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Title: European Summer School on Soil Survey Pedometrics tools for the analysis of soil typological maps and georeferenced soil databases


1
European Summer School on Soil Survey Pedometrics
tools for the analysis of soil typological maps
and georeferenced soil databases
  • J.J. Ibáñez (CCMA)
  • Centro de Ciencias Mediambientales
  • Consejo Superior de Investigaciones Científicas
    Madrid (Spain)

2
In Memory
  • Fridland (Russian Pedologist)
  • Hole and Campbell (American Pedologists)
  • Soil Pattern Analysis (Soil Maps)

3
Statistical Tools in Pedometrics
  • Typological Analysis of Pedotaxa and soilscapes
    This Dissertation
  • Continuum approach Geostatistics of single soil
    properties
  • Hybrid Methods Typological Analysis
    Geostatistics (into each soil polygon)
  • Other Methods Fractals, Non Lineal Dynamics,
    Fussy Sets, Neuronal Networks, Numerical
    Taxonomies

4
Premises Against the opinion of some
pedometrisians, without formation of the state of
the art in biological taxonomies, biodiversity
and conservation biology, we could consider soil
types or pedotaxa in similar fashion to biotaxa
(including ecosystems and habitats). There are
any epistemologic arguments against this
perspective. The continuum dilemma and the
spatial delimitation of natural bodies affect,
both to pedologic and biologic entities. We
recognize that, as is the case of ecosystems,
habitats and some species, the boundaries of
pedotaxa are fuzzy. Furthermore, the hierarchies
of biological taxonomies and the delimitation /
characterization of many biotaxa are also, at
least, fuzzy and arbitraries. We recommend to
these incredulous pedometrisians to consult the
abundant and suitable biological literature.
5
Introduction
  • Conceptual parallelisms
  • We need only count
  • 1. The different classes of objects
  • 2. The number of individuals or coverage in each
    classes
  • Biotaxa Pedotaxa
  • Ecosystems Soilscapes
  • Ecoregions Soil Regions
  • Biomes Pedomes
  • Ecosphere Pedosphere

6
Introduction
  • Data source
  • 1. Soil Surveys (Maps)
  • 2. Geographical Soil Databases (Inventories)
  • 3. Soil Information Systems

7
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8
The Taxonomic Vision of the Pedometrisians
Slatter Philosophy
9
From Soil Maps to Soil Resources Map
  • Measuring pedosphere surface (non trivial)
  • Measuring pedosphere volume (non trivial)
  • Pedosphere structure (typological approach)
  • Pedosphere properties (continuum appr.)
  • Measuring pedosphere vulnerability (risks)
  • Measuring pedosphere state (deterioration)

10
Introduction
  • Index and models of diversity have proved to be
    valuable in ecology. By using them, ecologists
    have studied the structure and organization of
    ecological entities at different scales (from
    plots to biomes).
  • These methodologies have been applied to the
    study of abiotic structures of landscapes on very
    few occasions, despite their potential.

11
Introduction
  • At the same time, this may be one of the ways to
    explore, quantify and compare the complexity
    abiotic landscapes in different areas and
    environments.
  • This is a type of spatial pattern analysis

