I am not sure I understand your question correctly, but I will give it a try.

First, a comments and a question on terminology:

- I believe that by fluvial power you mean stream power.

- Could it be that you mean sedimen transport rate and not necesarily erosion, since erosion is just the negative side of the sediment balance (sediment input minus sediment output).

The question arises because usually you would use the stream power to estimate a potential sediment transport rate, and thereafter you would balance inputs and outputs (hoy you do this is another issue) from your river reach to determine erosion, equilibrium or sedimentation.

How do you go from Stream Power to Sediment Transport Rate?

1. Estimate Actual Stream Power according to a given equation, e.g. the product of river discharge, slope and weight of water. Here the only challenge is to properly estimate the discharge.

2. Esimate the Critical Stream Power for some characteristic particle size on your channel or surface. There are a number of empirical relationships for this. Check the following for a brief discussion on it:

https://www.semanticscholar.org/paper/Stream-Power%3A-Origins%2C-Geomorphic-Applications%2C-and-Gartner/52236c6571b18f4c11bdd5436a47c4881baf66ee

3. Compare the Critical Stream Power with the Actual Stream Power. Either as a diference or a quocient, this would give you an index already, but not necesarily tell you if you have erosion, equilibrium or sedimentation going on.

4. Balance the sediment input and output of your river reach or surface to determine erosion, equilibrium or sedimentation. Usually you would do this through some kind of model, but I can well imagine the possibility of doing it analiticaly provided you make the proper assumptions. Be aware that the timeframe that you consider here, may be decisive on your end result. For example in a relatively small catchment a flood event lasting a few hours may cause erosion but if you consider a whole year, the stream may be sedimentating.

Finally, going back to your original question, this would recquire for you to go backwards on the approach presented (from 3. to 1. maybe), but this would require for you to have knowledge of the sediment balance at a given point in time, which is usually not the case. Geomorphologist take more or less this "backward approach" to estimate the size of some flows from sediment deposits.

Thank you so much Rafael Anleu for your detailed and comprehensive response. I appreciate your time.

For the Fluvial power, I mean stream power just like you rightly pointed out. I only wanted to determine the stream power per unit length along the main and few other major tributaries within the catchment. I do not necessarily want to determine erosion rate. I just wanted to use the stream power as a proxy for possible erosion variability along the rivers without going into detail sediment flux balance. You have provided very useful suggestions. Thank you for your time.

https://www.leg.state.mn.us/docs/2016/mandated/160722.pdf

I have not used this index, but this manual has portions addressing the index. The Index May serve as a comparative tool. The channel forming flow is bankfull, which for most stream types is about the 1.5 year storm. Since gullies have no active floodplain, there are added stresses. Stresses accumulate on the outside of meander bends, so in some instances installing groins, vanes, J hooks can move that stress toward the Thalweg and reduce bank stress. The inside of meander bends (point bars) release a portion of materials transported. If the size of substrate exceeds streams power to move it or bedrock armoring, the erosion rate is limited. And some materials are extremely erosive, and easily moved with low stream power.