Geometric spreading of an impulse antenna
Bob Jacobsen is correct -- the most likely problem is that there is not enough distance in your setup.

Here is an overview of near and far field in radar. Figure 1 in that article provides some nice intuition on the near field and the far field:


In the far field, we can see that the wave fronts all align with each other in the center line of the antenna pattern, so the signal at this centerline behaves the same as a signal we'd get from a point source. In this region spherical spreading loss applies and power will decrease as \$1 \over r^2\$ (amplitude will decrease as \$1 \over r\$).
In the near field (very close to the antenna) we can see that things are quite a mess. To my (limited) knowledge there is no useful, simple rule about power-versus-distance in this case.


From the above webpage, we see some approximations for where the near field and far field begin. You mention frequencies of 1GHz and 2GHz, which means wavelengths of roughly 0.3m and 0.15m. I'm guessing that your antennas shown in your picture have a dimension of roughly that size (i.e. your antenna aperture is a single wavelength or less, rather than multiple wavelengths large). So, from that page, the far field begins when \$r \gg \lambda\$.

In your setup you state that you are about 0.5m above the ground, which I would say doesn't qualify as "much greater than" your wavelength here. (Your experiment was performed with a distance of 1.6 and then 3.2 wavelengths. It is interesting to note that the 3.2 wavelengths case ended up closer to the expected answer.)

If possible, I would recommend repeating your experiment with maybe 5-10 wavelengths of distance. The tough part here is that you'd need that 5-10 wavelengths of distance in all directions -- I doubt the antenna pattern is very focused so you need to be sure that you're not getting coupling through side lobes that could be corrupting your measurement results.