ON MATHEMATIC FORMALIZATION OF SIMILARITY OF RECORDS OF ELECTRICAL AND SEISMIC SIGNALS
Rubrics: ORIGINAL ARTICLES
Authors:
1.
Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia
2.
Geophysical Center RAS, Moscow, Russia
3.
Geophysical Center RAS, Moscow, Russia
4.
Institute of physics of the Earth, Paris, France
5.
Laboratory of volcanology of University of Clermont-Ferrand, France
Type:
Article
Pages:
from 1
to 6
Status:
Published
Received:
25.12.2009
Accepted:
25.12.2009
Published:
25.12.2009
Language:
English
Keywords:
seismic and electrical signals, analytical dependencies, Lagrange interpolation
Abstract (English):
A possible way of analysis of records of seismic and electrical signals is an application of the classical interpolation theory 1-3, and also well-developed methods of the theory of information transmission 4,5. The authors had at their disposal seismic and electrical records. They represented tables of figures ndash; signals, recorded by corresponding devices at discrete moments of time. At that, for each record of a seismic signal wave there were two records of an electric signal a change of electrical resistivity of environment: one in orientation N-S, the other in orientation E-W.To pass from a discrete set of tabular data to its analytical representation the Lagrange interpolation procedure was used 1,2. This procedure to some extent amplifies the missing data, allowing to restore a continuous real-time signal according to an available discrete record to any desired degree of accuracy. It should be taken into account that any accompanying noise, related to the signal, would be also recorded. Thus, prior to signals' analysis, it would be desirable to remove any noise. Below Sx ndash; is a continuous seismic signal, and Ex ndash; a continuous electrical signal. According to 5 it can be recorded as 1 Here Si , Ei ndash; tabular values of measured seismic and electrical signals, a current coordinate recording time. The number records in interval is equal to N + 1. Supposing that out of the interval all nbsp;Si , Ei equal to zero, then series 1 can be replaced by final sums with summation of series from i = 0 to i = N assuming that electrical and seismic signals were recorded simultaneously, with a uniform pitch. According to 5, an analytical dependence can be constructed, expressing electric signal through a seismic one. A corresponding formula transfer function 5 can be represented in the following way The theory of Lagrange interpolation 1, 5 proves that function Ex can be reconstructed by function Sx with the help of transfer function 2 to any desired degree of accuracy in the following way Formula 3 has one obvious shortcoming ndash; it is rather complicated. It can be simplified, obtaining the following expression, connecting electrical and seismic signals. Parameters a and k could be chosen in such a way that function E*x will be very closely approximated to function Ex. There are different approaches to defining these parameters. For example, in the present work parameter a was defined graphically according to the condition of a maximum of the correlation coefficient between initial electrical record and its approximate representation 4. The fact that passing of wave P entails a drop in rock electrical resistivity is well known 6 and can be explained from a physical point of view. At the moment of passing of a tensile wave microfissures open and are filled with water fluid, followed by a drop in electrical resistivity in proportion to a degree of opening of micropores. Parameter of shift a in formula 4 in fact represents the time, required for filling micropores with water. References 1. Levin B. Y. 1956, Distribution of Finite Functions' Roots. M., Nauka, 682nbsp;p. 2. Akhiezer N. I. 1965, Lectures on the Theory of Approximations. M., GIFML, 407nbsp;p. 3. Kirillov A.A., Gvishiani A. 1982, Theorems and problems in functional analysis, New York-Heidelberg-Berlin, Springer-Verlag. Seria "Problem books in mathematics", 347 4. Gvishiani A., Dubois J. 2002, Artificial Intelligence and Dynamic Systems for Geophysical Applications. Springer-Verlag, Paris, 350 5. Yakovlev Y. I., Khurgin V. P. 1971, Finite Functions in Physics and Technology. M, GIFML, 408nbsp;p. 6. G. A. Sobolev, A. V. Ponomarev. Physics of Earthquakes and Precursors. M., Nauka. 2003. 270
