11.7 The Maximum-Entropy Approach  11. Special topics  11.5 Moments and associated

11.6 Log-normal distributions

Distributions encountered in meteorology are notoriously non-Gaussian, usually because they have more extreme events than expected for a Gaussian distribution. However, a distribution that is normal in the logarithm of a parameter often provides a much improved fit to observations. Another advantage of the log-normal distribution is that it is positive-definite, so it is often useful for representing quantities that cannot have negative values. Log-normal distributions have proven useful as distributions for rainfall amounts, for the size distributions of aerosol particles or droplets, and for many other cases.

The general form for a log-normal distribution is

$$P\left(\ln \bigl({{x}\over{x_0}}\bigr)\right) = {{1}\over{\si......2\pi}}} e^{-\left({{\ln^2(x/x_0)}\over{2\sigma_g^2}}\right)}$$ (11.39)

 

where $\ln(x_0)$ is the central value for the distribution and $\sigma_g$is the geometric standard deviation. This will have a Gaussian shape like that shown in Fig. 3.1 when plotted with a logarithmic abscissa. Figure 11.3 shows some examples of this distribution function, for different values of the geometric standard deviation, plotted with a linear abscissa. The shape differs if the plot is constructed using base-10 logarithms, because the meaning of the standard deviation changes and the normalization factor changes. The value of $\sigma_g^\prime$ applicable to a distribution in terms of base-10 logarithms is obtained from $\sigma_g$ for Naperian logarithms by the translation

$$\sigma_g^\prime = {{\sigma_g}\over{\ln(10)}} \ .$$ (11.40)

 

Figure 11.3:  The log-normal distribution function, for geometric standard deviations as labeled.

 

Because the distribution is skewed toward larger values, the mean value of x is larger than x₀. For the log-normal distribution constructed using Naperian logarithms,

$$\overline{x} = e^{\sigma_g^2/2} \ .$$ (11.41)

 

The variance in x is given in terms of x₀ and $\sigma_g$ by

$$\sigma_x^2 = x_0^2 e^{\sigma_g^2}(e^{\sigma_g^2}-1) =(\overline{x})^2 (e^{\sigma_g^2}-1) \ .$$ (11.42)

 

   
 11.7 The Maximum-Entropy Approach  11. Special topics  11.5 Moments and associated 

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