Seuront L. Fractals and Multifractals in Ecology and Aquatic Science

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284 Fractals and Multifractals in Ecology and Aquatic Science

turbulent conditions. It is stressed here that Equation (8.83) does not involve any assumption on

phytoplankton concentration being considered as a passive conservative tracer or a biologically

active tracer. Equation (8.83) is instead general and would lead to a steepening of any power

spectrum due to intermittency correction, whether it follows a k

−5/3

power law or not.

It is nally stressed that the divergences observed between Franks’s conclusions (Franks 2005)

and those presented here might also stem from the differences in the approaches used. First, the

observational platforms used by Franks and coworkers (Franks 2005; Franks and Jaffe 2008) return

horizontal 32 × 32 cm 2D uorescence distributions sampled vertically every 6 to 24 cm, signi-

cantly different from the uorescence data analyzed here that correspond to a time series recorded

from a xed depth at a rate of 2 Hz. Second, Franks’s data were collected in stratied waters 250

to 450 m deep 10 km offshore of San Diego (California) where chlorophyll a concentrations were

typically bounded between 0.1 and 0.6 μg l

−1

(Franks and Jaffe 2008, Figure 2). In contrast, the data

analyzed here were collected in the tidally mixed shallow coastal waters of the eastern English

Channel where chlorophyll a concentrations ranged from 6 to 30 μg l

−1

. It is then likely that the dif-

ferent spectral shapes returned by the analysis of those two data sets might also be related to their

intrinsic differences. Unambiguously demixing apples and oranges in an intermittent context would

then require further work through a thorough investigation of data sets collected following the dif-

ferent methods described above simultaneously in a range of marine environments.

8.4.2 Fr E q u E n c y di s T r i b u T i o n s a n d mu l T i F r a c T a l i T y

As stressed above (Section 8.1.2) and as implicitly seen from the sets of equations dening

most of the multifractal functions described in Section 8.2—that is, Equations (8.3), (8.21), and

(8.23)—multifractal sets and signals will return a power-law behavior in cumulative frequency dis-

tributions (Chapter 5). However, a cumulative frequency distribution exhibiting a power-law behavior

does not mean that the process being analyzed is multifractal. This is illustrated through the cumu-

lative probability distribution function, P(X ≥ x) ∝ x

−f

, and the probability distribution function,

P(X = x) ∝ x

−m

, used to characterize intermittent distribution, when 1 < m ≤ 3 (see Section 5.1 and

Equations 5.1 and 5.2). As discussed in Section 4.2.10.2 and Section 8.2.4, both nonintermittently

and intermittently distributed quantities can be described by the qth-order structure functions. The

function z(q) is linear for monofractal processes, that is, z(q) = qH, where H is the Hurst exponent as

dened in Chapter 4. In the case of multifractality, z(q) is nonlinear, concave, and takes the general

form z(q) = Aq + Bq

(see Section 8.3.2.2 and Equation 8.72). For a Lévy distribution, the structure

function exponent z(q) is dened as a bilinear model (Chechkin and Gonchar 2000; Nakao 2000):

z(q) = q/m (8.84)

for q < m (that is, q < f + 1), and

z(q) = 1 (8.85)

for q ≥ m (that is, q ≥ f + 1). In the special case of a Lévy ight, m = 2 (Shlesinger et al. 1996),

leads to z(q) = q/2 for q < 2, and z(q) = 1 for q ≥ 2. The cumulative probability distribution func-

tion estimated from a time series of the successive displacements of the calanoid copepod Temora

longicornis (Figure 8.2A) clearly exhibits a power-law behavior with f = 2.74 (Figure 8.17A); that

is, m = 1.74. The empirical function z(q) obtained from T. longicornis successive displacements is

subsequently compared with the functions z(q) theoretically expected for a Brownian motion and a

Lévy distribution characterized by m = 1.74 (Figure 8.17B). The nonlinearity of the empirical func-

tion clearly contrasts from the bilinear behavior related to Equations (8.84) and (8.85), showing that

a power-law signature in frequency distributions does not imply multifractality. In contrast, multi-

fractality is consistently related to a power-law signature in frequency distributions.

2782.indb 284 9/11/09 12:16:54 PM

From Fractals to Multifractals 285

8.5 Joint multiFractals

8.5.1 J

o i n T mu l T i F r a c T a l mE a s u r E s

Joint multifractals have been introduced by Meneveau et al. (1990) to assess the degree of correla-

tion between two simultaneously recorded multifractal elds such as the intermittent uxes e

and

discussed above. To my knowledge, this approach has only subsequently been applied once to the

joint analysis of crop yield and terrain slope (Kravchenko et al. 2000).

543210

2.5

2.0

1.5

1.0

0.5

0.0

0.1

P(X ≥ x) x

–φ

ζ(q)

0.01

0.001

0.1 110 100

Figure 8.17 (A) The cumulative probability distribution function estimated from a time series of the succes-

sive displacements of the calanoid copepod Temora longicornis shown in Figure 8.2A. A power-law behavior

with f = 1.74 (dashed line) is clearly visible for displacements ranging from 1 to 30 mm. (B) The struc-

ture function exponents expected for a Brownian motion z(q) = q/2 (dashed line), a Lévy ight, z(q) = q/2 for

q < 2 and z(q) = 1 for q ≥ 2 (open squares), and the empirical z(q) estimated from a time series of the successive

displacements of the copepod Temora longicornis shown in Figure 8.2A.

2782.indb 285 9/11/09 12:16:56 PM

286 Fractals and Multifractals in Ecology and Aquatic Science

Consider two multifractal measures m

and m

. The partition function M

(d ) (Equation 8.5) can

be rewritten as a joint partition function M

(d ) as:

Mff

ii12

() [()] [()]

()

δδδ

∑

(8.86)

where N(d ) is the number of boxes of size d,

() ()

δµδ

=∑

and

() ()

δµδ

=∑

; see Equation

(8.4). Note that when q

= 0 or q

= 0, Equation (8.86) reduces to Equation (8.5). As discussed for the

single partition function, low values of q

and q

characterize the low values of the rst and second

measures, while at high q

and q

the joint partition function depends mostly on high values of

and m

. From Equation (8.11), the mass exponent of order (q

, q

), t(q

, q

) comes from:

()

(,)

δδ

∝

−

(8.87)

From Section 8.2.2.1, the Hölder exponents a

and a

of multifractal measures m

and m

come as:

µδ

log()

log

(8.88)

and

µδ

log()

log

(8.89)

where d is the box size. From Equation (8.15), it comes that the number of cells N

(d ) with a

singularity strength (that is, Hölder exponent) within the ranges [a

, a

+ da

] and [a

, a

+ da

]

respectively for the rst and second measure scales with d as:

αα

δδ

()

(,)

∝

−

(8.90)

where f(a

, a

) is the joint singularity spectrum describing the abundance of cells with common

and a

values. The joint singularity spectrum f(a

, a

) and the joint mass exponent function t(q

, q

)

are connected via a double Legendre transform as:

112

(,)

dqq

(8.91)

212

(,)

dqq

(8.92)

f(a

, a

) = q

+ q

− t (q

) (8.93)

2782.indb 286 9/11/09 12:17:04 PM

From Fractals to Multifractals 287

As discussed in Section 8.2.2.1, the maximum value of the joint multifractal spectrum f(a

, a

) is

reached when q

= 0 and q

= 0, in which case f(a

, a

) equals the box-counting dimension of the

geometrical support of the measures m

and m

Meneveau et al. (1990) rst illustrated the applicability of joint multifractals to joint lognormal

and binomial distributions, and simultaneous experimental distributions of turbulent shear and tem-

perature gradients. A decade later, joint multifractals were used to assess the links between crop

yield and topography, and allowed to differentiate yield distributions corresponding to eld loca-

tions with high slopes and to make inferences about slope distributions that affect grain the most

(Kravchenko et al. 2000). A more general procedure based on joint moments is proposed hereafter

to study the joint properties of two multifractal measures.

8.5.2 Th E gE n E r a l i Z E d co r r E l a T i o n Fu n c T i o n s

8.5.2.1 deﬁnition

The joint correlation functions using structure functions introduced by Seuront and Schmitt (2005a,

2005b) are a continuation and development of the early study by Meneveau et al. (1990). Their

approach provides estimates of the fractal dimension of mixed singularities, instead of the scale-

invariant moment functions suggested hereafter. Another major difference between their joint mul-

tifractal formalism and the one provided here is that they did not normalize joint moments, as seen

below, to provide joint correlations. As such, the joint correlation function introduced hereafter

can be thought of as a more intuitive approach and as a high-order generalization of the standard

correlation between two variables X and Y. Note that joint moments for scaling structure functions

have also been proposed in the eld of econophysics, to study correlations for multiple assets, in

order to characterize their return distributions; see, for example, Muzy et al. (2001). However, the

nal objective of such a study is portfolio optimization, which is different from our analysis of the

generalized correlation between two multifractal elds.

Instead of random variables X and Y, consider two stochastic processes (ΔX

) and (ΔY

) charac-

terized by their pth- and qth-order structure functions as 〈 |(ΔX

〉 ≈ t

(p)

and 〈 |(ΔY

〉 ≈ t

(q)

. The

correlation between the two processes (ΔX

) and (ΔY

) then becomes a function of the scale and of

the statistical orders of moment p and q, expressed by the generalized correlation functions (GCF

hereafter) c(p, q) as (Seuront and Schmitt 2005a):

cpq

(,)

[|()||()|]

|( )| |( )|

〈∆ ∆〉

〈∆ 〉〈 ∆〉

ττ

µµτ

−rpq(,)

(8.94)

where for more generality we take p ≠ q and r(p, q) ≥ 0 (Seuront and Schmitt 2005a). The gener-

alized correlation exponents (GCE hereafter) r(p, q) are estimated as the slopes of the power-law

trends of c(p, q) vs. t in a log-log plot, and expressed as

rpqpqSpq

(,)()()(,)=+−

ζζ

(8.95)

where z

(q) and z

(q) characterize the multiscaling properties of the single uctuations 〈 |(ΔX

〉 and

〈|(ΔY

〉, and S(p, q) characterizes the multiscaling properties of the joint uctuations 〈 |(ΔX

|(ΔY

〉.

Both c(p, q) and r(p, q) are generalizations of the standard correlation function. In the special case

p = q = 1, Equation (8.94) indeed recovers the standard expression of the correlation coefcient

between (ΔX

) and (ΔY

). GCF and GCE hence express the correlation between |(ΔX

and |(ΔY

2782.indb 287 9/11/09 12:17:06 PM

288 Fractals and Multifractals in Ecology and Aquatic Science

together with their scale and moment dependence. Note that at a given scale t, if the uctuations

of the stochastic processes (ΔX

) and (ΔY

) are independent, r(p, q) = 0 for any combination of p and q.

Increasing values of r(p, q) would thus characterize increasing dependence between |(ΔX

and

|(ΔY

. Note that while independence implies uncorrelation, uncorrelation does not imply indepen-

dence. Uncorrelation corresponds to the relation r(1, 1) = 0 and implies independence only in spe-

cial cases such as for Gaussian processes. In the general case, this is no longer true: Independence

between the stochastic processes means that r(p, q) = 0 whatever the values of p and q. Figure 8.18A

–2.8

–2.9

–3.0

Log c (p, 2)

–3.1

–3.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.022

0.02

0.018

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

4.5

3.5

2.5

1.5

0.5

0.511.5 2 2.5 3 3.5 4 4.5 5

Log τ

Figure 8.18 (A) The generalized correlation functions c(p, q) vs. the time scale t in log-log plots, for the

grid-generated turbulent velocity time series and the in vivo uorescence time series shown in Figure 8.14A,D,

respectively. The function c(p, q) shown here has been estimated for a constant value of the statistical order

of moment q of velocity uctuations (q = 2), and various values of the statistical order of moment p of in vivo

uorescence (that is, p = 1, 2, and 3, from bottom to top). The slopes of the linear regression estimated over the

scaling ranges (dashed lines) provide estimates of the generalized correlation exponents r(p, q) (B). (Modied

from Seuront and Schmitt, 2005a.)

2782.indb 288 9/11/09 12:17:09 PM

From Fractals to Multifractals 289

shows the GCF, c(p, q), plotted in log-log plots vs. t, for the grid-generated turbulent velocity

time series and the in vivo uorescence time series, respectively (shown in Figure 8.14A,D). As

these time series have been independently sampled, they represent an archetypical example of two

independent multifractal processes. The very low values taken by the functions c(p, q) indicate the

absence of any correlation between the turbulent velocity and uorescence uctuations, |(ΔV

and |(ΔF

. This is conrmed and specied by the related values of the function r(p, q), which

remain statistically undistinguishable from zero, whatever the combinations of p and q values

(Figure 8.18B); see Box 8.2.

The main advantages of this technique are that (1) it makes no assumptions about the spectrum

or the probability distribution of either data set, (2) it takes into account the multiscale intrinsic

properties of intermittent processes, and (3) it allows testing for the phenomenology responsible

for the high intensity (rare and unexpected) uctuations observed in intermittent distributions,

considering their potential association with both high- and low-intensity uctuations character-

ized by high and low orders of moment. The generalized correlation functions and exponents

thus provide a general framework, as they express the correlation between the elds (ΔX

) and

Box 8.2 GEnERALIZED CoRRELAtIon FunCtIonS AnD

ExPonEntS In SPECIAL CASES

The function c(p, q) and the related scaling exponent r(p, q) can be used as an analysis tool

to study the couplings between two multifractal elds x (x = ΔX

) and y (y = ΔY

). To provide

some basis for discussion and interpretation of experimental results, some limit cases are

considered.

If x and y are independent r(p, q) = 0. On the other hand, in case of perfect proportionality

x = Ky, where K is a constant, or for random proportionality x = ky, where k is a random vari-

able independent on y, it is readily seen that

rpqpqpq

YYY

(,) ()() ()=+−+

ζζζ

(8.B2.1)

In particular, one may note that r(p, q) > 0 due to the convexity of the scaling functions z(p).

This relation can be directly tested to verify the proportionality hypothesis. Furthermore, the

shape of the surface obtained is symmetric in the p-q plane. In this specic case, the function

r(p, q) has the desirable advantage to reduce considerably the number of data points—that is,

r(p, q) values—needed to understand the relationship between the elds x and y.

Another very simple situation occurs when x = Ky

with b > 0 and K constant, or when x =

with k random and independent of y, then

(,)()()( )pq bp qbpq

=+−+

ζζζ

(8.B2.2)

r(p, q) in Equation (8.B2.2) is still positive but no longer symmetric in the p-q plane but in the

bp-q plane. The value of b may be rst estimated as the positive value such that

rp rbp(,0) (0,)=

(8.B2.3)

Using the values of b, this can be tested by verifying that r(p/b, q) is indeed symmetric in the

p-q plane. More generally speaking, the more r(p, q) is positive, the more the x = ΔX

and

x = ΔY

are dependent random variables.

2782.indb 289 9/11/09 12:17:13 PM

290 Fractals and Multifractals in Ecology and Aquatic Science

(ΔY

), together with their scale dependence and their moment dependence. This indicates if high-

intensity uctuations of one eld are highly correlated to low intensities of the other (or, in other

words, the gradients of two elds are proportional or inversely proportional) and the scales over

which these correlations occur.

8.5.2.2 applications

The potential effect of varying turbulent forcing on the local structure of physical and biological

parameters has been investigated using structure function analysis in Section 8.2.4.1 on the basis

of 24 time series of temperature, salinity, and in vivo uorescence (Figures 8.11 and 8.12). Those

results demonstrated that temperature and salinity uctuations remained similar under different

turbulent and tidal forcings, while in vivo uorescence (that is, phytoplankton biomass) clearly

exhibited more intermittent uctuations during ebb tide than ood tide and under conditions of low

turbulence. However, this does not provide any information on the nature of biophysical couplings

and their relation to turbulence and tide. This issue is investigated here using the generalized cor-

relation functions and exponents introduced in Section 8.5.2.1.

