Thesis_DinaElFilali_ApplicabilatyOfFractalGeometryToCharacterizeGullyFormation_2014

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Applicability of fractal geometry to
characterize gully patterns and formation
in a highly eroded catchment of SE Spain
BSc thesis by Dina el Filali
September 2014
Soil Physics and
Land Management Group

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Applicability of fractal geometry to characterize gully patterns and
formation in a highly eroded catchment of SE Spain
Bachelor thesis Soil Physics and Land Management Group submitted in
partial fulfillment of the degree of Bachelor of Science in International Land
and Water Management at Wageningen University, the Netherlands
Study program:
BSc International Land and Water Management
Student registration number:
9000707-240-030
YEI 80812
Supervisors:
WU Supervisor: Dr. Ir. Jantiene Baartman
Host supervisor: Dr. Joris de Vente
Examinator:
Prof. Coen Ritsema
Date:
14/07/201
Soil Physics and Land Management Group, Wageningen University

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"I am conscious that I am in an utterly hopeless muddle. I cannot think that the world, as
we see it, is the result of chance; and yet I cannot look at each separate thing as the result
of design."
Charles Darwin
"Where there is matter, there is geometry."
Johannes Kepler
“I show that behind their very wildest creations, and unknown to them and to several
generations of followers, lie worlds of interest to all those who celebrate Nature by trying
to imitate it.”
Benoit Mandelbrot

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Table of Contents
Preface................................................................................................................................................................ 6
Acknowledgements............................................................................................................................................ 6
Abstract .............................................................................................................................................................. 7
Introduction........................................................................................................................................................ 8
1.1. Gully and channel formation ............................................................................................................ 9
1.1.1. Drivers of gully development ....................................................................................................... 9
1.1.2. Factors controlling gully development ....................................................................................... 10
1.2. Fractals............................................................................................................................................ 10
1.2.1. Historical background................................................................................................................. 10
1.2.2. Definition and characteristics..................................................................................................... 11
1.3. Applicability of fractals in geomorphology ..................................................................................... 13
1.3.1. Basin morphometric analyses..................................................................................................... 13
1.3.2. Box-Counting method................................................................................................................. 14
1.3.3. Geomorphometry....................................................................................................................... 15
1.3.4. Fractal drainage basin analyses.................................................................................................. 15
2. Methods ................................................................................................................................................... 17
2.1. Study Area....................................................................................................................................... 17
2.2. Methodological design.................................................................................................................... 18
2.3. Data preprocessing and gathering.................................................................................................. 19
2.3.1. Preprocessing ............................................................................................................................. 19
2.3.2. Data gathering ............................................................................................................................ 19
2.4. Data processing............................................................................................................................... 19
2.4.1. Geomorphology.......................................................................................................................... 19
2.4.2. Land use...................................................................................................................................... 20
2.5. Data Analyses.................................................................................................................................. 20
2.5.1. Land use change ......................................................................................................................... 20
2.5.2. Morphological changes............................................................................................................... 20
2.5.3. Statistical analyses...................................................................................................................... 21
2.5.4. Fractal analyses .......................................................................................................................... 21
3. Results ...................................................................................................................................................... 23
3.1. Mapping and characterizing gully networks................................................................................... 23
3.2. Land use change.............................................................................................................................. 23

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3.2.1. Mapping and characterizing land use change ............................................................................ 23
3.2.2. Calculating land use change ....................................................................................................... 25
3.3. Geomorphological changes ............................................................................................................ 27
3.3.1. Length and sinuosity................................................................................................................... 27
3.3.2. Sinuosity and Strahler’s order .................................................................................................... 27
3.3.3. Basin Morphometry.................................................................................................................... 28
3.4. Statistical and fractal analyses........................................................................................................ 28
3.4.1. Self-similarity in channel profiles................................................................................................ 28
3.4.2. Fractal dimensions of channel network...................................................................................... 30
4. Discussion................................................................................................................................................. 31
4.1. Fractal geometry of gully systems .................................................................................................. 31
4.2. Changes in fractal geometry over time and the role of land use change ....................................... 32
4.3. Methodological uncertainties......................................................................................................... 33
5. Conclusion ................................................................................................................................................ 34
References........................................................................................................................................................ 35
Annex 1. Land use change effects on river channel .................................................................................. 39
Annex 2. Fractal relationship..................................................................................................................... 41
Annex 3. Walking Divider Fractal dimension calculating method ............................................................. 42
Annex 4. Morphometric parameters......................................................................................................... 43
Annex 5. Cross-section field area .............................................................................................................. 44
Annex 6. Land use change map ................................................................................................................. 45
Annex 7. Land use change table ................................................................................................................ 46
Annex 8. Group statistics and independent sample Test .......................................................................... 47
Annex 9. Correlation and regression (Width and Depth) .......................................................................... 48
Annex 10. Fractal Dimension Output data .................................................................................................. 49

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Preface
As part of the major BIL (Bachelor International land and Water management) and in the context of
acquiring international knowledge and expertise in the field of ILW (International Land and Water
management) students carry out an individual thesis research abroad during their BIL completion year.
For the past couple of years a consciousness nurtured within me, that our four-dimensional world, as Albert
Einstein once pointed out by adding the fourth dimension space-time, and which we refer to as our reality
can, regardless of the used tools, equations, variables and related parameters, only be approximated. Thus
when approximating or examining phenomena, they can either be studied in a reductionist way, based on
their consisting element/agents, or from a holistic point of view, based on the interaction of agents on
different levels of understanding. It has already been observed by many scientists that, of all the possible
rules to govern the interaction between agents, nature uses the simplest. Even more, these simple rules are
found on different levels of understanding, for example on a level on which neurons branch till the level on
which cities grow, and are used on diverse fields of science. Following this argumentation, an important
question is whether or not it would be possible, if understanding these rules to attain more insight in the
computation or simulation of landscape evolution and degradation processes such as soil erosion and even
more specific gully erosion?
Acknowledgements
This thesis report is the outcome of a journey during my BIL completion year as an attempt to describe the
natural phenomenon of gully formation from a holistic point of view using the concept of fractal geometry.
However this beautiful journey would have been impossible without my supervisors’ guidance, inspiration,
feedback and facilitation. I therefore would really want to thank dr. ir. Jantiene Baartman from Wageningen
University and dr. Joris de Vente at CEBAS-CSIC for making it all possible. Furthermore, last but not least, I
would also like to thank my lovely housemates, family and friends for their support and curious questions
throughout the whole processes.

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Abstract
Field-based studies indicate that gully erosion is an important soil degradation process, causing considerable
soil losses and producing large volumes of sediment. Gullies act as operative linkages for runoff and
sediment transport from uplands to low-lands, and have deteriorating offsite effects such as floods and
damage to infrastructures. Despite its large environmental and societal impacts, still little is known about
the initiation and further development of gully networks leading to an important knowledge gap in modeling
soil erosion processes and landscape evolution. In the early 1970s, mathematician Benoit Mandelbrot
introduced the fractal geometry of nature and stated that, every natural phenomenon exists of a fractal, a
simple shape that iterates itself defined by self-similarity. His concept of fractal geometry has since then
been used by numerous scholars on diverse fields. This research examined the possibility of applying the
concept of fractal geometry on the natural phenomenon gully formation, whereby the fractal characteristics
self-similarity, the degree to which an object is similar on different scales, and fractal dimensions,
determining the change in detail with change in scale, are used. Hence, a correlation is sought between the
concave profiles on different scales and change in fractal dimensions as a consequence of land use change in
the Carcavo basin situated in SE Spain. Results indicate self-similarity based on the correlation between the
variables width and depth of the concave profiles on different scales, from gullies all the way to valley.
Additionally when calculating the change of five fractal dimensions from 1956 and 2011, a decline in four
out of five fractal dimensions, was found as a consequence of the most estimated land use change,
reforestation. Therefore even though the change in fractal dimensions used as fractal characteristic of the
gully network have not changed significant, it can be concluded that reforestation as the highest arable and
percentage calculated land use change, determines a decline in the fractal dimensions of a gully network.

