# How can I downscale daily values of relative humidity?

I have a dataset (The ISIMIP interpolation of CMIP5 output) that contains daily values of the following variables:

 air_temperature (C)
air_temperature_max (C)
air_temperature_mmin (C)
relative_humidity (%)


There are other variables (wind, downwelling long/short wave radiation, precipitation) that I suspect would not be useful for a first order approximation)

What would be an appropriate way to downscale from daily to hourly values?

I have an algorithm to do this when daily $rh_{max}$ and $rh_{min}$ are given. This assumes a peak $rh$ at 10AM and uses cosine to cycle between max and min

$$rh_{scale} \leftarrow \dfrac{1}{2}(\cos(2 * \pi * (0:23 - 2) / 24) + 1)$$ $$RH \leftarrow rh_{min} + rh_{scale} * (rh_{max} - rh_{min})$$

Similarly, I get hourly temperature $t$ thus:

$$t = t_{min} + \dfrac{1}{2}(\sin(2*pi*(0:23 - 10) / 24) + 1) * (t_{max} - t_{min})$$

However, it is not clear how I can do this when mean RH is given. I know this is difficult, and there are likely many ways of bringing in other data sets, but I would like a simple first-order approximation that is better than assuming that RH is constant within each day.

• What do you want to use the hourly values for? You may wish to check out this publication: Kottayil, A., V. O. John, and S. A. Buehler (2013), Correcting diurnal cycle aliasing in satellite microwave humidity sounder measurements, J. Geophys. Res., 118(1), 101–113, doi:10.1029/2012JD018545. sat.ltu.se/members/ajil/publications/diurnal_cycle.pdf . I can expand this into an answer if I find the time. – gerrit Jul 9 '14 at 19:27
• Although that's for upper tropospheric humidity, where the diurnal cycle is different from near the surface. Where are you in the atmospheric column? – gerrit Jul 9 '14 at 19:54
• @gerrit surface ... I am working on simulating the terrestrial biosphere. – David LeBauer Jul 9 '14 at 20:16

Your approach to calculate an hourly temperature and relative humidity independently could be problematic, as these variables are dependent on one another. I'd be interested to see a plot of one such day of $t$ and $rh$ and then to calculate the water vapor mixing ratio or dewpoint from your values and see how it varies in response and whether it is realistic looking.
If all you have in mean $rh$ and and a min/max temperature, one approach might be to derive a mean temperature and then from that a mean water vapor mixing ratio. This mixing ratio can vary through evaporation, condensation, boundary layer mixing, etc, but for a rough approximation you could assume it is constant. If you assume a constant mixing ratio and a temperature profile (e.g. your sinusoidal assumption) you could calculate the $rh$ profile for the day. Note that you'll have to assure that your temperature curve does not cool below the dewpoint. You have the mean $rh$ as a crosscheck for this method and can use that to iteratively adjust your method for obtaining the water vapor mixing ratio.
If you can provide a $t_{max}$, $t_{min}$ and $\overline{rh}$ and a rough idea of where on earth your are looking at, I can run some numbers and provide more detail on this method.