Intensity Range Correction - How to choose a power value?

Question

I'm using lasrangecorrection from the lidR package to correct my Intensity values for effects of distance from sensor to target. I'm wondering how to best choose the f parameter (exponent for ratio of range to standard range). A power of 2.3 is recommended for vegetation, however this seems to flip the trend - instead of Intensity decaying as range increases, Intensity now increases as range increases (see images below). Currently, I'm using a trial and error framework to choose the power that will flatten the relationship. Is there a more efficient way to do this?

Relationship between Uncorrected Intensity & Range:

Relationship between Corrected Intensity & Range (power = 2.3):

Relationship between Corrected Intensity & Range (power = 1.6):

EDIT After the comment from @JRR, and examining an uncorrected intensity raster, it seems correction is not necessary.

Uncorrected Intensity Map With Estimated Sensor Positions Overlayed (mean intensity at 3m resolution):

Digital Elevation Model for Area of Uncorrected Intensity Image:

Canopy Height Model for Area of Uncorrected Intensity Image:

Answer 1

You must understand that lidR was designed mainly as a laboratory to conduct some research that were impossible or at least very hard to conduct with regular softwares. Your question typically falls in this scope. I think no one really knows. This type of correction has been largely discussed but little used because of the usual absence or sensor track data.

In your case I think (but it is hard to be sure with this scatter plot) that your intensities were already normalized by the data provider. There is probably little need of correction (it is easier to see on a map).

In a study I conducted (not published yet) I used the overlap to estimate the best f by searching the one that minimized the differences in overlaps. You could also refer to the reference mentioned in the documentation. They made pretty similar stuff.

Anyway, for almost every question like what is the best parameter of x my answer will almost always be: make a study and if you find a good answer publish your results. :-)