12
Information Lost (Map Generalization and Scales
13
The concept of diversity Huston (1995) "The
concept of diversity has two primary components,
and two unavoidable value judgments. The
primary components are statistical properties
that are common to any mixture of different
objects, whether the objects are balls of
different colors, segments of DNA that code for
different proteins, species or higher taxonomic
levels, or soil types or habitat patches on a
landscape. .Each of these groups of items has
two fundamental properties 1. the number of
different types of objects (e.g., species, soil
types) in the mixture or sample and 2. The
relative number or amount of each different type
of object. The value judgements are .whether
the selected classes are different enough to be
considered separate types of objects and
.whether the objects in a particular class are
similar enough to be considered the same type.
On these distinctions hangs the quantification
of biological diversity".
14
Preliminaries on Diversity Analysis From a
methodological point of view, the different ways
of measuring diversity may be grouped into three
categories - indices of richness number of
categories (e.g. biological species, communities,
pedotaxa, soilscapes, etc.) known to occur in a
defined sampling area - indices based on
proportional abundance of categories not only
the number but also their relative abundance (in
our case, the relative area occupied by each
pedotaxa) are taken into account - models of the
distribution of abundance of categories these
provide the most complete description but also
the least abridged. Hollow curves Measuring
conectance and pedocomplexity Species-area
relationships (Spars) Species-time relationships
(Divergent Pedogenesis) Species-energy
relationships Nested subset analysis Area
Selection Methods etc. .
15
The Hollow Curves
16
Hollow curves Everywhere
17
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18
Ranked-Abundance Plot of Island Areas in Km2
Hollow curve
19
Ranked-Abundance List (area in Km2) of
taxonomic pedotaxa Hollow curve
Pedotaxa
Area(Km2)
Ic
7303,65
Lc
5061,71
Rc
3267,15
Ie
2589,8
RO
1503,48
Jc
982,91
Be
751,21
Bc
654,07
Bk
556,63
Id
415,35
Lv
332,2
To
301,8
I
211,25
E
186,26
Bd
134,07
Lo
121,31
U
113,22
Vc
98,51
IRO
72,24
Bh
68,52
Oe
29,33
Gc
28,83
Re
19,5
20
Richness indexes
  • For many scientists the concept of diversity is
    restricted to an inventory of the number of
    different objects ("richness") present in a
    certain ecosystem, landscape, survey or taxonomic
    scheme.
  • It is possible to distinguish between
  • "numerical richness"
  • "species density".
  • Numerical richness is the ratio between the
    number of different soil taxa found and that of
    the soil profiles analyzed.

In soil surveys
  • Taxa density refers to the number of soil taxa
    per sampling area

21
Indices based on the proportional abundance of
objects. (Diversity sensu stricto)
  • Is the most frequent way of estimating diversity.
  • Diversity s.s. may be divided into two elements.
  • "richness"
  • "evenness"
  • The concept referring to the relative abundance
    of each object.
  • This distinction is logical, since for two
    different plots of land with the same area and
    identical richness, the most diverse will be that
    where the different types occupy equal area and
    are equally probable therefore.

22
A summary of the characteristics and performance
of a range of diversity statistics
23
Diversity concept (sensu stricto) (2)
  • Diversity indexes most used in ecology come from
    the Theory of Information. The most popular is
    the
  • Shannon Index (Shannon and Weaver, 1949)
  • Diversity is thus equated with the amount of
    uncertainty that exists regarding the species
    (objects) of an individual selected at random
    from a population. The more species there are,
    and the more even their representation, the
    greater the uncertainty and hence the greater the
    diversity.
  • Information content, which is a measure of
    uncertainty, is a reasonable measure of diversity

24
Diversity concept (sensu stricto) (3)
  • The mathematical expression of the Shannon's
    Index is
  • where
  • H' is the negative entropy "negentropy" or
    diversity of the population
  • pi is the proportion of individuals found in this
    ith taxa.
  • In fact, the true value of pi is unknown, and it
    is estimated by ni/N, where ni is the number of
    individuals of the object considered, and N the
    total number of individuals collected (it may
    also be the percentage of surface area occupied
    by this ith object).
  • The value of H' is the sum of the proportions of
    the individual objects multiplied by the negative
    logarithm of the proportion.
  • Information is maximum when the probabilities
    (proportional abundance) of all objects (S) are
    equal. It is then equal to the ln S. Information
    is 0 if there is only one possibility (one
    object), i.e. diversity is 0.

25
Example of a diversity analysis on the
composition of a hypothetical soilscape
diversity statistics and abundance distribution
model (best fit by chi-squared test). N Number
of observations S Richness H' Negentropy
index E Evenness Hmax Maximum negentropy.
26
Diversity concept (sensu stricto) (4)
  • The value of the Shannon Index is usually between
    1.5 and 4.5.
  • In nature it has not been possible to detect
    values larger than 5 bits although it is
    mathematically possible ( why?).
  • Although the Shannon index considers evenness, it
    is also possible to measur it independently.
  • For certain conditions of richness (S), maximum
    possible negentropy Hmax occurs in situations
    where all objects are equiprobable
  • H' Hmax lnS (2)
  • The relation between negentropy observed and
    maximum negentropy may be used, therefore, as a
    measure of evenness E
  • E H/Hmax H/ lnS (3)
  • The E index can take any value between 0 and 1,
    where 1 represents the situation in which all
    species or objects are equiprobable (e.g. when
    they occupy the same area) and tend to 0 when
    there is a highly non-uniform distribution of
    relative abundance (i.e. where one object
    dominates over all others).

