Approximation of the size distribution of marine particles by a sum of log-normal functions

A simple algorithm is presented that decomposes the size distribution of marine particles into a sum of log-normal components of the 0th order. The algorithm was applied to 412 particle-size distributions in a particle-diameter range of -0.5-200 pm measured by different researchers in various water bodies. A size distribution from this population may have from 1 to 6 log-normal components. The variability of the number of components reflects the variability in the shape of the size distribution and variations in the size range in which the data are available. The full-width-at-half-maximum of a component is approximately proportional to the peak diameter of the component. The maximum value of a component is approximately proportional to the inverse of the square of the peak diameter. Two standard components of the marine particle-size distribution were identified. The peak diameter (I&J, width parameter (a), and maximum value (FYI,,,), of the first component are, respectively, 0.66 pm, 0.673, and 1.34 x lo4 pm-’ cm-3. These parameters for the second component are 10.5 pm, 0.366, and 3.3 I.cm-’ cm-3. The size distribution of particles suspended in seawater is essential information in many marine sciences ranging from optics to sedimentology. Modern particle-sizing instruments make it relatively easy to determine particle size distribution at a high particle-size resolution. Analyzing the resulting large data sets is greatly aided by a method of approximating the experimental data that reduces the large number of parameters of the original data set (each data point can be regarded as a parameter). Besides reducing the number of parameters, a successful approximation technique has several advantages compared to the original representation of the data. Such an approximation technique can greatly simplify the classification of the size distributions (e.g. Kitchen et al. 1975) and give an insight into processes governing the formation of the distribution via linking the parameters of the approximation to physical models of such processes (e.g. Kiefer and Berwald 1992). An approximation technique that yields an analytical representation of the distribution makes possible analytical predictions of other physical properties of the particle population (e.g. the attenuation of light, Casperson 1977) as functions of the approximation parameters of the size distribution. Several functional representations of the size distribution of marine particles have been used to date. These representations include the power law (Bader 1970, also called the Junge distribution after Junge 1963), log-normal distribution (Jonasz 1987; Lambert et al. 198 I), Weibull distribution (Carder et al. 197 l), gamma function Acknowledgments This research was supported by contract W770 ll-4528/0 lXSK of the Defence Research Establishment Valcartier. We thank the anonymous reviewers for their comments. (Risovic 1993; Ulloa et al. 1992), and characteristic vet-, tors (Jonasz 1983; Kitchen et al. 1975, also referred to as principal components). Even a casual visual examination of the complex size distributions characteristic of biologically active waters (e.g. Jonasz 1983) leads to the conclusion that a simple function will not accurately approximate details of such distributions. Such size distributions frequently contain characteristic intermediary maxima linked to the presence of dominating phytoplankton populations (Chisholm 1992; Hood et al. 199 1; Jonasz 1983) or to the dynamics of the populations of nonliving particles (e.g. McCave 1983). Thus, a simple function, such as the power-law function, cannot realistically express such complex distributions at a fine size resolution, although the function can express a first-order approximation at a global size scale. In fact, the power-law function has been shown to be well founded in the dynamics of marine particles (Kiefer and Berwald 1992; Platt and Denman 1977). Jonasz ( 198 3, 1980) compared several functional representations of the size distribution of marine particles and suggested that a combination of the power-law function and a Gaussian function (or a sum of Gaussian functions) may be a versatile representation of the complex size distributions observed in biologically active surface waters of the ocean (e.g. Sheldon et al. 1972). Recently, Risovic (1993) presented an alternative composite model of a marine particle size distribution that is based on the combination of two generalized gamma functions. Van Andel(1973) and Spencer (1963) indicated that a sum elf log-normal functions may describe particle size distribution in sediments, but they pointed out computational difficulties involved in decomposing a particle size distribution into a sum of log-normal components. Fitting a combination of nonlinear functions, such as