Ionospheric Propagation Effects on GNSS Signals and New Correction Approaches - Global Navigation Satellite Systems: Signal,Theory and Applications - page 385

ray paths through the ionosphere on their way to a receiver and thus the TEC along a f 1 path
will be different from that along a f 2 path and also from that along the LoS path. Considering
this, TEC in Eq. (7) is separated into TEC LoS and Δ TEC bend where TEC LoS is the TEC along the
LoS and Δ TEC bend is the difference between TECs along a curved path and the LoS path. The
term Δ TEC bend represents TEC contribution due to ray path bending only, i.e., the second
and third order terms are not considered in TEC estimation by Eq. (7).

2.3 Ionospheric effects on GNSS observables
The observables are travel time or ranges which are deduced from measured time or phase
differences based on a comparison between received signals and receiver generated signals.
Thus, the ranges are biased by satellite and receiver clock errors, instrumental biases and
atmospheric effects, and therefore, called pseudoranges. The code pseudorange (Ψ) and
carrier-phase pseudorange (Φ) at a selected frequency can be described by observation
equations in units of length as

( )

( )

Ψ ρ =+ − + + + c dt dT d d d + + + dq dQ ε (10)

Igr A MP Ψ

Ψ

( ) ( )

Φ ρ =+ − − + + c dt dT d d d + + + + dq dQ N λ ε (11)

I A P Φ

Φ

where ρ is the geometric distance between a satellite and a receiver, c is the velocity of light,
dt and dT are the satellite and receiver clock errors, respectively, d I and d Igr are the
ionospheric effects on carrier-phase and code pseudoranges, respectively, d A is the
atmospheric (tropospheric delay) effect, ( d MP ) Ψ and ( d MP ) Φ are multipath effects on code and
carrier-phase pseudoranges, respectively, dq and dQ are the instrumental biases of the
satellite and the receiver, respectively, λ is the carrier wavelength, N is the integer carrier-
phase ambiguity, and ε Ψ and ε Φ are the rest errors. The carrier-phase pseudorange is
expressed in units of length (meters) instead of cycles. However, it can be expressed in
cycles dividing simply by the signal's wave length ( λ meter/cycle).

For simplicity we confine our interest to only ionospheric effects. Thus, the code and carrier-
phase pseudoranges can be simplified as

p q u

len len

=+ + =++++

(12)

Ψρ dd ρ d

Igr I I

2 3 4

f f f

p q u

=− + =− − − +

len len

(13)

Φρ dd ρ d

I I I

2 3 4

f 2 f 3 f

where f is the signal frequency. In case of GPS L1, L2 and L5 signals f = 1575.42, 1227.6 and
1176.45 MHz, respectively. To take into account the ray path bending on observables, the
term

len
I

d is introduced in Eqs. (12) and (13).

2.4 Multi-frequency combinations
2.4.1 First order ionosphere-free combination

As already mentioned, ionosphere is a dispersive medium, i.e., the ionospheric propagation
delay is frequency dependent. Therefore, one very popular way to get rid of ionospheric