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Introduction
Many studies have indicated that gully erosion represents an important sediment source in a variety of
environments and is considered as an important sign of land degradation (Vandekerckhove et al. 2003;
Shruthi et al. 2014). Gullies act as operative linkages for runoff and sediment transport from uplands to low-
lands (soil-loss), and have deteriorating off-site effects such as floods and damage of infrastructures
(Poesen, 2003). Poesen et al. (2003) state, that once gullies develop they increase the connectivity in the
landscape. Therefore, many cases of damaged waterways, either by sediment or due to transported
chemicals by runoff from agricultural field, relate to ephemeral gullying (Poesen et al. 2003). Consequently
as the existence of gullies influence the erosion intensity, the estimated soil losses of environments with
large gullies or dense gully networks such as badlands, are always higher than 50 t ha
–1
yr
–1
(Alonso-sarria et
al. 2011).
The quantification of gully erosion dynamics can assist in understanding gully formation and spatiotemporal
evolution (Shruthi et al. 2014). Insight on gully processes is not only important from a scientific or
geomorphological point of view, but also to enable land managers to develop sustainable planning
strategies for appropriate land utilization including both the stabilization of gullies as well as the prevention
of gully formation in erosion sensitive areas (Shruthi et al. 2014).
Although throughout the years several attempts have been made to develop models in order to either
predict gully sub processes or gully erosion in a range of environments, there are still no reliable models
available (Poesen et al. 2003). Thus in order to attain insight on the effects of environmental changes either
climatic or land use, there is still a need for further studies of gully erosion such as monitoring, experimental
and modeling studies (Poesen et al. 2003).
In an attempt to describe natural shapes and patterns, in the early 1970s, mathematician Benoit Mandelbrot
stated in his book The Fractal Geometry of Nature that, ”Clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line” (Mandelbrot
1982). According to Mandelbrot every natural phenomenon exists of a fractal, a simple shape that iterates
itself defined by self-similarity. Fractal geometry makes it possible to use a new vocabulary in order to
understand nature, determining that the similarity in a single tree is comparable to a whole rainforest.
Mandelbrot (1992) stated, “When you look at nature, you should not look at what you see, but what it took
to produce what you see. Instead of using formulas use pictures with the purpose of finding order in
disorder.”
Furthermore fractal geometry also appears to be a useful tool for the characterization of natural growth
patterns (Mandelbrot 1982). According to Barbera (1992), this statement is particularly true for the case of
complex non-deterministic patterns which are difficult (even conceptually impossible) to quantify using
Euclidean geometry. Especially for fluvial landscapes, fractal geometry appears to be useful (Barbera 1992).
By this means, Klinkenberg (1992) stated “As shown in many illustrations the inconsistency of the land
surface is important for determining the inconsistency of the phenomenon being studied and thus a single
variable as D (fractal dimension) may contribute significantly to the explanation of the perceived irregularity.
In that way, if by scaling, self-similarity is found it should be possible to simplify a natural
phenomenon”(Klinkenberg 1992).

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The question remains what is gully formation, what causes this natural phenomenon to occur and what are
the drivers behind it? To investigate if the theory of fractal geometry can also help answering these
questions we first need to know what are fractals exactly, and how can fractals help provide further insight
on gully formation? Therefore, the overall aim of this research is to provide further insight into the
occurrence and development of gully systems by applying the concepts of fractal geometry to the highly
dissected Carcavo catchment in SE Spain. Moreover, to evaluate the importance of natural versus
anthropogenic drivers of gully development, we compared the fractal dimension of a study catchment
characterized by important land use changes for different years (1956 and 2011). The specific objectives are
to: (i) Define the fractal geometry and occurrence of self-similarity within gully networks for different years
(1956; 2011); (ii) Quantify the relationship between environmental variables, historic land use changes and
the gully network; (iii) Explore the possibility of using fractal geometry to simulate gully network
development.
1.1. Gully and channel formation
1.1.1. Drivers of gully development
As stated by Alonso-Sarria et al. (2011) the main causes of the formation of badland areas, rills and gullies
are linked to the following factors: (1) Lithology and stratigraphy, (2) mineralogy and geochemistry of
sediments, (3) tectonics, (4) climatic characteristics, (5) scarce vegetation and (6) poorly developed soils.
Furthermore, the irregularity and change between different rock types and degree of compaction affects the
permeability and mechanical properties of rocks (cohesion, resistance to break, - disaggregate) leading to
highly varied erosion forms. Climatic conditions characterized by high intense rainfall events cause soil loss
and high temperatures enhance soil dissection and the formation of surface cracks (Alonso-sarria et al.
2011).
With regard to channel geomorphological processes, Boix-Fayos et al. (2007) stated that land-use changes
and erosion-control may encourage strong geomorphic responses in catchments which is reflected in the
morphological evolution of river channels. Recent studies have shown that deforestation and related
phenomena such as overgrazing and increasing agricultural pressure widen the channels and increase the
sediment supply (Wasson et al. 1998). Whilst when natural or planned reforestation occurs, channel
narrowing, pavement development, stream incision, change pattern from braiding to meandering,
establishment of vegetation around bars and decrease in sediment supply is perceived (Stott & Mount 2004;
Liébault et al. 2002). In addition, all these effects due to reforestation and afforestation are strengthened if
accompanied by erosion control measures such as the construction of check-dams (Trimble 1999). Hence
Boix-Fayos et al. (2007) summarize in a table (Annex 1) the main land-use changes and their effect on
channel morphology from recent publications.
Referring to land-use type, a preliminary field survey confirmed according to Lesschen et al. (2007) the
assumption that abandoned fields have more gully erosion compared to cultivated fields. This can be
described by the quicker concentration of runoff on abandoned land, due to crust formation and reduced
surface storage capacity (Lesschen et al. 2007). Lesschen et al. (2007) also state that land abandonment has
become one of the main changes in land use in Mediterranean countries due to changing European policies,
urbanization, globalization, desertification and climate change. However even though, semi-natural
vegetation after abandonment might increase the vegetation cover, improve soil properties, and decrease
runoff and erosion (García-Ruiz et al. 1996). Soil and water conservation structures, such as terraces, might
collapse due to lack of maintenance and piping and consequently increase erosion (Faulkner et al. 2003).

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Finally regarding soil type, according to Vandekerckhove et al. (2000) gullies occur more frequently in soil
types with high silt content and therefore all marls and Quaternary fills are considered to be potentially
vulnerable to gully erosion. One of the erosion processes linked to badlands (gully formation) is piping
(Romero Díaz et al. 2007). Piping processes are often perceived near man-made forms such as bench
terraces, small earth dams (around gully head-cuts) and in association with land-leveling measures (Marzolff
2011). In addition, piping is one of the geomorphologic processes that are most difficult to quantify as they
can barely be examined by conventional field survey methods (Marzolff 2011).
1.1.2. Factors controlling gully development
For the prediction of potentially vulnerable gully erosion areas, soil type, land use, climate and topography
are the control factors most often used (Poesen et al. 2003, Lesschen et al. 2007). For the factor topography
the topographic threshold concept is often used, in order to approximate where a gully might develop
(Lesschen et al. 2007). The topographic threshold concept is represented by a negative power law relation
, where S represents the slope, A the drainage area and the coefficients a and b depend on the
environmental characteristics (Lesschen et al. 2007). According to Poesen et al. (2003), the S-A relation
describesthe position in a landscape where ephemeral and permanent gully heads may develop. By
calculating the critical slope and selecting areas where the slope exceeded the critical slope, Lesschen et al.
(2007) derived potential areas subjective to gully erosion. These derived areas were characterized by
abandoned land, had erodible soils, and lay in a topographic position where gully heads might develop
(Lesschen et al. 2007).
1.2. Fractals
1.2.1. Historical background
Although Benoit Mandelbrot is often seen as the father of the
concept “Fractals”, he by himself quotes in his book Fractal
geometry in nature the seventeenth century English scholar
Richard Bentley: "All pulchritude is relative... We ought not to
believe that the banks of the ocean are really deformed,
because they have not the form of a regular bulwark; nor that
the mountains are out of shape, because they are not exact
pyramids or cones; nor that the stars are unskillfully placed,
because they are not all situated at uniform distance. These
are not natural irregularities, but with respect to our fancies
only; nor are they incommodious to the true uses of life and
the designs of man's being on earth (Mandelbrot 1982).“
Mandelbrot states that fractal geometry is not a straight "application" of 20th century mathematics, but in
fact is the result of a crisis mathematicians encountered when duBois Reymond in 1875 first described the
continuous non differentiable function constructed by Weierstrass (figure 1) (Mandelbrot 1982).
Figure 1: Weierstrass continuous Function

