Mitigation of higher order ionospheric effects on GNSS users in Europe

Abstract

Current dual-frequency GPS measurements can only eliminate the first-order ionospheric term and may cause a higher-order range bias of several centimeters. This research investigates the second-order ionospheric effect for GNSS users in Europe. In comparison to previous studies, the electron density profiles of the ionosphere/plasmasphere are modeled as the sum of three Chapman layers describing electron densities of the ionospheric F₂, F₁ and E layers and a superposed exponential decay function describing the plasmasphere. The International Geomagnetic Reference Field model is used to calculate the geomagnetic field vectors at numerous points along the incoming ray paths. Based on extended simulation studies, we derive a correction formula to compute the average value of the longitudinal component of the earth’s magnetic field along the line-of-sight as a function of geographic latitude and longitude, and geometrical parameters such as elevation and azimuth angles. Using our correction formula in conjunction with the total electron content (TEC) along the line-of-sight, the second-order ionospheric term can be corrected to the millimeter level for a vertical TEC level of 10¹⁸ electrons/m².

Appendix A

The integration of the Chapman function Eq. (8) yields the following sequence:

$$ {\int\limits_{\rm R}^{\rm S} {n^{\rm I}_{\rm e} (h)}} = N_{0} {\int\limits_{\rm R}^{\rm S} {\exp {\left(\frac{1}{2}(1 - z - \sec \chi \exp ( - z))\right)}}}{\rm d}h $$

(19)

where z = (h − h ₀)/H; applying differentiation and substitution

$$ \begin{aligned} \,&{\rm d}h = H{\rm d}z; {\rm R}^{\prime} = ({\rm R} - h_{0})/H; {\rm S}^{\prime} = ({\rm S} - h_{0})/H;\\ \,&{\int\limits_{\rm R}^{\rm S} {n^{\rm I}_{\rm e} (h)}} = \exp (0.5)HN_{0} {\int\limits_{{\rm R}^{\prime}}^{{\rm S}^{\prime}} {\exp ( - z/2)\exp ( - \sec \chi \exp ( - z)/2)}}{\rm d}z \end{aligned} $$

(20)

let x = exp (− z/2) and applying differentiation and substitution

$${\rm d}z = - (2/x){\rm d}x; {\rm R}^{\prime\prime} = \exp ( - \frac{{{\rm R} - h_{0}}}{{2H}}); {\rm S}^{\prime\prime} = \exp ( - \frac{{{\rm S} - h_{0}}}{{2H}}); $$

$$ \begin{aligned} {\int\limits_{\rm R}^{\rm S} {n^{\rm I}_{\rm e} (h)}} &= - 2\exp (0.5)HN_{0} {\int\limits_{{R}^{\prime\prime}}^{{S}^{\prime\prime}} {\exp ( - \sec \chi \cdot x^{2} /2)}}{\rm d}x \\ & \approx - 2\exp (0.5)HN_{0} \cdot \left. \frac{{\sqrt \pi} \cdot {\rm erfi}(x{\sqrt {- \sec \chi /2})}}{{2{\sqrt {- \sec \chi /2}}}} \right|^{{{\rm S}^{\prime\prime}}}_{{{\rm R}^{\prime\prime}}} \\ & \approx \exp (0.5){\sqrt {2\pi}}{\sqrt {\cos \chi}}HN_{0} \cdot \left. {{\rm erf}( - x{\sqrt {\sec \chi /2}})} \right|^{{{\rm S}^{\prime\prime}}}_{{{\rm R}^{\prime\prime}}} \\ & \approx 4.13HN_{0} {\sqrt {\cos \chi}} \cdot {\left[ {{\rm erf}( - {\rm S}^{\prime\prime}{\sqrt {\sec \chi /2}}) - {\rm erf}( - {\rm R}^{\prime\prime}{\sqrt {\sec \chi /2}})} \right]} \\ \end{aligned} $$

(21)

where erfi is the imaginary error function and erf is the error function. For GPS/GLONASS orbit height (∼20,200 km) or Galileo orbit height (∼23,200 km), and E, F₁ and F₂ layer scale heights and peak layer heights given in section 3, we estimate the error function term ${\rm erf}( - {\rm S}^{\prime\prime}{\sqrt {\sec \chi /2}}) - {\rm erf}( - {\rm R}^{\prime\prime}{\sqrt {\sec \chi /2}})$ in Eq. (21) to be ∼1 and hence Eq. (21) can be simplified as

$$ {\int\limits_{\rm R}^{\rm S} {n^{\rm I}_{\rm e} (h)}} \approx 4.13HN_{0} {\sqrt {\cos \chi}} $$

(22)