arXiv:cond-mat/0110476v1  [cond-mat.stat-mech]  22 Oct 2001
Lattice Model for Approximate Self-Aﬃne
Soil Proﬁles
A.P.F. Atman a,
J.G. Vivas Miranda b,
A. Paz Gonzalez b,
J.G. Moreira a,
aDepartamento de F´ısica, Instituto de Ciˆencias Exatas, Universidade Federal de
Minas Gerais, C. P. 702 30123-970, Belo Horizonte, MG - Brazil
bInstituto Universitario de Xeolox´ıa “Isidro Parga Pondal”, Universidade da
Coru˜na, La Coru˜na, Spain
Abstract
A modeling of the soil structure and surface roughness by means of the concepts
of the fractal growth is presented. Two parameters are used to control the model:
the fragmentation dimension, Df, and the maximum mass of the deposited aggre-
gates, Mmax. The fragmentation dimension is related to the particle size distribution
through the relation N(r ≥R) ∼RDf , where N(r ≥R) is the accumulative num-
ber of particles with radius greater than R. The size of the deposited aggregates
are chose following the power law above, and the morphology of the aggregate is
random selected using a bond percolation algorithm. The deposition rules are the
same used in the model of solid-on-solid deposition with surface relaxation. A com-
parison of the model with real data shows that the Hurst exponent, H, measured
via semivariogram method and detrended ﬂuctuation analysis, agrees in statistical
sense with the simulated proﬁles.
Key words: surface roughness, soil modelling, self-aﬃne proﬁles
PACS: 05.45.Df, 83.70.Fn, 92.40.Lg
Email addresses: atman@ﬁsica.ufmg.br (A.P.F. Atman), vivas@mail2.udc.es
(J.G. Vivas Miranda).
URL: www.ﬁsica.ufmg.br/ atman (A.P.F. Atman).
Preprint submitted to Elsevier Preprint
1 November 2018

1
Introduction
The concepts of the fractal geometry have been widely used to describe and
quantify irregularity in nature, and statistical self-aﬃne properties have re-
cently been identiﬁed in various Earth’s terrain [1] and proﬁles [2]. Several
works in soil science incorporates the fractal geometry to describe and model-
ing soil physical properties, soil physical processes and quantify the soil spatial
variability [3–6].
A very important property closely related to the fractal geometry is the soil
surface roughness, deﬁned as the conﬁguration of the soil microrelief. The soil
surface roughness exerts great inﬂuence on water inﬁltration, erosion and run-
oﬀeﬀects. Its quantiﬁcation is important for understanding the soil behavior
during degradation processes like rainfall erosion or abrupt changes such as
those induced by tillage [7]. In the last years, a considerable eﬀort was done to
simulate the soil structure; several models propose the simulation of the parti-
cle size distributions [4], the soil surface roughness [3,8], the morphology of the
pore-solid structure [9], etc. The major properties considered in these models
are the fractal dimension of the soil surface, D, the particle size distribution
(PSD), the pore size distribution and the surface roughness.
In this work we present a simple lattice model to simulate the soil structure and
reproduce the surface soil roughness. We basically use the ideas of the fractal
growth models [10] to generate an approximate self-aﬃne proﬁle, that exhibit,
as much as possible, a similitude with the fractal properties of real soils. It is
approximate in the sense that the scaling properties of these proﬁles are valid
only in a limited range of scales. Therefore, the model was validated by means
of a comparison between the fractal dimension estimated for simulated proﬁles
and for natural soil surfaces. The major improvements in our model are the
power law distribution of the sizes of the deposited particles or aggregates
and a random selection in the allowed morphologies of the aggregates. This
two features, not improved before by any model, are responsible for the soil
structure in our model. In the section II we present a detailed description of
the model, showing the particle size distribution and explaining the algorithm
for the random choice of the morphologies. We also present a brief summary
of the theory of growth surfaces. In the section III is presented the results
for several maximum aggregates sizes and diﬀerent fragmentation dimensions.
Finally, in the section IV we present our conclusions and perspectives.
2

1
10
R (= sqrt(Mmax/π))
10
0
10
1
10
2
10
3
10
4
10
5
N(r>R)
Fig. 1. Particle size distribution for a simulated soil with Mmax = 1000 and
Df = 3.0. The radius of the particles are calculeted via the expression: r2 = M/π,
that approximates the particles by discs with area M. Note the power law behavior
for the particles sizes.
2
The Model
The motivation in elaborate this model lies in aptitude to simulate some of
the majors properties of agricultural soil, like self-aﬃne proﬁles and porous
medium. The model have to allow the possibility to change some parameters,
as the aggregate size distribution, and the maximum mass of the aggregates.
These two parameters try to cover the structural variability found in natural
soils. The aggregate size distribution is assumed to have power law behavior
[5,6], and their exponent, Df, is one of the parameters of the model. The other
parameter, Mmax, is the maximum mass of particles utilized to generate the
aggregate, which is approximately related with the square of maximum ag-
gregate radius. In spite of others classical lattice models, where the particles
generated always have the same morphology, this model simulate the vari-
ability observed in natural soil, generating aggregates by a bond percolation
algorithm. All this characteristics have to be integrated in a very simple model
to allow the simulation of relatively large systems (L = 1000).
In this way, the major improvement to choose a lattice model is the simplicity
and the velocity to run the code, what possibilities the simulation of large
systems with great quantity of deposited particles (hundred millions). This
massive number of deposited particles are needed in order to avoid the eﬀect
of the constant attachment of micro random variability, represented by the
random morphology of the aggregates. Another advantage to work in the
lattice, is the theoretical background furnished by the fractal growth theory,
presented by Barab´asi and Stanley [10].
We consider the soil structure composed by a set of particles and aggregates
whose radius are power law distributed according to the following relation,
3

known as Turcotte’s empirical law [5],
N(r > R) ∼R−Df ,
(1)
where N(r > R) is the cumulative number of particles (or aggregates) with
radius r greater than R and Df is the fragmentation dimension of the particle
size distribution (PSD). There is some controversy in the literature about the
allowed range of the values of the Df. Tyler & Wheatcraft [11] argued that
Df < 3 because, under usual hypothesis (constant density, spherical particles,
etc), the mass distribution scales with M(r > R) ∼RD ∼R3−Df, where D
is the fractal dimension of the soil. Thus, considering a fractal distribution,
only the values Df < 3 have physical meaning. Martin & Taguas [4] present
several mathematical arguments supporting this conjecture, and show some
PSD simulations. However, there are several experimental studies, summarized
by Perfect & Kay [5] where the range of values of Df is 2.6 < Df < 3.5.
Gimenez et al [12] also aﬃrm that there is not any experimental relation
between the fractal dimension and the fragmentation dimension. We consider
that the usual hypothesis of constant density of the particles is not valid
when the radius of the aggregates grows up, due the presence of pores in the
structure. Thus, fragmentation dimensions greater than 3 are, in principle,
allowed.
According to the USDA classiﬁcation of soil texture [13], the basic particle
size classiﬁcation is:
• sand 50 < r < 2000µm
• silt 2 < r < 50µm
• clay r < 2µm.
The real soils can vary widely in the percentile of each range of sizes. The
study of Nemes et al proposes a standardisation of the classiﬁcation of the
European soils and shows several experimental data [13]. In the ﬁgure 1, we
present a typical PSD generated by the algorithm of the model. This PSD is
constructed considering that the particles are approximately circles; so, there
is a direct relationship between the mass number and the particle radius, that
is used to build the PSD.
The model has two parameters, the fragmentation dimension, Df, and the
maximum mass of the aggregates, Mmax. To every particle is selected a mass
number, M; when M > 1, the model choose a random conﬁguration for it;
this choice is one of the possible bond percolation clusters with a given size.
In the ﬁgure 2, we show some of the possible aggregate morphologies. These
two parameters try to cover, in a simple way, the structural variability found
in natural soils.
The deposit of each particle follows the model of the solid-on-solid deposition
with surface relaxation [10]. The initial position for the deposition is random
4

