Modeling of regional magnetic field applying spherical functions: theoretical aspect
Observation of geomagnetic field, measurement of its components values and establishment on their base models of geomagnetic field as well as geomagnetic mapping are the main trend of geomagnetic studies. Analytical model of geomagnetic field allows calculating the value of any component of geomagnetic field at any point of near-earth space and on the Earth. A new method has been proposed for construction of geomagnetic potential regional model. In accordance with Gauss’ fundamental studies, classical concept of geomagnetic potential became its record as endless series of Legeandre functions. A program of writing the series of geomagnetic potentials according to their spherical or ellipsoidal functions is used for most cases for modeling of global (normal) geomagnetic field with the length of a series equal to 9—13 harmonics. However in case when the sphere is not complete and only its part (a segment of the sphere or its trapezium) is taken into account the spherical Legeandre functions lose their orthogonality. In this connection for elaboration of regional field models different modifications of spherical harmonic analysis with application of spherical Legeandre functions even grade and real order are used. Such functions form an orthogonal by weight system of functions on arbitrary spherical trapezium but do not have recurrent correlation therefore we are to use for their calculation an arrangement as hypergeometric series. The area of determination of such functions is spherical segment. Working formulae have been obtained for constructing the model mentioned above. As front-end data for constructing a model of regional magnetic field the values of its components obtained at geomagnetic observatories have been used. It has also been suggested to make calculations of regional model of geomagnetic potential by a method proposed within the limits of a procedure deletion-computing-restoration. Initially a systematic product of components is found using a global model of geomagnetic field. Then abnormal values of components are calculated. A model of regional abnormal geomagnetic field is computed using basic functions. A parameter of Tikhonov’s regularizing is introduced to stabilize the solution.
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