Modern views on the determination of the figure of the earth - Physical Geodesy - page 333

For ξ and η , (8-152) becomes

ξ c = ξ c = η − ξ TI + δϕ normal , − η TI .

(8-153)

By means of (8-139) this may be written
ξ c = ξ

c = η

− ξ TI , − η TI . (8-154)

The interpretation of (8-154), however, is clear, simple, and rigorous: from
the dynamic deflections of the vertical at P , which are the very quantities
ξ and η , we subtract the effect of the topographic-isostatic masses, ξ TI and
η TI likewise at P . The vertical deflections so obtained, ξ c and η c ,thusdo
not really refer to the (co-)geoid; in reality, they refer to the earth's surface!
But what, then, means the normal plumb line curvature δϕ normal in (8-
153)? Does it not mean a reduction from the earth's surface to sea level? No,
in Eqs. (8-139) it only denotes the transformation between the geometrical
and the dynamical deflection of the vertical, both referred to the point P
of the earth's surface. This is also clear from Fig. 8.13, which illustrates the
formula

ε = ε + δ, (8-155)
extending (8-139) to an arbitrary azimuth, δ being defined by (8-141).
This interpretation of (8-153) or (8-154) as isostatically reduced deflec-
tions of the vertical at the earth's surface is exact, whereas the interpretation
of (8-8) as deflections at the cogeoid was only approximate. This is the de-
sired rigorous interpretation of our isostatically reduced vertical deflections.
This interpretation exactly corresponds to the modern view of gravity
reduction according to the theory of Molodensky. According to this view,
the isostatically (or in some other way) reduced gravity anomalies continue
to refer to the earth's surface. The classical gravity reduction (Sect. 8.2) had
comprised two procedures: mass transport and shift P → P 0 ; the new view
of gravity reduction only considers the mass transport; the problematic shift
P → P 0 is avoided.
Formally, a “normal free-air reduction”

∂γ
∂h h

F = − (8-156)

may be said to occur also in Molodensky's theory: normal gravity γ in the
new definition (8-128) of the gravity anomaly, where it refers to the telluroid
point Q , is computed by

γ = γ Q 0 + ∂γ

∂h h,

(8-157)

with h = Q 0 Q denoting the normal height of P . But instead of reducing
actual gravity g downward, from P to P 0 , now normal gravity is reduced