Fig. 2. A) Possible morphologies for M = 4. In this case, the morphologies are
the same observed in a popular game called “Tetris”. B) Some morphologies for
M = 400. Three morphologies are shown. Note the porous structure of the aggre-
gates, and its random shape. C) Soils proﬁles generated by the model. Above: typical
deposition in a lattice with L = 200, 20000 particles deposited with Mmax = 100
and Df = 3.0. Below: the same, with Df = 2.0. Note the variability of the soil
structure with the fragmentation dimension.
chose and the particle follows a straight line until touch the soil surface. Then,
the algorithm simulate a surface relaxation (without rotation) to the particles,
which only are adhered to the bulk when they reach the site with the local
minimum energy .
Considering the theory of growth surfaces, the solid-on-solid deposition be-
longs to the universality class of the Edwards-Wilkinson equation
∂h(x, t)
∂t
= ν∇2h + η(⃗x, t)
,
(2)
where h(⃗x, t) is the height of the site ⃗x at time t, and η(⃗x, t) represents a ran-
dom noise, that expresses the ﬂuctuations in the arrive of the particles and had
the following properties, < η(⃗x, t) >= 0 and < η(⃗x, t) η(⃗x′, t′) > = 2C δd(⃗x−
⃗x′) δ(t −t′).The δ is the usual Kronecker delta and C is a diﬀusion constant.
The width of the growth surfaces, w(L, t) then obeys a scaling relation
w(L, t) ∼Lαf(t/Lz) ,
(3)
where f(u) is a scaling function: f(u) ∼uβ for u << 1 and f(u) is a constant
for u >> 1; α, β and z are the roughness, the growth and the dynamical
exponents, respectively. They are known as the scaling exponents and are
5

10
0
10
2
10
4
10
6
10
8
number of deposited particles
0
0
1
10
100
w(L,t)
Nmax = 01
Nmax = 05
Nmax = 10
Nmax = 50
Nmax = 100
Nmax = 500
Nmax = 1000
0.15
0.25
0.35
0.45
0.55
slope
Df = 1.5
Fig. 3. Roughness of the simulated proﬁles, with several Mmax and Df = 3.0. Note
the variation of the slope between the two vertical dotted lines, shown in the small
window. The slope is measured with a rule with ﬁxed length, and varies the initial
point since the left vertical line to the right.
related by the scaling law
z = α
β
.
The EW scaling exponents are known exactly: α = 1/2, z = 2.
Another equation related to the deposition is the Kardar-Parisi-Zhang equa-
tion, that has a nonlinear term,
∂h(x, t)
∂t
= ν∇2h + λ/2(∇h)2 + η(⃗x, t)
.
(4)
To this equation the calculeted exponents are α = 1/2, z = 3/2.
So, we perform simulations to test the inﬂuence of the model parameters Df
and Mmax in the scaling exponents and fractal dimension of the soil proﬁles,
and compare the results with the experimental data disposable.
3
Results
We perform simulations in one dimensional lattice with length L = 1000 sites.
The results for the scaling exponents represent the average value in a set of
20 samples with the same parameters in each simulation, and diﬀerent depo-
sition sequence of the random aggregates. The number of particles/aggregates
6

deposited in each sample is N = 100000000. The behavior of the local width,
w(L, T), deﬁned by
w2(L, t) = 1
L
L
X
i=1

