Fractal river catchments

PJSchmid's brief introduction to fractals and multifractals

Most patterns in nature are so irregular, that, compared to Euclidean geometry, nature displays a very different level of complexity. The word fractal stems from the Latin word fractus, meaning broken, to describe natural objects and processes which are too irregular to fit into traditional geometric settings. The fractal approach has brought together a broad range of preexisting concepts from mathematics, physics, and biology, contributing to advancements in the sciences and technology.

The attraction of fractal geometry stems from its ability to analyse fragmented or irregular patterns and processes in nature that traditional Euclidean geometry fails to analyse. Fractality is often expressed by spatial or time-domain statistical scaling laws and is mainly characterised by the power-law behaviour of real-world physical systems. Mandelbrot (1977) introduced the concept of fractal sets, which enables an assessment of the amount of regularity in organisational structures in relation to the system's behaviour. Natural fractals fulfil theoretical and methodological criteria that include a high level of organisation, shape irregularity, self-similarity, scale invariance, iterative pathways, and a non-integer, fractal dimension. Although the definition of the term fractal is only tentative, a concept most often related to fractal geometry is that of functional, morphological and/or temporal self-similarity or self-affinity. A pattern or process is self-similar if it can be decomposed into smaller copies of itself, in a technical sense, on all scales. Random or natural fractals are statistically self-similar or self-affine, which means that their average properties are self-similar or self-affine over a restricted range of scales (Schmid 2000) . Within this definable scale-range, fractal objects are scale-invariant, where features look the same on increasingly smaller spatial or temporal scales. As complex patterns and processes often have fractal features, the fractal dimension can be used to estimate the degree of intricacy by evaluating how fast size increases relative to scale reductions. This rests on the assumption that the two quantities, size and scale, do not vary arbitrarily but rather are related by power-laws. Some power-law relationships and their scaling exponents appear to apply universally across diverse taxa and ecosystems (Schmid, Tokeshi & Schmid-Araya 2002).

Fractal geometry

River catchments in the Atacama desert

The word fractal was coined by Mandelbrot (1977) from the Latin word fractus, meaning broken, to describe objects which are too irregular to fit into traditional geometric settings (Fig. 1). Fractal geometry is an extension of conventional Euclidean geometry and occupies a borderline between linear geometry and complete randomness. A unique difference between fractal and Euclidean objects is that the length when measured depends on the scale, which denotes the resolution within the range of the measured quantity. Euclidean objects are defined by their constant length, regardless of the length of the measuring device, whereas fractal contours increase with increasing length of the measuring device. This is caused by the addition of more detail seen at larger magnification scales. This dependence of the measured length on the measuring scale is expressed as a fractal dimension measure and reflects the scale invariance and often also the complexity of the object. Mathematical and natural fractals are features or measures whose roughness and fragmentation look similar over a range of scales, although the underlying structures are often formed by random processes. Fractals describe patterns or processes devoid of translational symmetry that are self-similar or self-affine (Schmid 2000) and display power-law behaviour.

Power-laws and scaling exponents

Species-Area relationships  in an alpine gravel stream

The power-law function is a relationship of the form y = cxb (Fig. 2), which can also be expressed in double-logarithmic form as log y = log c + b log x (inset of Fig. 2). Scale invariance is the fundamental property behind many power-laws with non-integer scaling exponents observed in nature (Gisiger 2001). The topology of river basins as well as the hydraulic geometry of the drainage network and even the hydraulic response of the basin to different kinds of precipitation inputs display power-law relationships between the variables involved (Rodriguez-Iturbe & Rinaldo 2001). Power-law scaling also defines many patterns observed in ecology, ranging from individuals to the level of ecosystems (Margalef 1996). Many fundamental allometric characteristics of organisms scale with body-size as power laws with exponents often some simple multiple of 1/4 (Brown et al. 2002). These power-laws often hold for scales ranging over several orders of magnitude, sufficiently large to justify the search for unifying theoretical frameworks underlying these patterns and processes. Power-law behaviour, particularly non-integer power laws, is an area covering all fields of ecology (e.g., Margalef 1996;Kunin 1998; Harte et al. 1999; Schmid & Schmid-Araya 2007). Some of the power-laws in nature appear to apply universally to aquatic and terrestrial ecosystems and their organisms (Brown et al. 2002). Often more than 1000 different species can be found within a single, pristine riverine system. Its habitat diversity and complexity influence population dynamics and the structure of communities such as species diversity, dispersal, species abundance and body-size distributions. Some of these scale-related power-law relations are statistically self-similar or self-affine over a wide range of spatial or temporal scales. Particularly the multifractal concept can be used as a unifying concept integrating scale-dependence and complexity, both of which are central to our understanding of structural and functional processes in ecology . Recent studies on body-size scaling relationships conducted at local scales and across regions showed that both habitat and species-area relationships are scale-invariant, attributable to fractal and multifractal properties underlying species communities (Schmid & Schmid-Araya 2007).

