some aspects of the analysis of size spectra in aquatic ecology

Some Aspects of the Analysis of Size Spectra in Aquatic EcologyAuthor(s): Beatriz Vidondo, Yves T. Prairie, Jose M. Blanco and Carlos M. DuarteSource: Limnology and Oceanography, Vol. 42, No. 1 (Jan., 1997), pp. 184-192Published by: American Society of Limnology and OceanographyStable URL: http://www.jstor.org/stable/2838876 .

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184 Notes

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Submitted: 10 August 1995 Accepted: 8 April 1996 Amended: 20 July 1996

Limnol Oceanogr, 42(1), 1997, 184-192 ? 1997, by the American Society of Limnology and Oceanography, Inc

Some aspects of the analysis of size spectra in aquatic ecology

Abstract-The established approach to model seston size distributions involves the grouping of particles within logarithmic size classes and the examination of the relationship between density, or normalized biomass, and the characteristic sizes of the classes. Here we examine the distributional basis of the established approach and draw a connection between the biomass size spectrum and the Pareto distribution, a model widely used in other disciplines dealing with size-structured systems. We provide efficient estimators of the parameters and also suggest that datasets exhibiting significant departures from a smooth power function decline can be adequately modeled using a Pareto type II distribution.

Current instrumental developments in the analysis of individual particles have fostered research approaches based on individual particles (rather than on volume-averaged compound samples) as the units of analysis in biological oceanography (Legendre and Le Fevre 1991; Falkowski et al. 1991). This approach originated with the advent of au-

tomatic particle-sizers and the discovery of fundamental regularities in the size distribution of oceanic seston (Sheldon et al. 1972). These findings were followed by research, both experimental and theoretical, on the implications of the observed regularities in the size distribution of marine (e.g. Borgmann 1982, 1987; Rodriguez and Mullin 1986a,b) and freshwater seston (Sprules et al. 1983; Peters 1983, 1985) and benthos (Schwinghamer 1981; Hanson et al. 1989). The established approach to model seston size distribution involves grouping particles within logarithmic size classes and examining the relationships between their density, or biomass normalized to the width of each size class, and a characteristic value of the size classes (Platt and Denman 1978; Sprules and Munawar 1986). The distributional basis and implications of this procedure, however, have not been fully addressed (Ahrens and Peters 1991; Blanco et al. 1994).

Here we examine some mathematical properties of the so-called normalized biomass-size spectra (hereafter NB-SS) and discuss their effect on the inferences drawn from this model. We suggest that the description of plank-



Notes 185

tonic-or organismal in general-size distributions should be viewed in terms of distributional statistics and then show the relationship between the NB-SS approach and the Pareto distribution, a model widely used in other disciplines dealing with size-structured systems. Although the two approaches are not incompatible, we point out the weaknesses and dis- advantages of the NB-SS procedure and identify some new implications of the Pareto model. We then argue for the adoption of a more flexible model-the Pareto distribution of the second type. By using data from diverse aquatic ecosystems, we show that the Pareto model provides a better fit to the data and also yields a direct estimate of the allometric relationship between the size and abundance of system com- ponents.

Modeling biomass-size spectra in aquatic systems essen- tially amounts to describing accurately the size-frequency distribution of the particles. Once this distribution is defined, it can be integrated to calculate the probability that a particle taken at random (or fraction of all particles) will fall within a given size interval such that

prob(s1ower < S < Supper) =Ne

NT

rSupper

= f pdf(s)ds, (1) J lower

where s is size and pdf(s) is the probability density function of the underlying distribution. The number or abundance of particles (N) falling within this size range is simply the calculated probability multiplied by the total abundance (NT) of all particles [N = NTprob(s1ower < S < Supper)]. If the chosen measure of size is weight, then the biomass of particles within any size range can be easily calculated because the biomass and size of a single particle are then the same such that

Supper

biomass(slower<s<Supper) = NT i S pdf(s)ds. (2) Sower

If size is expressed differently, say as a linear dimension of the particles, then s in Eq. 2 can be replaced by the function [f(s)] relating weight to size. For simplicity, we assume that size is expressed as weight.

