Multifractal Modelling of Plankton Abundance in Coastal Waters Coastal Heterogeneity And Scaling Experiments  MR Claereboudt 1,2, WJS Currie 2, Y Tessier 3, JC Roff 2 Michel.Claereboudt@qiroq.ulaval.ca, wcurrie@uoguelph.ca, yves@balrog.physics.mcgill.ca, jroff@uoguelph.ca 1 GIROQ, Univ. Laval, Pav. Vachon, Ste-Foy, PQ, G1K 7P4 2 Dept. Zoology, College of Biological Science, University of Guelph, Guelph, ON, N1G 2W1 3 GIROQ, Physics Dept., McGill University, 3600 University St., Montreal, PQ, H3A 2T8

Introduction | Data Sets | Multifractals in 350 words | Double Trace Moment | Results | Simulations

Modified from a poster presented in 1996 at the 28th International Liège Colloquium on Ocean Hydrodynamics : Modelling hydrodynamically dominated marine ecosystems, Liège, Belgium.

The study of spatial biological variability in the ocean (often referred to as patchiness) has been recognized for many years and has received considerable attention. This research was mainly focused at the largest scales most conducive to the current sampling devices. However, an effective modelling of marine ecosystems requires the description of this spatial variability over scales ranging from cm to hundreds of km.

Multifractals extend to continuous measures, the fractal framework used to study sets of points. The idea underlying both fractals and multifractals is that of a scaling of properties. When integrated over space at a given resolution scale, the measure generates a field with values in every point of the space at that scale. The basic mechanism underlying multifractal fields is that of a multiplicative cascade through which a quantity (i.e. turbulence) is transferred from large to smaller and smaller scales resulting in complex patterns of heterogeneity in the spatial distribution of the measure.

The end result is that at the smaller scales, a large portion of the measure will be concentrated in small zones of the space. This can be mathematically translated as:

Pr is the probability of finding a value of the field j_l higher than a threshold l^g at a resolution of l. Unlike fractals, the scaling exponent l^g is not a constant, but now becomes a function of the threshold g; hence the idea of multiple scaling and multifractals.

A Seabird CTD (Conductivity-Temperature-Depth) profiler coupled with a Turner in-situ fluorometer and an Optical Plankton Counter (OPC) was towed at constant velocity and depth along horizontal profiles in the coastal region of the St. Lawrence Estuary. The sampling rate (4 Hz) combined with the towing velocity (1.6 ms^-1) gives a sampling resolution of approximately 0.4 m. Each transect was 4096 data points in length resulting in a total transect length of approximately 1800 m.

The measured fields were temperature, conductivity (salinity), dissolved oxygen, turbidity, fluorescence (chlorophyll-a) and zooplankton biomass. Click on the data series below for a clearer view of a typical transect.  

The multifractal equation (1) indicates that as the resolution of a measure increases, we are able to distinguish smaller and smaller areas of higher and higher values. Since the knowledge of a probability distribution of a process is equivalent to the knowledge of all of the statistical moments of this process we also have:

where á ñ indicates ensemble averaging (over several independent realizations) and q indicates the order of a statistical moment. In multifractals this correspondence is particularly simple; the exponents K(q) and c(g) are related through a Legendre transform (Parisi and Frisch 1985):

Through a reasoning similar to the Central Limit Theorum, the existence of a class of stable and attractive universal multifractals has been accepted. In this class of multifractals, the scaling function of the field K(q) and the codimension function c(g) can be described by 3 parameters (Schertzer and Lovejoy 1987) :

where (1/a + 1/a¢) = 1 and q > 0.

The double trace moment method was developed by Lavallée et al. (1991) to determine the codimension function c(g) and the scaling function K(q) of the field by exploiting universality to determine C₁ and a.

Consider the stationary multifractal process j_L(áj_L=1ñ) where the resolution L is the ratio of the largest to the smallest sampled scale. The basic idea is to generalize the application of statistical moments to the quantity ^gL^h by taking the nth power of the field j_l at the finest resolution l and then studying the scaling behaviour of the various qth statistical moments at decreasing values of the scale ratio L £ l :

This decreasing of the scale ratio is a degrading of the resolution obtained by averaging the values measured inside of each new larger scale. By plotting the two sides of this equation on log-log axes, the slope of the linear segments represents K(q,h) as a function of L.

Since we are considering normalized h powers, the K(q,h) scaling exponent function is related to the usual exponent by:

The usefulness of the double trace moment technique becomes apparent when it is applied to universal multifractals, since in this case we have the following (Lavallée et al. 1993):

For each q, Log(K) vs Log h will then be linear in the region of multiple scaling and allows an easy determination of a. For h = 1, K(q,1) is the intercept of the regression with log(h)=0 and thus:

If a measured field is multifractal, then its power spectrum has scaling (ie: a linear relationship). The first step in multifractal analysis is thus to identify the scaling region for each field.
click on the power spectra graphs for a better view

Temperature, turbidity (transmission) and oxygen fields are scaling over the whole range of measured scales (0.4m-2000m). The theory of isotropic 3D turbulence predicts a spectral slope of -5/3 for a passive scalar. The spectral slopes we estimated for both temperature and transmission are indeed very close to this expected value.

The other fields (salinity, phytoplankton, and zooplankton) show different scaling behaviours over different parts of the measured range of scales. For salinity, the break in scaling corresponds to the depth of the water column in this series of samples (8m). The breaks in scaling for the two biological fields occur at frequencies (spatial distances) that do not correspond to any particular physical features. We believe they correspond to particular length scales associated with biological activities such as swimming behaviour.

Summary

DTM multifractal analysis of above measured fields indicated that they were highly multifractal (a close to the upper limit of 2), but not very sparse (C₁ lies in the lower portion of its empirical range).

All measured fields are non-conservative (H ¹ 0) implying that the estimate of the mean changes with sampling scale.

a = 1.84 | C1 = 0.30

a = 1.84 | C1 = 0.05

a = 1.98 | C1 = 0.05

By reversing the process of analysis, simulations of one or more dimensional fields (assuming isotropism) can easily be produced. Shown here are simulations of multifractal fields with various values of a and C1 that are representative of those encountered during the sampling of oceanographic fields. These three 2D fields are conservative (the spectral slope is -1) and the mean level is 1. To obtain a non-stationary field with spectral slope B, the results must be fractionally integrated by multiplying the Fourier tranform by k^-H where H and b are related through equation 5. It can be easily seen that the graph on the far left with a low C1 (a sparse mean field) has much more coverage by high values (in red), and concurrently more coverage by low values (shown in white) compared to the other simulations.

References

Lavallée, D., D. Schertzer and S. Lovejoy. 1991. On the determination of the codimension function. Nonlinear variability in geophysics. D. Schertzer and S. Lovejoy. Amsterdam, Kluwer. pg 99:109.

Lavallée, D. S., S. Lovejoy, D. Schertzer and P. Ladoy. 1993. Nonlinear variability of landscape topography: multifractal analysis and simulation. Fractals in geography. Eds. L. De Cola and N. Lam. NewYork, PTR, Prentice Hall. pg 158-192.

Parisi, G. and U. Frisch. 1985. A multifractal model of intermittency. Turbulence and predictability in geophysical fluid dynamics and climate dynamics. Eds. M. Ghil, R. Benzi and G. Parisi. Amsterdam, North-Holland. pg 84-88.

Schertzer, D. and S. Lovejoy. 1987. Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades. J. Geophys. Res. (92) 9692-9714.

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