Expected range and adjusted range of hydrologic sequences

Because of the stochastic nature of hydrologic sequences such as streamflow the needed capacity of a reservoir is a random variable. Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range R n or adjusted range R n * of partial sums S i * = S i−1 * + ( X i − αω), where ω = μ = ε [ X i ] or ω = X ¯ n = ( 1 / n ) ∑ i = 1 n X i . Previous studies led to separate formulas for the expected range ε [ R n ] and the expected adjusted range ε [ R n *] for independent standard normal variables and for a ‘degree of development’ α=1. In a recent study it was demonstrated that for any 0 ≤ α ≤ 1 and for any underlying distribution of the streamflow variables X i , both the expected range and the expected adjusted range can be obtained from a single formula by changing an appropriate parameter. This paper elaborates on this previous study. Among other things it graphically exhibits the ‘transient’ nature of the general formulas for the expected adjusted range.