Diagnostics of the high-precise ballistic measured gravity acceleration by methods of non-classical errors theory

Lviv Polytechnic National University
International University of Economics and Humanities named after Academician S. Demianchuk

The purpose of the investigation is to show the necessity of using modern ideas about the law of error distribution for observations involved in the categories of the “Non-classical error theory of measurements” (NETM) in the process of performing high-precision ballistic definitions of gravitational acceleration. These definitions are characterized by large volumes, which according to the H. Jeffreys’ theory, professor at the University of Cambridge, automatically takes them beyond the bounds of the classical concepts about the errors of measurements law. These outdated views about the distribution law of errors of large volume measurements are the main obstacles to improve the methodology of these highly precise and important definitions. The research methodology is provided by the NETM-procedures that was designed to control the probabilistic from of the statistical distribution of absolute high-precise ballistic measurements g with large sample volumes based on H. Jeffreys’ recommendations and on the principles of hypothesis testing theory. The main result of the research is to carry out NETM-diagnostics of a metrological situation with the ballistic gravimeter FG-5 after some improvements of the program of the observation. This method of diagnostics is based on the use confidence intervals to the estimates of asymmetry and kurtosis of the obtained samples of measurements g with the following application of the Pearson’s -test to determine the significance of the deviations of its distribution from the established norms. In accordance with the categories of the NETM, such norms are the Gauss’s and Person-Jeffreys’s laws, since only they ensure the non-singularity of the weight function of the sample, and therefore the possibility of obtaining non generate estimates g during the mathematical processing of measurements. Scientific novelty: using the possibilities of the new important tool in the field «Data analysis» using the NETM to improve the technique of the high-precise measurements g, which are performed in a complicated metrological situation with the necessity of taking into account a number of non-stationary sources of systematic errors. The practical significance of the research is in use of NETM-diagnostics of the probabilistic form of the distribution of measurements g in order to improve the methodology of these highly precise determinations. The investigation seeks reasons for the deviations of errors distributions from established norms providing metrological literacy of the high-precise large-scale measurements.