A possible way of analysis of records of seismic and electrical signals is an application of the classical interpolation theory 1-3, and also well-developed methods of the theory of information transmission 4,5. The authors had at their disposal seismic and electrical records. They represented tables of figures ndash; signals, recorded by corresponding devices at discrete moments of time. At that, for each record of a seismic signal wave there were two records of an electric signal a change of electrical resistivity of environment: one in orientation N-S, the other in orientation E-W.To pass from a discrete set of tabular data to its analytical representation the Lagrange interpolation procedure was used 1,2. This procedure to some extent amplifies the missing data, allowing to restore a continuous real-time signal according to an available discrete record to any desired degree of accuracy. It should be taken into account that any accompanying noise, related to the signal, would be also recorded. Thus, prior to signals' analysis, it would be desirable to remove any noise. Below Sx ndash; is a continuous seismic signal, and Ex ndash; a continuous electrical signal. According to 5 it can be recorded as 1 Here Si , Ei ndash; tabular values of measured seismic and electrical signals, a current coordinate recording time. The number records in interval is equal to N + 1. Supposing that out of the interval all nbsp;Si , Ei equal to zero, then series 1 can be replaced by final sums with summation of series from i = 0 to i = N assuming that electrical and seismic signals were recorded simultaneously, with a uniform pitch. According to 5, an analytical dependence can be constructed, expressing electric signal through a seismic one. A corresponding formula transfer function 5 can be represented in the following way The theory of Lagrange interpolation 1, 5 proves that function Ex can be reconstructed by function Sx with the help of transfer function 2 to any desired degree of accuracy in the following way Formula 3 has one obvious shortcoming ndash; it is rather complicated. It can be simplified, obtaining the following expression, connecting electrical and seismic signals. Parameters a and k could be chosen in such a way that function E*x will be very closely approximated to function Ex. There are different approaches to defining these parameters. For example, in the present work parameter a was defined graphically according to the condition of a maximum of the correlation coefficient between initial electrical record and its approximate representation 4. The fact that passing of wave P entails a drop in rock electrical resistivity is well known 6 and can be explained from a physical point of view. At the moment of passing of a tensile wave microfissures open and are filled with water fluid, followed by a drop in electrical resistivity in proportion to a degree of opening of micropores. Parameter of shift a in formula 4 in fact represents the time, required for filling micropores with water. References 1. Levin B. Y. 1956, Distribution of Finite Functions' Roots. M., Nauka, 682nbsp;p. 2. Akhiezer N. I. 1965, Lectures on the Theory of Approximations. M., GIFML, 407nbsp;p. 3. Kirillov A.A., Gvishiani A. 1982, Theorems and problems in functional analysis, New York-Heidelberg-Berlin, Springer-Verlag. Seria "Problem books in mathematics", 347 4. Gvishiani A., Dubois J. 2002, Artificial Intelligence and Dynamic Systems for Geophysical Applications. Springer-Verlag, Paris, 350 5. Yakovlev Y. I., Khurgin V. P. 1971, Finite Functions in Physics and Technology. M, GIFML, 408nbsp;p. 6. G. A. Sobolev, A. V. Ponomarev. Physics of Earthquakes and Precursors. M., Nauka. 2003. 270
Keywords:
seismic and electrical signals, analytical dependencies, Lagrange interpolation
seismic and electrical signals, analytical dependencies, Lagrange interpolation
1. Akhiezer, Lectures on the Theory of Approximations, 1965.
2. Levin, Distribution of Finite Functions' Roots, 1956.
3. Sobolev, Physics of Earthquakes and Precursors, 2003.
4. Yakovlev, Finite Functions in Physics and Technology, 1971.
5. Gvishiani, Artificial Intelligence and Dynamic Systems for Geophysical Applications, 2002.
6. Kirillov, Theorems and Problems in Functional Analysis, 1982.
