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TOWARDS FOUR-DIMENSIONAL MODELING OF
GEOSPATIAL PHENOMENA: AN INTEGRATION OF
VOXEL AUTOMATA AND THE GEO-ATOM THEORY
by
Anthony Jjumba M. Sc., Simon Fraser University, 2009
B.Sc., Makerere University, 2000
Thesis Submitted in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy
in the
Department of Geography
Faculty of Environment
Anthony Jjumba 2015
SIMON FRASER UNIVERSITY
Summer 2015
ii
Approval
Name: Anthony Jjumba Degree: Doctor of Philosophy Title: Towards Four-Dimensional Modeling of Geospatial
Phenomena: An Integration of Voxel Automata and the Geo-Atom Theory
Examining Committee: Chair: Geoff Mann Associate Professor, Geography
Dr. Suzana Dragićević Senior Supervisor Professor, Geography
Dr. Shivanand Balram Supervisor Senior Lecturer, Geography
Dr. Margaret Schmidt Supervisor Associate Professor, Geography
Dr. Kirsten Zickfeld Supervisor Assistant Professor, Geography
Dr. Hao (Richard) Zhang Internal Examiner Professor, School of Computing Science
Dr. Andrew Crooks External Examiner Assistant Professor, Department of Computational Social Science George Mason University
Date Defended/Approved: June 15, 2015
iii
Abstract
Geographic Information Systems (GIS) are widely used in both the management of
spatial data as well as in the study and analysis of spatiotemporal processes. However,
contemporary spatial data models are based on the principles of traditional two-
dimensional cartography that simplify space to the horizontal Cartesian plane. This is
partly due to the limitations in the way spatial phenomena are generally conceptualized
and represented in GIS databases using the vector and raster geospatial data models. Over
the last two decades, various methods of modeling the third dimension, incorporating the
temporal component, and linking the field and object perspectives have been proposed
but they have received little integration in GIS software applications. The main objective
of this dissertation is to develop modeling approaches that can represent dynamic spatial
phenomena in the four-dimensional (4D) space-time domain (three-dimensional space
plus time) using a theoretical geospatial data model and the principles of complex
systems and geographical automata theory. Using the theory of complex systems and the
geo-atom concepts, this dissertation proposes and implements a voxel-based automata
modelling approach for the study and analysis of 4D spatial dynamic phenomena. The
data are structured using the geo-atom model, a theoretical geospatial data model that
links the object and field perspectives of space and explicitly models the four dimensions
of the space-time continuum. The results from implementing voxel-based automata
indicate their usefulness in simulating dynamic processes, the management of geographic
data, and the development of three-dimensional landscape indices and spatiotemporal
queries. The significance of the study is that it provides a robust platform that
demonstrates the potential of the voxel automata and geo-atom spatial data model to
represent spatial dynamic phenomena in 4D.
Keywords: voxel automata; geo-atom; four-dimensional GIS; geospatial modeling; 3D; spatial indices; complex systems
iv
Dedication
Dedicated, with deep gratitude, to my wife and children
for their love and encouragement.
v
Acknowledgements
This research has been funded through a Discovery Grant from the Natural Sciences and
Engineering Research Council (NSERC) of Canada awarded to Dr. Suzana Dragićević.
Additional funding was provided by Simon Fraser University through the President's PhD
Research Stipend and two Graduate Fellowships. I am very grateful for this financial
assistance.
I am deeply indebted to Dr. Dragićević for her support and guidance throughout my
graduate studies. The countless hours spent reviewing and contextualizing my work are
not lost on me but without her excitement about 4D GIS in addition to her patience and
understanding of my personal circumstances, this thesis would not have been completed.
I would also like to thank Drs. Schmidt, Balram and Zickfeld for their feedback and
comments which have enabled me to clarify and corroborate many issues in this thesis.
My colleagues in the Spatial Analysis and Modelling Lab, your cheer and companionship
was much appreciated. To my family and friends (you know who you are), thank you
very much. Finally, and with much affection, thank you Lára for your support and
patience.
