Ground GNSS Ionosphere Sounding - GNSS Remote Sensing: Theory, Methods and Applications - page 82

where N e (,', h ) is the ionospheric electron density, , ' and h are the longitude,
latitude and height, respectively. To obtain N e , 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. ( 4.27 ) along the ray path i can be approximately written as a finite
sum over the pixels j as follows:

X

M

STEC i D a ij n j (4.28)

j D 1

where a ij is a matrix whose elements denote the length of the path-pixel intersections
in the pixel j along the ray path i , and n j 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:

Y D Ax C " (4.29)

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 a ij . The unknown electron densities x can be estimated by the
ionospheric tomographic reconstruction technique. Many tomography algorithms
are used in different ways, e.g. singular value decomposition (Wall et al. 2001 ),
correlation function (Ruffini et al. 1998 ), and algebraic reconstruction technique
(Gordon et al. 1970 ). One of the most common approaches is the algebraic recon-
struction technique (ART), which was first introduced in Computerized Ionospheric
Tomography (CIT) by Austen et al. ( 1988 ). 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 ;Tsaietal. 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, the MART performs a multiplicative modification
in each iteration, and thus the inversion results are always positive. Therefore,
MART has the advantage over ART in determining the electron densities that avoid
unreasonable negative values and is the one used in this study. Basically, the MART
algorithm is iterated cyclically:

D x j :

k a ij

y i
h a i ;x k

x kC1
j

(4.30)

i

where y i is the i th observed STEC in a column of m measurements, x j is the j th
resulted cell electron density in a column of n unknowns, a ij is the length of link i
that lies in cell j , k is the relaxation parameter at the k th iteration with 0 < k < 1,
and the inner product of the vectors x and a i is the simulated STEC for the i th path.