About the possibility of fractional differentiation application in gravimetry and magnetometry
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- Category: Geology
- Last Updated on 25 December 2013
- Published on 19 December 2012
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Authors:
P.A. Minenko, Dr. Sci (Phys.-Math.), Associate Professor, Krivorozhsky Teacher Training Institute of the State Higher Educational Institution «Krivorozhsky National University», Professor of the Department of Computer Science and Applied Mathematics, Krivoi Rog, Ukraine
V.P. Minenko, Geologist, unemployed, Krivoi Rog, Ukraine
R.V. Minenko, Master Degree on Mathematical Sciences, unemployed, Krivoi Rog, Ukraine
Abstract:
Purpose. To increase explorative possibilities ofgravimetryand magnetometry through the transformation of the field by fractional differentiation.
Methodology. Forore and hydrocarbon deposits prospecting we use solutions of inverse problems of the field of magnetic and gravimetric surveys. However, the solutions are unstable and ambiguous. For increase of geological pithiness of the solutions of the inverse problems we use derivatives of the higher order integer of gravitational and magnetic potential. However, prospecting possibilities of a field and its first-order derivative on depth strongly differ and become worse with depth, and the second-order and third-order derivatives of the field do not allow determining objects at a depth more than 1000 m at all. In order to increase the prospecting possibilities it is possible to take some fractional derivatives of the field between zeroth order and the first order. Moreover, derivatives of the integral orders are functions linearly dependent to each other. And each new derivative almost doesn’t add any geological information. But the close derivatives of the fractional order, on the contrary, are linearly independent and provide unambiguity of the solutions of the inverse problems, and improve their prospecting possibilities. The fractional analysis is already known for more than 200 years and widely applied in other fields of science and techniques. In practice fractional derivative fields were calculated by authors for the field using formulas of Grunwald-Letnikov.
Findings. For quantitative interpretation of the field we didn’t have corresponding solutions of directproblems of gravimetry and magnetometry because of the lack of the general formula of potential of derivative of nthorder. Authors have deduced the formula and have solved some direct problems for fractional derivatives of the field for the purpose of their application in practice.
Originality. The theory of gravitational and magnetic potential has been expanded on the area of fractional orders of differentiation. This gives us the additional tool to increase possibilities of magnetic and gravitational survey with prospecting purposes.
Practical value. Opportunities and mathematical tools for more substantial and enhanced research of separate geological sites by methods of gravimetry and magnetometryhave been created.
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