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“Fractal” is a word that Mandelbrot established in
order to bring a range of subjects together under one
title, which enhanced the development of “pure”
mathematics (Mandelbrot 1982). This development
was seen as a great revolution of ideas that separated
the classical mathematics of the 19
th
century,
characterized by the regular geometric structures of
Euclid and the continuous changing dynamics of
Newton from the modern mathematics of the 20
th
century (Mandelbrot 1982). The revolution itself was
forced by the encounter of natural structures that did
not fit the patterns of Euclid and Newton (Mandelbrot
1982). As a consequence new structure (sets) were established such as Peano’s space-filling curve (figure 2),
that were initially regarded as “pathological”, as a “gallery of monsters” (Mandelbrot 1982). These irregular
sets however provided better representation of many natural phenomena, than the figures gained by
classical geometry, and were able to be studied using the framework fractal geometry (Falconer 2003).
1.2.2. Definition and characteristics
In 1967 Mandelbrot conceived and developed, a new geometry of nature and used this in diverse fields,
describing many irregular and fragmented patterns around us which eventually lead to theories identifying a
family of shapes he calls fractals (Mandelbrot 1982). Mandelbrot came up with the word “fractal” based on
the Latin adjective “fractus” and its corresponding verb “frangere” which means “to-break” or create
irregular fragments (Mandelbrot 1982).
Falconer (2003) stated, that the definition of a “fractal” should be regarded in the same way as a biologist
regards the definition of “life”. He stresses that there is no solid and firm definition of the word “life” (nor
“fractal”) but rather a list of properties characterizing a living thing such as the ability to reproduce-,
transfer-, change-, and exist, all to a certain extent based on its environment (Falconer 2003). As a result,
even though most living species contain most of these characteristics, there are living species that are
exceptions to each one of them (Falconer 2003). Thus corresponding to fractals, it is best to regard them as
a set (of data) having fractal-properties rather than searching for a precise definition of the word, that might
eliminate and or prevent interesting cases (Falconer 2003).
While a range of mathematical definitions can be used for different kinds of fractal geometric applications,
the following equation (1) is often used, whereby the number of objects (the fractured fragments), with a
characteristic linear (integer) dimension , a constant proportionality defined as C and the fractal
dimension D (Turcotte 1997).
Figure 2: Peano's space-filling curve

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The Euclidean dimension (also known as the Hausdorff
dimension) D, in figure 3 of a point is equal to zero, that of a
line fragment is one, of a square two and of a cube three,
thus when the fractal dimension is an integer it is also equal
to the Euclidean dimension. However when the fractal
dimension is not an integer but a ratio, the above
mentioned equation (1) is rewritten into equation (2), as a
(natural) logarithm (Turcotte 1997).
( ⁄ )
( ⁄ )
( ⁄ )
( ⁄ )
Equation 2
One of the most important fractal properties is self-similarity, which according to mathematics is retained
when an object is exactly or approximately similar to a (scaled) part of itself (Mandelbrot 1982). Self-
similarity is visible in many real world objects such as trees and coastlines (Lam & Cola 1993). The fractal
(“fractured”) dimension as mentioned in the previous paragraphs, is a ratio providing an index of a fractal
pattern or sets, calculating (and comparing) their irregularity in the change of detail of a pattern by the
change in scale (i) at which it is measured (Mandelbrot 1982). Scale invariance is an exact example of self-
similarity and a necessary condition to apply the above mentioned equations (1) and (2), since the length
scale enters a power to the law relation, a fractal relation (Turcotte 1997). Scale invariance allows one to
“extrapolate from properties observed at one scale to the properties of a scale which has not been
observed” (Klinkenberg 1992).
1.2.2.1. Fractal sets
In the past century, different fractal-sets have been developed in order to bring under one umbrella a broad
range of former concepts from pure mathematics to the most empirical aspects of engineering (Turcotte
1997). A set is a collection of distinct objects which exists of elements (data) that can be anything from
numbers, letters, and ratio’s, or even other sets. A fractal set, is a spatial set demonstrating regular scaling
relationships between the number of basic elements and their measure (size, density, intensity etc.)
including temporal and dynamic phenomena such as stream networks (Lam & Cola 1993).
Equation 1
Figure 3: Hausdorff dimension

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One of the most famous sets is the Cantor set (see figure 4).
This set it created by dividing a line segment having a
Euclidean dimension ( ) of one, into three equal fragments
by which only the middle part is removed and two third of
the fragments is kept (C, i.e. ratio kept). This division, , can
be carried out constantly for eternity, resulting in shorter
and shorter line lengths that can be characterized by a
fractured dimension between 0 and 1 (Turcotte 1997).
The condition of scale invariance is also conducted in the
Cantor set (figure 4), by which the iteration is set up to the
fourth order, i= 4. Because in the zero-order (level 0) the
first line segment is divided into three equal unit lengths,
r=3 and N is equal to the remained segments which is 2, the
first-order cantor set (level 1) is used as the generator for
higher-order sets (Turcotte 1997). Each of the two remaining, fractured line segments at first order are
replaced by an in scale-reduced version of the generator to attain the second-order set, and so on for higher
orders. This means that if “n” iterations are executed, the line length at the n
th
iteration, , is related to the
length at the first iteration, , by the ratio ( ⁄ ) = ( ⁄ ) . Cantor’s set is often referred to as “dust”,
since in the ( ⁄ ) = ( ⁄ ) , the next limit can be approached and , by which zero is the
Euclidean dimension of a point, and thus results in an infinite set of clustered points.
1.3. Applicability of fractals in geomorphology
In the field of geomorphology, the concept of fractal geometry has been used to describe numerous forms,
patterns and processes, such as stream patterns, river networks, water erosion, fluvial land-sculpting and
topography (Chase 1992, Lam & Cola 1993). Different methods have been used for the calculation and
determination of fractal properties. According to Lam and Cola (1993), quantifying the fractal dimension D
of geographic data is the first step in understanding spatial complexity based on the fundamental fractal
relationship (annex 2). Lam and Cola (1993) explain a couple of fractal measurement methods for curves
such as the Walker Divider (annex 3) and methods more sensitive for irregularity as the Box-Counting
method.
1.3.1. Basin morphometric analyses
Basin morphometric analyses refers to the quantitative assessment of the earth surface characteristics such
as basin parameters (Talukdar 2002). This approach was given input by Horton (1945). Horton's law of
stream lengths suggested that a geometric relationship existed between the numbers of stream segments in
sequential stream orders. The main objective of morphometric analysis is to determine the drainage
characteristics in order to explain the overall evaluation of the basin (Talukdar 2002). Therefore according to
Talukdar (2002) basin geomorphic characteristics have been believed to be important indices.
Important drainage basin characteristics (indices) are drainage density , stream frequency and
drainage intensity . Horton (1945) has introduced drainage density as expression for indicating the
closeness of channel spacing. The drainage density is influenced by environmental factors such as climate,
lithology, relief ratio and vegetation (Pd & Dc 2013). Low drainage density is representative for areas with
highly permeable subsoil material, dense vegetation and low relief (Pd & Dc 2013). High drainage density on
Figure 4: Cantor set

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the other hand is typical for impermeable subsurface material, sparse vegetation and mountainous relief (Pd
& Dc 2013). Stream frequency is directly related to lithological characteristics and resembles the number of
stream segments per unit area (Horton 1945b).
Throughout the 20
th
century different models for assigning stream order have been introduced as illustrated
in figure 5. However in most studies Strahler’s system, a slight modified version of Hortons system, is often
used because of it’s simplicity (Talukdar 2002). Strahler’s order implies the smallest unbranched channels
are assigned 1
st
order, whereby two channels of 1
st
order give a stream of 2
nd
order and two 2
nd
order
stream form a 3
rd
order stream.
Figure 5: Stream Order models
The number of streams of u
th
order are named Nu whereby Lu states for the stream length, the average (or
mean) length of streams in u
th
orders. Stream length (Lu) is calculated by dividing the total length of all
streams of u
th
by the number of streams in that order (Talukdar 2002). Other important stream
characteristics are the Bifurcation ratio (Rb ) and the Mean stream Length ratio (RL). As the Bifurcation Ratio
depends on relief and structural development, it helps interpreting the run off behavior as well as shape an
idea of the drainage basin (Strahler 1952). A high bifurcation ratio indicates structural distortion in the basin
area (Talukdar 2002). River channels continuously wander from a straight line path, with curves and knicks in
platforms perceivable on different scales of view, from the short channel reach to full stream length (Snow
1989). The ratio of wandering (meandering) of streams is another important stream characteristic entitled
sinuosity, the extent to which actual channel length (between the beginning and the end) of a channel
diverges from the long length (straight line from beginning till end).
1.3.2. Box-Counting method
The box counting method can be used for branching
fractals like stream networks as well as for
discontinuous fractals such as dusts and galaxies.
According to Foroutan-pour et al. (1999) it is an easy
method to use, automatically computable, and
applicable for patterns with or without self-similarity.
In this method, each studied image containing a fractal
pattern is covered by a sequence of grids of
descending sizes and for each of the grids, two values
are recorded: the number of square boxes intersected Figure 6: Box-Counting method

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by the image, N(s), and the side length of the squares, s.
Figure 6, visualizes the measurement of Britain’s coastline with descending box sizes. The decrease in box-
size increases the count and therefore the length of the coastline. When plotting ( ( against
( ⁄ ) the regressio n slope D of the straight line is formed. This regression line indicates the degree of
complexity or fractal dimension between 1 and 2 ( (Foroutan-pour et al. 1999). An image with a
fractal dimension equal to 1 or 2 (integer) is regarded as completely differentiable. The linear regression
equation used by Foroutan-pour et al. (1999), to estimate the fractal dimension was defined as follows, by
which N(s) is proportional to ( ⁄ ) :
( ( ( ⁄ ) ( ( ) Equation 3
1.3.3. Geomorphometry
Geomorphometry is the quantitative description of geometrical features on the land surface. Klinkenberg
(1992) investigated the potential role that “fractal parameters” might have, when these parameters are
used as geo-morphometric indices. Correlations between the fractal dimension (D) and 24 morphometric
parameters, related to five different morphometric characteristics (elevation, gradient, aspect, profile
convexity and plan convexity) were evaluated (Annex 4) (Klinkenberg 1992). However, no strong correlation
between the traditional used morphometric parameters and break distance or the fractal dimension (D) was
found. Klinkenberg (1992) suggested that the fractal dimension captures some aspects of the surface
irregularity, which is unique and that the true geomorphic implication of D will become apparent after more
studies. Nevertheless, the conclusion was that the fractal dimension D can be related to lithology and to
surface processes (Klinkenberg 1992). In addition, Klinkenberg (1992) states that fractals may provide
justifying means to diverse systems of landscape formation since fractal phenomena canto be studied using
second-order statistics (vario-grams), re-normalization group transformations, and because fractal behavior
can be explained in terms of spatial autocorrelation and iteration (Klinkenberg 1992).
1.3.4. Fractal drainage basin analyses
The applicability of fractal characteristics such as fractal dimension on river channels, drainage basins and
landscape has been investigated by many authors. Itit has been shown that fractal modeling can be applied
to characterize the evolution of stream systems in terms of geological constraints (Cheng et al. 2001,
Schuller et al. 2001).
According to Schuller et al. (2001) hydrologists are interested in calculating two fractal dimensions for
channels, the fractal dimension of an individual channel, d, and the fractal dimension of a stream(channel)
network, D. The fractal dimension of an individual stream is a measure of its irregularity; it is a measure of
the extent to which a stream (channel) meanders. The fractal dimension of an individual stream length can
be computed when determining the slope of the line formed when the mainstream length is plotted against
the Euclidean mainstream length on a log–log graph (Schuller et al. 2001).
On the other hand, the fractal dimension of a river network is a measure of the ability of a network to fill a
plane, and rises from the branching nature of a river network and from the sinuosity of individual streams
(Schuller et al. 2001). Mandelbrot (1982) and Tarboton et al. (1988) believe that if a stream network were
truly space-filling, as is the case with a topologically random-networks, one could expect a fractal dimension

16
of 2.0, when computing the river network (i.e. the Euclidean dimension of a plane). However most studies
have concluded that river-networks are not space-filling, as at a certain level the stream stops when a hill-
slope begins (Schuller et al. 2001). In addition, mechanisms governing the overland flow, channel flow and
erosion prevent establishment of a space-filling network (Schuller et al. 2001). Therefore, the fractal
dimension of a stream network is generally expected to be less than 2.0, and it varies from one location to
another (Schuller et al. 2001)
In 1957, Hack introduced the power-law relationship between the mainstream length (L) of a river and the
drainage area (A) as follows . A similar relationship also exits between the total stream length (∑
per catchment and drainage area ∑ ( , by which is the exponent with
( (Cheng et al. 2001 . If stream networks obtain “Space-filling” properties and are free of
geological constraints (Cheng et al. 2001). On the contrary, if ( the evolution of the stream
network is constrained geologically. The value of D as the ratio of the total stream length over drainage area,
can be calculated by rewritting ∑ as follows: ⁄
( ⁄
(√ ) , consequently can be
obtained with the following equation:
√
Equation 4
In 1989, Barbera & Rosso related Hortons’s laws of bifurcation ratio
(
(
and average stream length
ratio
(
(
, to the fractal dimension of stream networks as follows:
Equation 5
By which, the value of D has been interpreted by many authors as a possible measure indicating the degree
of randomness in the evolution of stream network or lack of geological constraints (Cheng et al. 2001).
Likewise related to Horton’s laws Khanbabaei et al. (2013) presented the fractal dimension of a drainage
area by the following equation 6, whereby stands for the bifurcation ratio and for the drainage area.
Equation 6

17
2. Methods
2.1. Study Area
For this research the Carcavo basin located in the region of Murcia, Southeast Spain (figure 7) is chosen as
study area, since this area is representative for semi-arid catchments that are vulnerable to erosion and
characterized by a dense drainage network (Lesschen et al. 2009). The region of Murcia belongs to the
structurally transformed Betic ranges in the SE part of the Iberian Peninsula, forming important relief
features (Alonso-sarria et al. 2011). These relief features, ridges, have been formed by tectonic activity and
subsequently shaped by climate-induced processes of erosion and sedimentation resulting in badland areas
(Alonso-sarria et al. 2011).
Figure 7: Study Area
The Carcavo basin is located about 40 km northwest of the city of Murcia in Southeast Spain, near the town
of Cieza (UTM 4228000 N; 630000 W; European_1950 datum zone 30N) (Lesschen et al. 2007). The semiarid

18
A. Literature
study
B. Data pre-
processing
C. Data
gathering
D. Data
processing
E. Data
analyses
Study the concept fractals and its applicability on gully formation (i.e. study
phenomenon gully). Choose suitable study area based on literature study.
Define spatial extend (basin drainage border) of study area (using DTM) of Ortho-
imagery (1956-2011)
Landuse maps
(1956-2004)
Define landuse
change (1956-
2004)
Calculate areal
and percentage
landuse change
Delineate gully network
Define geomorphological change (1956-2011): Channel
length, sinuosity, stream order
Drainage basin
characteristics:
RB, RL, Dd, Fz, Id
Statistical
Analyses
Fractal Analyses:
fractal dimensions
Measure cross-
section dimensions
Define width
depth for
different scales (in
field + desk)
Self-similarity:
correlate variables
(width & depth)
climate of this area beholds a mean annual rainfall between 260 and 275 mm and mean annual potential
evapotranspiration values higher than 850 mm (CONESA-GARCÍA & GARCÍA-LORENZO 2009). Both the
intensity and rainfall volume are concentrated in a short period, especially in spring and autumn with
rainfalls ranging from 100 till 300 mm in a view hours (CONESA-GARCÍA & GARCÍA-LORENZO 2009). The
basin covers an area of 30 km
2
and an altitude ranging between 220 and 850 masl (Lesschen et al. 2007).
Most soils in the area are thin (Leptosols), weakly developed (Regosols) and mainly characterized by their
parent material (Calcisols and Gypsisols)(Lesschen et al. 2007). Within the framework of reforestation and
soil conservation programs large parts of the degraded land were reforested with pine trees (Pinus
halepensis Mill.) in the 1970s, and 36 check-dams were constructed over the period from 1969 till 1977
(Castillo et al. 2007), Lesschen et al. 2007).
2.2. Methodological design
This research examines the applicability of fractals to the Carcavo
basin, using two different 2D perspectives, based on the notion that
the study area can be interpreted as a spatial 3D context (figure 8). In
the X,Y-plane (orthogonal) the change in geomorphological basin
characteristics are determined for the years 1956 and 2011 and the
change in areal and percentage land use type is calculated from 1956
till 2004. Subsequently using geomorphological characteristics (length,
sinuosity, drainage area), the change in morphometric parameters
and of five fractal dimensions are calculated. In the Y,Z and X,Z-plane
(side views), the correlation between the width and depth is sought
using cross-sections on different spatial scales in order to determine
scale-invariance (self-similarity) between the gullies and its
surrounded landscape. Figure 9 visualizes the methodological design
of this research.
Figure 8: 3D context of study area Carcavo
Figure 9: Methodological design

19
2.3. Data preprocessing and gathering
2.3.1. Preprocessing
After choosing the study area, the basin border (drainage area) of Carcavo was extracted from a DTM
(Digital Terrain Model) file in ArcGis with a 4 m resolution (ESRI 2013). Using the mosaic tool, the different
Orthophoto-images overlaying the study area were converted to maps (files) for the years 1956 and 2011.
The border shape file was subsequently used as input in order to determine the spatial extend of the used
GIS data, Ortho-imagery (1956; 2011) in raster format and Land use maps (1956; 2004) in vector format
(ESRI 2013).
2.3.2. Data gathering
2.3.2.1. Mapping and characterizing gully networks
In order to characterize the gully networks from the Ortho-images (1956; 2011) two shape-files were
created. Using the editor toolbox in ArcMap the gully networks were delineated manually based on the
observed interpretation from the Ortho-images of the years 1956 and 2011.
2.3.2.2. Measuring profiles on different scales
For the measurement of gully dimension two techniques were used. On basin scale the cross-sections
(width, depth) of the valley, the river, side river and creeks were generated from the DTM file using the
interpolate tool in ArcMap in meters. After determining a suitable gully field survey area (annex 5) based on
the interpreted gullies from Ortho-image and logistics (connection by car), the cross-sections (width, depth)
were measured in the field using a clinometer and a 50 meter long measuring tape. This measurement has
been done 6 times on smaller and smaller scale. On the smallest scale, 4 cross-section of a gully were
measured in field.
2.4. Data processing
2.4.1. Geomorphology
To calculate the sinuosity of the characterized gullies (channels) several steps were taken in ArcMap using
the table tools (ESRI 2013).
1) Add field in the tables of the characterized gully networks (shape files) + field name (=X_EndPoint)
2) Calculate geometry (of field) Calculate X Coordinate of every end point of every delineated line
representing a gully/channel.
3) Add field in the tables of the characterized gully networks (shape files) + field name (=Y_EndPoint)
4) Calculate geometry  Calculate Y Coordinate of every end point.
5) Add field + Field name (=X_StartPoint)
6) Calculate Geometry  Calculate X coordinates of every start point
7) Add field + Field name (=Y_StartPoint)
8) Calculate Geometry  Calculate Y coordinates of every start point
9) Add field + Field name (=d_Long)
10) Calculate field (long channel length√| | |( |
11) Add field + Field name (= Sinuosity)
12) Calculate field ⁄