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Table of Contents
Approval ............................................................................................................................. ii Abstract .............................................................................................................................. iii Dedication .......................................................................................................................... iv Acknowledgements ..............................................................................................................v Table of Contents ............................................................................................................... vi List of Tables ..................................................................................................................... ix List of Figures ......................................................................................................................x
Chapter 1. Introduction .................................................................................................1 1.1. General Introduction ...................................................................................................2 1.2. Geospatial Data Models ..............................................................................................5
1.2.1. Raster Data Model .........................................................................................6 1.2.2. Vector Data Model ........................................................................................6 1.2.3. Unified Models ..............................................................................................6 1.2.4. The Geo-Atom Data Model ...........................................................................7 1.2.5. The Voxel-Based Data Model .......................................................................7
1.3. Representing Dynamic Spatial Phenomena in four dimensions .................................8 1.3.1. An Overview of Temporal Dimensional Representation ..............................8 1.3.2. Complex System Theory for Three and Four-Dimensional
Representations .............................................................................................9 1.4. Spatiotemporal Queries .............................................................................................11 1.5. Research Problem and Objectives ............................................................................12 1.6. Datasets and Software Tools .....................................................................................14 1.7. Structure of the Thesis ..............................................................................................15 References ..........................................................................................................................16
Chapter 2. Voxel-based Geographic Automata for the Simulation of Geospatial Processes ..................................................................................22
2.1. Abstract .....................................................................................................................22 2.2. Introduction ...............................................................................................................23 2.3. Literature Overview ..................................................................................................25 2.4. Methods.....................................................................................................................28
2.4.1. The Formalism for Voxel Automata ...........................................................28 2.4.2. Voxel Space .................................................................................................29 2.4.3. The 3D Neighbourhood ...............................................................................30 2.4.4. Voxel State ..................................................................................................31 2.4.5. Transition Rules ..........................................................................................32 2.4.6. Time-steps ...................................................................................................35
2.5. Simulation of Spatial Processes in 4D ......................................................................35 2.5.1. Binary Voxel Automata ...............................................................................35
Sandpile Model ................................................................................................................... 35
vii
Object Movement in Space ................................................................................................. 37 Dispersal of Objects in Space ............................................................................................. 38 Landslide Processes ............................................................................................................ 39
2.5.2. Multiclass Voxel Automata .........................................................................42 Advection with Multiple Classes ........................................................................................ 42 Particle Segmentation with Multiple Classes ..................................................................... 43
2.6. Conclusions ...............................................................................................................44 References ..........................................................................................................................46
Chapter 3. Integrating GIS-Based Geo-Atom Theory and Voxel Automata to Simulate the Dispersal of Airborne Pollutants ...................................53
3.1. Abstract .....................................................................................................................53 3.2. Introduction ...............................................................................................................54 3.3. Theoretical Background ............................................................................................56
3.3.1. Modelling the Space-Time Domain ............................................................60 The Geo-Atom Structure for a 3D Voxel Automaton ........................................................ 61
3.3.2. Simulating the Dispersal of Airborne Pollutants in 4D ...............................62 3.4. Methods.....................................................................................................................63 3.5. Transport due to Particle Advection .........................................................................64
3.5.1. Transport due to Particle Diffusion .............................................................65 3.5.2. The Elements of the 3D Voxel Automaton .................................................66
Transition Rules for Modeling the Dispersal of Pollutants................................................. 69 3.5.3. Model Implementation ................................................................................70
3.6. Simulation Results and Discussion ...........................................................................70 3.7. Conclusions ...............................................................................................................81 References ..........................................................................................................................83