The examination of size distributions is not restricted to biological oceanography, and the modeling of the size distribution of objects is a major goal of many disciplines dealing with size-structured systems. The description of size distributions is an important aspect of research in many fields, including geophysics (Carder et al. 1971), marine geology (Bader 1970), biogeochemistry (Wells and Goldberg 1994), geography (Korcak 1938; cf. Mandelbrot 1982), earth sciences (Richter 1958; cf. Winiwarter and Cempel 1992), astronomy (Winiwarter 1983), engineering (Cempel 1991; Gaston 1993), economy (Pareto 1897; Simon and Bonini 1958), sociology (Auerbach 1913), scientiometrics (Price 1967), and even semiotics (Zipf 1949). Scientists working in these disciplines have long struggled to find the best model to represent the size distributions of their study objects. Surprisingly, the distributions appropriate for such widely

200

.P- 150

4- 100

00

~ 0

0 * t * * * * * W N 4 1 * * * N - I

m ~~~~~n C . I:X 0 f3 . . c; C: .N 6 1 Z

t6 cr "It C; 0 H tn, N

XoN t 'IT X e>n tn 00 = 0

a > en

Size classes (cm)

Fig. 1. Size data for east Atlantic Ocean fishes (MacPherson and Duarte 1994) showing long right-tail distribution.

differing objects and systems are all characterized by very long right tails, a feature also ubiquitous to aquatic populations (Fig. 1). An examination of the models used to describe size distributions in these disciplines also shows a remarkable convergence toward the use of a common underlying distributional model (Table 1) of the general (cumulative) form

Ns= aS (3)

where N_, is the number of objects greater than a threshold size (S), and a and /3 are the fitted parameters describing the total number of objects in the dataset and the logarithmic rate of decline in number of objects with size, respectively. Although several distributions can display such a power function decline under certain parameter specifications (e.g. the exponential and power functions distributions), the above model corresponds to the Pareto distribution, first introduced a century ago for the analysis of income distribution in society (Pareto 1897). The Pareto distribution has a probability density function defined as

pdf(s) = ckcs-(c+l) (c > 0, s - k > 0), (4)

where s is size and c and k are the distribution's shape and scale parameters, respectively. The parameter k is a constant that represents either the size of the smallest object (for the standard range of the Pareto variate, i.e. k < s < ??) or a compounded measure of the observed range of size. The latter is more relevant to our case since no organism is of infinite size. The parameter c is an empirical constant that describes the decline of probability as size increases. Figure 2 shows the shape of the Pareto distribution for various values of c.

If we assume for a moment that this distribution adequately represents the size distribution of particles, then the number of particles and biomass within any size range can be calculated by integration after substituting this probability density function into Eq. 1 and 2 as



186 Notes

Table 1. Models used to describe size distributions in different disciplines, showing remarkable similarity among them.

Discipline Subject Model Definition Reference

Economy Incomes n = aS-y n = number of people Pareto 1897 (cf. Wini- having the income ?S; warter and Cempel a, y = const. 1992)

Sociology Cities within a country S(j) = aj-13 j = rank number, S(j) Auerbach 1913 (cf. Wini- size of the object warter and Cempel ranked j; a, f8 = const. 1992)

Biology Species within genera S(j) = aj- j = rank number, S(j) Willis and Yule 1922 (cf. size of the object Winiwarter and Cem- ranked j; a, f3 = const. pel 1992)

Geography Aegean Islands N(a) = const. a-B N(a) = number of islands Korcak 1938 (cf. Has- of area >a; B = const. tings and Sugihara

1993) Semiotics Words within texts and babbling pn = An-B n = rank number, pn = Zipf 1949 (cf. Winiwart-

of babies size of the object er 1983) ranked n; A, B = const.

Earth sciences Energy release of seisms and p(Se ? S) = (S/SO)-7 S = size, S0 = min size, Richter 1958 (cf. Wini- gaps between them Se = measured values warter and Cempel

of a symptom (size); 1992) y = const.

Economy Business firms P(S' ? S) = (S/SO)-" S = size, S0 = min size, Simon and Bonini 1958; Se = measured values Roehner and Wini- of a symptom (size); warter 1985 (cf. Wini- y = const. warter and Cempel

1992) Marine geology Cosmic and terrestrial dust, ses- N = K(xIxo)-c N = number of particles Bader 1970

ton, and fine sediments > a given size x; K, c = const.

Ecology Patches of vegetation prob(A < a) = const. a-B a = area, A = measured Hastings et al. 1982 value of area; B= const.

Ecology Parasites on a host, individuals pn = A(n + m)-B n = rank number, Pn = Winiwarter 1983 within species, genera within size of the object families ranked n, m = parame-

ter influencing the distribution for small n only; A, B = const.

