**Yu. Dubovenko, Cand.
Sci. (Phys.-Math.), Senior Sci. **

**E-mail:
nemishayeve@ukr.net **

**Subbotin Institute of
Geophysics, NAS of Ukraine **

**Apt. 304; 32, Palladin
Ave., Kyiv-142, Ukraine **

**UNIQUNESS
OF APPROXIMATION CALCULATIONS FOR MULTILAYERED DENSITY INTERFACES **

**The goals of the paper
are to obtain mathematical constructions for geological objects, such
as synclines and anticlines; to substantiate the uniqueness of the
inverse problem when renovating analytical models for the
horizontally layered geological media with several density interfaces
in contact surfaces predefined by Chorniy; and to try the techniques
developed for their iterative calculation. A combination of these two
models develops a new and more accurate approach to gravimetric
inverse problems for the contact interface. This becomes necessary to
improve standard fit procedures when solving inverse problems in
gravity and magnetic fields. The inverse problem of the density
interface in the horizontally layered geological media with several
density interfaces is confined to the solution of the nonlinear
integral equation that describes the contact surface restricted by
the given constant asymptotes within the planar region. Still, this
makes computation more complicated because of the problem of
equivalency solutions. Two field separation theorems are proposed for
this model – one for several 1-connected volumes and another
one for the non-crossed layers. The theorems of uniqueness are built
on the theorems of field separation enabling the solution of the
inverse problem by the summary external gravity field of n objects
(ore bodies, layer interfaces etc.) through the solution of the
inverse problem for separate objects – by the appropriate field
values from these geological objects. The numerical schemes for the
definition of the initial approximation of the density interface in
the multilayered geological media are stated. These algorithms
formally coincide within the first iteration. There are also proposed
analogical techniques based of the Chebyshev iteration construction
for the iterative specification of the behavior of the contact
asymptotes. There were modeled synthetic initial approximations of
synclines and anticlines by these algorithms. An alternative calculus
method for it is pointed out, which is based upon the definition of
the different moments of the interface curves. For the integral
calculation there is obtained an appropriate expression in the finite
quadratures. Modeling data show that new analytical constructions for
the calculation of the multilayered contact interfaces within their
Newtonian numerical approximation converge more quickly in comparison
with classic techniques for the contact definition. Their
invariability for the big dimension field data should be tested on
the real measurements. No attempts to apply rough approximations were
successful: convergence was considerably less than in previous cases,
and, besides, there was a rather ambiguous geological maintenance. **

**Key words: potential
theory, analytical model, contact problem, classes of density
interfaces, gravity fields separation, modeling.**