Chapter 4. Spatial Indices for Measuring Three-Dimensional Patterns in Voxel-Based Space .....................................................................................90
4.1. Abstract .....................................................................................................................90 4.2. Introduction ...............................................................................................................90 4.3. An Overview of Spatial Indices for 2D Space ..........................................................93 4.4. Methods.....................................................................................................................97
4.4.1. Number of Bloxels ......................................................................................97 4.4.2. Bloxel Surface Area ....................................................................................98 4.4.3. Expected Surface Area ................................................................................98 4.4.4. Volume ........................................................................................................99 4.4.5. Fractal Dimension .....................................................................................100 4.4.6. Lacunarity ..................................................................................................100 4.4.7. 3D Moran's I ..............................................................................................101
4.5. Implementation of the Spatial Indices for 3D Space ..............................................103 4.5.1. Hypothetical Results ..................................................................................103
Number of Bloxels ........................................................................................................... 103 Bloxel Surface Area.......................................................................................................... 104 Expected Surface Area ..................................................................................................... 105
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Fractal Dimension ............................................................................................................ 106 Lacunarity ......................................................................................................................... 108
4.5.2. Complexity of Static Phenomena ..............................................................109 4.5.3. Complexity of Dynamic Phenomena ........................................................112
4.6. Conclusions .............................................................................................................114 References ........................................................................................................................114
Chapter 5. A Representation and Analysis of a Process' Spatiotemporal Dynamics Using a 4D Data Model .........................................................121
5.1. Abstract ...................................................................................................................121 5.2. Introduction .............................................................................................................121 5.3. Theoretical Background ..........................................................................................124
5.3.1. Two-Dimensional Representation of Geospatial Phenomena ...................124 5.3.2. Four-dimensional Representation of Geospatial Phenomena ...................125
5.4. Methods...................................................................................................................128 5.4.1. Conceptualization of Space and Time .......................................................128 5.4.2. Spatiotemporal Queries .............................................................................129
Simple Spatiotemporal Queries ........................................................................................ 130 Spatiotemporal Range Queries ......................................................................................... 131 Spatiotemporal Behavior Queries ..................................................................................... 132
Rate of Change .......................................................................................................... 132 Location Descriptor Queries ...................................................................................... 132
Temporal Relationship Queries ........................................................................................ 133 5.4.3. Case Study: A Model for Snow Cover Retreat .........................................133
5.5. Implementation and Results ....................................................................................135 5.5.1. Queries for Snow Cover Retreat Model ....................................................136 5.5.2. Simple Spatiotemporal Queries .................................................................137 5.5.3. Spatiotemporal Range Queries ..................................................................138 5.5.4. Spatiotemporal Behavioral Queries ...........................................................139
Rate of Change Query ...................................................................................................... 139 Location Descriptor .......................................................................................................... 140
5.5.5. Temporal Relationship Queries .................................................................141 5.6. Conclusion ..............................................................................................................143 References ........................................................................................................................144
Chapter 6. Conclusion ................................................................................................150 6.1. Summary of Findings ..............................................................................................151 6.2. Perspectives on Further Research Work .................................................................153 6.3. Thesis Contribution to the Body of Knowledge .....................................................154 References ........................................................................................................................155 Appendix A. Co-authorship Statement ...................................................................157
ix
List of Tables
Table 4.1. The calculated 3D indices for the bloxel representations in Figure 4.6 ............................................................................................................110
Table 4.2. 3D Indices for the bloxels extracted from a binary vegetation dataset ......................................................................................................111
x
List of Figures
Figure 2.1. Various configurations for 3D neighborhoods ..........................................31
Figure 2.2. Types of voxel states (a) binary and (b) multistate automata ....................32
Figure 2.3. The relative position of the voxels in a 3D Moore neighborhood .............34
