mountaineous regions due to the negligence of the term cos
Treatment of the
Molodenskij integral equation for the surface density layer in mountaineous areas
remains still an open question.
Evaluation of the height anomalies
We have learnt
and ya,yb are the values of normal gravity on the equator and at the poles.
Integrating 1/y-times the derivative above we get finally
- R sin 2 H " -0. 17 11 sin 2H
where H is the height above the ellipsoid.
Evidently, this correction has to be
p 6 r.
P a r.
HereS is the maximum inclination of the tellurooid at P or, which is the same, the
angle between the normals tor. and to the ellipsoid.
Hence on the ou
div f = Vf = Vcfw_UVW) = VUVW + UL\W
div g = Vg = Vcfw_WVU) = VWVU + WL\U
Applying the Gauss formula successively to f and g we get
W 8n dS
Jffvcfw_VWVU + WL\U) dV .
Substracting the second from the first formula yields:
\ cff s
The classical approach in geodesy is based on the use of two different
reference surfaces- the ellipsoid of rotation and the geoid.
To establish the
relation between the two surfaces is one of the main tasks of geodesy.
old problem of
Molodenskij has derived the following approximate expression for
the proof of which we are not going to give here:
where llg refers to the telluroid cfw_or the terrain), y and
refer to the telluroid,
are the components of the deflection o
We may note that when defining
the telluroid or quasigeoid we do not
have to postulate any hypotheses.
surfaces are purely conventional
(mathematical as opposed to physical in
case of the geoid) and are not even meant
to represent any physical prop
4) Molodenskij equation for disturbing potential
In order to get rid of the volume term, let us apply the 3rd Greens
identity to the normal potential U as we know it from classical geodesy.
up with Molodenskij equation for U in following form: