Nonlinear Processes in Geophysics Projection of compact fractal sets : application to diffusion-limited and cluster-cluster aggregates

The need to assess the three-dimensional fractal dimension of fractal aggregates from the fractal dimension of two-dimensional projections is very frequent in geophysics, soil, and atmospheric sciences. However, a generally valid approach to relate the twoand three-dimensional fractal dimensions is missing, thus questioning the accuracy of the method used until now in practical applications. A mathematical approach developed for application to suspended aggregates made of cohesive sediment is investigated and applied here more generally to Diffusion-Limited Aggregates (DLA) and Cluster-Cluster Aggregates (CCA), showing higher accuracy in determining the three-dimensional fractal dimension compared to the method currently used.


Introduction
Fractal aggregates occur very commonly at many scales in geophysical observations such as, for instance, in cosmology (e.g., galaxy distribution and patterns at large scales, Coleman and Pietronero, 1992;Thieberger and Spiegel, 2000), physics of the atmosphere (e.g., clouds, aerosols, contaminant dusts, Friedlander, 1977;Shaw, 2003), soil sciences (e.g., soil water retention properties, Perrier et al., 1996), and sedimentology (e.g., flocculation of cohesive sediment, Kranenburg, 1994;Winterwerp, 1999;Maggi, 2007).Determining the three-dimensional (3-D) capacity dimension of geophysical fractal aggregates is particularly beneficial to scale geometrical properties (e.g., effective density, porosity, strength, mass density distribution, etc.) with a characteristic length scale (Meakin, 1991).However, direct assessment of the 3-D capacity dimension is rare in practice due to technical limitations of observational instruments.More likely, 2-D projections can be assessed from optical measurements.The Correspondence to: F. Maggi (fmaggi@berkeley.edu)2-D capacity dimension has been used up to now to estimate the 3-D capacity dimension as follows.Given a 3-D fractal set (aggregate) S 3 ⊆ R 3 and its 2-D projection S 2 ⊆ R 2 obtained by application of the linear transformation (projection) P : R 3 → R 2 the capacity dimensions d 0 of S 2 and S 3 are assumed to relate as d 0 (S 2 )= min{2, d 0 (S 3 )} (Vicsek, 1992;Meakin, 1998).
The inverse form (1) is commonly used to estimate d 0 (S 3 ) from d 0 (S 2 ).However, there are three major limitations using Eq. ( 1): (i) it assumes that S 2 and S 3 are indefinitely extended sets; (ii) for d 0 (S 2 )=2 the 3-D capacity dimension d 0 (S 3 ) is undetermined; (iii) for projections P : R n →R m with n>m, the generalized dimensionality d q (Grassberger and Procaccia, 1983) is dimension-preserving only for 1<q≤2, whereas for the capacity dimension d 0 , which equals d q for q=0, the transformation P is not dimension-preserving (Hunt and Kaloshin, 1997;Thieberger and Spiegel, 2000).
These aspects rather suggest that, for compact (finite and closed) sets like the ones obtained by empirical observation, d 0 (S 3 ) =d 0 (S 2 ) also when d 0 (S 2 )<2.This analysis questions the applicability of Eq. (1) to geophysical observations.Errors introduced using Eq.(1) were shown in Maggi and Winterwerp (2004) when applied to kaolinite aggregates formed in water suspensions.In the same instance, an empirical relation that used the perimeter-based fractal dimension d P was proposed to compute d 0 (S 3 ) from projections S 2 , thus circumventing the use of the 2-D capacity dimension d 0 (S 2 ) in favor of alternative information not belonging to the generalized dimensionality d q .This approach yielded the following (Maggi and Winterwerp, 2004) Published by Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union.2), for different resolutions =L/ , where L is the length scale of the aggregate and is the box size of the -covering. where is the perimeter-based fractal dimension (Meakin, 1998), with P and A the dimensionless perimeter and area of S 2 , respectively, and a( ) and b( ) defined as known for any given resolution =L/ , with L the size of S 2 and S 3 , and the box size of the -covering of S 2 (Maggi and Winterwerp, 2004).Equation ( 2) is not resolution-limited as far as the disequality (4 − 4) ≤ (1+ √ 4d 0 (S 3 )−3)/2 is satisfied (Maggi and Winterwerp, 2004).A representation of Eq. ( 2) is given in Fig. 1 for various values of , showing that the accuracy in estimating d 0 (S 3 ) from d P (S 2 ) increases when d P (S 2 ) can range over the full interval [1, 2], that is, for increasing .
The aim of this work is threefold.First, we derive the condition that any relationship between d 0 (S 2 ) and d 0 (S 3 ) must satisfy for compact fractal sets.Second, we show that both Eqs.(1) and (2) satisfy such a condition.Third, we show that Eq. (2) produces better results than Eq. ( 1), and that its validity extends to DLA (diffusion-limited) and CCA (clustercluster) aggregates, typologies which occur commonly in a variety of situations in geophysics.

Method
2.1 Relationship between d 0 (S 2 ) and d 0 (S 3 ) A general relationship between d 0 (S 2 ) and d 0 (S 3 ) can be derived as follows.For a compact fractal aggregate S 3 ⊂ R 3 of length scale L, the volume, V , scales as V =L d 0 (S 3 ) with 0 ≤ d 0 (S 3 ) ≤ 3 the 3-D capacity dimension of S 3 (Meakin, 1991).In analogy, the projected area, A, scales as A=L d 0 (S 2 ) with 0 ≤ d 0 (S 2 ) ≤ 2 the 2-D capacity dimension of the projected set S 2 of S 3 .Under the assumption that L is the length scale for both S 2 and S 3 , the volume V must be (3) Substituting the scalings for V and A into Eq.( 3) we obtain This inequality sets the condition that must be respected whatever the function relating d 0 (S 2 ) and d 0 (S 3 ).Condition (4) can be generalized for projections P : R n → R m of compact fractal sets in any embedding dimensions n>m as (5) Note that d 0 (S n )=d 0 (S m ) + (n − m) represents the relationship between d 0 (S n ) and d 0 (S m ) of an aggregate in R n and its cross-section in R m (Vicsek, 1992).

Application to diffusion-limited and cluster-cluster aggregates
Geophysical fractal aggregates produced by various kinematic processes generally show different geometrical organization.For instance, DLA aggregates, which are formed by attachment of individual particles that move in space by diffusion, are featured by radial development of stellar-like branchings.Instead, CCA aggregates, which are formed by clusters of particles that recursively attach to each other and move in space by diffusion, display clustered masses connected by bridges, forming (sometimes closed) rings.Fully developed DLA and CCA aggregates usually have different capacity dimensions in R 3 , that is d 0 (DLA)≈ 2.5 and d 0 (CCA)≈ 1.8, respectively (Vicsek, 1992).However, projections P of those aggregates may have similar capacity dimensions d 0 (S 2 ) in R 2 , thus preventing to discerning aggregate type and 3-D capacity dimension from the capacity of S 2 .A conceptual example is given in Fig. 2, where two sets B and C of different outlook (ramified the former and ringshaped the latter) have identical capacity dimension d 0 but different perimeter-based fractal dimension d P in R 2 .For this test, we have constructed 10 DLA aggregates and 10 CCA aggregates of 1000 seeds according to the algorithms described in Vicsek (1992) and Meakin (1998) (Fig. 3).Next, 2-D projections in the three principal directions have been computed.From these we have calculated the average d P (S 2 ) and d 0 (S 2 ), terms used in Eqs. ( 1) and ( 2) to compute d 0 (S 3 ).When Eq. ( 2) has been used, an additional computation has been carried out to make sure that the disequality (4 −4)≤ (1+ been compared to the ones obtained by construction (effective d 0 (S 3 ) in Fig. 4).
The results in Fig. 4 show that application of Eq. ( 2) always gives better estimations of d 0 (S 3 ) from the projections compared to Eq. ( 1).The loss of information induced by P on the sets affects their capacity, but preserves the information at the boundary of the projections.Because the perimeter-based fractal dimension d P (S 2 ) is representative of the boundary segmentation rather than the set capacity, Eq. ( 2) allows better prediction of d 0 (S 3 ).These results show that the geometrical organization of the boundary of the sets S 3 can be considered P-invariant, and that this aspect can be fruitfully exploited to estimate d 0 (S 3 ) of real geophysical fractal aggregates.

Exceptions
Two particular cases exist for which Eq. ( 2) is biased: regular solids with porous structure (sponge-like) and Euclidean solids with fractal surface (rock-like).
For the former (sponge), consider as hypothesis a regular cube with a high level of fine porosity so that d 0 (S 3 )<3 (Fig. 5a).The projected image S 2 appears fully massive and its contour does not present segmentation.Consequently, d P (S 2 )=1 for → ∞, and from Eq. ( 2) we obtain d 0 (S 3 )=3, which contradicts the hypothesis.
For the latter (rock), consider as hypothesis a fully massive solid with capacity dimension d 0 (S 3 )=3 and with a fractal (rough and irregular) surface (Fig. 5b).Its projected image S 2 is fully massive but the contour shows a high level of perimeter segmentation.As a consequence, d P (S 2 )>1 and from Eq. ( 2) we obtain d 0 (S 3 )<3, which again contradicts the hypothesis.

Summary
A simple method to compute the three-dimensional capacity dimension from two-dimensional projections of compact fractal sets is tested on DLA and CCA aggregates, and compared to the approach more commonly used in geophysics.The method presented here makes use of the perimeter-based fractal dimension and does not obey the mathematical limitations that apply to the capacity dimension and the generalized dimensionality.The numerical simulations presented here show that this method produces better predictions of the three-dimensional capacity dimension in the two aggregate typologies (DLA and CCA) analyzed here compared to the current state-of-the-art method.

Fig. 1 .
Fig. 1.Representation of the 3-D capacity dimension d 0 (S 3 ) as a function of the perimeter-based fractal dimension d p (S 2 ), Eq. (2), for different resolutions =L/ , where L is the length scale of the aggregate and is the box size of the -covering.

NonlinFig. 2 .
Fig. 2. Examples of sets with identical capacity dimension d 0 and different perimeter segmentation.Computation of the perimeterbased fractal dimension d P has been done neglecting the inner voids, according to Maggi and Winterwerp (2004).

Fig. 3 .
Fig. 3. Examples of DLA and CCA aggregates.Their 2-D projections are shown beside the 3-D views.

Fig. 5 .
Fig. 5. (left) regular solid with porous (fractal) structure and massive (non-fractal) projection and, (right), Euclidean solid with fractal surface and massive projection with highly segmented contour.Light gray means porous structure, while dark gray means Euclidean structure.