Probabilistic modelling of crossing in small samples and application of runs to hydrology

Şen Z.

Journal of Hydrology, vol.124, no.3-4, pp.345-362, 1991 (Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 124 Issue: 3-4
  • Publication Date: 1991
  • Doi Number: 10.1016/0022-1694(91)90023-b
  • Journal Name: Journal of Hydrology
  • Journal Indexes: Scopus
  • Page Numbers: pp.345-362
  • Istanbul Medipol University Affiliated: No


The first part of this paper presents exact probability distribution functions (PDF) of upcrossings in stationary first-order Markov processes for finite sample lengths. In the derivation of these PDF, the enumeration technique is used and the correctness of the results obtained is checked with the total probability of all the possible upcrossings, which has to equal unity for any sample length. Numerical calculations of these PDF on digital computers led to the condition that the expected number of crossings is linearly proportional to the truncation level probability. Hence, the proportionality factor is a function of the first-order autorun coefficient, which is intimately related to the first-order autocorrelation coefficient. The PDF are valid for stationary and dependent processes irrespective of their underlying probability distribution functions. In the second part of the paper, various analytical expressions concerning run properties, such as the mean positive and negative run lengths, run sums, and number of crossings, are presented. Their applications to a set of serially dependent annual flow sequences are performed. Good agreements are obtained between the derived expressions for calculating run properties and their counterparts from observed sequences. © 1991.