# One option of constructing three-dimensional distribution of the mass and its derivatives for a spherical planet Earth

2016;
: pp. 36 – 44

1
Lviv Polytechnic National University
2
Department of Cartography and Geospatial Modeling of Lviv Polytechnic National University
3
Department of Cartography and Geospatial Modeling of Lviv Polytechnic National University
4
Department of Cartography and Geospatial Modeling of Lviv Polytechnic National University

Purpose. To build a three-dimensional function of the mass distribution of the Earth's interior according to the parameters (Stokes constant to the second order inclusive) of the external gravitational field of the Earth without considering the minimum deviation from its known density models in geophysics. Methodology. The classic methods of constructing mass distribution use only the Stoke’s constants zero and second orders. In iterative methods of determining the distribution models the reference model of density is taken for zero approximation which is agreed upon by Stoke’s constants up to the second order inclusive. Further, the coefficients of potential expansion to a certain order are taken into account, but their contribution to the function of mass density does not investigate. This research provides an attempt to obtain such an estimation. The proposed method is approximate, but in the iterative process a function of the density is not only used, but also its derivatives. Bringing the order moments of density toward the controlled values (values that are defined on the surface of a sphere) makes it possible to analyze the process of successive approximations. Results. In contrast to the second-order model, which describes the global gross irregularities, the obtained distribution function gives a detailed picture of the placement density anomalies (deviation of three-dimensional functions from the average on the sphere – “izoden”). Analysis of maps at different depths
(2891 km core-mantle, 5,150 km of the inner-outer core) allow making preliminary conclusions about global redistribution of mass due to the rotating component of gravity across the radius: its dilution along the axis of rotation and accumulation of rejecting it. This is particularly evident for the equatorial regions. On the contrary, there is minimum deviation in the polar  regions of the Earth, which also have their own justification since the value of the rotation force decreases when moving away from the equator. The function of mass distribution, which is constructed using the proposed method, describes the mass distribution better. Originality. This research is in contrast to the classical results which have been obtained from the Adams-William’s equations for the derivatives of the density of one variable (depth), and make attempted to obtain derivatives using Cartesian coordinates. Using the gravitational field parameters up to second order increases the order of approximation of the distribution function of the masses of three variables from two to four through the possibility of restoring the planet's mass distribution by its derivatives. At the same time, in contrast to previous research, geophysical information accumulated in the reference PREM model is used, therefore, features of the internal structure are taken into account. Practical significance. The received function of mass distribution of the Earth can be used as a zero-order approximation when used in the presented algorithm Stokes constant of higher order. Their applications give the possibility to interpretate of the global anomalies of the gravitational field, and explore the geodynamic processes deep inside the Earth.