hi(t) −h(t)
2
,
(5)
with the model parameters are shown in the ﬁgure 3. Note the roughness satu-
ration at N ∼10000000. The fractal dimension of the ﬁnal proﬁle is estimated
using the Hurst exponent H. The Hurst exponent is associated to the fractal
dimension via the relation D = 2 −H. There are several ways to measure H.
At this work, we use the detrended ﬂuctuation analysis (DFA), improved by
the ﬁrst time by Moreira et al [15], and the semivariogram method, that uses
a height-height correlation function.
The DFA consists in measure the roughness around the mean square straight
line [15]. The roughness W(L, ǫ, t) at the scale ǫ, is given by
W(L, ǫ, t) = 1
L
L
X
i=1
wi(ǫ, t)
(6)
and the local roughness wi(ǫ, t) is deﬁned by
w2
i (ǫ, t) =
1
2ǫ + 1
j=i+ǫ
X
j=i−ǫ
{hj(t) −[ai(ǫ)xj + bj(ǫ)]}2
(7)
where ai(ǫ) and bi(ǫ) are the linear ﬁtting coeﬃcients to the displacement data
on the interval [i −ǫ, i + ǫ] centered at the site i. Self aﬃne proﬁles satisfy the
scaling law
W(ǫ) ∼ǫH
(8)
that is used to measure H. The semivariogram method estimates the spa-
tial variability, through the calculus of the semivariance as a function of the
distance between points. This function is can be estimated by:
γ(ǫ, t) =
1
2n(ǫ)
n(ǫ)
X
i=1
[hi(t) −hi+ǫ(t)]2 ,
(9)
where hi(t) the height in the location i at time t, and n(ǫ) represents the
number of pairs of points which are separated by ǫ. In the case of self-aﬃne
proﬁles γ exhibit a power law behavior,
γ(ǫ) ∼ǫ2H ,
7

1
10
100
1000
Mmax
0.30
0.40
0.50
0.60
0.70
0.80
H
10
-1
10
0
10
1
Df
0.2
0.3
0.4
0.5
0.6
0.7
H
1
10
100
1000
Mmax
0.15
0.25
0.35
0.45
β
10
-1
10
0
10
1
Df
0.1
0.2
0.3
0.4
0.5
β
A
B
C
D
Mmax = 100
Mmax = 100
Df = 3.0
Df = 3.0
Fig. 4. Summary of the exponent values. A) and C): Behavior of the Hurst expo-
nent with the model parameters. Note that there is not a remarkable trend to the
Hurst exponent with any model parameters. The range of experimental values is
0.3 < H < 0.7. B) and D): Behavior of the growth exponent with the model param-
eters. Note the dependence of the growth exponent with the average radius size: as
the radius increase, the growth exponent increases too. The dependence of β with
the fragmentation dimension is inverse.
and, in the same manner as in the DFA method, the H exponent can be
utilized to calculate the fractal dimension.
The semivariogram method is especially useful in cases where the data are
not regular spaced, and for this reason, is widely used in estimating fractal
dimension of soil surfaces [5].
The range of the H values is shown in the ﬁgure 4A and 4B. In the ﬁgure
4C and 4D, we show the dependence of the growth exponent with the model
parameters. We conclude that the increase of the maximum particle mass
Mmax have similar eﬀects to the decrease of the fragmentation dimension: both
introduces a nonlinear correlation in the system, expressed by the increase in
the value of the growth exponent β, changing the universality class of the
deposition. Nevertheless, the fractal dimension of the surface do not alters
considerably with the model parameters.
4
Conclusions and Perspectives
In this work, we present a new model to simulate the soil structure and its
surface roughness. Two parameters are used to control the simulated proﬁles:
the maximum mass of the particles and the fragmentation dimension. The
major improvements of this model are: the random conﬁguration allowed to
the particles or aggregates and the power law distribution of its radius. These
two features are not present in any model discussed in the literature until to-
8

day, and permits the reproduction of the variability observed in natural soils.
The results obtained shows a good agreement for the exponent H calculated
from the simulations and measured experimentally [7]. We also observe a de-
pendence of the growth exponent β with the maximum mass allowed to the
particles: as the Mmax grows, the value of the β exponent grows from the EW
value (β = 1/4) to the KPZ value (β = 1/3). So, the increase of the averaged
particle radius corresponds to the introduction of nonlinear correlations into
the system.
Next, we expect increase the system size and control better the shape of the
deposited particles, in order to perform simulations closer to the real soils. We
also intend simulate the rainfall eﬀect over the simulated structure, to verify
the dependence of the Hurst exponent with the rain [14].
This work was sponsored by the brazilian agencies CNPq, FAPEMIG and
FINEP, and by the spanish agency AECI.
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