A new stochastic model of episode peak and duration for eco-hydro-climatic applications

Long series of observations on environmental processes provide a baseline record to better gauge recent episodes relative to prior ones. In this context, episodes are consecutive observations either above or below a reference level. Such episodes can be quantified in terms of three random variables: duration (the number of time intervals in the episode), magnitude (the sum of all process values for a given duration), and peak value (the absolute maximum reached by the process within a given episode). In this paper we present a new stochastic model for the bivariate distribution of episode duration and maxima (peak values). The model follows naturally (in the mathematical sense) from the definition of episodes, it properly reflects the randomness of duration, and it answers an explicit call made by several authors for a theoretical (rather than empirical) modeling of episode parameters. The model, which is based on the stochastic theory of random maxima, is called BTLG because it is Bivariate and has Truncated Logistic and Geometric marginals. Such mixture of discrete and continuous components reflects the discrete nature of duration and the continuous nature of the peak value. A similar approach was used for our previously published bivariate stochastic model of episode duration and magnitude [Biondi, F., Kozubowski, T.J., Panorska, A.K., 2005. A new model for quantifying climate episodes, Int. J. Climatol., 25, 1253–1264; Kozubowski, T.J., Panorska, A.K., 2005. A mixed bivariate distribution with exponential and geometric marginals. J. Stat. Plann. Inference, 134, 501–520]. The BTLG model was applied to a 2300-year long dendroclimatic record from the eastern Sierra Nevada headwaters of the Walker River, between California and Nevada. After testing the sensitivity of this proxy record as a moisture (water-year precipitation) indicator, positive (wet) and negative (dry) episodes (467 each) were stochastically modeled. Maximum likelihood estimates of model parameters, as well as expected and observed distributions, were computed, showing an overall good fit. The practical use of the model was illustrated by computing the likelihood of extended dry and wet episodes, including the 1930s drought and the early 1900s pluvial. Our results show that the chance of an extreme dry period is much greater than the chance of an extreme wet spell.