GNSS Atmospheric and Ionospheric Sounding – Methods and Results - Global Navigation Satellite Systems: Signal,Theory and Applications - page 370

single layer model (SLM) ionosphere approximation was used. SLM assumes that all the
free electrons are contained in a shell of infinitesimal thickness at altitude H (generally 350
km above the Earth). A mapping function is used to convert the slant TEC into the vertical
TEC (VTEC) as shown:

Fz ( ) =− (1 R cos(90 − + z ) ( R H )) (11)

where R is Earth radius, H is SLM height, and z is satellite zenith angle. When using the
above mapping function, F(z), one can obtain VTEC values at the ionosphere pierce points
(IPPs). The GPS-derived TEC can correct ionospheric delay for microwave techniques and
monitor space weather events.

3.2 3-D ionospheric tomography reconstruction

The STEC is defined as the line integral of the electron density as expressed by:

R

= ∫

satellite

STEC N (,,) λϕ h ds (12)

e

R

receiver

where λϕ is the ionospheric electron density, λ, ϕ and h are the longitude,

latitude and height, respectively. To obtain N , the ionosphere is divided into grid pixels
with a small cell where the electron density is assumed to be constant, so that the STEC in
Eq.(4) along the ray path i can be approximately written as a finite sum over the pixels j
as follows:

e Nh

(,,)

M

= ∑ STEC a n (13)

i ijj

j = 1

where i a is a matrix whose elements denote the length of the path-pixel intersections in the
pixel j along the ray path i , and n is the electron density for the pixel j . Each set of STEC
measurements along the ray paths from all observable satellites at consecutive epochs are
combined with the ray path geometry into a linear expression:

YAx ε = + (14)

where Y is a column of m measurements of STEC, x is a column of n electron density
unknowns for cells in the targeted ionosphere region, and A is an m × n normal matrix with
elements i a . The unknown electron densities x can be estimated by the ionospheric
tomographic reconstruction technique. Many tomography algorithms are used in different
ways, e.g. algebraic reconstruction technique (Gordon et al., 1970). One of the most common
approaches is the algebraic reconstruction technique (ART), which was first introduced in
Computerized Ionospheric Tomography (CIT) by Austen et al. (1986). This is an iterative
procedure for solving a linear equation. A modified version of ART is the so-called
multiplicative ART (MART), where the correction in each iteration is obtained by making a
multiplicative modification to x (Raymund et al., 1990; Tsai et al., 2002). The ART generally
produces estimates of the unknown parameters by minimization of the L2 norm, while the
MART follows maximum entropy criteria and thus underlies different statistics. In addition,