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then Eq. (11-3) gives
δT =
L
(∆ g ) ,
(11-5)
as it should be.
The same principle works also with deflections of the vertical ξ, η at the
earth's surface, both as input data and as output results (Sects. 8.14 and
10.2).
The underlying isostatic model is in principle arbitrary. For practical
purposes it should provide a good approximation (small residuals δT )and
be computationally convenient.
We see, however, a change of perspective. Collocation is no longer applied
to the “real” anomalous gravity field as in (11-5) but to the residual field, re-
moving the field generated by the assumed topographic-isostatic model. The
model is arbitrary, but the derived quantities must be computed in a rigorous
consistent fashion. (Consistency for the quantities computed by collocation
as guaranteed by a correct covariance propagation; see Sect. 10.2.)
This change of perspective may not seem important because it is just
a change of nomenclature: what formerly was importantly called “isostatic
anomaly” is now degraded to a miserable “residual”. However, the remove-
restore principle permits also the use of other approximate fields to remove
trends; especially one of the numerous existing “earth (gravity) models” (EM
or EGM) consisting of spherical-harmonic expansions of the potential T up
to degree 180 or higher.
Therefore, we “remove” from the observations - gravity anomalies,
gravity disturbances, deflections of the vertical, etc. - the effect EM com-
puted from the earth model used, and after collocation “restore” the effect of
the EM on the result. The mathematics is the same as in (11-4) and (11-5):
δT EM =
L
( δ EM )
(11-6)
and
δT =
L
( ) .
(11-7)
We have only slightly generalized from ∆ g to .
Now we proceed an important step further. The remove-restore principle
has only two requirements:
1. the removed auxiliary potentials must be harmonic, precomputable,
and used in a mathematically consistent way: what is removed in the
input must be restored in the output;
2. in the usual case of linearity, two or more different auxiliary potentials
may be used (removed-restored) simultaneously.
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