Astronomy Chemical elements within stars Pn = A(n + m)-B n = rank numberpn = Winiwarter 1983 or in the entire cosmos size of the object

ranked n, m = parameter influencing the distribution for small n only; A, B = const.

Astronomy Masses in the solar system or in Pn = A(n + m)-B n = rank number, Pn = Winiwarter 1983 the Satum system size of the object

ranked n; m = parameter influencing the distribution for small n only; A, B = const.

Engineering Vibrational amplitudes in me- P(Se ? S) = (S/So)-Y S = any symptom (size) Cempel 1991, 1992, chanical systems P(Se ? 5) = exp-(SISO) Y of technical condition 1993

P(Se ? S) = 1 -exp-(SISO)-Y of a mechanical sys- P(Se ? S) = 1 + y - y(S/S0) tem, So = min size, Se

= measured values of a symptom (size), P(SO > S) = probability of operating in good conditions; y = const.

Biogeochemistry Colloids in seawater N = N- N= number of particles Wells and Goldberg 1994 > diameter d; k, 13 = const.



Notes 187

1.6 -

1.4

1.2 c-1b k==

ci

-o -0 0.6 c 0.5

0.4

0.2

0 1 2 3 4 5 6

Size (S)

Fig. 2. The probability density function of a Pareto variate for different values of shape parameter c for k = 1.

rSupper N(slower<s<Supper) = NT cf -(c+ Ods

Slower

= NTkC(SjpCper- Sioier) (5)

and

r Supper

biomass(sower<S<Supper) = NT J ckcs-cds Slower

NTckc - c (S5p~ - -C+ lupper Jlower J

(6) respectively. Note that the only difference between Eq. 5 and 6 is the exponent associated with the size variable s, which is more negative in Eq. 5 by exactly one unit.

The description of seston size distribution was prompted by the advent of the electronic particle counter and sizer. This instrument gave the number and volume of particles within logarithmic (base 2) size classes. Scientists adopted the grouping of particles in log2 size classes in their analyses even when the sizes of individual particles were measured directly under the microscope (e.g. Sprules et al. 1983; Gasol et al. 1991; Gaedke 1992). The size (S) dependence of particle abundance (N) and biomass (B) within each of the log2 size classes was then examined using regression analysis of the form N (or B) = a(S)b. However, this model was found to be flawed by the differential width imposed by the logarithmic nature of the size classes, with small size classes containing organisms of roughly similar sizes (e.g. 1-2 rnm) and larger classes comprising organisms ranging meters in size (cf. Blanco et al. 1994). The normalized spectrum, where the biomass in different size classes is scaled to the width of the size class, was proposed to solve this problem (Platt and Denman 1978; Sprules and Munawar 1986). The NB-SS has been adopted since as the model of choice to examine plankton size spectra (Ahrens and Peters 1991; Rojo and Rodriguez 1994).

What kind of biomass-size spectrum (normalized or un-

1 0 4 I I, I IIH H I I I It ll 11111 I ,, I 111111 I I I le1|1 I I I I I 11 i0

0 0 Slope = -0.5 0 o c=1.5 1000 0 0 k=l

100 1 0

10 - o~~~~~~~~~~

I 10 100 1000 104 105 lo, 107

S lower Fig. 3. Expected normalized biomass-size distribution for a the-

oretical Pareto size distribution with c = 1.5 (Eq. 5).

normalized) would we observe were the particle sizes truly distributed according to a Pareto law? This can be examined by using the Pareto distribution (Eq. 6) integrated over equal logarithmic (any base) size classes (i.e. by assuming Suppd Siower = J, a constant) and taking logarithms yielding

ln[biomass(S<ower<S<JS1ower] = (-c + 1)ln(Siower)

+ ln[NTckc(J-c+l - 1)] - ln(-c + 1). (7)

The slope of this Pareto-based biomass-size spectrum can then be calculated by differentiating Eq. 7 with respect to ln(Slower), which is equal to the constant - c + 1. In other words, if we plot the resulting biomass against Xiower (or Xupper) on a double logarithmic scale, the resulting plot will be a straight line with slope equal to - c + 1 (Fig. 3 shows a hypothetical Pareto distribution with c = 1.5). Therefore, the slope of the unnormalized biomass-size spectrum is an estimator of -c + 1, from which we can simply calculate the parameter c of the underlying Pareto distribution. Normal- izing (sensu Platt and Denman 1978)-the division of the biomass within a size class by the width of the size class- is done to estimate the average probability density within each size range so as to obtain, in simple terms, a picture of what the true histogram looks like. In the present context, normalizing has the net result of subtracting one from the slope of the unnormalized spectrum, thereby providing directly an estimator of the exponent -c of the biomass-size distribution (see Eq. 6). Thus, fitting straight lines to NB-SS has corresponded, apparently unwittingly, to assuming that the particle sizes are distributed according to a Pareto distribution. More remarkably, the slope of the normalized spectra is an unbiased (although inefficient; see below) estimator of the exponent of the "biomass distribution" (Eq. 6). The underlying Pareto size distribution has an exponent one unit more negative at -c - 1 (Eq. 4, Table 2).

The foregoing discussion illustrates how the normalized biomass-spectrum approximately correctly depicts how the biomass is distributed as a function of size. Given that for a



188 Notes

Table 2. Summary of exponents from the different representation of the size spectra in relationship to the parameter c of the underlying Pareto size distribution.

Size spectrum representation Exponent

Total biomass over log classes -(c - 1) Biomass distribution -c Abundance distribution -(c + 1)

broad range of organisms the exponent of the NB-SS of seston is often in the neighborhood of -1, particle biomass declines as the reciprocal of particle size (i.e. the biomass of small particles is much larger than that of big particles). This allometric decline does not in any way invalidate the claim that the biomass over logarithmically widening size classes will be independent of size (although not necessarily equal or constant; Harris 1994), i.e. a flat Sheldon-type spectrum. There is no contradiction between the two last state- ments. However, we think that some confusion has arisen in the interpretation of how biomass is related to size because of the use of logarithmic size classes.

A more statistically sound way would have been to group the individual weights into arbitrary but equal arithmetic size classes, construct frequency or biomass histograms from those, and then express them on a double-logarithmic scale. It can be shown that, at the limit (AS -> 0), the resulting plot of log biomass vs. log size would converge toward the usual. logarithmic normalized biomass-size spectrum, thereby demonstrating that biomass distribution is negatively related to size as a power function with an exponent close to -1. As an analogy, the difference in interpretation is equiv- alent to claiming that data truly originating from a log-normal distribution are in fact distributed symmetrically simply because when we plot the frequencies with logarithmic size classes, the resulting histogram seems symmetrical. If the question is about the distribution of X, and X comes from a log-normal distribution, then it is asymmetrically distributed irrespective of the way we choose to visualize the histogram. Incidentally, any distribution can be made to seem approximately flat upon appropriate nonlinear scale transformation (of which the logarithmic progression is one), so the flatness of the Sheldon-type spectrum should not be overinterpreted.

Misconceptions about the interpretation of the NB-SS, however, extend beyond a misleading inference about the allometric scaling of biomass distribution. Normalized biomass spectra are often interpreted as a plot of numerical abundance vs. size (Platt and Denman 1978). Although they do represent the abundance (because biomass divided by individual average weight equals numerical abundance) within discrete log size classes as a function of a representative value of the size classes (usually the minimum size, but cf. Blanco et al. 1994), it is inappropriate to interpret these spectra as correctly representing the continuous size distributions of particles. To obtain such a distribution, one would have to further "normalize" the abundance-size spectrum, similar to Blanco's superspectrum (Blanco et al. 1994). More simply, one can use the above equations to infer how particle abundance and size are related. We emphasize that the slope of the NB-SS does not describe the allometric scaling of

abundance but rather corresponds to the allometric scaling of biomass to size (Table 2).

Modern sizing instruments (e.g. flow cytometers, ad- vanced electronic and laser particle counters) and micro- scopic techniques are now able to produce estimates of the size of each individual particle examined. Grouping these individual values into logarithmic size classes therefore represents a substantial loss of information by reducing the de- grees of freedom available for statistical analyses. With the computers now available, this thwarting of statistical power cannot be justified on the basis of data reduction needs. Moreover, estimates of the number of particles in classes of larger particles are generally based on fewer data points than are those in size classes of small particles, resulting in an uneven weight of the individual observations when constructing normalized size spectra; that is, individual observations of large particles have a disproportionate weight on the analysis compared to those of small particles. Even the choice of the logarithmic base will influence the ensuing results on at least two counts. First, it will create more or fewer empty classes. Although empty classes are a common observation in planktonic and benthic ecosystems (e.g. Echevarria et al. 1990; Rodriguez et al. 1987; Sprules and Munawar 1986; Schwinghamer et al. 1981), they are usually ignored as if they represented missing values and not the absence of organisms. Second, such a choice will affect the uncertainty on the slope estimates (Blanco et al. 1994). Fig- ure 4 illustrates the substantial changes in the slope estimate (and its associated uncertainty) of the normalized size spectrum for the size distributions of sestonic particles from northwest Mediterranean surface waters (B. Vidondo unpubl. data) with different logarithmic bases. Thus, estimating the parameter of the underlying Pareto distribution by lumping observations into equal logarithmic classes, normalizing, and finding the slope of the resulting log biomass-log size relationship is, from a statistical point of view, a suboptimal procedure indeed. It is analogous to estimating the mean of a normal distribution by constructing a frequency histogram and fitting it to the equation (l/2orr2)05exp[(x - u)2/2or2] with some nonlinear algorithm.

There are two well-studied methods to estimate the parameters k and c of a Pareto distribution (Eq. 4) that make use of all the individual observations-the least-squares and the maximum likelihood estimators (Johnson and Kotz 1970). However, because much significance is given to the exponent parameter c in ecological studies, simulation studies (not shown) suggest that the least-squares estimator is less sensitive to sampling and sizing errors than is the maximum likelihood estimator. The least-squares estimator of c, derived from the cumulative distribution function of Eq. 4, can be obtained simply by plotting the probability that the size (s) of a particle taken at random will be greater than size S [prob(s ' S)] as a function of S on a double-logarithmic scale. In practice, the term prob(s ' S) is calculated for each particle simply as the fraction of all particles larger than or equal to itself (Ns?sINT). If the particles are distributed according to a Pareto model, this graph will display a straight line. An ordinary least-squares regression line passed through these points will produce all the necessary statistics to evaluate the parameters of the underlying Pareto



Notes 189

-0.8

co .0

o -

M -1.2

0.12

0.100

0.08 a) 0/

0.0

0.04

0.02

0 ,, I 1 I

0 2 4 6 8 10 Logarithmic base of size classes

Fig. 4. Variation of the slope and its standard error (SE) of the normalized size spectrum with changes in the logarithmic base used to build size classes for the size distribution of sestonic particles from northwest Mediterranean surface waters (B. Vidondo unpubl. data).

distribution. The slope is an efficient and unbiased estimator of the parameter c (Eq. 4), and the parameter k can be estimated as the antilog of the ratio intercept/slope. In the process, each particle contributes one point on this plot and therefore all of the information contained in the observations is used.

By using this approach we were able to obtain very precise and robust estimates of the parameters of the underlying Pareto distribution in several planktonic datasets (Fig. 5), without having to group the data into different logarithmic size classes or to deal with empty size classes. The nearly perfect r2 (0.99) clearly indicates that the Pareto distribution approximates these data extremely well, not unlike the excellent fit with which the Pareto describes the distributions of income, cities, biological species within genera and within families, words, business firms, earthquakes, chemical elements, vibration amplitudes in mechanical systems, etc. (cf.

