Here are some helpful publications for your purposes:

- Attached you find my paper on determining GPS differential receiver biases. It goes into detail about the method through which TEC is calculated from GPS observables, as well as the various challenges that we are faced in trying to get reliable TEC from GPS observables.

- The following goes into more detail about the higher order ionospheric effects on GPS signals and describes the Taylor expansion of the Appleton-Hartree equations that was undertaken for GPS use (we only typically use the first order term in TEC calculations)http://www.sciencedirect.com/science/article/pii/S0273117710001183

Hope this helps.

http://www.sciencedirect.com/science/article/pii/S0273117710001183

Article Determining Receiver Biases in GPS-derived Total Electron Co...

Dear Adnan,

I would strongly suggest you to take a look at Chapter 9 of the IERS Conventions (2010), available at:

http://www.iers.org/SharedDocs/Publikationen/EN/IERS/Publications/tn/TechnNote36/tn36_132.pdf;jsessionid=D2372AF71AB1030C0EA33EBC1D3369BA.live2?__blob=publicationFile&v=1

There, you can see how the formulation of the index of refraction by Applenton-Hartree is expanded into a second-order Taylor approximation under one assumption that applies for GNSS signals (basically, that the signal frequency is bigger than the plasma frequency associated to the free electrons of the ionosphere). From there, the ionospheric dependent terms in the carrier phase are derived. The first order approximation would correspond to the TEC contribution, and the rest of expressions should be accounted for when working with high precision navigation and/or working on temporal series.

A full consideration of all this terms is carried out in the paper:

Hernández-Pajares, M., À. Aragón-Ángel, P. Defraigne, N. Bergeot, R. Prieto-Cerdeira, and A. García-Rigo (2014) Distribution and mitigation of higher-order ionospheric effects on precise GNSS processing, J. Geophys. Res. Solid Earth, 119, 3823–3837, doi:10.1002/2013JB010568

You can also check the fourth chapter of:

Budden, K. G. (1985). The propagation of radio waves: the theory of radio waves of low power in the ionosphere and magnetosphere. Cambridge University Press.

There, you have a step by step construction of the Appleton-Hartree formula that could give you a more profound understanding of the refraction index.