The nature of the dependence between temperature and phytoplankton distributions has

been assessed through the generalized correlation functions, c(p, q) and the related generalized

correlation exponents, r(p, q), between temperature and uorescence time series, for each of

the 24 time series mentioned above. Figure 8.19 shows the GCF, c(p, q), plotted in log-log plots

versus the time scale t, for simultaneously recorded temperature and uorescence time series

for ebb tides (Figure 8.19A) and ood tides (Figure 8.19B), as well as for temperature and uo-

rescence time series taken at different moments of the tidal cycle, and a fortiori independent

(Figure 8.20A). Both the power-law behavior of the functions c(p, q) over the whole range of

available scales, and the positive values taken by the GCE, r(p, q), indicate a form of depen-

dence between temperature and uorescence uctuations. In addition, the values of the func-

tions c(p, q) are smaller during ood tide (Figure 8.19A) than during ebb tide (Figure 8.19B),

suggesting a differential correlation between temperature and phytoplankton biomass uctua-

tions controlled by tidal processes. On the other hand, the weak values taken by the functions

c(p, q) estimated between independent temperature and uorescence time series (Figure 8.20A)

indicate a low correlation between temperature and phytoplankton biomass uctuations, 〈(ΔT

)

〉

and 〈(ΔF

)

〉. This is conrmed by the related values of the functions r(p, q), which remain close

to zero, whatever the combinations of p and q values (Figure 8.20B).

These observations were rened comparing the functions r(p, q) obtained between temperature

and uorescence time series in different tidal and turbulent conditions. Figure 8.21 shows the func-

tions r(p, q) obtained for all combinations of p and q values (between 0.5 and 5) with 0.1 increments

for three levels of turbulence (10

−4

, 10

−5

, and 10

−6

⋅s

−3

) during ebb and ood tides, respectively. In

both case, the correlation between temperature and uorescence uctuations increases with increas-

ing hydrodynamic conditions (Figure 8.21A,B,D,E) and is weaker, even nil, in low turbulent con-

ditions (Figure 8.21C,F). On the other hand, the decorrelation observed between temperature and

phytoplankton uctuations during both ebb and ood tides under weak turbulent conditions sug-

gests an increase in the biological contributions to the control of phytoplankton biomass distribution

and conrms previous observations (cf. Figure 8.12). Phytoplankton uctuations then appear inde-

pendent from the temperature uctuations under the lowest turbulence levels investigated here, that

is, 5 × 10

−7

⋅s

-3

(Figure 8.21C,F). This conrms the differential physical control suggested under

strong turbulent conditions from the analysis of the shape of the function z

(q), and its comparison

with the function z

(q) (see Figure 8.12).

More specically, the overall shape of the functions r(p, q) indicates that large phytoplankton

uctuations are associated, under strong enough turbulent conditions, to strong temperature gradi-

ents, and vice versa. This tendency seems to reect, over a slightly wider range of scales, ndings

2782.indb 290 9/11/09 12:17:14 PM

From Fractals to Multifractals 291

of Desiderio et al. (1993), who observed the occurrence of 0.1 to 0.2-meter-thick uorescent lay-

ers just above local temperature gradients. As this suggests a proportionality relationship between

temperature and phytoplankton uctuations, Equation (8.B2.2) has been veried testing the validity

of Equation (8.B2.3) over a wide range of b values (Box 8.2). Using b values ranging between 0.05

and 5 (with 0.05 increments), we then showed that Equation (8.B2.3) is veried for four of the six tur-

bulence levels investigated during ebb tides, i.e., e = 10

−4

⋅s

–3

with b = 0.90, e = 5 × 10

−5

⋅s

–3

with

b = 0.85, e = 10

−5

⋅s

–3

with b = 0.80, and e = 5 × 10

−6

⋅s

–3

with b = 0.78, and two of the six turbu-

lence levels investigated during ood tides, that is, e = 10

−4

⋅s

–3

with b = 0.96 and e = 5 × 10

−5

⋅s

–3

Log τ

Log c (2, q )

