Although I don't have the ambition to fully answer the question (I'm not an expert in either acoustics or fluid mechanics), I'll try and share my point of view of things with everyone. Hopefully, someone will catch on and be able to get somewhere from there.

You suggest determining three main parameters – height, volume flow and distance – by analysing the recording of the sound. First of all, you will have some serious difficulty finding the distance. Although, in a perfect scenario, the sound intensity should follow the inverse-square law with the distance, it will be largely affected by the terrain in practice. Consider standing hidden behind a large rock or in a direct line of sight of the waterfall, with both distances being equal. In the first case the sound will definitely be noticeably weaker. The sound will also probably differ when you stand in a marrow valley or on an open plane.

So let's skip the distance. Suppose we have a fine spot close to several waterfalls, each similar and in similar conditions. Now, we are just interested in the height and flow of the waterfall. I agree with you that the sound energy will be given by the gravitational energy of the falling water – but for some friction, possible change in kinetic energy of the water and loss due to turbulence of the flow. Now, the first thing I am not sure about is, how are these losses affected by the height or the flow of the waterfall? Will the losses change linearly with these parameters or is the relation more complex? And are there other factors that can play role?

Second important question is, how is the sound of the waterfall affected by its height and flow? Does one of them have a stronger impact on the spectrum than the other one? Or does a waterfall with a height $h$ and flow $Q$ have the same sound as a waterfall with height $2h$ and flow $Q/2$? In my ignorance, it is as well possible that the spectrum will not be affected at all and they will sound the same, just with different intensity.

To my (lack of) knowledge, these questions cannot answered by some simple theoretical predictions and the only possible way to find a solution might be to directly measure the sound of several waterfalls and compare it. But as I said, I'm not skilled in this field and might as well be completely wrong.

Finally, as far as practical realization is concerned, it will be very difficult to measure sound of different waterfalls in the same conditions. The sound will be strongly affected by reflection (if you have rocks all around, the reflection will be stronger than in the middle of a forrest), or other noise sources (such as the water flow below or above the waterfall). But one moght be able to eliminate these in post-processing if one knows the conditions in which the recording was taken.

Before I finish, I remind once again that I am not an expert in the fields of acoustics or field dynamics and it is possible that my argumentation is wrong. But I believe that this post can start a discussion that will lead to a conclusive result.

Although I don't have the ambition to fully answer the question (I'm not an expert in either acoustics or fluid mechanics), I'll try and share my point of view of things with everyone. Hopefully, someone will catch on and be able to get somewhere from there.

You suggest determining three main parameters – height, volume flow and distance – by analysing the recording of the sound. First of all, you will have some serious difficulty finding the distance. Although, in a perfect scenario, the sound intensity should follow the inverse-square law with the distance, it will be largely affected by the terrain in practice. Consider standing hidden behind a large rock or in a direct line of sight of the waterfall, with both distances being equal. In the first case the sound will definitely be noticeably weaker. The sound will also probably differ when you stand in a marrow valley or on an open plane.

So let's skip the distance. Suppose we have a fine spot close to several waterfalls, each similar and in similar conditions. Now, we are just interested in the height and flow of the waterfall. I agree with you that the sound energy will be given by the gravitational energy of the falling water – but for some friction, possible change in kinetic energy of the water and loss due to turbulence of the flow. Now, the first thing I am not sure about is, how are these losses affected by the height or the flow of the waterfall? Will the losses change linearly with these parameters or is the relation more complex? And are there other factors that can play role?

Second important question is, how is the sound of the waterfall affected by its height and flow? Does one of them have a stronger impact on the spectrum than the other one? Or does a waterfall with a height $h$ and flow $Q$ have the same sound as a waterfall with height $2h$ and flow $Q/2$? In my ignorance, it is as well possible that the spectrum will not be affected at all and they will sound the same, just with different intensity.

To my (lack of) knowledge, these questions cannot answered by some simple theoretical predictions and the only possible way to find a solution might be to directly measure the sound of several waterfalls and compare it. But as I said, I'm not skilled in this field and might as well be completely wrong.

Finally, as far as practical realization is concerned, it will be very difficult to measure sound of different waterfalls in the same conditions. The sound will be strongly affected by reflection (if you have rocks all around, the reflection will be stronger than in the middle of a forrest), or other noise sources (such as the water flow below or above the waterfall). But one moght be able to eliminate these in post-processing if one knows the conditions in which the recording was taken.

Before I finish, I remind once again that I am not an expert in the fields of acoustics or field dynamics and it is possible that my argumentation is wrong. But I believe that this post can start a discussion that will lead to a conclusive result.

Edit: As was pointed out to me by BrianWa in the comments, the height of the waterfall will affect the speed of the droplets that fall down and this will certainly affect the spectrum of the sound. However, I believe that this change will be observable only until such a height that the air resistance during the fall compensates the gravity and the droplet does not increase the speed anymore. All waterfalls higher than this threshold will thus probably have the same (or very similar) sound spectrum.