R. Minenko1, MSc, E-mail: firstname.lastname@example.org,
P. Minenko1, Dr. Sci. (Phys.-Math.), Prof.,
Yu. Mechnikov2, Geologist
INVERSE LINEAR PROBLEMS IN GRAVIMETRY: IN SEARCH FOR SUSTAINABLE SOLUTIONS
1Kryvyi Rih National University, 54 Gagarina Ave., Kryvyi Rih, 50086 Ukraine,
2Kryvyi 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.
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. M│nenko P., Minenko R., (2014). Obernen│ l│n│jn│ zadach│ grav│metr│┐ ta magn│tometr│┐ z utochnjujuchimi │terac│jnimi 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. V│snyk 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).