Exactly Solvable Record Model for Rainfall

Daily precipitation time series are composed of null entries corresponding to dry days and nonzero entries that describe the rainfall amounts on wet days. Assuming that wet days follow a Bernoulli process with success probability $p$, we show that the presence of dry days induces negative correlations between record-breaking precipitation events. The resulting nonmonotonic behavior of the Fano factor of the record counting process is recovered in empirical data. We derive the full probability distribution $P(R,n)$ of the number of records $R_n$ up to time $n$, and show that for large $n$, it converges to a Poisson distribution with parameter $\ln (pn)$. We also study in detail the joint limit $p\to 0$, $n\to \infty$, which yields a random record model in continuous time $t=pn$.

Figure 1

The Fano factor of the record process obtained from simulations (symbols) is compared to the analytic limit function $F(t)$ in Eq. (14) (full line). Note that the numerical estimates start at $F_1=1-p$.

Figure 2

Blue squares show the Fano factor of precipitation records estimated from daily rainfall amounts at 144 German weather stations. For comparison, the full line shows simulation results obtained from the Bernoulli model with the average rainfall probability $p=0.5$. A similar analysis covering the time period 1961 to 2013 can be found in [28].