how the dielectric constant (real and imaginary part) are derived from the S11 measurement?

Good afternoon,

I'm a PhD student and I'm writing a paper about the use of Keysight 85070D dielectric probe kit (an open-ended coaxial cable connected to a network analyzer) to contaminated soil.

I did not find, in the application notes, any information about how (mathematically) the real and the imaginary dielectric permittivity are calculated after the signal detected by the network analyzer. Inversion equations I mean, something like that.

In literature, some authors described these equation (e.g. Xu et al 1991), here their abstract:

"Several formulas for the calculation of the admittance of open-ended coaxial probes from the complex permittivity, E, of the measured medium and the dimensions of the probe are presented. The inverse problem, calculation of c from the measured admittance of the probe, is also studied in detail. These formulas are discussed and compared in terms of numerical calculations. New universal diagrams of solutions of these problems are presented which are both convenient and useful for practical measurements. Some distinguishing features of the relationship between E of the measured medium and the admittance of the probe are discussed".

I would need this information to complete my work. Someone knows where i can find it?

Thank you so much!
Sincerely

Andrea Vergnano