

Ionospheric Delay

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Title Ionospheric Delay
Level Advanced
Year of Publication 2011



A medium where the angular frequency ω and the wave number k are not proportional, is a dispersive media (i.e., the wave propagation speed and thence, the refractive index depends on the frequency). This is the case with the ionosphere where ω and k are related, in a first approximation, by [Crawford, 1968][1]:

\omega^2=c^2k^2+\omega_p^2 \qquad \mbox{(1)}

where c is the propagation speed of a signal in the vacuum and [Davies, 1989][2]

\omega_p=2\pi\, f_p \qquad\mbox{with}\qquad f_p=8.98\sqrt{N_e} \qquad\mbox{in Hz} \qquad\mbox{(2)}


The previous equation (1) is named the Relation of Dispersion of the ionosphere, and ωp is called the critical frequency of the ionospheric plasma, in the sense that signals with ω < ωp will be reflected and signals with ω > ωp will cross through the plasma [Davies, 1989] [4].

The electron density in the ionosphere changes with the height having a maximum of N_e\simeq 10^{11} - 10^{12} e^-/m^3 around 300-500\, km. Thence, according to the expression (2), electromagnetic signals with f>f_p \simeq 10^6Hz will be able to cross the ionosphere. This is the case of GNSS signals which frequencies are at the order of 1 GHz = 109 Hz. Radio signals with frequencies under fp will be reflected in the ionosphere.

From equation (1), and taking into account that ω = 2πf and the definition of Phase & Group Velocity

v_{ph}=\frac {\omega}{ k}~,\qquad v_{gr}=\frac {d\omega}{d k} \qquad\mbox{(3)}

it follows:

v_{ph}= \frac{c}{\sqrt{1-\left(\frac{f_p}{f} \right)^2}} \qquad\mbox{(4)}

Thence, according to (5),

n_{ph}=\frac c{v_{ph}} \qquad ~, \qquad n_{gr}=\frac c{v_{gr}} \qquad\mbox{(5)}


n_{ph}=\sqrt{1-\left(\frac{f_p}{f} \right)^2}\simeq 1-\frac{1}{2}\left(\frac{f_p}{f} \right)^2= 1-\frac{40.3}{f^2}N_e \qquad\mbox{(6)}


 n_{ph}=1-\frac{40.3}{f^2}N_e \qquad\mbox{(7)}

Differentiating the equation (1) with respect to k and taking into account (3), (5) and the approximation (1-\varepsilon^2)^{-1/2}=1+\frac{1}{2} \varepsilon^2, yields the group refractive index: [footnotes 2]

 n_{gr}=1+\frac{40.3}{f^2}N_e \qquad\mbox{(8)}


\Delta=\int_{_{\mbox{straight line}}}{(n-1)\,dl} \qquad\mbox{(9)}

the difference between the measured range (with the frequency f signal) and the Euclidean distance between the satellite and receiver is given by:

\Delta_{ph,f}^{iono}=-\displaystyle\frac{40.3}{f^2}\int{N_e\, dl} \qquad \qquad
\Delta_{gr,f}^{iono}=+\displaystyle\frac{40.3}{f^2}\int{N_e\, dl} \qquad\mbox{(10)}


The differences \Delta_{ph,f}^{iono} and \Delta_{gr,f}^{iono} are called the phase and code ionospheric refraction and, the integral is defined as the Slant TEC (STEC):

STEC=\int{N_e\, dl} \qquad\mbox{(11)}

Usually, the STEC is given in TEC units (TECUs), where 1\,TECU=10^{16}e^-/m^2 and the ionospheric delay If (at the frequency f) is written as:

I_f\equiv \Delta_{gr,f}^{iono}= \alpha_f\,STEC \qquad\mbox{(units: meters of delay)} \qquad\mbox{(12)}


\alpha_f= \frac{40.3\cdot 10^{16}}{f^2}\; m_{\mbox{signal delay}_{(\mbox{at frequency}f)}}/_{TECU} \qquad\mbox{(where f is in Hz)} \qquad\mbox{(13)}


Figure 1: Figure at left: Vertical Total Electron Content in units of 0.1 TECUs\simeq 1.6\,cm of delay in the GPS L1 signal at 19UT of the 26th of June of 2000. The plot at right shows the Solar flux evolution during the last solar cycles.
IONO Delay TECupci00178 19UTv01.png
IONO Delay Solar.png


Advanced comment: Second order ionospheric effect

The previous results of equation (12) are many often referred as the First Order Ionospheric effect, being denoted as I1_f\equiv I_f. That is:

I1_{ph,f}=-\alpha_f STEC=-I1_{gr,f} \qquad\mbox{(14)}

As commented before, the equation (1) is a simplification of the Relation of Dispersion of the ionosphere given in [McCarthy,D. and Petit,G., 2009][3], where the dependence of the refraction index with the magnetic field {\mathbf B} has been neglected. If such terms are taken into account, higher order ionospheric terms appears (but, they mean less than the 0.1\% of the total effect).

The Second Order Ionospheric term, its implementation and impact on geodetic estimates is analysed in detail in [Hernandez-Pajares et al., 2007][5].Its effect on carrier (I2ph,f) and code (I2gr,f) measurements at the frequency f is given by:

I2_{ph,f}=-\displaystyle \frac{7527\, c}{2 \,f_1^3}\int_{rec}^{sat} N_e B \cos{\theta}\, dl \qquad\mbox{(15)}

I2_{gr,f}=-2\;I2_{ph,f} \qquad\mbox{(16)}

where c is the speed of light. B is the module of the earth magnetic field {\mathbf B} and θ the angle between B and the propagation direction. The units are in the International System of Units (SI).

In global geodetic computations, I2 mainly affects to the satellite clocks estimate (cm level) and orbits (few mm), but the impact on receiver positions is usually smaller than 1 mm (static positioning).

For more information, see:


  1. ^ The approximation \sqrt{1-x^2}\simeq 1-\frac{1}{2}x^2 can be used when |x| \ll 1. Notice that taking f = 109 Hz as the frequency of the GNSS signals, and being f_p\simeq 10^7 Hz, thence, x=f_p/f\simeq 10^{-2}, which justifies the approximation.
  2. ^ Notice that ngr > nph and, thus, from (5) it follows that vgr < vph. Moreover, as nph < 1, thence v_{ph}=\frac{c}{n_{ph}}>c. That is, the phase travels faster tan the speed of light, but no information is carried, so no the relativity principle is violated.


  2. ^ a b [Davies, 1989] page 94: N_e=1.24\cdot 10^{10} f_p^2, with fp in megahertz and Ne in electrons per meter cubic.
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