**R.
Minenko**^{1}**,
MSc****,
****E-mail:
maestozo.1_pavel@mail.ru****,**

**P.
Minenko**^{1}**,
Dr. Sci. (Phys.-Math.), Prof.****,**

**Yu.
Mechnikov**^{2}**,
Geologist**

**INVERSE
LINEAR PROBLEMS IN GRAVIMETRY: IN SEARCH FOR SUSTAINABLE SOLUTIONS**

^{1}**Kryvyi
Rih National University,****
****54
Gagarina Ave., Kryvyi Rih, 50086 Ukraine****,**

^{2}**Kryvyi
Rih Geophysical Department****,
****2
Geologichna Str., Kryvyi Rih, 50001 Ukraine**

**The
paper aims at determining the causes of the change in density for
ILPG unjustified solutions, providing a theoretical proof, and
building a method for solving a real ILPG reproduction of the density
distribution in the anomalous body along its vertical axis. Inverse
problems in gravimentry and magnetometry are clearly and technically
incorrect, for various optimization criteria give different
solutions, and they can be substantially different in some areas of
the interpretation model. Besides, when stability of solutions is
checked, there is often revealed a mismatch: small errors in the
field in many places cause large changes in density in the blocks
located under these points. The paper gives coverage of scientific
findings that contribute to inverse linear problems. Namely, Acad.
Strakhov postulates stable and geologically meaningful ILPG solution
will only be obtained through methods of constrained optimization,
and develops an iterative method of least squares of the residuals.
Acad. Starostenko develops iterative correction for solving linear
algebraic equation. Doc. Minenko proves a theorem stating equality
area map projection field and interpretation model to map the fields
makes a prerequisite for ILPG sustainable solutions. Acad. Strakhov's
iterative method of least squares for residual field is further used
by Doc. Minenko to develop a filtering iterative method of simple
iteration adjusted by Acad. Starostenko through optimizing iterative
least sums of the squares of corrections to the density of rocks. The
finding is a guaranteed method of iterative optimized sustainable
solutions for ILPG multilayer interpretation model, in which each
horizontal layer is densely packed by cuboid-shaped blocks of
different unknown density. Still, the main drawback of the method is
it does not ensure absolute geological or physical equivalency
between ILPG density values of each block model and real values of
rock massif density. Doc. Minenko develops a two-step procedure for
finding ILPG sustainable and meaningful solutions. Further solutions
being achieved (meaning iterative refinement of the problem being
made following the equalizing of the initial conditions of the
iterative process in the second stage in all layers of the model), we
obtain the density distribution, which coincides with one in
anomalous bodies of the theoretical model. This means that the main
reason for the density reduction in the ILPG solution with depth in
the first stage is lack of control over the residual distribution
field at each iteration and point during their conversion into
iterative corrections for all blocks of the models below the pitch
dot.**

**Keywords:
gravimetry, inverse problem, iterative method, iterative correction,
optimization criterion, refinement.**

**References****:**

1.
Minenko P.A., (2005). Teoreticheskoe obosnovanie preobrazovanija
modelej reshenija nekorrektnoj linejnoj zadachi gravimetrii v
korrektnuju s optimizaciej iteracionnogo processa na osnove
uslovno-jeskstremal'nyh kriteriev. Teorija i praktika geologicheskoj
interpretacii gravitacionnyh i magnitnyh anomalij: materialy 32-j
sessii mezhdunarodnogo nauchnogo seminara im. D.G. Uspenskogo
(29.01-01.02.2005). *Perm*,
115-118. (In Russian).

2.
Minenko P.A., (2006). Isledovanie kristalicheskogo fundamenta
lineynonelineynymi metodami magnitometrii i gravimetrii.
*Geoinformatika*,
4, 41-45. (In Russian).

3.
Minenko P.A., Minenko R.V., (2012). Uproshhennye algoritmy reshenija
obratnyh zadach gravimetrii filtracionnymi metodami. *Geoinformatika*,
2(42), 27-29. (In Russian).

4.
M³nenko P., Minenko R., (2014). Obernen³ l³n³jn³ zadach³ grav³metr³¿
ta magn³tometr³¿ z utochnjujuchimi ³terac³jnimi popravkami vishhogo
porjadku. *Visnyk
of Taras Shevchenko National University of Kyiv: Geology*,
1(64), 78-82. (In Ukrainian).

5. Starostenko V.I., Kozlenko V.G., Kostjukevich A.S., (1986). Sejsmogravitacionnyj metod: principy, algoritmy, rezultaty. V³snyk AN URSR, 12, 28-42. (In Russian).

6.
Strahov V.N., (1990). Ob ustojchivyh metodah reshenija linejnyh
zadach geofiziki. II. Osnovnye algoritmy. *Izv.
AN SSSR. Fizika Zemli*,
8, 37-64. (In Russian).