There have been a number of studies which show that we can expect more extreme rainfall with a global warming (e.g. Donat et al., 2016). Hence, there is a need to increase our resilience to more rainfall in the future.

We can say something about how the rainfall statistics will be affected by a global warming, even when the weather itself is unpredictable beyond a few days.

Statistics is remarkably predictable for a large number of events where each of them is completely random (welcome to thermodynamics and quantum physics).

The normal distribution has often been used to describe the statistical character of daily temperature, but it is completely unsuitable for 24-hr precipitation. Instead, the gamma distribution has been a popular choice for describing rainfall.

I wonder, however, if there is an even better way to quantify rainfall statistics.

I have played around with the gamma distribution in an attempt to model daily rainfall statistics and its dependency on a set of physical factors. Without much success.

However, then I noticed that most daily rain gauge appeared to be almost exponentially distributed if I only included the rainy days (e.g. setting the threshold for a wet day at 1 mm).

When I plotted the histogram for rainfall on wet days with a log-y axis, I would mostly get a straight line of dots (see a typical example below).

The nice thing with the exponential distribution (which is a particular case of the gamma function) is that it only requires one parameter to specify the mathematical curve: it’s the inverse of the mean value .

I then used Bayes’ theorem to account for dry and wet days, where the probability for rainfall was taken to be the wet-day frequency .

The advantage of this approach is that I now had two parameters which were easy to estimate: the wet-day mean precipitation (or mean rainfall intensity) and the wet-day frequency .

Furthermore, it turned out that is often closely connected to the wind direction, and can easily be predicted based on circulation patterns or sea-level pressure anomalies.

It was harder to find a systematic influence on , as it is likely affected by several factors, including the air moisture (which depends on temperature) and cloud top heights.

The total precipitation is the product of , where is the number of days.

In other words, and tell me many things I needed to know about the rainfall statistics (there are other aspects too, such as the mean duration of dry/wet spells, the spatial extent, and whether it comes as rain, sleet, snow or hail).

The equation for estimating the probability for a rain event with amounts exceeding can be written as (using 1-CDF for the exponential distribution):

(1)

I have called it the “rain equation”, both because the name has not been taken and because it can provide many answers concerning rainfall.

It can address questions about the likelihood of heavy rainfall and whether it is due to an increase in the number of rainy days (e.g. due to changes in circulation) or because the rains have become more intense.

It is also on par with the normal distribution – in both cases, they are not meant to provide accurate probabilities for extreme events far out in the tails.

However, they are both capable of quantifying the probability of more moderate values, which can be illustrated in the figure below:

The rain equation captures long-term changes as well as inter-annual variations. In this example, I used the annual wet-day mean precipitation and frequency estimated from the observations themselves to show its potential.

It can also be assessed against observations in a more systematic way, as in Figure 2:

A correlation of 0.98 is quite impressive, however, the rainfall is not perfectly exponentially distributed (Benestad et al., 2012). It nevertheless provides a means to address climate change connected to a change in either or .

We have used the rain equation in an attempt to downscale seasonal and decadal forecasts for precipitation (Benestad and Mezghani, 2015).

One thing that puzzles me, however, is that I cannot see this equation being used very much, despite the fact that it is so simple, seems so obvious, and can demonstrate impressive capabilities.

I would have thought it is an old formula. Perhaps one that has gotten out of fashion, but is documented in old papers that are not yet digitized and easy to google. Perhaps with a different name. Or have I missed something?

References