20
Hydroflow Brazilian software was used for the stream order calculations. This software assigns stream order
for vector data based on two input data: channel network (shape files containing the characterized gullies of
1956 and 2011) and basin borders (shape file extracted from DTM). The flow direction is defined by the
outflow point which is supposed to be the only vector line that intersects with the basin border (Labgis n.d.).
2.4.2. Land use
Before calculating the land use change, the land use maps of 1956 and 2004 were assigned equal numerical
values for different land use types. The land use maps contained vector data and are derived from the
Erosion and conservation of Soils group at CEBAS-CSIC. After assigning grid codes (numerical values) to the
different land use types of the vector data, the shape-files are converted into raster data-files whereby the
grid-code (land use type, numerical value) is the Cell-value and the cell size is 1m
2
. Using the tool raster
calculator in ArcMap a land use change map was then obtained. The attribute table of this land use change
map visualizes the grid-codes, as the product of both land use maps, and the amount of cells each grid-code
beholds.
Raster Calc Equation: (
2.5. Data Analyses
2.5.1. Land use change
As the cell-size of the land use change map is equal to a square meter, the counted cells of each land use
change are also equal to the changed areal in square meters. By this means the areal land use change of one
land use in 1956 to a land use in 2004 can be considered. In order to calculate the percentage land use
change the following equation is used:
( ⁄ )
2.5.2. Morphological changes
Several important morphometric basin parameters were examined for 1956 and 2011 using the delineated
gullies and channels (table 1). The calculations of the parameters were carried out in ArcMap and Excel (ESRI
2013; Microsoft Corporation 2010).

21
Table 1: Morphometric parameters
Morphometric
Parameter
Methods/formulas References
Stream Order ( )
Hierarchical rank
Strahler
(1952)
Stream length ( ) Horton (1945)
Mean Stream
length
Total stream length divided by total number
of streams
Strahler
(1952)
Stream length
ratio ( )
(
(
Where = total length of stream segment
of order u, and total length of streams
in next higher order.
Strahler
(1952)
Bifurcation ratio
( )
Where = total number of stream
segment of order u, and total number
of streams in next higher order.
Strahler
(1952)
Drainage Basin
Area ( )
Total area of drainage basin derived from
ArcMap (Table info)
ESRI (2013)
Drainage density
( )
Horton (1945)
Stream Frequency
( )
Horton (1945)
Drainage Intensity
( )
2.5.3. Statistical analyses
To analyze the significant change in channel characteristics, length and sinuosity, an independent sample
Test was carried out in SPSS (SPSS INC 2008). In this way the variables are grouped by years (1956 and 2011)
and the null hypothesis (H0) stating that the Mean (variability) of the variables Sinuosity and Channel length
of the two groups is equal, was tested (i.e. there is no significant change in sinuosity nor channels length).
2.5.4. Fractal analyses
2.5.4.1. Self-similarity and scale invariance channel profile
In order to examine the self-similarity on different scales, the width and depth values of the profiles were
entered into SPSS as two variables. Two different analyses were carried out on the variables of 10
measurements (cross-sections) in order to examine the correlation between the width and depth, Bivariate
correlation (1) and a linear regression (2) using an Anova test in SPSS (SPSS INC 2008).
2.5.4.2. Fractal dimensions
Five fractal dimensions were calculated for 1956 and 2011. By exporting the shape-files of the delineated
channel network as images the fractal dimension was calculated of the years 1956 and 2011 using the box-

22
counting method in ImageJ (Rasband n.d.). The fractal dimension of an individual stream was calculated by
determining the slope on the log-log plot of the actual length per stream segment versus the Euclidean
stream length (straight line from beginning till end) in SPSS. Subsequently in excel the change (1956-2011) in
fractal dimensions of the channel networks, drainage area and channels were calculated. Table 2 illustrates
the calculated fractal dimension and their methods/formulas.
Table 2: Used Fractal Dimensions methods and formula’s
Methods/formula’s References
Fractal dimension D, of
exported delineated
gullies as images
(1956 and 2011)
Box-counting method using ImageJ (Rasband n.d.)
Fractal dimension
individual stream, d
Determine slope of LOG-LOG plot
(average length versus Euclidean
length)
(Schuller et al.
2001)
Fractal dimension
channels, √
Schuller et al.
(2001)
Fractal dimension of
channel network
(branching),
Tarboton, Bras,
and Rodriguez-
Iturbe (1988)
Fractal dimension
drainage area,
Khanbabaei,
Karam, and
Rostamizad
(2013)

23
3. Results
3.1. Mapping and characterizing gully networks
Figure 10 shows two maps containing delineated gullies in ArcMap (1956; 2011) and the Carcavo basin
border..
Figure 10: Delineated gullies (1956; 2011)
3.2. Land use change
3.2.1. Mapping and characterizing land use change
Figure 11 shows the land use maps of 1956 and 2004. Unfortunately as can be perceived not the entire
Carcavo basin is covered with land use data. Table 3 shows the land use change matrix indicating the grid
codes for changed land use classes. This matrix displays all the possible cell-values (grid-codes) the land use
change map could have had, based on the “raster calc equation” described above (in paragraph 2.5.1.). Yet
out of all the possible land use change combinations, only the in color highlighted cell values are found in
the land use change map. Annex 6 contains the obtained land use change map of which only the attribute
table is used for further analyses (annex 7).

24
Figure 11: Land use (1956; 2004)
Table 3: Land use change matrix, based on using gridcodes of land use types
Land use 2004
Olive/
almond
Cereals Peaches Grapes
Abandoned
olive/
almond
Abandoned
cereals
Semi-
natural
vegetation
Semi-
natural
forest
Reforestation
Built-up
area
Water
1956 Grid Code 1 2 3 4 5 6 7 8 9 10 11
Cereals 10 101 102 103 104 105 106 107 108 109 110 111
Olive/
almond
20 201 202 203 204 205 206 207 208 209 210 211
Gullies
1956
30 301 302 303 304 305 306 307 308 309 310 311
High
density
forest
1956
40 401 402 403 404 405 406 407 408 409 410 411
Low
density
forest
1956
50 501 502 503 504 505 506 507 508 509 510 511
Medium
density
forest
1956
60 601 602 603 604 605 606 607 608 609 610 611
Pasture
1956
70 701 702 703 704 705 706 707 708 709 710 711
Shrubland
1956
80 801 802 803 804 805 806 807 808 809 810 811

25
3.2.2. Calculating land use change
3.2.2.1. Areal land use change
In order to obtain insight on the areal land use change a land use change matrix was generated (Table 4). This shows the amount of square meters of each land
use in 1956 reformed into a land use in 2004 (i.e. the counted cells annex 6). In Excel (Microsoft Corporation 2010) the cell values are edited in order to
visualize their values using the edit formatting rule and thus highlighted in dark green the highest areal land use change from 1959 and in red the lowest areal
land use change that took place. As can be noticed in dark green, the highest areal land use change that took place was Pasture land replaced by Reforestation
(5219821 m
2
) and the lowest in dark red High Density forest replaced by Reforestation (49m
2
)
Table 4: Areal land use change (m
2
)
Land Use (m
2
) 2004 Olive/almond Cereals Peaches Grapes
Abandoned
olive/almond
Abandoned
cereals
Semi-
natural
vegetation
Semi-
natural
forest
Reforestation
Built-
up
area
Water
SUM
m
2
1956 Grid_Code 1 2 3 4 5 6 7 8 9 10 11
Cereals 10 3302172 1655519 209427 254771 943699 767550 19368 1362861 35992 5251 8556610
Olive/almond 20 251521 55005 40391 54972 35130 205193 2102 1212 645526
Gullies 1956 30 165443 42722 1463 16647 34367 864807 138555 2590167 48082 3902253
High density forest
1956
40 26737 130424 49 157210
Low density forest
1956
50 4601 2783 199332 272843 1130121 18535 1628215
Medium density
forest 1956
60 253927 1239864 453569 1947360
Pasture 1956 70 130353 14685 32634 40225 945785 81615 5219821 814 6465932
Shrubland 1956 80 74766 6252 4186 8663 32180 2025508 444982 2535275 1633 33033 5166478
SUM m
2
3928856 1719178 0 215076 367720 1093645 5138618 2362781 13497056 40541 106113 28469584
Cell Value=