27
Some limitations of diversity analysis
  • Estimates of diversity can be made on information
    from several scales or different intensity
    sampling. For example, as a consequence of map
    generalization, considerable information is lost
    (e.g., taxa with scarce spatial representation
    are not usually considered). Thus, the results
    obtained varies according to the quality and
    quantity of the available information .
  • The figures obtained for richness and entropy
    indices are determined by the maximum number of
    taxa which the classification used allows.

28
Why Drainage Basins
  • According to the Horton Laws size-frequency
    distributions of drainage basins are conform to
    power laws.
  • They are also fractal structures

29
The fractal nature of river networks and the
Horton Laws (2)
  • For many purposes and increase of the order (or
    rank) of a basin could be understood like an
    increase of complexity.
  • Many branching systems follow Horton Laws. For
    example
  • Mammalian circulatory and respiratory systems
  • Plant vessel-bundle vascular system
  • Plant radical systems
  • In these disciplines the Strahler classification
    also has been used with profusion.

Horton Laws are an example of space-filling
fractals. This is a natural and common structure
for ensuring a very fast and economic flow (and
sometimes exportation) of matter.
In space filling fractals energy dissipated to
distribute resources is minimized.
30
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31
The fractal nature of river networks and the
Horton Laws (3)
  • Horton Law of Stream number
  • Nk RB(s-k) (1)

Where Nk the number of streams of order K Rs
the branching ratio (a constant characteristic of
the basin) S the order of the basin
Horton Law of Stream Length Lk RL(k-1)L1
(2)
Where Lk average lengh of Kth order stream RL
stream lengh ratio, a constant for any given basin
Horton Law Stream Slopes S'k Rs (1-k) S'1 (3)
Where S'k is the average slope of the Kth
order streams Rs (1-k) is the stream slope
ratio ( a constant)
Where A'K is the average area drained by each
Kth order stream RA(k-1) the stream area ratio
Schumm Law of Stream Area A'K RA(k-1)A'1 (4)
Here Ak is defined to include not only the area
drained directly by the particular stream but
also that drained by all its tributaires of lower
order
32
The fractal nature of river networks and the
Horton Laws (4)
  • The fact that the variables Nk, L'k, S'K and A'K
    are all related to stream order by either a
    direct or an inverse geometric series suggest
    that they are also related to each other by
    simple power laws (Allometric Growth Law). For
    example, by eliminating K between equations 2 and
    4, we find
  • L's/L'1 (A'K/A1)n (5)
  • Where n ln RL / ln RA (6)
  • Hack Law of Mainstream Lengh-Basin Area
  • L CAn' or empirically L 1.4A2/3 (7)
  • Where
  • L mainestream lengh
  • C a constant with experimental figure ? 1.4 and
    n' ? 0.6
  • Therefore the river open system grows
    allometrically according to the general power law
  • Y axb

33
Pedodiversity at scale 1 20.000 (Drainage basins)
34
Pedodiversity and Diversity of Plant Communities
in the Iberian Peninsula according to Basins Rank
35
Pedodiversity and Diversity of Plant Communities
in the Iberian Peninsula according the to size of
the basins
36
Diversity, drainage basins in geomorphology and
pedology (1)
  • Fluvial patterns are filling fractal trees (see
    the nature of Horton Laws)
  • The evolution of fluvial systems (increase of
    fractal dimension of fluvial courses) induces an
    increase in complexity of pedogeomorphological
    landscapes and biological ones..
  • Fluvial incision (chronosequences) also increases
    negentropy and richness of geomorphological and
    soil landscapes.

37
Diversity, drainage basins in geomorphology and
pedology (2)
  • In high structured drainage basins, diversity and
    richness of soils and geomorphologic units
    increases from top (mainly erosive) to bottom
    (mainly depositional).
  • There are close connections between the land's
    fractal dimension and its geomorphologic and
    pedological negentropy (Shannon diversity Index).
  • Are Drainage basins and their soil and
    geomorphologic cover "dissipative systems" ?.
  • Increase of soilscapes, geomorphologic landscapes
    and plant landscapes diversities, in
    correspondence with drainage basins
    hierarchisation, may be scale invariant.

