Precipitation storm property distributions with heavy tails follow tempered stable density relationships

Precipitation storm properties, especially high impact and low probability extremes in storm intensity, duration, and frequency, are important for hydrologic engineering design/control and hydrologic response to changing climate. This study reveals that the property distributions of precipitation storms in the southwestern United States exhibit heavy tails, which follow a tempered stable density function, one possible universal density for hydrologic variables. The precipitation time series from four regions in different states are decomposed as sequences of storm intensity, duration, and frequency, where the underlying anomaly or heavy tailed distribution for each characteristic is explored. Analysis shows that the average storm intensity, storm duration, and interstorm period distribute as a rescaled and tempered stable density function with a variable index. The probability distribution is also space dependent, likely due to climatological variation, which can be represented by a space-dependent index and/or truncation parameter in the tempered stable density function. These sequences, or the cumulative rainfall, therefore might be treated as a realization of a three-stage random walk process, where each stage contains unique tempered stable random variables. This finding leads to a distributed-order, fractional-derivative model with exponentially truncated power-law densities to quantify precipitation storm properties, while the practical application, of which, remains to be shown.