Ionospheric Propagation Effects on GNSS Signals and New Correction Approaches - Global Navigation Satellite Systems: Signal,Theory and Applications - page 398

simplified as 0.6577 N m TEC . Thus, the third order residual term can be approximated by the
first term only as (using Eqs. 9 and 19)

534.27

Δ = s N TEC (38)

3 m

22
12

ff

The third order term Δ s 3 will be measured in meters when f is measured in Hz and the
maximum ionization N m and TEC in m -3 and electrons/m 2 , respectively.

4.3 New approaches for correcting LoS propagation assumption errors
4.3.1 Excess path length correction

As we have seen in the section 3.2, the excess path length d I len can be computed by Eq. (33).
There is another formula published by Hoque & Jakowski (2008) for the excess path length
computation.

( )

− 5 2

7.5 × 10 exp − 2.13 β TEC

len
I

d =

(39)

( )

1/8

4

fHhm

where d I len is measured in meters, TEC is in TEC units, frequency f in GHz, atmospheric
scale height H and maximum ionization height hm in kilometers and elevation β in radians.
Comparing both formulas we see that Eq. (33) requires only TEC and elevation information
as inputs whereas Eq. (39) additionally requires ionospheric parameters H and hm .
However, these parameters are not easy to estimate in practical cases.

Both the correction formulas are derived based on simulation studies using Chapman
profiles for the ionosphere. The Chapman profile (Rishbeth & Garriott, 1969) has been
proved very useful for modeling ionospheric correction. It describes the electron density
distribution n e as a function of height h in the ionosphere as

e nh

() = Nm exp(0.5(1 − − z exp( ))) − z (40)

where Nm is the maximum ionization and z = ( h - hm )/ H in which hm is the height of
maximum ionization and H is the atmospheric scale height.

We have found that the correction by Eq. (33) shows the best performance for the
atmospheric scale height H = 70 km. However, when the scale height is too low (e.g., H = 60
km) or too large (e.g., H = 80 km), its performance degrades especially at low elevation
angles (see Fig. 9). Our present investigation shows that its performance can be improved by
taking into account the d I len dependency on the rate of change of TEC with respect to the
elevation angle. In order to find their dependencies, the excess path length has been
computed by the ray tracing program considering Chapman profiles with different H = 60
and 80 km. The signal frequency f = 1227.6 MHz, parameters hm = 350 km and Nm =
4.96×10 12 m -3 are kept constant in each case. The total electron content in the vertical
direction is found 123 and 164 TEC units, respectively. The obtained d len , TEC, and the first
and second order TEC derivatives with respect to the elevation angle dTEC / dβ and