Geoscience Reference
In-Depth Information
where
∂S
(
r, ψ
)
/∂ψ
is expressed by the second equation of (6-32) or the
second equation of (6-36). The linear approximation of (8-74) is evidently
equivalent to (8-53).
This indirect procedure, downward continuation to sea level and again
upward continuation to ground level or above, has the advantage that only
the conventional spherical formulas are needed; yet at the same time the
irregularities of the earth's topography are fully taken into account. The
downward continuation of ∆
g
need be performed only once; the resulting
anomalies ∆
g
harmonic
may be stored and used for all further computations.
Just as ∆
g
is related to ∆
g
harmonic
by analytical continuation, so are
ζ
and
N
harmonic
, the height of a “harmonic geoid”. A final and hopefully instructive
and not too dicult review will be found in Sect. 8.15.
An elementary explanation from daily life
Generally, “analytical continuation” means continuation by the same math-
ematical formula: Taylor series, Laplace equation, or even an elementary
explicit equation.
Let us illustrate the meaning of analytical continuation by means of an
almost trivial example from everyday life (Fig. 8.7). A person is driving a
car along a road which at first is completely straight; at point
B
, however,
it suddenly turns into a circular curve. Thus, our person first drives along
the straight segment of the road. Unfortunately, he is tired and sleepy just
when the straight road suddenly turns into a circular curve. Thus, our sleepy
driver fails to turn the steering wheel and goes straight ahead, the car leaving
the road. Fortunately, the slope is mild, the driver immediately takes control
again and manages to bring the car to a stop at
C
without major damages.
The driver (one of the authors of this topic) has even found the experience
an excellent example to illustrate analytical continuation in his courses!
The gravitational potential corresponds to the road
ABC
,which,after
some idealization, can be considered
“piecewise analytic”
, consisting of the
straight line
AB
and the circular arc
BC
. The transition from the straight
line to the circle is continuous and differentiable at
B
, but the curvature
C
C'
B
A
Fig. 8.7. An illustration of analytical continuation