1. Arnautov, G. P., Koronkevich, V. P., & Stus, Yu. F., (1982). The Interferometer of the absolute lazers ballistic gravimeter. Institut avtomatici i elektrometrii SO AN USSR, Novosibirsk, Preprint 196. 37 p.
2. Bessel, F. W. (1818). Fundamenta astronomiae. Konigsberg.
3. Bessel, F. W. (1838). Untersuhungen uber die Wahrscheinlichkeit der Beobachtungs-fehle. Astronomische Nachrichten, b. 15, 369.
4. Bolshev, L. N., & Smirnov, N. V. (1983). Tables of Mathematical Statistics. Moscow: Science. (in Russian).
5. Borodachev, N. A. (1950). The Main Questions of the accuracy of the Theory of Manufacture. Editor A. N. Kolmogorov. Moscow - Leningrad: AS USSR Publ., 360 p, [In Russian].
6. Bruevich, N. G. (Editor). (1973). Production Accuracy in the Mechanic and Instrument engineering.
7. Cramér, H. (1946). Mathematical methods of statistics. 1946. Department of Mathematical SU.
8. Doolittle, C. L. (1910). Results of Observations with the zenith telescope and the Wharton reflex zenith tube. The Astronomical Journal, XXVI, 608, Albany.
9. Doolittle, C. L. (1912). Results of observation with the zenith telescope and the Wharton reflex zenith tube. The Astronomical Journal, 27, 133-138.
10. Dvulit, P., & Dzhun, I. (2017). Application of methods of the non-classical error theory in absolute measurements of Galilean acceleration. Geodynamics, (22), 7-15.
11. Dzhun, I. V. (1969). Pearson Distribution of type VII in the errors of Observations of Latitude Variations. Astrom. Astrofiz. 2, 101 ̵ 115.
12. Dzhun, I. V. (1974). Analysis of parallel Latitudinal Observations performed under the general program. Extended abstract of Cand. Degree of Phis. - Math. Sci.: spec. 01.03.01 "Astrometry and Celectial Mechanics". Kyiv: Institute of mathematics of AS USSR.
13. Dzhun, I. V. (1983). Fluctuations in Weight of Individual Measurements of the Gravity Acceleration and the Way of their Account for ballistic Observations Processing. In Repeated Gravity Observations: Theory and Results. Moscow: MGK Prezidiume AS USSR, Neftegeofizika Publ., 46-̵ 52.
14. Dzhun, I. V., Arnautov G. P., Stus Yu. F., Shcheglov S. N. (1984). Feature of the Dis-tribution Law for the Results of Ballistic Measurement of the Gravity Acceleration. Repeat Gravimetric Observations: Theory and Results. Moscow: MGK Prezidiume AS USSR, Neftegeofizika Publ., 87-̵ 100.
15. Dzhun, I. V. (1992). Mathematical Treatment of Astronomical and Space-Based Information in non-Gaussian Observation Errors. Extended Abstract of Doctoral Dissertation in Physics and Mathematics. Main Astronomical Observatory of the National Academy of Sciences of Ukraine, Kyiv.
16. Dzhun, I. V. (2012). Distribution of errors in multiple large-volume observations. Measurement Techniques, 55, 393-396., Springer.
17. Dzhun, I. V. (2015). The Non-classical Errors Theory of Measurements. Rivne: Estero Publ., 168 [in Russian].
18. Dzhun, J. V. (2017). A new importnat tool in the field of intelligent data analysis. Alcide De Gasperi University of Euroregional Economy in Jozefow. Intercultural Communication, 1/2, 162-175.
19. Eddington, A. S. (1933). Notes on the method of least squares. Proceedings of the Physical Society, 45(2), 271.
20. Fedorov, E. P. (1963). Nutation and forced motion of the Earth's pole from the data of latitude observations. Oxford, New York, Pergamon Press.
21. Gauss, C. F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium (Vol. 7). Perthes et Besser.
22. Gauss, C. F. (1823). Theoria combinationis observationum erroribus minimis obnoxiae (Vol. 1). Henricus Dieterich.
23. Geary, R. C. (1947). Testing for Normality. Biometrika, 34, 209 ̵ 242.
24. Hammond, J. A., & Faller, J. E. (1971). A laser-interferometer system for the absolute determination of the acceleration due to gravity. Precision Measurement and Fundamental Constants; Proceedings, 343, 457.
25. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., & Stahel, W. A. (1986). Robust statistics (pp. 29-30). New York: Wiley.
26. Hulme, H. R., & Symms, L. S. T. (1939). The law of error and the combination of observations. Monthly Notices of the Royal Astronomical Society, 99, 642.
27. Idelson, N. I. (1947). Method of Least Squares and the Theory of Math. Treatment of Observations). [In Russian]. Geodezizdat. Moscow - Leningrad.
28. Jeffreys, H. (1938). The law of error and the combination of observations. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 237(777), 231-271.
29. Jeffreys, H. (1939). The law of error in the Greenwich variation of latitude observations. Monthly Notices of the Royal Astronomical Society, 99, 703.
30. Jeffreys, H. (1998). The theory of probability. OUP Oxford.
31. Lucacs, E. A. (1942). A Characterization of the normal Distribution. Annals of Mathematical Statistics. 13, 91-93.
32. Newcomb, S. (1886). A generalized theory of the combination of observations so as to obtain the best result. American journal of Mathematics, 343-366.
33. Ogorodnikov, K. F. (1928). Procedure for Reducing Observations by introducing Mean Weights in application to Statistical Study of Stellar Motions, Astron., Jurn., 5(1), 1-̵ 21.
34. Pearson, K. (1902). On the Mathematical Theory of Errors of Judgment with special Reference to the Personal Equation. Philosophical Transactions of the Royal Society of London. Ser. A., 198, 235-̵ 296.
35. Sakuma, A. (1973). A permanent station for the absolute determination of gravity approaching one microgal accurace. Proc. Symposium on Earth's gravitational field and secular variations in position. University of N. S. W., Sidney. p. 674-684.
36. Student. (1927). Errors of routine analysis. Biometrika, 151-164.
37. Tukey, J. W. (1960). A survey of sampling from contaminated distributions. Contributions to probability and statistics, 448-485.
38. Tukey, J. W. (1962). The future of data analysis. The annals of mathematical statistics, 33(1), 1-67.