Self-similarity

Self-similar water cascades

The concept of self-similarity forms the basis of mathematical and some natural fractals, but it is not a prerequisite to applying the fractal approach. As shown in Fig. 3, self-similar objects are those whose component parts resemble the whole object. The reiterations of irregular details occur at progressively smaller scales (Fig. 3). A pattern or process is self-similar if it is invariant with respect to any transformation in which all the coordinates are scaled by the same factor. Self-similar features remain invariant under changes of scale, displaying scaling symmetry, thus, rescaling is generally isotropic or uniform in all directions. Natural patterns or processes are statistical self-similar if its statistical properties remain scale-invariant over a specified range of spatial or temporal scales. A self-similar or statistically self-similar pattern or process is represented by a fractal dimension that remains constant for each subpart of the whole. The properties of fractal objects imply that a fractal, for instance, with a dimension of 1.6 is more than a line but less than a plane.

Self-affinity

Size distribution of lakes display self-affine fractality.

Self-affine objects are a union of rescaled copies of itself. The reiterations of irregular details occurs at progressively smaller scales - although not uniform in all directions. Consequently, rescaling is anisotropic or dependent on direction, which means in a 2-dimensional context that the horizontal axis scales different from the vertical axis. Examples include the size distribution of lakes, such as those shown in Fig. 4, river networks, the scaling of watercourses, topographic transects etc. that are self-affine fractal features in a landscape. Other examples include the movement trace of an organism's foraging activity that may depict a self-similar pattern in 2-dimensional space, but the search path is self-affine if the trace is plotted as a function of time. Self-affine or statistical self-affine fractals are defined by local dimensions, non-uniform scaling and care must be exercised in choosing an appropriate method to estimate the fractal dimension of a self-affine pattern. Some self-affine curves are not necessary univalued functions and, therefore, are best assessed using multifractal approaches.

Dimensions

Self-similar Lophosoria quadripinnata (J.F. Gmelin) C. Christensen. Each pinna looks similar to the fronds of the fern.

The dimension of a point is 0, of a line and a plane are 1 and 2, respectively, and organisms move and feed in 3-dimensional space. Conventionally, integer dimensions are used such as exponents on length: surface = length2 or volume = length3. These integer dimensions are inadequate in describing complex natural forms, such as those in Fig. 5, and dynamical processes that do not have specific scales of length and time. A dimension gives a precise parameterisation of conceptual or visual intricacy and is formalised mathematically as the intrinsic dimension of a topological space. This dimension is called Lebesgue Covering dimension (topological dimension) and it measures how an object fills space. With increasing space-filling the dimensionality increases. To adequately define the topological characteristics of dynamical structures and processes, fractional power dimensions are necessary.
Fractal geometry allows to measure objects in a non-integer or fractional way when the unit of measurement changes, hence the term fractal. Depending if the patterns or processes are self-similar or self-affine, different mathematically defined notions of [mono]fractal dimension are used. The fractal dimension, `D`, is a value larger than the topological dimension, `d_T`, and it is a number that characterises the way in which the measured length between given points increases as the scale decreases.

