Relationships Among Water Storage Variables

Among a large number of random variables related to water storage for given sample size n, the most studied are the range $R_n\text{,}$ deficit $D_n\text{,}$ and negative run‐sum $V_n\text{.}$ A simulation of 5,000 samples for various n was performed for three assumed population processes: normal independent, normal dependent, and gamma independent. It was found by simulation, and verified by an analytical derivation, that the range and deficit are equal $(R_n=D_n)$ in 50% of generated samples, while in the other 50% the range is greater that the deficit $\left(R_n>D_n\right)\text{.}$ Probabilities of $D_n=V_n$ range from zero to more than 0.60; they decrease as n and skewness $C_s$ increase, but increase as the lag‐1 autocorrelation coefficient increases. The case of $R_n=D_n=V_n$ has similar patterns. The ratios of sample means for these three storage‐related variables, either for 5,000 samples or only the samples for which the values of these variables are not equal, exhibit these general patterns: (1) Ratios $D_n/R_n$ increase with n converging to an asymptotic value of 0.7854 in case of 5,000 samples; (2) for $R_n>D_n$ samples, the ratios $D_n/R_n$ are much smaller than for the entire set of 5,000 samples with $R_n⩾D_n;$ (3) ratios $V_n/D_n$ and $V_n/R_n$ are much smaller than $D_n/R_n\text{.}$ The pairwise lag‐0 correlation coefficients of $R_n\text{,}{D}_n\text{,}$ and $V_n$ for 5,000 sample estimates show that corr $\left(R_n\text{,}{D}_n\right)$ changes around 0.45 as n increases in the case $C_s=0;$ this correlation is small $(0-0.25)\text{,}$ but increase with n for $C_s=3\text{.}$ For $R_n>D_n$ samples, corr $\left(R_n\text{,}{D}_n\right)$ becomes negative, oscillating around $-0.07\text{.}$ corr $\left(D_n\text{,}{V}_n\right)$ shows a decrease with an increase of n for $C_s=3\text{,}$ but increases with an increase of autocorrelation, corr $\left(R_n\text{,}{V}_n\right)$ decreases with an increase of n, changes slightly with autocorrelation and with skewness.