A unified probability distribution of second order difference of global streamflow

Hydrological processes, as part of a natural system, are highly complex and chaotic. By analyzing long time series of streamflow data from ~ 4800 hydrologic stations, it is interesting to find out that probability density functions in second order difference (SOD) of the streamflow data are fat-tailed and bell-shaped curves, and their cumulative distribution functions (CDFs) follow an S-shaped curve (S-curve). We found that t-distribution is a good approximation for S-curve, which uses the degree of freedom (DF) to control the tail thickness. We also found that DF of more than 80% of stations are gathered in the range between 5 and 8. Analysis of the S-curves in seven large river basins indicated that the S-curves can vary with time and space, which is regarded as a good indicator for identifying natural and anthropogenic changes. This study provides a symmetrical, identical and concise probability distribution to describe global streamflow under changing environment.