References

Brown, J.H. et al. 2002. Phil. Trans. R. Soc. Lond.B 357, 619.
Gisiger, T. 2001. Biol. Rev. 76, 161.
Harte, J., Kinzig, A. & Green, J. 1999. Science 284, 334.
Kunin, W.E. 1998. Science 281, 1513.
Mandelbrot, B.B. 1974. J. Fluid Mech. 62, 331.
Mandelbrot, B.B. 1977. Fractals: Form, Chance, and Dimension. W.H. Freeman, San Francisco.
Margalef, R. 1996. BioSystems 38,141.
Rényi, A. 1955. Acta math. hung. 6, 285.
Rodriguez-Iturbe, I. & Rinaldo, A. 2001. Fractal River Basins. Chance and Self-Organization. Cambrige University Press.
Schmid, P.E. 2000. Adv. Ecol. Res. 30, 339.
Schmid, P.E., Tokeshi, M. & Schmid-Araya, J.M. 2002. Proc. R. Soc. Lond. B 26, 2587.
Schmid, P.E. & Schmid-Araya, J.M. 2007. Body size and scale invariance: multifractals in invertebrate communities. In Hildrew, A.G., D.G. Raffaelli & R. Edmonds-Brown (eds) Body Size: The Structure and Function of Aquatic Ecosystems. Cambridge University Press, Cambridge. 140.

Fractals have immeasurably widened our ability to describe and understand nature. Fractal analysis provides us with models of reality much more realistic than Euclidean geometry with its integer exponents and smooth shapes. Fractals are characterised by a fractional scaling exponent, the fractal dimension, which describes the irregularity and intricacy of the object. Fractal scaling requires that the size-frequency distribution of objects has a power-law dependence on size over a wide range of scales.

Fractals are shapes whose roughness and fragmentation remains essentially unchanged at smaller scales. Monofractals or single fractals are homogeneous fractals, that means they are represented by an irregularity depicted by a single scaling exponent and fractal dimension. Simplified, a fractal dimension of a monofractal object can be calculated as the quotient of the log-transformed size of the object and the log-transformed measuring scale.

Mathematical or deterministic fractals

One-scale Cantor set

Subsets of mathematical fractals are infinitesimally subdivisional, each subset, however small, containing no less detail than the complete set. Mathematical fractals exhibit exact self-similarity across all spatial or temporal scales. An example of a set with non-integral dimension is the one-scale Cantor set (Fig. 6). This set is produced from the unit interval [0,1] by successively removing the middle one-third of each of the equal-sized line segments remaining (Fig. 6). This process is iterated ad infinitum. The result is an infinite number of clustered points in the interval between 0 and 1. The total length of the line segments in the `n`th iteration is `epsilon_n` = `(2/3)^n`. The number of line segments after the `n`th iteration is `N(epsilon) = 2^n`, each of length `epsilon` = `(1/3)^n`, for `n = 1,..., N`. Defined by the decrease of the coverage with each iteration, the Kolmogorov capacity dimension of the Cantor set is given as `D_0 = - lim_[epsilon   ->  0] [log N(epsilon)] / [log epsilon] = log 2 / log 3 = 0.631`. Thus, for Cantor sets the fractal dimension is in the range of `0 < D < 1` and, as it is only composed of points, it can be described as a 'dust'. It follows that the topological dimension of the Cantor set is `d_T = 0`.

Natural or random fractals

Mycelium growth on decaying logs of Eucryphia cordifolia Cavanilles

Unlike mathematical fractals, natural or random fractals are only self-similar (scale-invariant) in a statistical sense, because all natural structures are truncated at certain scales. By relating the size of the random variations to the scale, enlargements of small parts of an object have the same statistical distribution as the whole set. Often, random sets - such as epilithic biofilm patches, the outline of lichen, Mycelium (Fig. 7), bacterial culture growth, detritus particle, etc. - are statistically self-similar not only for a given value of the scaling ratio `epsilon`, but for all scaling ratios above some lower and below some upper cut-off value. Thus, randomness and self-similarity are not mutually exclusive concepts. Natural fractals look qualitatively the same over a range of scales, but the dimension of these fractal objects is restricted to this specific scale range. Depending on the natural pattern or process to be assessed, and if it is self-similar or self-affine, different estimators of the fractal dimension are used such as those based on e.g., variational method, curve-length method, root-mean-squares method, Poincaré sections method, rescaled-range analysis, Fourier and wavelet power-spectrum (e.g., Schmid 2000).

