GPR: Hyperbola or Parabola?

When I was first introduced to GPR, the person in charge of the academic course said that in GPR data we have hyperbolic signatures “as you know from reflection seismic, so we do not go into it”. But that was a poor explanation, especially considering that reflection seismic was done after and at very basic level.

I did my little research at the time and found the explanation: the arch shape or inverted U, as I call it when I talk to people not in the sector, is simply caused by the fact that an antenna irradiates the material with a conic pulse. It means that an object can be detected not only when directly below the antenna but also before and after.

But what is that “arch” that we get as response?

If we have a closer look at what GPR does when we scan, we see that the wave paths for two different antenna positions can be mathematically linked using Pythagoras’ Theorem as per the sketch below.

A is the distance run by the antenna, we can call it x and it represents the different positions of the antenna on the surface.

B is the path of the wave run in a time t0 at velocity v, when the antenna is directly above the target.

C is the path of the wave run in a time t at velocity v, when the antenna is not directly above the target.

v and to are constant values so the only variables are t and x.

What does the final equation tell us? Let’s have a look at some high school math about conic sections (or simply conics).

If we consider a double cone and we slice it with planes at different angles, we get three different curves called ellipses, parabolas and hyperbolas (a circle is special type of ellipse). The simplified equation for each of them is also reported.

The equations we got from GPR is the hyperbola one: it is easy to recognise because it is the only one with the minus sign. That’s why GPR response has hyperbolic signatures.

So what does that equation we derived from GPR tell us? That is a hyperbola and that the velocity and time (depth) are the parameters controlling the hyperbola shape (more or less wide, more or less pronounced): that’s why high velocities (lower dielectric media) produce wider hyperbolas as well as later (deeper) targets. That’s why softwares use hyperbola fit to derive the velocity of the signal.

All these considerations would not work with a parabola.

I found the parabolic terminology to be present in official reports, in PhD thesis and peer reviewed articles. Long time ago I dealt with a person calling the reflections parabolas: when I heard that for the first time, I thought I didn't understand anything about GPR, but then I asked the reason of that term and the answer was “I don’t have academic education and clients are more familiar with parabolas”. You don’t need to have academic education to read a manual and people not in the sector shouldn’t be treated as ignorant. The word "parabola" is more familiar for them? Maybe because of TV antennas…I wonder why they are called parabolas.