It is certain that the intensity of rainfall available to you that it calculated for a period of time may be the day or year or hour, in the sense that there is a head of rain water collected by the rain apparatus in a certain time. mm/day, ....... so on..
Search to find the detailed rainfall collections if you want more detailed rainfall intensity data. If all available is monthly average in mm, take the monthly average and divide by number of hours in that month. So if you had 300 mm in April, take 360 mm/(30 days * 24 hours) = 0.5 mm/hour average monthly intensity. If you knew it rained on only 10 days, the average rate on those days would be 1.5 mm/hour. The value of recording rain gauges is that detailed rainfall intensity rates can often be calculated. If rainfall in tipping bucket standard gauge was recorded every 15 minutes, and the highest 15 minute value was 12 mm, the highest 15 minute intensity has a rate of 48 mm/hour.
Thank you so much for this answer Prof. How do I make reference to this? Better still, how scientific is this suggestion since it seem to give me the results I was expecting?
Rainfall rates= rainfall volume in the period that rain occur. No need to have a reference, Of course, if you want to talk "generally" on the weather or precipitation of your study area. If you are going to use rainfall rates in details (mm/min, mm/h,..) and apply it in some formula to understand other variables in environment, then you may want to measure it with more precision.
Here, for monthly rain averages, you may don't know of daily rain events sizes. For example, rainfall rates for a monthly rain average of 50 mm per 5 rainy days is different from this average for 10 rainy days. So, you can not easily speak about rainfall rate without knowing exactly the time/period of rainfall.
If you need a reference, I suppose you might find a reference in standard hydrology or meteorology methods books or handbooks. But I agree a reference is probably not needed. But you need to clarify the basis of the calculations. Rainfall intensity rates can be for many time periods, such as 15 minutes, 30 minutes, 1 hour, 2, 3, 6, 12, 24 hour, 2, 3, 6 days, etc. and as taken from what time period, such as 2005 -2018 for a specific location (latitude, longitude or other identifier).
I need to reclarify, if rainfall was only measured monthly in rain gauge, then only monthly, quarterly, yearly intensities are probably reportable. Generally, these intensity rates based on monthly data probably are not going to mean much for erosion, infiltration, runoff or other calculations. If you had a monthly station record at one site, but an array of recording raingauge sites in adjacent areas or vicinity, you might be able to better estimate intensities at site of interest through comparison and extrapolation.
Numerical differentiation of daily or 30 minutes accumulated rainfall data will help according to me, to get to rainfall rates. I operate a meteo station (Antwerp, Belgium) in the Wunderground network which measures accumulated and rainfall rate. At the moment a severe drought period is setting in, just like last year (2018).
RF is generally recorded as monthly average in mm, take the monthly average and divide by number of hours in rainy days in particular month. So if you had 600 mm monthly average RF in May with 20 rainy days and 11 dry days, take 600 mm/(20 days * 24 hours) = 0.8 mm/hour average monthly RF intensity.
If you have mm of rainfall recorded every day or any time steps, then your rate of rainfall is mm/day or mm/(recording interval). If its daily there are several techniques to downscale it to hourly. Please find the attached paper that presents one of the methods to generate hourly precipitation from daily.https://andrewsforest.oregonstate.edu/sites/default/files/lter/pubs/pdf/pub3636.pdf
I think it might be hard to transfer the rainfall amount (mm) into rainfall intensity (mm/h) if no other temporal recording was available. The traditional method was used the standard rain gauge usually having diameter of 20 cm. The rainfall amount could be read directly, or calculated with the precipitation volume (mL or L) by dividing the orifice area (cm2). If record the starting and ending time of rain events, the rainfall duration could be easily gotten. Then, rainfall intensity could be computed with rainfall amount (mm) and duration (h). Currently, automatic recoding is commonly applied by using the tipping bucket rain gauges with different resolution of 0.1 mm, 0.2 mm, 0.5 mm, etc. Rainfall amount and duration could be directly read by connecting it to the computer. It provides a convenient practice for recording rainfall characteristics. However, it has systematic errors for miss the recording of inflow between the tips. The larger for the tip volume, the relatively bigger error of the recording. Therefore, it needs calibration. Dr. Iida et al. (2012) have published a research paper for calibrating the tipping bucket rain gauges at Hydrological Processes (doi: 10.1002/hyp.9462). Hope it helps for your question.