Figure 2.4. Voxel automata simulation outcomes for the sandpile model (a) for a run up to 3000 iterations and (b) for a series of successive critical moments when the form of the sandpile significantly changes its shape in one iteration. ................................................................................37
Figure 2.5. Objects moving towards a central region of attraction, simulation outcomes for fish moving towards a feed patch as an example .................38
Figure 2.6. Simulation outcomes of a reproduction and dispersal of an organism in a fluid medium. ......................................................................39
Figure 2.7. Simulation of a landslide scenario using voxel automata for time T=50, T=500 and T=1800 as well as T=250 and T=500 for a portion of the study area at a larger scale. .................................................41
Figure 2.8. Simulation outcomes of multistate voxel automata of particle dispersion using (a) Moore and (b) von Neumann 3D neighborhoods ............................................................................................43
Figure 2.9. Voxel automata simulation outcomes for segmentation of particles in a moving fluid ........................................................................................44
Figure 3.1. Representation of the 3D voxel automata neighborhood: (a) 3D Moore neighborhood, (b) face neighbors within this neighborhood, (c) diagonal neighbors and (d) double-diagonal neighbors .......................67
Figure 3.2. A two dimensional representation of the wind directions in (a) the horizontal plane and (b) the vertical plane .................................................69
Figure 3.3. Voxel automata simulation of the propagation of pollutants from a single non-continuous point source in 3D space over time .......................72
Figure 3.4. Measuring pollutant concentration values:(a) the positions, located at 10, 20, 24 units from the point source, at which voxel values are recorded; (b) the line graph of the concentration values as recorded at the three positions for the scenario depicting the propagation of pollutants from a single non- continuous point source ..............................73
Figure 3.5. Voxel automata simulation of the propagation of particles from a single continuous point source over 3D space and time ............................75
xi
Figure 3.6. Concentration values measured at 10, 20, and 24 voxel units downwind from the point source for the scenario depicting the simulation from a single continuous point source .....................................76
Figure 3.7. Voxel automata simulation of the propagation of particles in an uneven landscape from (a) a single non-continuous point source (b) a continuous point source .....................................................................77
Figure 3.8. Graphical representations of the concentration values measured at 10, 20, and 24 voxel units for (a) a non-continuous point source and (b) a continuous point source scenario ................................................78
Figure 3.9. Voxel automata simulation of the propagation of pollutants from (a) a single continuous point source under randomly fluctuating westerly wind speed magnitude and (b) multiple continuous point sources under randomly fluctuating northerly, southerly and westerly wind speed magnitudes ...............................................................79
Figure 3.10. Graphical representations of the concentration values measured at 10, 20, and 24 voxel units for the scenario with random fluctuations of the wind speed magnitudes ................................................80
Figure 3.11. Pollutant propagation from a continuous point source in a non-homogenous landscape that represents a hypothetical urban setting .........81
Figure 4.1. Moran's I values for the different 3D voxel state configurations depicting (a) negative spatial autocorrelation for a perfectly uniform pattern (b) no spatial autocorrelation for a random pattern and (c) a strong positive correlation for a clustered pattern. ....................102
Figure 4.2. Different patterns of three voxels arrangements in a bloxel: (a) in a row, and (b, c) at right angles ..................................................................105
Figure 4.3. The calculated fractal dimension values of various bloxels: a) voxels arranged as a single row are effectively a linear 1D object, (b) arranged contiguously in a single plane, whereas (c) and (d) are 3D bloxels. ...............................................................................................107
Figure 4.4. Fractal dimension values obtain for various complex bloxels .................107
Figure 4.5. Lacunarity measure of a bloxel with a) smooth surface b) symmetrically arranged gaps (uniformly distributed), and c) randomly distributed gaps on the surface ................................................108
Figure 4.6. A hypothetical landscape showing accentuated sample 3D bloxels ........110
Figure 4.7. A hypothetical binary distribution of voxels in the landscape: black voxels representing a plant species of interest and white the other vegetation cover .......................................................................................111
Figure 4.8. Simulations results of a dynamic process at selected time-steps T representing iterations 1, 500, 1000, 2000 and 3000. ..............................113
xii
Figure 4.9. Graphical representation the 3D indices over time a) fractal dimension and lacunarity, and b) volume, surface area and expected surface area indices. On the y-axis is the scale for the index value and the number of iterations on the x-axis ...........................113
Figure 5.1. Three snapshots of a field process in voxel space representing the attribute and temporal value updated at every time-step using the geo-atom model .......................................................................................129
Figure 5.2. The data query procedure narrows the search by sequentially limiting the spatial extents, attribute values and temporal bounds. .........130
Figure 5.3. A voxel automata simulation of snow cover change as a function of elevation for three selected time periods t1, t2 and t3, which could be days or weeks depending on the atmospheric temperature. ................135
Figure 5.4. Snapshots of the results from the snow cover retreat process where T1, T3, T6, T10 represent 1 week, 3 weeks, 6 weeks and 10 weeks, respectively, from the start of the simulation...........................................137
Figure 5.5. Selected voxels for the simple and spatiotemporal range queries (a) query for 14.0 cm and 15.0 cm thickness of snow at time-step 5, and (b) all active voxels with snow at time-step 5 ...................................138
Figure 5.6. Results for spatiotemporal range queries .................................................139
Figure 5.7. Output for a pre-defined boundary selection ...........................................140
Figure 5.8. Sequential occurrence of events: (a) all voxels with snow depth 20 cm at any point between weeks 1-10, (b) voxels with snow depth 20 cm at week 3, (c) voxels with snow depth of 20cm at week 5 ...........142
1
Chapter 1. Introduction
The study of the occurrences, relationships and distributions of spatiotemporal
phenomena in topographic landscapes is important to our understanding of the real world.