26
3.2.2.2. Percentage land use change
In order to obtain insight into the ratio of each land use change from 1956 till 2011, a percentage matrix is calculated and illustrated in table 5. In this table the
cell values are edited in excel in order to visualize the highest percentage ratios of land use change in dark green and the lowest in white. As illustrated, the
highest ratios of land use change are found in the columns of Semi/natural forest and Reforestation. Remarkable is that 82,96% of all High density forest is
replaced by Semi-natural forest; that 80,73% of all pasture land is replaced by reforestation and that 66 38 % of all the Gullies (1956) are Reforested whilst
22,16% are replaced by Semi/natural vegetation.
Table 5: Percentage land use change
Land use 2004 Olive/almond Cereals Peaches Grapes
Abandoned
olive/almond
Abandoned
cereals
Semi-
natural
vegetation
Semi-
natural
forest
Reforestation
Built-
up
area
Water SUM
(%)
1956 Grid Code 1 2 3 4 5 6 7 8 9 10 11
Cereals 10 38,59 19,35 2,45 2,98 11,03 8,97 0,23 15,93 0,42 0,06 100.0
Olive/almond 20 38,96 8,52 6,26 8,52 5,44 31,79 0,33 0,19 100.0
Gullies 1956 30 4,24 1,09 0,04 0,43 0,88 22,16 3,55 66,38 1,23 100.0
High density
forest 1956
40 17,01 82,96 0,03 100.0
Low density
forest 1956
50 0,28 0,17 12,24 16,76 69,41 1,14 100.0
Medium density
forest 1956
60 13,04 63,67 23,29 100.0
Pasture 1956 70 2,02 0,23 0,50 0,62 14,63 1,26 80,73 0,01 100.0
Shrubland 1956 80 1,45 0,12 0,08 0,17 0,62 39,20 8,61 49,07 0,03 0,64 100.0
Cell Value=

27
3.3. Geomorphological changes
3.3.1. Length and sinuosity
As visualized in the boxplots below (figure 12 and 13) both the channel length and the sinuosity of the gullies
have declined from 1956 till 2011.2011. In order to quantify the decline significantly an independent sample
Test was carried out in SPSS for both variables (length and sinuosity) in order to estimate the significant
change. Additionally as presented in Group Statistics (annex 8) the Mean, Standard Deviation and Standard
Error Deviation have declined, for channel length and sinuosity, whilst the number of gullies (N) increased.
In the Independent Samples Test, the Levene's Test for Equality of Variances shows that the P-Value for
both tests is equal to 0,000. This implies that the null hypothesis can be rejected, that therefore the
variances are unequal, indicating that the variables sinuosity and channel length have declined significant.
3.3.2. Sinuosity and Strahler’s order
As can be perceived in figure 14, the mean sinuosity of the
channels increases with increasing Strahler’s order, while
fromyears 1956 to 2011 the mean sinuosity for each order
decreaseddecreases.
Figure 12: Boxplot gully length (1956; 2011) Figure 13: Boxplot Gully sinuosity (1956; 2011)
Figure 14: Sinuosity - Strahler's Order

28
3.3.3. Basin Morphometry
Table 6 shows the changes in (geo) morphological drainage basin properties from 1956 till 2011. Remarkable
are the decrease in channel length, sinuosity and drainage density, whilst the number of channels,
bifurcation ratio, mean stream length ratioRatio, Stream Frequency and drainage intensity have increased.
Table 6: Geomorphological changes (1956-2011)
Morphometric and channel
parameters
1956 2011 Change Change
(%)
Average sinuosity channels 1,084 1,072 -0,012 -1,1
Count N channels Strahler’s
order (u)
1 913 1075 162 18
2 208 214 6 3
3 15 10 -5 -33
Total N count of channels (basin) 1136 1299 163 14
Sum stream Length uth
order
(Lu) in m
1 154022 160411 6389 4
2 64254 55174 -9080 -14
3 5333 3822 -1511 -28
Total Sum Lu (basin) in km 223,6 219,4 -4,2 -2
Mean Stream Length uth
order
(Lu) in m
1 168,7 149,2 -19,5 -12
2 308,9 257,8 -51,1 -17
3 355,5 382,2 26,7 8
Total mean channel length (basin) in
m
196,8 168,9 -27,9 -14
Bifurcation Ratio (Rb) Rb2,1 4,4 5 0,6 14
Rb3,2 13,9 21,4 7,4 54
Rbmean 9,1 13,2 4,1 45
Mean stream Length ratio (RL) RL2,1 1,83 1,73 -0,10 -5
RL3,2 1,15 1,48 0,33 29
RLmean 1,49 1,61 0,11 8
Drainage Basin Area ( ) in m2
31756675
Drainage Density (Dd) 7,04 6,91 -0,13 -2
Stream frequency (Fz) 35,77 40,90 5,13 14
Drainage intensity (Id) 251,88 282,61 30,73 12
3.4. Statistical and fractal analyses
3.4.1. Self-similarity in channel profiles
3.4.1.1. Channel Profiles
Figure 15 shows a side-view from the Carcavo basin where the concave shape is visible in ArcScene, whilst
figure 16 shows the outcome of the interpolate tool, a diagram with the width and depth in meters (ESRI
2013).

29
3.4.1.2. Self-Similarity
The outcome of the width and depth of the 10 measurements are summarized in table 7. For testing the
fractal property of self-similarity in channel profiles, an Anova-Test was carried out in SPSS (SPSS INC 2008),
in order to determine the correlation between the width and depth variables of the measured channel
profiles on different scales. Annex 8 contains the outcome of the bivariate correlation and linear regression
using an Anova test in SPSS. As shown in annex 8, rR and R square are almost equal to 1 (high), meaning that
there is a correlation between the variables width and depth. The line describing the correlated relationship
is visualized in figure 17, using the function curve-estimation in SPSS, and is defined by the following
equation where D = Depth (in meters) and W = Width (in meters): . The coefficients of
this equation are taken from annex 9, the coefficients table.
Figure 15: Side-view Valley Figure 16: Cross-section valley
Table 7: Cross-section measurements
Cross-
Section
Width Depth
Valley 6.500,00 600,0
Main River 450,00 68,00
Side River 350,00 60,00
Creek 350,00 30,00
Creek 180,00 26,0
Creek 0,00 1,00
Gully1.1 8,40 1,00
Gully1.2 7,60 2,00
Gully1.3 4,70 1,70
Gully1.4 2,40 0,85
Figure 17: Curve-fit (Width-Depth correlation)

30
3.4.2. Fractal dimensions of channel network
In annex 10, the graphs describing the fractal relationship between box-sizes and binary count of pixels
resembling the whole image are presented in figure 23 (1956) and figure 24 (2011). In addition, Annex 10
also contains the log-log plots describing the fractal relationship between actual stream length and the
Euclidean length (straight line) of 1956 (figure 25) and 2011 (figure 26). As visualized in table 8, all the fractal
dimensions, except the fractal dimension of the drainage area have decreased.
Table 8: Fractal dimensions (1956; 2011)
1956 2011 Change % Change
D Box Counting method 1,973 1,957 -0,016 -0,8
d
Fractal dimension
individual stream
1,023 1,018 -0,005 -0,5
Fractal dimension
channel network
(branching)
5,538 5,418 -0,120 -2,2
Fractal dimension
drainage area
0,128 0,149 0,022 16,8
Fractal dimension
channels
1,426 1,424 -0,002 -0,2

