Pergamon Adv. SpaceRes. Vol. 14, No. 12, pp. (12)61-(12)64, 1994 Copyright © 1994 COSPAR Printed in Great Britain. All rights reserved. 0273-1177/94 $7.00 + 0.00 IONOSPHERIC MAPPING BY REGIONAL SPHERICAL HARMONIC ANALYSIS" NEW DEVELOPMENTS G. De Franceschi, A. De Santis and S. Pau Istituto Nazionale di Geofisica, Via di Villa Ricotti 42, 00161 Roma, Italy ABSTRACT The method of Spherical Cap Harmonic Analysis (SCHA) /1/ has been applied/2/to the critical frequency of the F2 layer (foF2) for mapping and modelling it over Europe. The model was based on longitudinal expansion in Fourier series, and fractional Legendre colatitudinal functions over a spherical cap including Europe. Here a new and simpler technique, previously developed for the regional modelling of the geomagnetic field /3/, is introduced and described. The basic improvement of the new method, called Adjusted Spherical Harmonic Analysis (ASHA), implies the use of conventional Spherical Harmonic (SH) functions after the colatitude interval is adjusted to that of a hemisphere. Examples are shown de~ling with the application of ASHA to retrospective mapping and modelling of the monthly medians of foF2 over Europe. INTRODUCTION The knowledge of the spatial and temporal behaviour of the critical frequency, foF2, of the F2 layer can considerably help in the prediction of the proper ionospheric conditions for radio communica- tions through the ionosphere. The global CCIR model/4,5/is available for such a purpose. Such a model is based on data taken all over the world. In Europe, owing to the high density of the iono- spheric stations, one can determine a regional model based only on the data taken over the region of concern with the advantage of a better fit with available experimental data and a greater detail in terms of minimum wavelength the model portrays. A new technique for modelling and mapping the most evident features of the foF~ is described in/2/and/6/as Spherical Cap Harmonic .kn~ysis (SCHA). Even though SCHA was first introduced to deal with the 3-dlmensional geomagnetic potential/1/, it can also be applied to any general function f(,X, 0) defined over a spherical cap. The part which is dependent on longitude can be expressed in terms of Fourier series. The colat- itude part can be expanded in terms of new fractional Legendre functions, P~n,, characterized by a non-integer harmonic degree, which alternatively satisfies one of the two conditions imposed by Haines/1/on the Legendre functions at the edge of a cap with half-angle 00: dP~(cosOo)/dO = 0 o, Pn~(cosOo) = 0 (1) The values of nk, one at a time determined from (1), are usually ordered by an integral index k starting at zero. Finally the expansion takes the form K k f(~, 0) = E E (g~ cos mA + h~* sin mA). P~ (cos 0). (2) k=0 m=0 The spherical expansion is developed over a cap-like region, using a new reference system having origin at the Earth's centre and pole at the centre of the spherical cap. Expressions for the new Legendre functions of non-integer degree, and the needed coordinate transformations as well, are (12)61