3 Ionospheric models

In order to decide between possible ionospheric models which might be used to fit the GPS TEC data, we compared the zenith distance variation of TEC and angular refraction as predicted by spherically-stratified models employing actual ionospheric profiles, parabolic layers, and a simple, uniform layer. We found that, with appropriate values for the ionospheric height and layer thickness, a simple, uniform-layer model for the ionosphere provides a prediction of the TEC and of ionospheric refraction that is as accurate as models involving more complicated profiles of the electron density with height for elevations greater than 15$^\circ$. See, for example, Thompson et al. (1986, p. 445) for a comparison of such models. A uniform model has thus been adopted. This conclusion differs from that of Spoelstra (1983) who found that the inclusion of vertical profiles of ionospheric density improved his estimates. This is probably because Spoelstra was forced to work with top- and bottom-side sounder observations in order to estimate the TEC while we are able to determine the TEC directly from the GPS observations.

However, horizontal gradients of the ionospheric density are extremely important for the prediction of refraction and TEC. Under typical conditions, the refraction caused by horizontal gradients exceeds that caused by the spherical (Earth curvature) component for source elevations above 10$^\circ$. This is illustrated by an example of the data from an individual satellite pass as shown in Fig. 1 where it is evident that the TEC in the south considerably exceeds that in the north at similar elevations.

Figure 1: An example of the raw data obtained in this project. These data were obtained by the center receiver from satellite PRN #6. The lower panel shows the elevation (solid line) and azimuth divided by 4 (dashed line). In this case the satellite was acquired in the northwest at an elevation of 30$^\circ$, passed nearly through the zenith and was lost in the south at an elevation of 10$^\circ$. The dots in the upper panel are the raw values of the tau-TEC in TEC Units (TU). The solid line is the phase-TEC which has been offset so that it agrees with the tau-TEC at high elevation. The dotted lines indicate the values that would have been obtained with an ionosphere having no horizontal gradient and vertical TEC values of 5, 10, 15, and 20 $\rm {TU}$. The dashed line in the upper panel represents the average tau-TEC after the satellite and receiver offsets have been applied to the data and the dash-dot-dash line is the tau-TEC as predicted by our model ionosphere (which was fitted to the data from all the visible satellites)

Such data show that a uniform spherical component alone cannot fit the TEC data. Therefore, we have adopted an ionospheric model which has a constant electron density over a thickness, d, centered at a height, h, and which has a constant horizontal gradient of electron density at an arbitrary azimuth. For simplicity we have used fixed values of 175 km and 400 km for d and h, respectively. This simplification does not adversely affect the accuracy of the model, as it is easily shown that the calculated parameters of the model are only weakly dependent upon the height and thickness of the refractive layer. In fact, the GPS data can be fit equally well with an infinitesimally thin layer. We retained a finite thickness layer because it is more physically plausible and also because we were guided in the choice of a model by angular refraction data. A very thin layer does not give a satisfactory fit to the available refraction data.

Therefore, data are fitted to a three parameter model: the vertical TEC at the location of the receiver, the magnitude of the horizontal gradient at this location and the azimuth of this gradient. When fitting the averaged tau-TEC data once every few minutes we find that this simple model usually fits the data to better than ten percent. The accuracy of the fit is demonstrated in the upper panel of Fig. 1.

At a wavelength of one meter, the ionospheric Faraday rotation is expected to be at most a few turns in the daytime and less than one turn at night. The rotation depends only on the magnitude of the ionospheric electron density and upon the angle between the source direction and the Earth's magnetic field. Since our model fits the TEC data well, and the Earth's magnetic field is known accurately, we should expect to be able to correct for the Faraday rotation to $< 10\hbox{$^\circ$ }$ at a wavelength of 1 m, and $< 1\hbox{$^\circ$ }$ at a wavelength of 20 cm. Travelling ionospheric disturbances will cause a smaller-scale spatial gradient in the observed plane of rotation of the polarized emission - this phenomenon is probably responsible for the residuals in the fits to our simple model. Under normal ionospheric conditions, these disturbances will differentially rotate the plane of polarization across the VLA by only a few degrees in the largest configuration at a wavelength of one meter, and thus are not expected to limit the applicability of our method.

On the other hand, ionospheric refractive effects and interferometer phase correction at meter wavelengths present far greater difficulties because these effects are very large. At a wavelength of one meter, ionospheric delays of hundreds of wavelengths are common in daytime. Also, the refractive effects depend more critically upon the horizontal gradient of the TEC than on the vertical TEC itself. With TEC data from only a limited number of satellites we cannot expect to model these gradients with great accuracy. The lines of sight to the satellites puncture the ionosphere at points separated by about 1000 km so we should expect to only model ionospheric structures on similar scale sizes; we cannot hope to model the smaller-scale, $\sim$100 km, structures. These smaller structures also cause important refractive effects and cannot be modeled unless the observed radio source and one of the satellites are nearly coincident in the sky and we sample the smaller scale structures in this area of the sky with receivers separated at appropriate spacings.