31
4. Discussion
4.1. Fractal geometry of gully systems
With regard to self-similarity, the studied watershed Carcavo is interpreted as a spatial three dimensional
context shaped by a variety of processes such as: tectonic uplifting, fluvial land sculpting processes, erosion,
abrasion, soil deposition, human impact and environmental characteristics further influencing these
processes along with specific gully sub-processes such as piping or head-cut retreat. The outcome of these
processes, is a landscape (catchment) in which concave shapes are perceivable on different scales from
valley (scale) till gully (scale). In this study only the width and depth variables of the concave repeating form
(on different scales) are used. Based on the width and depth, as a correlation on different scales is found, it
might be possible for the Carcavo Basin to interpolate further width-depth development using the equation
presented in paragraph 3.4.1.2. ( ; whereby D=Depth and W=Width). However in order
to attain further insight on self-similarity of gullies with regard to their spatial environment, based on the
concave shape visualized in cross-sections, other concave-shape characteristics should be taken into
consideration.
As to the fractal dimension of stream network , according to Cheng et al. (2001) when it is close to 2, it
implies statistically speaking the stream network in the entire area satisfy a space-filling property or are free
of geological constraints. However the results of this study shows a fractal dimension of a stream network
equal to 5,538 in 1956 and 5,418 in 2011. The high , value might be explained by examining the count of
channels of different orders that subsequently influence the Bifurcation ratio. As viewed in table 6, the
count of number channels of the first order, are over three times higher than the second order and thus
result in high bifurcation ratios. Unlike Pal & Debnath (2013) of whom the mean bifurcation ratio diverged
from 3.44 till 8, the mean bifurcation ratio of this study was estimated at 9,1 for 1956 and 13,2 for 2011. The
high count of 1
st
order channels (gullies) can either be the outcome of not connecting the channel lines
together, or on the other hand is the outcome of interpreted gullies not, from an Ortho-image perceivable,
connected to a stream network but randomly occurring in the catchment. For example if the streams are
really not connected, but are isolated streams not connected to the main stream. In addition unlike Cheng et
al. (2001) and Pal & Debnath (2013), the obtained channel network was not extracted from DTM-files, but
gained through manual delineation and interpretation of channels from Ortho-imagery (1956-2011).
Also with respect to the fractal dimension of the Drainage area , the result of this research shows a
strange value. Different than Pal & Debnath (2013), the value is very low and approximated 0,128 for
1956 and 0,149 for 2011. The abnormal low values might be caused by the estimation of the drainage area.
The drainage area was extracted from the DTM-file, whilst the channels were manually interpreted. By this
means, the drainage area might be approximated larger than the actual assumed drained area by the
delineated channels.
According to Cheng et al. (2001) the value represents the degree of randomness in the evolution of the
stream network, whereby D=1 corresponds to linear stream patterns which are associated with geological
and structural constraints, and D=2 to “area-like” patterns (i.e. space filling patterns) which are considered
free of geological constraints and behold a random property. The fractal dimension of the channels over
drainage area was estimated at 1,426 in 1956 and 1,424 in 2011. This means that stream (channel)-
network of the Carcavo basin is linear and the evolution is constrained geologically.

32
The fractal dimension of an individual stream, d, was estimated at 1,023 and 1,018 for 1956 and 2011
respectively. As the value of d is almost equal to 1 (dimension of a linear line), the streams can be regarded
as linear, constrained geologically and in addition do not contain space-filling properties (Schuller et al.
2001). The low values can be explained by the small values in sinuosity (figure 14 and table 6) defining the
ability of a stream to wander and thus fill a plane (2 dimensional).
The box-counting method provides an indication of
complexity, the basis of a fractal dimension D. It refers to the
change in detail or number of parts something is made out
of, with change in scale. Since for both years the fractal
dimension of the stream (channel) network is almost equal to
2 (1,973 for 1956 and 1,957 for 2011), the network can be
assumed as highly complex, space-filling and not geologically
limited in further evolution (Cheng et al. 2001). However this,
the value of 2, contradicts the interpretation of the other
values and might also be the result of the, by the software
automatically chosen bounding area, see figure 18 (Rasband
n.d.). A small bounding area encloses more details and
therefore results in higher complexity (i.e. higher fractal
dimensions).
4.2. Changes in fractal geometry over time and the role of land use change
With regard to the change in fractal dimensions (table 8), small declines (beneath 1 percent) have been
calculated for the D (box-counting method), d (Fractal dimension individual stream) and DL (Fractal
dimension channels). The decline in fractal dimension of channel network (branching) was estimated at
2.2%, while on contrary the fractal dimension of the drainage area increased with 16,8%.
As presented in table 4, the highest ratios of land use change that took place from 1956 till 2011 are found
in the uses Semi/natural forest and Reforestation. It is remarkable that 66,38 % of all the gullies (1956) were
reforested whilst 22,16% were replaced by Semi/natural vegetation. As briefly explained in the introduction,
according to table 6 (annex 1) summarized by Boix-Fayos et al. (2007), reforestation along with erosion
works (such as check-dams) cause the following changes in stream evolution: channel narrowing, change
from braiding to meandering, decrease in mean annual flow, channel incision, decreases flood peaks,
sediment yield becomes steady, decrease in hill-slope sediment production, settlement of bars of vegetation.
The decline in fractal dimension of the individual stream d, can therefore be explained by the change in land
use type. As reforestation increases channel incision, decreases the wandering of a channel and thus
decreases the fractal dimension d. The decrease in fractal dimension determined by using the box-counting
method D, can either be explained by incision that decreases complexity (wandering) or with settlement of
vegetation decreasing the total stream length (table 6). Similar for the fractal dimension of channels and
channel network (branching) , as reforestation causes settlement of bars and vegetation, the total stream
length decreases. Whereby, the decrease in total stream length can also be explained by decrease in
sinuosity, as a result of stream incision. Finally, the decline in fractal dimension of the drainage area can
be explained by the decrease in total stream length as a consequence of reforestation, by which the
drainage area is assumed not to have changed.
Figure 18: Bounding box

33
Eventually when examining table 6, the following changes in basin characteristics can be observed: decrease
in channel length, sinuosity and drainage density, while the number of channels, bifurcation ratio, mean
stream length ratio, Stream Frequency and drainage intensity have increased. Noticeable is the decline in
the important drainage characteristic, drainage density (Dd), which was estimated at 2% from 1956 till 2011.
As the drainage density did not change significantly, it might also clarify the insignificant change in the
fractal dimensions.
4.3. Methodological uncertainties
Along this research, the choices of techniques have influenced the outcome (accuracy) of the data. During
primary data gathering by which gullies were manually delineated, it was often difficult to detect the
boundaries of the gullies. Whereby, it was sometimes more difficult to separate the darkness of a shrub
from the shadow of a gully. Therefore the error during delineation was larger than the actual possible
difference in gully-formation between the years 1956-2011. Especially since the Ortho-image of 1956 was in
black and white while the Ortho-image of 2011 was in color. In addition to the delineation of channels
(gullies), inaccurately connecting the lines, decreases the ability of the software Hydroflow to assign higher
Strahler’s stream orders (Labgis n.d.). This eventually results in overestimated bifurcation ratios and
subsequently high channel network fractal dimension.
With regard to the land use maps used as secondary data, due to time constraints the land use map of 2004
could not be up-dated and altered using the Ortho-imagery of 2011. This would have been needed to be
able to use consistent data for the channel networks as well as the land use change. In this study it was
assumed the land use did not change significantly in 7 years on a 55 years interval. In addition to the used
land use maps, as shown in figures 11 and 23 not the whole studied drainage area (extracted from DTM) was
covered with land use data.
With respect to the drainage area boundaries, it was assumed not to have changed. Therefore the extracted
boundaries of the DTM file from 2011, was used for both years. Additionally different methods were used,
for drainage area approximation than for channel and gully estimation. This might have influenced the
outcome of the fractal dimension of drainage area Da and fractal dimension of channel DL.
Finally, even though the box-counting analysis is an appropriate method of fractal dimension estimation for
images with or without self-similarity (Foroutan-pour et al. 1999). According to Foroutan-pour et al. (1999),
the technique, including processing of the image and definition of the range of box sizes, requires a proper
implementation to be effective in practice. Especially defining the largest and smallest requires extreme care
(Foroutan-pour et al. 1999). Therefore, further research should be carried out, on the box-counting method
used parameters, such as range of box-sizes and grid position.

34
5. Conclusion
This study examined the possibility of seeking self-similarity, between the concave shapes of cross-sections
throughout the Carcavo basin on different scales (valley till gully). Results show a linear correlation between
the variables width and depth defined by the following equation: ( ; whereby D=Depth
and W=Width). In addition this study also examined the possibility of using the fractal dimension in order to
obtain insight on the gully development, as a consequence of land use change (from 1956 till 2004),
influencing erosion processes and altering drainage basin characteristics. Even though the estimated fractal
dimensions might, due to methodological uncertainties, have strange values, a decline in four of the five
fractal dimensions is found on basin level. The decline in fractal dimensions might be explained by the land
use change analyses. This revealed that the largest percentage land use change that took place from 1956
till 2004 was reforestation, which leads to less hillslope erosion and more active channel incision. Hence
even though the change in the fractal dimensions do not show a significant decline, it can be concluded that
reforestation (in the Carcavo basin), calculated as the highest (percentage) land use change, decreases the
fractal dimensions of the gully network.
However, in order to predict gully behavior, on the basis of land use change, the catchment could have
perhaps been divided in smaller sub-catchments in order to determine the correlation between fractal
dimension change with respect to a land use change. Additionally, in order to simulate gully behavior by
means of an algorithm, the fractal generator (the basis of fractal pattern), that can be repeated iteratively,
also other factors influencing the occurrence and development of gully formation should be taken into
consideration (Lithology, Area-Slope threshold, climatic characteristics etc.).
In conclusion land evolution processes such as erosion and more specific gully erosion are natural processes
governed by the universal laws of physics. Therefore by enclosing all the factors influencing gully
development, an algorithm (fractal generator rules) might be found describing the fractal behavior of gully
systems. Thus, since we are only capable of approaching the universal truth, further insight on the rules
used by the universe to manifest itself on different levels of understanding can help us not only in viewing
natural phenomena from a more holistic point of view, but it can also perhaps unlock chains for solutions
and technologies in order to tackle the problems that humans are confronted with nowadays such as soil
loss, sedimentation of water courses, water and arable land-scarcity, pollution, and drought.