1. Byzov D. D., Cidaev A. G. Metodika postroienia 2D plotnostnoi modeli vierhniei mantii s uchetom uslovia izostaticheskoi kompensacii na glubinie [Method for construction of a 2D density model of the upper mantle in conditions of isostatic compensation at different depths]. Uralskii geophizicheskii vestnik [Ural Geophysical messenger]. 2015, no. 1(25), pp. 33–36.
2. Bullen K.Ye. Plotnost' Zemli [The density of the Earth]. Мoscow: Mir, 1978. – 437 p.
3. Garkov V. N., Trubicyn V. P. Phizika planetarnyh nedr [The physics of planetary subsurface]. – М.: Nauka, 1980, 448 p. – (Glav. red. Phiz.-mat. Lit.).
4. Korbunov A. I. Metod funkcionalnyh predstavlenii pri reshenii obratnyh zadach gravimetrii [A functional representations method in solving inverse problems of gravimetry]. Phizika Zemli [Physics of the Earth]. 2015, vol. 4, pp. 3–14.
5. Mashimov M. M. Teoreticheskaia geodezia: Spravochnoie posobie [Theoretical Geodesy: Reference Guide]. Мoscow: Nedra, 1991, 268 p.
6. Meshcheriakov G. A. Ispolzovanie stoksovyh postoiannyh Zemli dlia utochnenia jeio mehanicheskoi modeli [Using Stoke's constants of the Earth to refine its mechanical model]. Geodezia, kartographia i aerophotosiomka [Geodesy, cartography and aerial photography], 1975, vol. 21, pp. 23–30.
7. Meshcheriakov G. A., Fys M. M. Opredelenie plotnosti Zemnyh nedr riadami po biortogonalnym sistemam mnogochlenov [Determination of the density of the Earth's interior by rows of biorthogonal systems of polynomials] Teoria i metody interpretacii gravitacionnyh i magnitnyh anomalii [Theory and methods of interpretation of gravity and magnetic anomalies]. 1981, pp. 329–334.
8. Meshcheriakov G. A., Fys M. M. Trehmernaia i referencnaia plotnostnyie modeli Zemli [The three-dimensional and reference density models of the Earth]. Geophizicheskii Jurnal [Geophysical Journal]. Kyiv, 1986, vol. 8, issue 4, pp. 68–75.
9. Meshcheriakov G. A. Zadachi teorii potenciala i obobshchenaia Zemlia [Tasks of potential theory and generalized Earth]. Мoscow: Nauka, 1991, 216 p. (Glav.red. Phiz.-mat. Lit.).
10. Meshcheriakov G. A., Zazuliak P. M., Kulko O. V., Fys M. M., Shtabliuk P. I. Variant mekhanicheskoi modeli nizhnei mantii [Option of mechanical model of the lower mantle]. Trudy III Orlovskoi konferencii "Izuchenie Zemli kak planet metodami astronomii, geophiziki i geodezii" [Proceedings III Orel conference "Study of the Earth as a planet by astronomy, geophysics and geodesy methods"]. 1994, pp. 172–177.
11. Molodenskii S. M., Molodenskii M. S., Begitova T. A. 3D-modeli medlennyh dvijenii zemnoi kory I verhnei mantii v ochagovyh zonah seismoaktivnyh oblastei I ih sravnenie s vysokotochnymi dannymi nabliudenii [3D-model of the slow movements of the crust and upper mantle in the focal zones of seismically active areas and compare them with the high precision data of observations]. Phizika Zemli [Physics of the Earth]. 2016, no. 5, pp. 25–50.
12. Moritc G. Figura zemli: Teoreticheskaia geodezia i vnutrenniee stroienie Zemli [The figure of the Earth: Theoretical Geodesy and internal structure of the Earth]. Kyiv, 1994, 240 p.
13. Pankov V. L., Garkov V. I. O raspredilenii plotnosti v nedrah zemli [About the density distribution in the Earth's interior]. Zemnyie prilivy I vnutrieniee stroienie Zemli [Earth tides and the internal structure of the Earth]. 1967, pp. 44–61.
14. Fys M. M., Fotca R. S., Sohor A. R., Volos V. O. Metod znahodjennia gustyny rozpodilu mas planet z urahuvanniam stoksovyh stalyh do chetvertogo stepenia [Method for planets density distribution construction with using of stoke's constants to fourth order]. Geodynamika [Geodynamics]. 2008, vol. 1(7), pp. 25–34.
15. Fys M. M., Zazuliak P. M., Cherniaha P. G. Znachennia ta variacii gustyny u centri mas elipsoidalnyh planet [The value and variations of density in mass centers of ellipsoidal planets]. Kinematika i phizika nebesnyh tel [Kinematics and physics of celestial bodies]. 2013, vol. 29, no. 2, pp. 62–68.
16. Fys M. M., Holubinka Yu. I., Yurkiv M. I. Porivnialnyi analiz formul dlia potencialu ta joho radialnyh pohidnyh trysharovyh kuliovyh ta elipsoidalnyh planet [Comparative analysis of formulas for the potential and its radial derivative three-layered spherical and ellipsoidal planets] Suchasni dosjaghnennja gheodezychnoji nauky ta vyrobnyctva [Modern achievements in geodetic science and industry]. 2014, vol. I (27), pp. 46–51.
17. Fys M. M., Yurkiv M. I., Brydun A. M. Nablygenyi metod pobudovy pohidnyh ta znachennia rozpodilu mas nadr planet na prykladi spherychnoi planet Zemlia [Approximate method for construction of derivatives and values of mass distribution of the planet in case of spherical Earth]. New technologies in geodesy, land management, forest inventory and nature management [Novi tehnologii v geodezii, zemlevporiadkuvanni ta pryrodokorystuvanni]: Materials VIII Intern. Scientific and Practical Conference, 6–7 October, 2016, Uzgorod, Ukraine, pp. 47–52.
18. Cerklevych A. L., Zaiatc O. S., Fys M. M. Gravitaciini modeli tryvymirnoho rozpodilu gustyny planet zemnoi grupy [Earth group planets gravitational models of 3-d density distributions]. Geodynamika [Geodynamics]. 2012, vol. 1 (12), pp. 42–53.
19. Cherniaha P. G., Fys M. M. Novyi pidhid do vykorystannia stoksovyh stalyh dlia pobudovy funkcii ta ii pohidnyh rozpodiliv mas planety [The new approach for using of stoke's constants to build functions and its derivatives mass distribution of planets]. Suchasni dosjaghnennja gheodezychnoji nauky ta vyrobnyctva [Modern achievements in geodetic science and industry]. 2012, vol. II (24), pp. 40–43.
20. Shcherbakov A. M. Obiemnoie raspredilenie plotnosti Luny [Volumetric density distribution of the Moon]. Astronomicheskii vestnik [Astronomical messenger]. 1978, vol. ХІІ, no. 2, pp. 88–95.
21. Yatskiv Ya. S. Nutacia v sistieme astronomicheskih postoiannyh [Nutation in the system of astronomical constants]. Kyiv, 1980, 59 p. (Preprint AnN USSR;ITF–80–95P).
22. Dzewonski A., Anderson D. Preliminary reference Earth model. Physics of the Earth and Planet Inter. 1981, no. 25, pр. 297–356.
23. Martinenc Z., Pec K. Three – Dimensional Density Distribution Generating the Observed Gravite Field of planets: Part II. The Moon. Proc. Int. Symp. Figure of the Earth, the Moon and other Planets. Czechoslovakia, Prague. 1986, no. 1, pp. 153–163.
24. Moritz G. Computatson ellipsoidal mass distributions Department of Geodetic Science, The Ohio State University. 1973, no. 206, p. 20
25. Willamson E. D., Adams I. H. Density distribution in the Earth. Journal of the Washington Academy of Sciences. 1923, vol. 13, No 19, pp. 413–428.