Particle fragmentation

Organic particle size-distribution

In soil and aquatic systems, organic particles such as detritus (Fig. 8), serve as food and habitat for a variety of small-sized uni- and multicellluar organisms. The size-frequency distribution of those particles often closely follows a power-law function, suggesting the fractal nature of particle fragmentation-processes (Schmid & Schmid-Araya 2007). The size-frequency distribution of objects or fragments in a statistically self-similar system is defined as `N_P(E >= epsilon) = c epsilon^-delta` (Mandelbrot 1983; Schmid 2000), where `N_P` is defined as the number of organic particles/particle aggregates larger than or equal to a specific size `epsilon`, `c` is a constant of proportionality and `delta` is the fragmentation fractal dimensionality of the particle-size distribution. This fractal dimensionality implies that similar, often biological, mechanisms govern the particle breakdown across a wide range of scales (Schmid & Schmid-Araya 2007).

References

Mandelbrot, B.B. 1983.  The fractal geometry of nature. W.H. Freeman & CO, New York.
Schmid, P.E. 2000.  Adv. Ecol. Res. 30, 339.
Schmid, P.E. & Schmid-Araya, J.M. 2007.  Body size and scale invariance: multifractals in invertebrate communities. In Hildrew, A.G., D.G. Raffaelli & R. Edmonds-Brown (eds) Body Size: The Structure and Function of Aquatic Ecosystems. Cambridge University Press, Cambridge. 140.

Fractality in nature often manifests itself not just in terms of a single fractal dimension (monofractal) but as many interwoven fractal sets, all of them associated with their own scaling exponent, resulting in an entire spectrum of exponents (multifractal). While the applicability of a monofractal implies scale-invariance, multifractal applicability does not. The pertinence of multifractal statistics to natural patterns and processes provide important clues to the underlying physical processes.

Introduction to multifractals

The multifractal approach is an extension of the monofractal concept to include intricate, often convoluted, structures such as heterogeneous fractals. One of the properties of multifractals is a spectrum of dimensions of which the capacity dimension `D_0` is only one. Multifractal statistics offers a framework to analyse and quantify intricate systems such as those in nature. Multifractals are found in all fields of the sciences as diverse as, for instance, quantum mechanics (e.g., Richardella et al. 2010), biochemistry (e.g., Dewey 1997), aquatic sciences (e.g., Schmid & Schmid-Araya 2007; Seuront 2010) and are widely applied in economy, technology and engineering (e.g., Li & Mao 2012; Ţălu & Stach 2014).

Pore-particle binary image of a streambed sediment cross-section.  Pore-particle cross-section with one of several superimposed square lattices.
To understand multifractals in the context of aquatic systems, let us consider a simple example. The pore-space distribution in streambed sediments serves as habitat for many different species and varies considerably among different locations. These variations have important implications, for instance, for environmental predictions, functional relations in food-webs, water quality assessment, and natural resource management. The image of a streambed cross-section is composed of pixels of pore and particle size-classes (Fig. 9). In the monofractal case, a square of a superimposed lattice (Fig. 9) is counted as occupied by pores, regardless of whether it is entirely covered by pore-class pixels or just by a single pore-class pixel. In contrast, the multifractal approach accounts for the pore area or volume contained in each square or circle (Schmid & Schmid-Araya 2019). Thus, we divide the image of the sediment cross-section into `n` squares of size `epsilon`. For each square `i` of size `epsilon` the fraction of pore space in that square is calculated as `p_i = m_i` / `M`, where `m_i` is the number of pore-class pixels, that is the pore area or 'mass', contained in a square and `M` is the total number of pore-class pixels in `n` squares of size `epsilon`.

Multifractal analysis

Multifractal pore-space distribution from a cross-section of frozen streambed sediments extracted from the Afon Goch in Wales