For analysis in geographic information systems (GIS), the phenomena need to be
appropriately described and their location geo-referenced (Burrough, et al., 2005).
Additionally, geographic phenomena are intrinsically dynamic (Jakle, 1971) and, thus,
need an additional temporal description to characterize their dependence on time
(Langran, 1992). Yet, the geospatial data models used to represent spatiotemporal
phenomena are largely two-dimensional in their conceptualization of space. Both the
temporal component and the third spatial dimension (for example, elevation, depth,
intensity) are technically referenced as additional attributes of the horizontal location.
The temporal component is often displayed as a series of snapshots in many GIS
applications (Langran and Chrisman, 1988; Dragicevic and Marceau, 2000; Keon, et al.,
2014; Yi, et al., 2014), and GIS technology is still dominated by the concept of the paper
map which can be attributed to the simplicity it offers in spatial representation.
The two-dimensional spatial data models, which are generally accepted as the alternative
methods of representing space, are derived from the field and object perspectives of the
real world (Goodchild, 1989; Burrough and McDonnell, 1998). In the field perspective,
each location in space is mapped to a value selected from an attribute domain while for
the object perspective, space is perceived as a region populated with discrete entities,
each having an identity, geometry and other added attributes corresponding to the
phenomenon under study. Depending on the purpose and circumstance of the modeling
exercise, spatial phenomena are generally represented using either of these two
2
contrasting perspectives (Couclelis, 1992). However, the representation of certain
phenomena appears to combine aspects of both of these two perspectives and this duality
of the field and object perspectives is valuable and well articulated (Worboys, 1995). One
well described case that involves both the field and object perspectives is the familiar
concept of a density field, which is created after determining the number of objects and
the extent of the area around them. The density field value is not uniquely defined for any
given location. To calculate the density, it is necessary to define the area around the
objects, and depending on the size of the selected area, different values can be obtained.
In other words density is a field based variable whose values at each location cannot be
separated from the size and shape of the spatial objects used to create them (Peuquet, et
al., 1998).
Thus, the broader purpose of this dissertation is to model various spatiotemporal
phenomena using a geospatial data model that explicitly represents time as well as the
three dimensions of space. The geospatial data model that is used links both the field and
object perspectives of space. It is applied within the context of simulations that
encompass three-dimensional space plus time as an additional dimension. This is a novel
and ongoing research effort for a multi-dimensional representation framework that
provides the ability to enhance the understanding of geographical reality. Spatial Data
that comprehensively represent space and time in the four dimensions of the space-time
continuum can be queried for spatiotemporal relationships such as duration and
frequency of incidents or events and their corresponding impacts in order to provide
improved insights into the evolution of the of dynamic spatial phenomena.
1.1. General Introduction
Geographic phenomena are characterized by entity flows and the movement of objects
but their representation in GIS applications is often simplified to a static two-dimensional
map view (Goodchild, 2010). In those instances where there is no change in the geo-
referenced location, the phenomena are at least characterized by variable changes in the
3
values of the attributes used to describe them. For synthesis and illustration, the
phenomenons are oftentimes depicted as geographic systems comprised of interacting
spatial entities. The relationships and interactions between these constituent entities,
which are ultimately responsible for driving the dynamics of the systems, occur along a
temporal continuum as events, activities and processes (Yuan and Hornsby, 2008).
However, the integrated representation of space and time in GIS software applications
that is central to geographic analysis and modeling has not been fully developed despite
extensive research since the 1980s (Peuquet, 1984; Langran and Chrisman, 1988).
