2.3.3. Analysis of Statistical Distribution

The statistical distribution of rainfall data is an important component of hydrological and hydrometeorological studies, such as in intensity–duration–frequency (IDF) relationships and design storms. The quantile-quantile (Q-Q) plot is a graphical tool for determining whether the two datasets, i.e., ground- and satellite-based, have similar distributional shapes. The technique is conducted by plotting quantiles (or percentiles) of the two datasets versus one another and comparing the plot with a 45◦ reference line. Accordingly, the Q-Q plot is a scatter plot, with the points falling approximately along the reference line standing for a common distribution for the two datasets. On the contrary, the greater the departure from the reference line, the greater the evidence for refusing this assumption. It is worth noting that the quantiles of a dataset are the points below which a certain proportion of the data lies. For example, in a classic standard normal probability distribution with a mean of 0, the 0.5 quantile (or 50th percentile), 0 means that half the data are not exceeding 0.

There are also analytical methods, such as the chi-square and Kolmogorov–Smirnov 2-sample tests, that are used for assessing if two sets of quantiles follow the same distribution. However, the Q-Q plot is favorable as it provides more insight into the nature of the difference between two datasets than analytical methods. Although the Q-Q plot is only a visual check rather than an air-tight proof, it helps to observe if the assumption is plausible and, otherwise, which data points at which quantile cause the violation of the assumption.

The Q-Q plot can easily show the under/overestimation of a dataset, i.e., satellite, compared to the rain gauge, between percentiles of the datasets. Additionally, many distributional aspects, including shifts in location, shifts in scale, change in symmetry, tail behavior, and the presence of outliers, can be discerned. The behavior of the tail of the Q-Q plot can be important for extreme hydrology studies. The tail refers to data points associated to statistically rare incidents, such as values above the 95th or 99th percentile of the datasets.