Log c (2, q)

0.0

0.6

0.4

0.2

0.0

–0.2

–0.4

–0.6

–0.8

–1.0

–1.2

–1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Log τ

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

–0.5

–1.0

–1.5

–2.0

–2.5

–3.0

Figure 8.19 The generalized correlation function c(p, q) vs. the time scale in log-log plots, for temperature

and uorescence time series simultaneously recorded during ebb tide (A) and ood tide (B). The function

shown here have been estimated for a constant value of the statistical order of moment q of temperature uc-

tuations (q = 2), and various values of the statistical order of moment p of in vivo uorescence (that is, p = 1,

2, and 3, from bottom to top). The slopes of the linear regression estimated over the scaling ranges (dashed

lines) provide estimates of the generalized correlation exponents r(p, q). (Modied from Seuront and Schmitt,

2005b.)

2782.indb 291 9/11/09 12:17:17 PM

292 Fractals and Multifractals in Ecology and Aquatic Science

with b = 0.91. Figure 8.22 shows the relation r(p, 0) vs. r(0, bp) corresponding to the functions

r(p, q) shown in Figure 8.21A,B with b = 0.90 and b = 0.80, respectively. The correlation shown

between temperature and phytoplankton uctuations under high turbulent conditions is then

related to a power-law dependence relationship of the form ΔF

∝

(ΔT

)

–1.80

–1.90

Log c (2, q)

–2.00

–2.10

–2.20

0.0

1 234

0.04

0.03

0.02

0.01

q (ﬂuorescence)

p (temperature)

0.5 1.0

1.5 2.0

Log τ

2.5 3.0 3.5

Figure 8.20 The generalized correlation function c(p, q) vs. the time scale t in log-log plots, for tempera-

ture and uorescence time series simultaneously recorded at different moments of the tidal cycle, and a priori

independent (A). The function c(p, q) shown here have been estimated for a constant value of the statistical

order of moment q of temperature uctuations (q = 2), and various values of the statistical order of moment p of

in vivo uorescence (that is, p = 1, 2, and 3, from bottom to top). The slopes of the linear regression estimated

over the scaling ranges (dashed lines) provide estimates of the generalized correlation exponents r(p, q) shown

in (B). (Modied from Seuront and Schmitt, 2005b.)

2782.indb 292 9/11/09 12:17:19 PM

From Fractals to Multifractals 293

8.6 intermittency and multiFractals:

biological and ecological imPlications

Intermittency can have implications in a wide range of biological and ecological processes in

terrestrial and aquatic ecosystems. Based on the marine background of the author, this section

explores a few areas of marine sciences that could be signicantly affected by intermittency.

Specically regarding probabilistic arguments derived from the multifractal framework described

1.50

1.30

1.10

0.90

0.70

0.50

0.30

1.20

1.00

0.80

0.60

0.40

0.20

0.00

1.20

1.00

0.80

0.60

0.40

0.20

0.00

0.90

0.80

0.70

0.60

0.50

0.40

0.10

0.30

0.20

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0.08

0.06

0.04

0.02

0.00

12345 12345

q (ﬂuorescence) q (ﬂuorescence)

p (temperature)

12345

p (temperature)

12345

p (temperature)

12345

p (temperature)

12345

p (temperature)

Figure 8.21 The generalized correlation exponents r(p, q), shown as a function of both p and q, which char-

acterize temperature and in vivo uorescence uctuations, respectively. The functions r(p, q) correspond to

three different levels of turbulence:

e = 10

−4

⋅s

−3

(A,D), e = 10

−5

⋅s

−3

(B,E), and e = 10

−6

⋅s

−3

(E,F) inves-

tigated during ebb tide (left panel) and ood tide (right panel). (Modied from Seuront and Schmitt, 2005b.)

2782.indb 293 9/11/09 12:17:24 PM