GNSS Atmospheric and Ionospheric Sounding – Methods and Results - Global Navigation Satellite Systems: Signal,Theory and Applications - page 371

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:

λ a

⎛ ⎞

k j

y

⎜ ⎟

k + 1 k i

x = x . (15)

j j

⎜ ⎟

k

ax ,

i

⎝ ⎠

where yi is the ith observed STEC in a column of m measurements, xj is the jth resulted cell
electron density in a column of n unknowns, aij is the length of link i that lies in cell j,

λ is

the relaxation parameter at the kth iteration with 0 < < , and the inner product of the

vectors x and ai is the simulated STEC for the ith path. The electron density matrix x is
therefore corrected iteratively by the ratio of the measured STEC and the simulated STEC
with a relaxation parameter of λ until the residual does not change (see Figure 10). This
relaxation parameter value is chosen from experience in which the best λ value is identified
where the solution converges quickly with a reasonable number of iterations and the
residuals are a minimum. Here λ =0. 01 has been chosen for all iterations. In addition, it is
noted that any iterative algorithm requires an initial condition before the iteration begins.
Due to the poor STEC geometry, the initialization could be extremely important for the
tomographic reconstruction. In practice, the closer the initial condition is to the true electron
density distribution, the more accurate the reconstruction will be. Here the latest IRI-2007
model (http://nssdcftp.gsfc.nasa.gov/models/ionospheric/iri/iri2007) is used as an initial
guess for the reconstruction iteration.

1

Reconstruction solution

No
(iteration)

Yes

0 k +1

x y − A . x < ε

k

k +

x

x

λ a

⎛ ⎞

k ij

y

⎜
⎝

i
x

⎟
⎠

k

〈 . a 〉

i

Fig. 10. Flow chart of MART