38
Pedodiversity at Global Level
39
Pedodiversity at Continental Level
40
Aegean Archipelago Correlation between two
hierarchical levels (FAO)
41
Pedodiversity and Richness in the Aegean
Archipelago
42
Pedodiversity in plots of different soilscapes
(Terrace chronosequence in the Henares River
43
Edafodiversidad a pequeñas Escalas Cronosecuencia
del Río Henares
44
Genetic Pedodiversity
45
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46
Diversity statistics for observed horizon
sequences and their absolute and relative
frequency for the Abal soil series Belgium
(original data from Van Orshoven et al., 1991).
47
Measuring Complexity and Connectance Graphos
48
Measuring Complexity two approaches
49
Pedocomplexity and Connectance (Henares River
dataset)
50
Beta (Differentiation) Diversity
51
Example of the application of how inventory
diversity and differentiation diversity concepts
could be applied in 1250000 scale natural region
soil map (the hypothetical are covered by the
soil map is 1 Millon ha)
52
Cascade hierarchy of diversities
53
Beta Diversity Analysis Pedodiversity at Global
Level
54
Beta Diversity Analysis Pedodiversity at Global
Level
55
Abundance Distribution Models
56
Abundance Distribution Models
  • Abundance distribution models (e.g., geometric
    series, logseries, lognormal distributions, gamma
    distributions, power laws, broken stick models,
    etc.) are statistical tools which ecologists have
    applied for decades with the object of analyzing
    the intrinsic regularities of various ecological
    entities.
  • It would be interesting to apply these techniques
    in pedology with the object of detecting the
    similarities and differences between the entities
    of both natural resources.
  • The same is true respect other abiotic resources
    such as, geology, geomorphology, climatology, and
    so on.
  • The two former will give approximately straight
    lines when log abundance is plotted against rank
    whilst the latter shows up as a straight plot
    when abundance is plotted against log rank

57
Rank-Abundance Plots
58
Rank-Abundance Plots
59
Abundance Distribution Models Geometric Series
60
Summary of the most popular abundance
distribution models in ecology Taking into
account the underlying fragmentation mechanisms
of each model, the broken-stick model is the
statistical realistic expression of an
intrinsically uniform distribution and in the
opposite the geometric and log series models the
statistical expression of an uneven distribution.
The log normal model is in most respects
intermediate between broken-stick and geometric
or log series extreme models.
61
Geometric series model (1) This model (Motomura
1932) is based in the assumption that one entity
is subdivided recursively following a fixed rule
for a given fraction k. If the entity is
idealized by the unit segment, the first fragment
measures k, the next one is a fraction of what
remains, then it measures k(1-k) and so on. It is
obtained the series k, k(1-k), k(1-k)2,. ,
k(1-k)i-1. It is a geometric series for the first
under the hypothesis that the abundance of the
taxa are proportional to the numbers of the above
series, they correspond to the ranked-abundance
list as proportions. For a community of NT
individuals, NT Ck k(1-k)i-1 is the size of the
i-th most abundant taxa, where Ck is a constant
which value is being ST the total number of
taxa, in order to ensure that
62
Logseries model (1) Suppose that the mechanism of
the geometric series model stems from the fact
that the taxa arrive at successive uniform time
intervals and proceed to preempt a fraction of
the remaining niche before the arrival of the
next, then, the randomization of the time
intervals leads to a log series distribution
(Fisher 1943). According to the log series
distribution the frequency of species with N
individuals is for N1,2, where
and are constants. Then, it is a
discrete distribution.
63
Geometric vs. Logarithmic series
64
Abundance Distribution Models Lognormal Series
65
Log normal model (1) It is a mixture of
geometric and broken stick models. There are
recursivity and randomization. The segment of
unit length is broken, at a randomly chosen
point, into two parts. One part is chosen
randomly and independently of its length, and is
broken into two parts chosen one point of its
points randomly. One of the three resulting parts
is chosen randomly and independently of its
length, and is broken into two parts chosen one
point of its points randomly. And so on. Then,
the distribution of the lengths of all the parts
tends to log normal form.
66
Log normal model (2) If we now assume that the
species in a multy-speciose community have
divided up some limiting resource among
themselves in this manner, and that the abundance
of each taxa is proportional to its share of the
resource, then the taxa-abundance distribution is
log normal. This fragmentation process has been
applied previously in soil sciences in connection
with aggregates and particle-size distribution
(Hatch 1933 Kolmogorov 1941, among others). In
this case, the density function is where
and are the random variable mean and
standard deviation of It is also used the
parameter ? to characterize the distribution.
67
Abundance Distribution Models Broken Stick model
68
Summary of the most popular abundance
distribution models in ecology Broken-stick
model In this case the unit segment is broken,
randomly and at once, among the number of
observed taxa for taxa, points are located
randomly in the unit interval so that the
predicted sizes are proportional to the sizes of
the obtained subintervals. As before, it may be
computed the abundance distribution. In this
case, the density function is a negative
exponential with parameter for
69
Diversidad Area en el tiempo y diferentes
geoambientes
70
Distribution Abundance Models
  • Geométric Series
  • Logarithmic Series
  • Lognormal Distr.
  • Broken Stick Model
  • Others
  • Power Law
  • Fractals
  • ? Small samples
  • ? Environmental Perturbations
  • ? Larger non-perturbed Communities
  • ? SPARs and SPITs
  • ?Ecological Dynamics
  • ? Evolutive Dynamics
  • ?General System Dyn.
  • ? Fit Problems

