Geoinformatika 2018; 1(65) : 41-47

SOLUTION OF NOT CORRECT INVERSE PROBLEMS FOR MAGNETIC SURVEYS OF LARGE SCALE

P.A. Minenko¹, R.V. Minenko², Yu.P. Mechnikov³, I.V. Plishko⁴

¹ Krivorozhsky State Pedagogical University, 54, Gagarin Ave., Kryvyi Rih, 50086, Ukraine
² State institution of higher education “Kryvyi Rih National University”, Vitalyy Matusevych str.,11, Kryvyi Rih, 50027, Ukraine
³ Krivorozhskaya Geophysical Party, 2, Geologicheskaya Str., Kryvyi Rih, 50045, Ukraine
⁴ State Enterprise “Ukrchermetgeologia”, 4, Zhenevskaya Str., Krivoy Rog, 50000, Ukraine e-mail: presto2presto@karbon.com.ua

Purpose. The research was conducted with the view to develop a method of obtaining a stable as well as geologically meaningful solution to inverse magnetometry problems for large-scale surveys, using multilayer models of the geological environment; to compare the solutions of inverse problems with an exact and inaccurately known depth of the lower boundary of the deposit.

Design/methodology/approach. To solve ill-posed inverse problems of magnetometry, a discrete (grid) analogue of the solution to an integral equation of the first kind is used, which is reduced to solving a system of linear or nonlinear algebraic equations with functional coefficients and a right-hand side complicated by various kinds of errors. Since geological bodies have an arbitrary shape and are inhomogeneous in magnetic properties, the functional coefficients are also calculated with errors. We solve these systems, as it done in ordinary statistics, in almost all cases by the method of least squares. The requirement of stability of the solution is achieved by the fact that above each block of
the grid model there must be at least one point with the field measured in it. And, conversely, under each point of the field there must be at least one block of the grid model of geological bodies. With a rectangular or square network of field measurements, the size of the field map and the size of the grid model of geological bodies must coincide.

Findings. The solutions of inverse problems for a real field are given for the 2 models: with the exact and inaccurate (presumed) location of the lower boundary of the structure of the deposit. In the geological sense, the content of the solution has been only for the inverse problem for a single-layer model. For a two-layer model, the physical parameters of the upper block are determined more accurately than for the lower one. With more layers, the solution for the lower layers is equivalent, in which the difference from the real increases with an increasing depth of the block. It is also possible to solve inverse problems with different depths for blocks of each layer, but then they need to be set by
a separate array. In any case, a stable solution of the inverse problem can be obtained; but whether it is meaningful in the geological sense, depends on the availability of some part of the a priori data.

Practical value /implications. As a result of the carried out experimental studies on the site of the Petrovskoye deposit, it is established that, if the area of the mass model and the area of the field map are equal, the inverse problem is always solved steadily. In the case of a single-layer or multilayer mass model, the inverse problem does not have a meaningful solution in the geological sense since the average intensity of the magnetization over the block is an equivalent solution. With the high detail of the partition of the mass model into blocks, it is possible to obtain practically real maps and sections of the magnetization distribution of rocks reflecting the main regularities in the distribution of magnetic ores in the structure of the deposit. In the presence of a priori data on the depths of the location of the boundaries of blocks and the content of total and magnetic iron in certain blocks of ore deposits, the accuracy of solving inverse problems has substantially increased.

Keywords: magnetometry, solution of inverse problem, iterative method, least-squares method, discrepancy of the field.

The full text of papers

References:

1. Minenko P.A., Minenko R.V. Modifikatsiya metoda regulyarizatsii v OLZG dlya poiskovykh rabot v kristallicheskikh porodakh Ukrayinskoho krystalichnoho shchyta. Scientific Bulletin of National Mining University, 2006.no. 9, pp 34-39 [inRussian].
2. Minenko P.A. Іssledovaniye kristallicheskogo fundamenta lineyno-nelineynymi metodami magnitometrii i gravimetrii. “Geologorazvedka”, Rossiyskiy geofizicheskiy zhurnal, 2007, no. 45-46. pp. 60-64 [in Russian].
3. Minenko R.V., Minenko P., Mechnikov Yu. Problema poshuku zmistovnykh rozv’yazkнv obernenykh linнinykh zadach magnitometrii kompleksuvannyam interpretatsiynykh modeley. Visnik of Taras Shevchenko National University of Kyiv, Geology, 2015, no. 2(69), pp. 87 — 95 [in Ukrainian].
4. Tikhonov A.N., Arsenin V.YA. Metody resheniya nekorrektnykh zadach. Moscow.: Nauka, 1979, 286 p. [in Russian].
5. Tikhonov A.N., Dmitriyev V.I., Glasko V.B. Matematicheskiye metody v razvedke poleznykh iskopayemykh, Moscow: Znaniye, 1983, 64 p. [in Russian].