Ground Penetrating Radar formula

[PeterPeter]

TL;DR

Derivation of reflection coefficient?

I am interested in the physics of ground penetrating radar.
1) Does anyone know where I can find a derivation of the formula in the attached jpg for the energy reflected? K=the dialectic constant.
2) An intuitive explanation of why the dialectic constant is important in determining the energy reflected. Obviously it must have something to do with the motion of the free elections.
Thanks in advance
Jerry

Source of jpg:

Interpreting GPR Data: The Basics Part 1 by Greg Johnston

 

[jedishrfu]

Perhaps this:

https://wiki.seg.org/wiki/Reflection_coefficient

 

[tech99]

In this formula, notice that the SQRT sign could be placed ahead of the whole expression. Then we can see that K1-K2 is the portion of incident power reflected and K1+K2 is the total incident power, so the formula is trivial and not very enlightening. Reflection Coefficient is specified in terms of the fields (or voltages) rather than the power, so we take a square root.
When a wave is traveling in free space, the electric and magnetic fields have a fixed ratio of 377. When the wave enters another medium, such as moist soil, the main things that happen are that the permittivity and conductance increase. Permittivity is associated with capacitance. This causes the electric field to be weaker than in free space, and the ratio of electric to magnetic fields is altered. This is equivalent to the case of a transmission line where there is a change in characteristic impedance. The wave entering the ground cannot suddenly adjust to the new impedance, and part of it is reflected back.
Water has a very high permittivity which is caused by the asymmetrical shape of the molecule, making it polar.

 

[PeterPeter]

In this formula, notice that the SQRT sign could be placed ahead of the whole expression. Then we can see that K1-K2 is the portion of incident power reflected and K1+K2 is the total incident power, so the formula is trivial and not very enlightening. Reflection Coefficient is specified in terms of the fields (or voltages) rather than the power, so we take a square root...

Are you saying that the energy is proportional to the SQRT of the dialectic constant, K?