71
Observed and expected values of the cumulative
distribution functions of pedotaxa area
Pedotaxa Area
Observed
Expected
Log-Normal
Logarithmic
Geometric
Broken Stick
20
1
1,727
8,987
0,199
0,423
29
2
2,511
9,894
1,641
0,610
30
3
2,594
9,977
1,773
0,631
69
4
5,255
12,025
5,006
1,426
73
5
5,478
12,164
5,224
1,505
99
6
6,774
12,915
6,407
2,017
114
7
7,420
13,262
6,955
2,307
122
8
7,740
13,429
7,218
2,460
135
9
8,227
13,677
7,611
2,706
187
10
9,863
14,476
8,875
3,662
212
11
10,513
14,783
9,362
4,105
302
12
12,372
15,643
10,736
5,618
333
13
12,885
15,879
11,115
6,110
416
14
14,041
16,414
11,979
7,362
557
15
15,505
17,108
13,112
9,278
655
16
16,281
17,489
13,741
10,470
752
17
16,916
17,810
14,277
11,548
983
18
18,072
18,424
15,317
13,756
1504
19
19,665
19,363
16,967
17,298
2590
20
21,242
20,474
19,077
20,917
3268
21
21,762
20,907
19,979
21,889
5062
22
22,519
21,631
21,678
22,790
7304
23
22,962
22,127
23,101
22,974
72
Distribution functions of the models and
observed values of pedotaxa area
73
Ranked-Abundance List of the models
and of observed values
(Logarithmic scale)
74
Diversity and Area For many conservation
biologists, one of the most important empirical
observations in ecology is that larger areas
contain more species than smaller areas. For
understanding how communities are organized or
what the consequences of reserve design are for
the maintenance of biodiversity, any relationship
is more informative. Yet an understanding of
what determines species-area curves remains
elusive at the date.
75
F(N) Total number of taxa with size
in excess of N (km2) Species or Peodotaxa
accumulation curves (Aegean Isles
76
The theory of island biogeography The theory of
island biogeography (MacArthur 1960, 1965,
MacArthur and Wilson 1963, 1967) has been for
decades the keystone in biodiversity analysis on
islands, as well in conservation biology. This
theory rest on the assumption that the number of
species residing in a habitat is the result of an
equilibrium between immigration and extinction.
A goal of the theory is the explanation of the
dependence of species richness upon such factors
as area and the proximity and magnitude of
sources of immigrants.
77
The Predictions of the theory of island
biogeography Two predictions of the Theory of
Island Biogeography, among others, are (i)
The most widespread distribution abundance
patterns of the species assemblages into island
(as in mainland) communities is the canonical
lognormal distributions (i) The species
numbers increase with the area according to a
power law. Much interest has centred on the value
of p, the slope of the ln S versus ln A
regression. the pedorichness-area relationships
conform to a power law whose exponent is around
the value of 0.25. An impressive body of field
data has been accumulated to support the validity
of the equations and assumptions of the theory.
Observed values often fall in the range 0.2-0.4
78
Habitat heterogeneity However the mechanism
leading SPARs are at best obscure. There are
currently other alternatives, which differ from
that of island biogeography. One of them
focussed on habitat-heterogeneity a larger
area encompasses more diverse habitats, which
support more species. Thus, habitat
heterogeneity approach would argue that the
number of species reflects the range of habitats
included in the area.
79
Richness-area curves (1) The dependencies of the
number of species (or the number of other
different taxonomic categories) on the area
(referring in particular to islands and other
analogous habitats of patchy spatial occurrence),
that are explored in ecological literature, may
be classified into two kind of models. Those
that could be approximated by a linear relation
between log S and log A (log S p log Aq for
certain constants p and q ) and those which
approximation is a linear relation between S and
log A (S a log Ab for certain constants a and
b ). First relation is a power function with
exponent p and prefactor ( ). Second
relation is simply a logarithmic function of S in
terms of A (or an exponential function of A in
terms of S ). The first will be called the
power model and the latter the logarithmic model.
80
Richness-area curves (3) The log normal and the
broken-stick models yield power curves. For the
log normal model the exponent is related with
parameter by the expression (May 1975)
(eq. 1) For the
broken-stick model . Let us remark
that for the lognormal model, the smallest
exponent is obtained for ,
that is to say, for the so called canonical
lognormal model. The geometric and log series
model yield logarithmic expressions for the
richness-area curves. It holds (May 1975) that
(eq. 2) Summarizing, while a
logarithmic expression directly results from the
log series distribution of species-abundance
combined with the assumption that population size
scales linearly with habitat area, the canonical
log normal distribution with the same assumption
leads to a power expression.
81
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82
Pedodiversity-Area Relationships
  • Continents, Drainage, Isles, Basins at Different
    Scales and Plots

