Jp4.16 an Iterative Method for Generating Scaling Log-normal Simulations

It has been shown that the single-point probability density functions of the data of certain cloud properties, such as cloud optical depth, are positively skewed and resemble the probability density functions of a log-normal random variable (e.g. Barker et al., 1996). There is also evidence that, for certain spatial scales, the two-point statistics are such that the energy spectrum Ek satisfies a power-law relation, with scale invariant exponent β lying between 1 and 3, i.e. Ek ∝ k−β , 1 < β < 3, where the energy spectrum is the modulus squared of the Fourier transform of the data, and k is the wave number. A Gaussian random field with any desired energy spectrum may be easily generated using Fourier-space filtering due to the fact that the Gaussian distribution is stable (i.e., the sum of any number of Gaussian random variables is a Gaussian random variable). A log-normal simulation may then be generated by exponentiating this Gaussian random field. However, a log-normal field generated in this manner will not have the same energy spectrum as the original or ‘mother’ Gaussian field. In fact, in general, the form of the energy spectrum will also be affected (e.g. if the energy spectrum of the Gaussian field satisfies a power-law, then the log-normal field will not). Ideally, it would be possible to specify a target energy spectrum, and have a method to analytically compute the energy spectrum of a mother Gaussian field that will lead to a log-normal field with this target energy spectrum. This approach is taken by Evans et al. (1999) in the case where an auto-correlation function, which depends only on the distance between the field values, can be specified. Thus, the technique of Evans et al. (1999) is limited to the simulation of stationary fields. In the case of scaling random fields, stationarity can be satisfied by restricting β < 1, or alternatively, by introducing an integral scale (a “scale break”) beyond which the variance does not increase. (see, e.g. Davis et al., 1994; Frisch, 1995). Below, we outline a technique that can be used to generate non-stationary scaling log-normal discrete random fields with scale invariant exponents in the range 1 < β < 3 (i.e. discrete random fields with log-normal singlepoint statistics and energy spectra that satisfy Ek ∝ k−β, 1 < β < 3). The random fields generated using this new technique are useful for the simulation of cloud data, when, for instance, the origin of a possible scale break