## 1. Introduction

[2] Modern single frequency satellite navigation and positioning systems require the use of precise and high resolution ionosphere correction models. On the other hand, precise measurements of modern space-geodetic techniques, such as the Global Navigation Satellite Systems (GNSS) allow the study of ionosphere variations with an unprecedented accuracy. Usually the ionosphere is assumed to be a single layer in a constant height and the vertical total electron content (VTEC) is modeled globally in most cases by a spherical harmonic expansion up to specific degree for fixed time intervals. The VTEC maps from Center of Orbit Determination in Berne (CODE) [*Schaer*, 1999] computed as one of five VTEC solutions within the International GNSS Service (IGS) [*Hernández-Pajarez et al.*, 2009] are a prominent example. However, spherical harmonic models are not optimal for representing data of heterogeneous density and quality. As an alternative we introduce a global VTEC model which is based on compactly supported localizing B spline functions allowing the handling of data gaps in case of unevenly distributed data, such as data from terrestrial GNSS stations. This allows us to work within an Earth-fixed reference system (in contrast to Sun-fixed reference systems [*Dettmering*, 2003] used by IGS) and to parameterize not only the horizontal position but also the time dependency of VTEC.

[3] In this paper we will compare two different model approaches, especially the ability to compensate and handle areas without data; that is, data gaps will be investigated. We distinguish between (1) a *classical* approach which is based on a system of three-dimensional (3-D) base functions defined as the tensor product of spherical harmonics for longitude and latitude as well as polynomial B spline functions for the time (section 3.1) and (2) an *alternative* approach which is constructed as a system of 3-D base functions defined as tensor products of trigonometric B spline functions for the longitude and two sets of polynomial B spline functions for latitude and time, respectively (section 3.2).

[4] For providing the reader with the theoretical background, in section 2 the basic formulae for 1-D and 2-D expansions are introduced; this is done briefly for the spherical harmonic expansion and the polynomial B spline expansion, but more detailed for the trigonometric splines since the latter are so far not used very often in ionospheric modeling. Based on these foundations, we construct in section 3 our two approaches mentioned above in detail and apply them to two different simulated data sets in section 4. The main objective of the paper is the comparison of the two approaches with respect to a global approximation of VTEC and the handling of data gaps. These two issues are discussed in section 4.2.