The starting point in the analysis of multifractals is to define a weighted measure as `z_i(q,epsilon) = p_i^q`, where `z_i` represents the proportion of pore-space in squares of size `epsilon`. The `q` values are the statistical moments of the measure. Based on the weighted measure, we can define the generating function as `z(q,epsilon) = sum_[i=1]^[n(epsilon)] z_i(q,epsilon)`. Multifractality may be intrinsic to these gravelly sediments if the log-log transformed relation of the power-law function `z(q,epsilon) prop epsilon^[-tau(q)]` is highly significant for each of various `q`- moments. `tau(q)` is the Rényi exponent that can then be expressed in the following equation as `tau(q)= - lim_[epsilon -> 0] [log z(q,epsilon)]/log epsilon`.
Various formalisms have been developed to depict the multifractal properties in terms of the singularity spectrum, `f(alpha)` that describes the fractal dimension `f` of a subset with a given Lipschitz-Hölder exponent `alpha` or Rényi's generalised dimensions `D_q` (e.g., Schroeder 1996; Schmid & Schmid-Araya 2007). Using a Legendre transformation from the variables `q` and `tau` to the variables `alpha` and `f` we can define the relation between `tau(q)`, `alpha` and `f(alpha)` as `alpha(q) = - d/[dq] tau(q)` and `f(alpha(q)) = q alpha(q) + tau(q)`, respectively (e.g., Halsey et al. 1986), where `f(alpha(1)) = alpha(1) = D_1` (see below). A singularity spectrum with a wide parabolic shape, as shown in Fig. 10 for the pore-space distribution, is typical for habitats exhibiting multifractal properties.
The generalised dimensions, which are based on the concept of generalised entropies (Rényi 1955), can be expressed as `D_q = [tau(q)]/(q-1) = lim_[epsilon -> 0] [log z(q,epsilon)]/[(q-1) log epsilon]` for all `q` - moments `!= 1`. For `q` = 1, the dimension is given as `D_1 = [d tau(q)]/[d q] = lim_[epsilon -> 0]sum_[i=1]^[n(epsilon)] [z_i(1,epsilon) log z_i(1,epsilon)]/log epsilon`. Thus, the ecologically most relevant dimensions are `D_0`, the Kolmogorov capacity, `D_1`, the information and `D_2` the correlation dimension (Hentschel & Procaccia 1983). `D_1` provides information about the degree of heterogeneity in the distribution of the measure and `D_2` is associated with the correlation function and quantifies the average distribution density of the measure. In the monofractal case, `D_0` is similar or equal to `D_1` and `D_2`, while multifractals exhibit `D_0 > D_1 > D_2`. The pore-space distribution of the sediment cross-section displays a multifractal spectrum with `D_0` = 1.68, `D_1` = 1.34 and `D_2` = 1.29 (Fig. 10). We can define the spectrum width as the difference between `D_2` and `D_0` (Schmid & Schmid-Araya 2019). A wide spectrum width implies a heterogeneous distribution in pore space in this streambed section.

Mathematical multifractals

Two-scale Cantor set

While a Cantor set, generated from segments of equal length, is defined by a single scaling factor, multifractals are described by two scaling factors, one for the supporting fractal and one for the probability weights. To understand the logic of mathematical multifractals, let us modify the Cantor set (Fig. 11). The set has a length `l_0 = 1` with a probability weight `p_0 = 1`. The basic difference between the construction of a mono- and multifractal set is that the initial set is divided into segments of unequal lengths. Applying the generator of a two-scale Cantor set consists of splitting the bar into two rescaled lengths, for instance, one of length `l_1 = 0.50` and the other of `l_2 = 0.25` with the middle piece cut out. Each of the segments has a probability, the larger segment has a probability of `p_1 = 0.67` and the smaller one of `p_2 = 0.33`.

Singularity spectrum and generalised dimensions of two-scale Cantor set

To analyse this fractal object we can define the generating function as `Z(tau,q) = sum_[i=1]^[N(l)] p_i^q / l_i^tau`, where `N` is the total number of segments with size `l_i` and probability `p_i` of the `i`th segment. At the first level of the modified Cantor set the sum of the generator is given as  `Z_1(tau,q) = n_1 p_1^q / l_1^tau + n_2 p_2^q / l_2^tau`, where `n_i` is the number of segments at level `i`. To assess the extremes of the singularity spectrum, the minimum and maximum Lipschitz-Hölder exponent is given as `alpha_min` = `log p_1 / log l_1` and `alpha_max` = `log p_2 / log l_2`, respectively. Thus, the Lipschitz-Hölder exponent of this two-scale Cantor set ranges from  `alpha_min = 0.585` to `alpha_max = 0.792`, with `f_max = f(alpha(0)) = 0.694` (Fig. 12). The segment distribution of this modified Cantor set displays a spectrum of generalised dimensions with `D_0` = 0.694, `D_1` = 0.689 and `D_2` = 0.683 (Fig. 12).