35
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39
Annex 1. Land use change effects on river channel
The table below summarized by Boix-Fayos et al.( 2007) shows the main land-use changes and their effect on channel morphology from recent publications.
Table 9: Land use change and responses

41
Annex 2. Fractal relationship
Simplification of the fractal relationship as: (𝑒 . In this equation, COUNT
stands for the N (number of objects) and STEP the size of the measure or denominator . D represents the
degree to which COUNT increases with increasing resolution. Thus when using scale-dependency
(𝑒 and substituting equation 1 in 2, equation 1 can be attained
(𝑒 . This equation (3) shows a linear relationship when using
the logs of the variables, as ( ( ( (𝑒 (Lam & Cola
1993). Hereby, the fractals dimension and b describes the slope of the line estimating the
relationship between ( and ( as shown in the log-log plots of figure 13 (Lam &
Cola 1993).
Figure 19: log-log fractal relation plots

42
Annex 3. Walking Divider Fractal dimension calculating method
Figure 19 illustrates the outcome of the input curve of a circle. Since a circle does not contain self-similarity
and hence is not a fractal the log-log plot does not show a linear relationship and the fractal dimension is
almost 1 (Euclidean and not a fractal). Figure 20, on the other hand where the Koch snowflake was chosen
as the input shows a linear relation on the log-log plot and a fractal dimension D of 1.2026 (Lam & Cola
1993).
Figure 20: Output from walking-divider applied on circle Figure 21: Output from walking-divider applied on fractal Koch
Snowflake

43
Annex 4. Morphometric parameters
Table 10: Morphometric parameters used in Klinkenberg ’s
Parameter Definition
Elevation
Mean Mean elevation; a measure of the "bias" (Evans, 1984 )
Range Max-min elevation; the most commonly used measured of relief
Standard deviation Standard deviation of the elevations: a statistic which Evans (1984) feels better represents relief
than range
Skewness Skewness of the elevations, measures the degree to which the distribution follows a normal
distribution: a statistic which Evans (1984) feels avoids the dependence of the hypsometric integral
on extreme values; a measure of the "massiveness" of relief
Kurtosis Kurtosis of the elevations, measures the extent of the tails in the distribution of elevation; another
measure of the "massiveness" of relief
Coefficient of dissection Reflects the distribution of the landmass with elevation (Strahler, 1952 ); youthful landscapes should
have values above 0.65, over mature landscapes should have values below 0.35. The formula used is
𝑑 =
𝑒𝑣 𝑚𝑒 𝑒𝑣 𝑚
𝑒𝑣 𝑚 𝑥 𝑒𝑣 𝑚
= (𝐻𝑦𝑝 𝑚𝑒 𝑐 𝑒 )
Gradient
One of the two components of the linear slope vector; the first vertical derivative of elevation
Mean Mean gradient: a widely used summary statistic for surface roughness (Evans, 1984); an important
component of the "gradient" factor identified by Evans (1984)
Standard deviation Standard deviation of the gradient
Skewness Skewness of the gradient, a measure of the "massiveness" of relief
Kurtosis Kurtosis of the gradient, another measure of the "massiveness" of relief
Gradient classes Percent of the gradients less than a specified angle (2 ° , 5 °, 10 °, 15 °, 25 ° and 45 ° )
Aspect
One of the two components of the linear slope vector, the first horizontal derivative of elevation
Mean Mean aspect (which ranged from - 180 ° to + 180 ° )
Standard deviation Standard deviation of the aspects
Skewness Skewness of the aspects
Kurtosis Kurtosis of the aspects
Profile convexity
Rate of change of gradient measured down the slope of maximum gradient; the second vertical
derivative of elevation
Mean Mean profile convexity
Standard deviation Standard deviation of the profile convexity; an important component of the "gradient" factor
identified by Evans (1984)
Skewness Skewness of the profile convexity
Kurtosis Kurtosis of the profile convexity; an important component of the "gradient" factor identified by
Evans (1984)
Plan convexity
Rate of change of aspect (a measure of countour curvature); the second horizontal derivative of
elevation
Mean Mean plan convexity
Standard deviation Standard deviation of the plan convexity, reflects the contour crenulations: a statistic Evans ( 1984 )
feels could be a possible replacement for drainage density
Skewness Skewness of the plan convexity; a measure of the "bias" (Evans, 1984)
Kurtosis Kurtosis of the plan convexity

44
Annex 5. Cross-section field area
Figure 22: Field area

45
Annex 6. Land use change map
Figure 23: Land use change map 1956-2011

46
Annex 7. Land use change table
Table 11: Attribute table - Land use change map
VALUE COUNT VALUE COUNT
101 3302172 409 49
102 1655519 501 4601
104 209427 506 2783
105 254771 507 199332
106 943699 508 272843
107 767550 509 1130121
108 19368 511 18535
109 1362861 607 253927
110 35992 608 1239864
111 5251 609 453569
201 251521 701 130353
205 55005 702 14685
206 40391 705 32634
207 54972 706 40225
208 35130 707 945785
209 205193 708 81615
210 2102 709 5219821
211 1212 710 814
301 165443 801 74766
302 42722 802 6252
304 1463 804 4186
305 16647 805 8663
306 34367 806 32180
307 864807 807 2025508
308 138555 808 444982
309 2590167 809 2535275
311 48082 810 1633
407 26737 811 33033
408 130424

47
Annex 8. Group statistics and independent sample Test
Channel length (1956-2011)
Sinuosity (1956-2011)
Group Statistics
Year N Mean Std. Deviation Std. Error Mean
Sinuosity
1956,00 1136 1,0844913509683E0 ,10113292080705 ,00300056738633
2011,00 1299 1,0725277698396E0 ,08096855016478 ,00224652774859
Independent Samples Test
Levene's Test for
Equality of Variances
t-test for Equality of Means
95% Confidence Interval of the
Difference
F Sig. t df
Sig.
(2-tailed)
Mean
Difference
Std. Error
Difference
Lower Upper
Sinuosity
Equal variances
assumed
12,401 ,000 3,239 2433 ,001 ,011963 ,00369384956 ,0047201656 ,019206997
Equal variances
not assumed
3,192 2168,33 ,001 ,011963 ,00374837186 ,0046128041 ,019314358
Group Statistics
Year N Mean Std. Deviation Std. Error Mean
Length_m 1956 1136 196,84 269,381 7,992
2011 1299 168,90 234,353 6,502
Independent Samples Test
Levene's Test for
Equality of
Variances
t-test for Equality of Means
95% Confidence Interval of
the Difference
F Sig. t df
Sig.
(2-tailed)
Mean
Difference
Std. Error
Difference
Lower Upper
Length(m)
Equal variances
assumed
14,171 ,000 2,736 2433 ,006 27,934 10,208 7,917 47,952
Equal variances not
assumed
2,711 2266,470 ,007 27,934 10,303 7,729 48,139

48
Annex 9. Correlation and regression (Width and Depth)
Correlation
Width_m Depth_m
Width (m) Pearson Correlation 1 ,998
**
Sig. (2-tailed) ,000
N 10 10
Depth (m) Pearson Correlation ,998
**
1
Sig. (2-tailed) ,000
N 10 10
**. Correlation is significant at the 0.01 level (2-tailed).
Regression
Model Summary
R R Square Adjusted R Square
Std. Error of the
Estimate
,998 ,997 ,996 11,554
The independent variable is Width_m.
ANOVA
Sum of Squares df Mean Square F Sig.
Regression 304367,343 1 304367,343 2279,886 ,000
Residual 1068,009 8 133,501
Total 305435,352 9
The independent variable is Width_m.
Coefficients
Unstandardized Coefficients
Standardized
Coefficients
t Sig.B Std. Error Beta
Width_m ,091 ,002 ,998 47,748 ,000
(Constant) 8,067 3,955 2,040 ,076

49
Annex 10. Fractal Dimension Output data
Output box-Counting method
Figure 24: Fractal Dimension slope 1956 Figure 25: Fractal Dimension slope 2011
Output fractal relationship individual stream
Figure 26: log-log plot Euclidean line(d-long) vs. actual Length (1956) Figure 27: log-log plot Euclidean line (d-long) vs. actual Length (2011)

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