## 1. Introduction

[2] The Global Positioning System (GPS) has been used as an ionospheric research tool for almost 2 decades [*Yunck et al.*, 2000] and offers near-global coverage combined with continuous operation. Networks of fixed ground-based receivers have been established for many different scientific applications, and daily observations from these receivers are available via the Internet. *Davies and Hartmann* [1997] discuss the opportunities presented by GPS to study the ionosphere and compare the use of GPS with polar-orbiting satellites such as the U.S. Navy Navigation Satellite System (NNSS).

[3] The slant total electron content (sTEC) is defined as the integrated value of electron concentration along the line of sight from the transmitter to the receiver. Transionospheric radio signals suffer a phase advance and a group delay, both of which are dependent on the signal frequency. Since GPS satellites transmit on two frequencies, L1 (1575.42 MHz) and L2 (1227.6 MHz), TEC can be inferred by measuring either the phase advance or the group delay of the L1 signal with respect to the L2 signal [*Lanyi and Roth*, 1988]. This permits spatial maps of ionospheric TEC to be generated. An important application of such TEC maps is in the estimation of the ionospheric error in single-frequency GPS navigation solutions [*Harris et al.*, 2001].

[4] Local TEC maps can be produced using a single receiver station [*Coco et al.*, 1991; *Ciraolo and Spalla*, 1997]. A simple way to map TEC over wide geographical areas is to approximate the whole ionosphere, which extends from about 80 to over 1000 km, by a thin shell at a fixed altitude [*Mannucci et al.*, 1998]. A problem with this approach is that the ionosphere is a highly variable medium, both temporally and spatially, and the vertical distribution of ionization cannot be represented accurately at all times and locations by a thin shell.

[5] An alternative approach is to invert the TEC observations to yield the spatial distribution of electron concentration represented in three-dimensional voxels of constant electron concentration. Slant or vertical TEC can then be calculated along any direction by integration through the grid along any required path without the need to interpolate across a shell or apply any mapping functions to convert between slant and vertical TEC. This technique is applied here using the inversion algorithm of *Spencer and Mitchell* [2001], known as the Multi-instrument Data Analysis System (MIDAS). MIDAS carries out a full mathematical inversion in three spatial dimensions and one time dimension using an algorithm extended from the least squares linear matrix inversion applied by *Fremouw et al.* [1992] to two-dimensional ionospheric tomography.

[6] In the case of two-dimensional inversions (tomography) the ionization distribution is assumed to be stationary during the time of a low-Earth-orbit satellite pass. Initially it seemed that GPS satellites could not be used for ionospheric imaging because their transit time is many hours, too long to image a moving ionosphere. Nevertheless, multiple GPS satellites are in view at any time, and hence a three-dimensional inversion is conceptually possible. However, this snapshot set of measurements results in a sparseness of data, whereas a time-dependent inversion allows a great increase in the quantity and angular coverage of measurements that can be used in each inversion. For this reason a four-dimensional determination of the ionization distribution has been implemented. The algorithm of *Fremouw et al.* [1992] is extended into a time-dependent inversion by incorporating a priori information about the evolution of the electron concentration during a specified period of time, typically 1 hour, assuming that the change in electron concentration within a voxel with time is linear. Details of the MIDAS inversion technique are given by *Mitchell and Spencer* [2003]. In the present paper, TEC mapping using the MIDAS four-dimensional inversion is compared with the thin shell approach for both simulations and experimental case studies.