Lacunarity

Mean scale-related lacunarity values for different lattice orientations

Lacunarity is a concept introduced by Mandelbrot (1983) to describe the distribution of gap sizes in a fractal sequence. The word stems from lacuna which is the Latin word for ditch or gap. Lacunarity analysis derives from fractal mathematics and allows to determine the texture associated with patterns of spatial dispersion across multiple scales in one- to three dimensions. Lacunarity is a counterpart to fractal dimension as it depicts and quantifies aspects of patterns that exhibit scale-dependent changes in structure. In addition to the multifractal spectrum, the pore-space may be fully characterised by its lacunarity (Fig. 13). Here, we estimate lacunarity from the variations in pixel density given as `Lambda(epsilon,gamma)=(sigma/mu)^2` in each image for different scales `eta`and orientations `gamma` of a square lattice. Although there are several approaches to assess lacunarity, we define here the grand mean lacunarity, `Lambda(lambda)`, as the deviation from translational and rotational invariance by depicting the size distribution of voids in a sediment structure across a range of scales and orientations as `Lambda_epsilon(lambda)=sum(Lambda(epsilon,gamma))/N_epsilon` (Fig.13)and `Lambda(lambda)=sum(Lambda_epsilon(lambda))/N_gamma` where `Lambda_epsilon(lambda)` is the mean lacunarity across scales and `N_epsilon` and `N_gamma` are the total number of different scales and lattice orientations, respectively. Thus, lacunarity can be estimated as the average across the spatial scale range covered by the size and shape composition of a 'habitat structure'. Habitat structures which may display identical monofractal dimensions are distinguishable by the differences in their lacunarity values. Higher lacunarity values (Fig. 13) characterise irregular and patchy arrangements of voids (variant fractals), while habitats with lower lacunarity are indicative for more regularly spaced voids, often combined with translationally and rotationally invariant fractal features. Lacunarity displays transitions from statistically self-similar to self-affine patterns, and together with multifractality, may best define all elements of habitat complexity (Schmid & Schmid-Araya, 2019; Schmid-Araya & Schmid, 2019). The mean lacunarity of the pore-space distribution of the streambed sediments (shown in Fig.9) is `0.30+-0.01` SE, which pertains to a patchy arrangement of different pore sizes.

References

Dewey, T.G. 1997 Fractals in molecular biophysics. Oxford University Press, Oxford.
Halsey, T. C. et al.1986.  Phys. Rev. A, 33, 1141.
Hentschel, H. & Procaccia, I. 1983.  Physica D, 8, 435.
Li, D. & Mao, J.- F. 2012. PIER, 126, 399.
Mandelbrot, B.B. 1983.  The fractal geometry of nature. W.H. Freeman & CO, New York.
Rényi, A. 1955. Acta Math. Hung. 6, 285.
Richardella, A. et al. 2010. Science 327, 665.
Schmid, P.E. & Schmid-Araya, J.M. 2007.  Body size and scale invariance: multifractals in invertebrate communities. In Hildrew, A.G., D.G. Raffaelli & R. Edmonds-Brown (eds) Body Size: The Structure and Function of Aquatic Ecosystems. Cambridge University Press, Cambridge. 140.
Schmid, P.E. & Schmid-Araya, J.M. 2019. Application of multifractal analyses to habitat complexity research in stream systems: a cross-scale approach. In: Tokeshi, M. (eds) Habitat complexity in aquatic systems: ecological perspectives.
Schmid-Araya, J. M. & Schmid, P. E. 2019. Habitat complexity and meiofauna: a review and new insights on food-search dynamics of organisms across habitat patches of varying complexity. In: Tokeshi, M. (eds) Habitat complexity in aquatic systems: ecological perspectives.
Schroeder, M. 1996.  Fractals, chaos and power laws. Minutes from an infinite paradise.W.H. Freeman and Company, New York.
Seuront, L. 2009. Fractals and multifractals in ecology and aquatic science. CRC Press, Taylor & Francis Group, NY.
Ţălu, Ş. & Stach, S. 2014. Polym. Eng. Sci. 54, 1066.

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