y = 5.8429 + -1.0043x R2 = 0.94 y 5.0594 + -0.95178x R2 = 0.96

E 0

3

A c=l.2 2 : 6 .10

l 3 4 56 Log PatceVlm0 m) LgPril oue(3

N

0 z o surface 70 m

0~~~~~~~~ A ~~~~~~c =1.229 c =1.106

~~~~ ~~~k = 0.028 k = 0.051 W t C 0.99 e = 0.99

L,c -2 0 C .2 -3

LI -4 p suriace 70 m

-5 .. 1 2 3 4 5 1 2 3 4 5 6

Log Particle Volume (tin) Log Particle Volume (gin)

Fig. 5. The Pareto distribution plot for size distribution of sestonic particles in waters from the surface and 70-n depth of the northwest Mediterranean (B. Vidondo unpubi. data).

Winiwarter and Cempel 1992). However, deviations from a perfectly straight line are still apparent for very small particles.

Acknowledging that the underlying distribution of pelagic particles is of the Pareto type may also pave the way for a more general explanation for the observed size distribution of aquatic organisms in nature. At the general level, this may be done by carefully examining the inferences drawn from similar Paretian models proposed in the other disciplines dealing with size-structured systems (Table 1). More specif- ically, however, Pareto variables have several properties that may be unusual or unfamiliar to many ecologists. For ex- ample, it is well known in statistical theory that the expected value of a Pareto variate with exponent -2 or shallower (i.e. c < 1) is undefined. In our context, this means that the average size of organisms (and hence biomass) is unstable, i.e. estimates of the mean will not converge as the number of observations increases (Peters 1994). It is striking and perhaps not aleatory that most size-frequency distribution of aquatic organisms hover around this critical value (corre- sponding to a NB-SS slope of -1). Similarly, power law distributions such as the Pareto are intimately related to fractals (Schroeder 1991), a theory that is providing fertile ground to several applications in ecology (Meltzer and Has- tings 1992). The negative of the exponent of a Pareto distribution (c + 1) provides a direct estimate of the fractal dimension of the process under study (Schroeder 1991; Has- tings and Sugihara 1993). Systems with a fractal structure lack an inherent characteristic scale, suggesting that a single process operating on all scales may be underlying the observed size structure. Whether this view is compatible with the prevailing notion of a directional biomass flow up the size spectrum (Borgman 1982, 1987; Platt and Denman 19078) rman to,- be% furtherexloed



190 Notes

Researchers examining biomass-size spectra have often speculated about the existence of secondary ecological scal- ings whereby organisms within common functional groups (phytoplankton, zooplankton, fish) display steeper biomass- size spectra than expected from the overall trend (Thiebaux and Dickie 1992; Dickie et al. 1987; Boudreau et al. 1991; Peters 1991). Apart from the possibility that such steeper lines might be the artifactual result of using ordinary least- squares regression on datasets covering different ranges in the independent variable (Prairie et al. 1995), such secondary scaling may also be a real feature of the size structure of aquatic ecosystems (e.g. Gasol et al. 1991). We suggest that this hypothesis could be tested by estimating and comparing the parameters of the Pareto distribution estimated for each group. Following Winiwarter and Cempel (1992), this should be achieved by first standardizing the data to the minimum size within each group, i.e. by dividing all the size observations by the size of the smallest organism (se) within that group (s' = sls,). Significant differences between the within and among functional groups c parameter of the Pareto distribution would provide strong evidence for such secondary scaling.

The Pareto model, just like the NB-SS, may exhibit a lack of fit in certain datasets in which the abundance or biomass of small particles is lower than that predicted by a straight line (e.g. Ahrens and Peters 1991; Wells and Goldberg 1994). Gasol et al. (1991) aptly handled such cases by nonlinear curve-fitting, although, as they pointed out, the ecological meaning of the polynomial coefficients is far from obvious. Instead, we suggest that such lack-of-fit problems may be overcome by using a Pareto distribution of the second type (see Johnson and Kotz 1970), whose probability density function is given by

pdf(s) = c(K + D)c(s + D)-(c+). (8)

Eq. 8 differs from the original Pareto model (Eq. 4) only by the additive constant D. Estimators for the parameters K, c, and D of this distribution can be derived again by using its cumulative distribution function and are obtained by regress- ing log[prob(s ? S)] on s by means of an iterative nonlinear regression algorithm with the model

log[prob(s ? S)] = c log(K + D) - c log(S + D). (9)

Although somewhat more complicated to compute, application of this distribution provided excellent fits to a dataset on macrophytes poorly modeled by the ordinary Pareto distribution (Fig. 6) and would undoubtedly provide an even more perfect fit to the Mediterranean seston data (Fig. 5). The Pareto II has the additional advantage that it is more general and flexible because the ordinary Pareto can be considered a special case of the second type when D = 0. Al- though the parameters D and c may not have the same in- tuitive meaning as with the Pareto I, they can also be compared among systems.