83
Power laws fit of pedotaxa of the Aegean
Archipelago (1.000.000)
84
Power Law Fits of Pedotaxa and Vascular Plants In
Brithish Isles
85
Drainage Basins at 11.000 Scale in Spain
pedodiversity-area relationships and
biodiversity-area relationships
86
Spars for Drainage basins at the scale 120.000
87
Pedosphere Power Laws and different Scales From
14M to 1 500m
88
Result of power law fits of pedotaxa-area
relationships for the six tested datasets
89
Nested Subset Analysis
90
Nested Subsets and taxa-range size distribution
  • A common pattern of taxa distribution has been
    termed nested subsets by ecologists
  • Species present in any particular biological
    assemblage tend also to be present in larger
    assemblages.
  • The same is true for the so called species-range
    size distribution
  • Locally abundant species tend to be widespread,
    whereas locally rare species tend to be narrowly
    distributed

91
Nested Subset Matrix
92
Non-Perfect Nested Matrix
93
Results of the nested subset analysis for the
Aegean Archipelago
94
Results of Nested Calculator for the six tested
datasets
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Hollow curve and power law for pedo-taxa range
size distribution in Aegean Archipelago
Pedotaxa number of presences in the more of 120
analyzed islands
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Conclusions The regularities detected in the
pedological entities tested are surprisingly
similar to those in ecological literature. Both,
diversity-area relationships, abundance
distribution models, nested subsets, taxa
range-size distributions appear to follow the
same trend in ecology and pedology. Thus, for
example, our results suggest that the biological
assumptions underlying the Theory of Island
biogeography could not be the cause of the last
relation and the generalized value of the 0.25
for the exponent. Thus, in contrast with the
generalized ecologist opinion, neither power laws
for SPARs, Nested subsets, fractals structures,
and log-normal distributions for DEMs etc, do not
appears to be a property of the structure of the
communities and the spatial distribution of the
species respectively.
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Other Typological Pedometrics Methods
  • Will be analysed in the dissertation on
    Biodiversity Loss

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Power Laws and Fractals
99
Complementarity Methods for the Design of
Networks of Natural Soil Reserves
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The End Fin
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Why Landmasses
  • According to Mandelbrot (among others), the
    size-frequency distribution of landmasses fit to
    a power law
  • size-frequency distribution of islands in a given
    archipelago is an example for small landmasses
  • That is a problem of fragmentation

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Why Drainage Basins
  • The surface fragmentation of landmasses in
    drainage basins is an example of fractal
    structures
  • The channels are also filling fractal trees

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Background (1) Williamson (1981) found an
almost perfect linear relationship between the
logarithm of the number of geological types and
the logarithm of the mapped area (a power law)
using a geological map. He concluded that
Should this pattern of environmental variation
be found to be common, then at least some
species-area variation found in plants could be
ascribed reasonably directly to the environment
in which they are growing, while that of animals
might be ascribed either to the environment, or
to the plants, or to both (see also Rafe et al.
1985).
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Background (2) Cody (1983) observed that bird
diversity on islands in the Gulf of California
increase in a stepwise fashion as island area
increased. He suggested that this was controlled
by the geomorphology of drainage basins. On
different-sized islands larger islands could
support higher-order rank channels and therefore
a greater range of riparian habitats than smaller
islands (see also Ibáñez et al. 1990 and 1994).
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