Austral. J. Statist., 15 (2 ) , 1973, 136

LETTER TO THE EDITOR

DOUGLAS SHERMAN~ Australia% Defence .Scienti$c .Service, Department of .Supply, Aeronautical Research Laboratories, Port Helbourne, Victoria

The paper by Professor Hasofer (Austral. J. Xtatist., Nov., 1970) on the derivative and upcrossings of the Rayleigh process, is capable of an interesting extension to the non-central chi distribution with two degrees of freedom.

Hasofer assumes that over the long term the horizontal velocity components U(t) , V ( t ) are normally distributed with zero mean, and so the horizontal wind speed is Rayleigh-distributed. Using these assumptions, it is possible to study the wind over short periods (of the order of an hour) during which maximum gusts occur, but it is necessary to have a good insight into the auto-correlation function of the wind speed (see, e.g., Hasofer and Sharpe (1969)).

An alternative approach which may be more immediately useful, is t o consider one of the velocity components, U(t ) , to have a non-zero mean value, say, over a short period. Then if U and V are independent, normally distributed variables, and the wind speed W(t) is given by

W2 = uz+ v2 W 2 will have a non-central chi squared distribution with two degrees of freedom, and W will have a corresponding non-central chi distribu- tion, with the probability density (see, e.g., Ruben (1960))

where h, is the variance of U ( t ) and V ( t ) . Then Hasofer’s argument still holds to show that the derivative

of W(t) is independent of W ( t ) , and has a normal distribution with mean zero and variance A2, where A2 is the variance of U‘(t) and V‘(t) .

The joint probability density of W ( t ) and W’( t ) is

and the mean frequency of upcrossings of a level W is given by

W,f(W,W‘)dW‘

References Hasofer, A. M., and Sharpe, K. (1969).

Ruben, H. (1960).

‘‘ The analysis of wind gusts.’’ Auatralian

“ Probability content of regions under spherical normal dis- Meteorological Magazine, 17, No. 4.

tributions. I.” Annals of Mathematical Statistics, 31, 598-618.

Manuscript received December 18, 1972.