Finally, we stress that there will always be datasets for which neither the Pareto (I or II) nor the NB-SS will appro- priately describe the size and biomass distribution of aquatic organisms (e.g. multimodal distributions). For such cases, attempting to force the data to these specific distributions is ill-founded and can be highly misleading.

2.5 l

.S * *~~~~~~~~~ 2

1.5 *N

E~~~~~~~~~~~~~ 2 z 0

0.5

-3 -2 -1 0 1 2 3

y = 1.433 xLog (-0.102 + 1.232) - 1.433 x Log (x + 1.232) R2 = 0.99 0

-0.5 N

X ~ ~ ~~~

A -2.5 A -1.5

0 -2.0

co

o' -2.5

3.0 macrophytes

3.5 -3 -2 -1 0 1 2 3

Log Weight (g FW)

Fig. 6. The size distribution of submersed macrophytes in Lake Memphremagog (Canada; C. M. Duarte unpubl. data) plotted as cumulative probability plot (@). Dash line corresponds to best-fit line assuming an underlying Pareto size distribution and solid line corresponds to a Pareto II distribution.

Our examination of the statistical basis of biomass-size spectra has revealed peculiarities and caveats in the currently accepted procedure. Although the slope of the NB-SS does provide an estimate of the allometric exponent describing the decline of biomass with increasing body size, the wide- spread conclusion, based on a slope of -1, that biomass is independent of size is misleading. Additionally, we showed that the NB-SS approach is methodologically deficient. We suggest that the distribution of particle sizes in aquatic systems should be considered as Paretian variates, either ordinary or modified (type II). This change in approach should prove beneficial by rendering the results more general and applicable to a wider range of datasets, as well as making



Notes 191

them more precise thus allowing more powerful tests of hy- potheses. It may also prove a unifying approach to the study of size-structured systems across other disciplines.

Beatriz Vidondo

Centro de Estudios Avanzados de Blanes-CSIC Cami de Santa Barbara s/n 17300 Blanes, Girona, Spain.

Yves T. Prairie

Departement des sciences biologiques Universite du Quebec a Montreal Case postale 8888, succ. Centre-Ville Montreal, Quebec H3C 3P8

Jose M. Blanco

Station Zoologique, B.P. 28 06230 Villefranche-sur-Mer, France

Carlos M. Duarte

Centro de Estudios Avanzados de Blanes-CSIC Cami de Santa Barbara s/n 17300 Blanes, Girona, Spain

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Submitted: 21 April 1995 Accepted: 1 April 1996

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Charcoal analysis in marine sediments

Abstract-A technique is described for measuring charcoal in small samples (5 mg) of marine sediments to quantify the contribution of charcoal to the total organic carbon loading of marine sediments. Charcoal is measured as elemental carbon by gas chromatography after acidification with hot concen- trated nitric acid in situ within aluminum sample cups to re- move calcium carbonate and refractory carbon such as coal, pollen, and humic acids. The in situ acidification eliminates sample loss during sequential decarboxylation and oxidation and provides a precise (?2.2% of the measured value) and rapid (-50 analyses per week per analyst) means to measure charcoal in marine sediments. The absolute detection limit of the charcoal determinations is 0.70 ,ug C (3 times mean blank value) and the relative detection limit is 0.01%.

The records of total organic carbon (TOC) routinely re- ported from marine sediments include a mixture of carbon from terrestrial (e.g. charcoal and noncharcoal) and marine (e.g. primarily algae) sources. Studies that assess the relative contributions of marine and terrestrial sources to the TOC loading of marine sediments usually target bulk C: N ratios, carbon isotopes, and biomarkers. An analytical technique is described herein that uses elemental analysis to quantify one component of terrestrial carbon-the amount of charcoal

produced by terrestrial biomass burning and deposited in the oceans. The burning of carbonaceous matter produces charred particles that can be transported long distances via winds and rivers to coastal, deltaic, and ocean environments where they may become preserved in the sediments as a means of long-term carbon storage. At present, charcoal's contribution to marine sediments is largely ignored, although charcoal particles have been identified in Late Pleistocene marine sequences of the tropical Atlantic Ocean (Verardo and Ruddiman 1996) and in marine sediments from- the Pa- cific Ocean at -65X 106 yr (Herring 1985). The ocean acts as a passive collector of charcoal, thereby making marine sediments an excellent natural repository from which to study the transfer of charcoal from terrestrial to marine environments.

As used here, charcoal refers to the pryolized remains of terrestrial biomass (e.g. trees, grasses, plants) in the elemental state (Smith et al. 1975). Extracting this entirely terrestrial component from the TOC of ocean sediments leaves a "residual" organic carbon signal (e.g. TOC minus charcoal) that is a mixture of marine organic carbon and noncharcoal terrestrial carbon. This distinction improves our understanding of true marine organic carbon burial from the marine stratigraphic record and forces a reevaluation of marine pro-



some aspects of the analysis of size spectra in aquatic ecology

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