Correction to: “Methodology and consistency of slant and vertical assessments for ionospheric electron content models” and to “Consistency of seven different GNSS global ionospheric mapping techniques during one solar cycle”

1 Correction to: J Geod https://doi.org/10.1007/s00190-017-1032-z and https://doi.org/10.1007/s00190-017-1088-9

In the original publication of the articles, “Methodology and consistency of slant and vertical assessments for ionospheric electron content models” and “Consistency of seven different GNSS global ionospheric mapping techniques during one solar cycle”, a common typo affecting the text only (not the computations) has been recently noticed. It compromised the definition of the scaling factor from Global Navigation Satellite Systems ionospheric delay to electron content, which is clarified in this erratum.

In equation 1 of Hernández-Pajares et al. (2017) and in equation 8 of Roma-Dollase et al. (2018), the observed Slant Total Electron Content difference $\varDelta S_o$, among two line-of-sights GNSS transmitter-receiver at different times t and elevations above the horizon E, is related with the corresponding difference of geometry-free (ionospheric) combination of carrier phases in length units, $L_I=L_1-L_2$, by means of the inverse of the scaling factor $\alpha$, i.e. equation 1.

$$\begin{aligned} \varDelta S_o = S_o(t) - S_o(t_{E_{\mathrm{max}}}) =\frac{1}{\alpha }\left( L_I(t) - L_I(t_{E_{\mathrm{max}}}) \right) \end{aligned}$$

(1)

However there is a typo in the detailed dependence and value of $\alpha$ given in both papers Hernández-Pajares et al. (2017) and Roma-Dollase et al. (2017), mainly consisting in the missing frequency dependent factor. The right and complete expression of $\alpha$, which has been the one used in the computations of both papers (Hernández-Pajares et al. 2017; Roma-Dollase et al. 2018), is given in next equation 2,

$$\begin{aligned} \alpha =\frac{q^2}{8\pi ^2m_{\mathrm{e}}\epsilon _0}\left( \frac{1}{f_2^2}-\frac{1}{f_1^2}\right) \simeq 0.105~\frac{{\mathrm{m}}}{\mathrm{TECU}} \end{aligned}$$

(2)

being q and $m_{\mathrm{e}}$ the charge and mass of the electron, respectively, $\epsilon _0$ the dielectric constant in the vacuum, $f_1$ and $f_2$ the carrier frequencies corresponding to $L_1$ and $L_2$; and the right hand approximation in equation 2 is valid for Global Positioning System frequencies, being 1 TECU = ${10}^{16}~{\mathrm{m}}^{-2}$ (see Hernández-Pajares et al. (2010, 2011)).