GIS models are a limited representation of the complex reality on the earth’s surface
(Goodchild, 1992). Geographic information about four-dimensional space, as it pertains
to topographic landscapes and the geographic processes operating therein, cannot be
adequately displayed using two-dimensional GIS technology. This problem was realized
from the earliest days of the development of the field of geographic information science
(GIScience) as a discipline of study (Goodchild, 1992; Hazelton, et al., 1992) and
significant developments regarding the integration of time in spatial representation have
been put forward. In relational databases, time is considered as an intrinsic part of the
data and time-stamps can be added to a table, a relation or to a tuple (Snodgrass and Ahn,
1986). Although this technique is widely used in aspatial databases and is popular in GIS
databases, it is inadequate for space-time analysis (Yuan, 2007). Nonetheless, its wide
application in time-stamping of data layers as outlined by Langran and Chrisman (1988)
or to individual spatiotemporal objects as proposed by Worboys (1994) has been useful in
the visualization of spatiotemporal changes.
Dynamic phenomena are specifically characterized by continual change as they evolve in
space and time. For example, a simulation model developed to adequately represent the
transport and dispersal of airborne particles can be achieved with the aid of a four-
dimensional spatiotemporal model. Such a model would allow for the querying of the
phenomenon’s space and time topological relationships and it would serve as a building
block in the construction of simulations for what-if scenarios. Moreover, the results could
be employed in a GIS for further analysis.
4
It is generally accepted that the study of geographic phenomena, characterized by
interactions between distinct component parts, is well suited to the application of
simulation modeling using the theory of complex systems (Manson, 2001; O'Sullivan,
2004; Batty, 2005; Holland, 2006). In this context, the geographic phenomena are
conceptualized as complex systems that have a variety of interacting component parts,
and the objective of simulation modeling is to characterize interrelationships between
them over time in an artificial environment. These systems exhibit geographic complexity
and their constituent parts are sometimes organized into different sub-systems (Manson
and O'Sullivan, 2006). They are intrinsically four-dimensional in nature because the
interactions between the component parts take place in topographical landscapes for
extended periods of time. Due to the nonlinear relationships between them, they tend to
exhibit characteristics of self-organization, adaptation, feedback loops, emergent
behavior and bifurcations (Forrest, 1990; Holland, 1995). These properties are
characteristic of complex systems and provide a framework for designing models with
the assumption that the emergent characteristics of a system are best understood by
examining the inter-relationships between the component parts (Parker, et al., 2003).
The results from simulation models that rely on a two-dimensional data framework are
limited in their use for spatiotemporal analysis because time-stamped data layers are used
to animate the chronological changes of the study area. The animations are easy to
understand and they provide a global system state at each time-step, however, as pointed
out by Langran and Chrisman (1988), this aggregate presentation of the outcomes hides
the rich underlying spatiotemporal topological relationships that would explain the
process. This problem is unavoidable when a two-dimensional data framework is used. A
four-dimensional data model could provide a framework for interrogating space and time
relationships. However, a major challenge of implementing such model in a GIS pertains
to the manner in which dynamic phenomena are conceptualized (Couclelis, 2010).
5
1.2. Geospatial Data Models
In conventional GIS software applications, the features on the earth's surface are
described, organized and stored as a collection of thematic layers with the aid of
geospatial data models. Different thematic layers of the same geographic location are
usually overlaid on top of each other to provide a comprehensive representation of all the
features in the area. Although they provide a static representation of reality, spatial data
models allow for a logical consistency that enables a computer to represent real entities as
graphical elements (Goodchild, 1992). The two broad perspectives used to conceptualize
reality and thereafter transform it into geographic data are, namely, the field-based and
object-based views (Sinton, 1978; Galton, 2001; Worboys and Duckham, 2004). The
field-based perspective conceptualizes real world entities as a continuous spatial
distribution over a geographical region, while the object-based perspective forces the real
features of the natural world into discrete, identifiable entities across geographic space.
Inevitably, these two approaches have led to two distinct methods of spatial analysis
whereby a particular focus on spatial operations for discrete objects has been reserved for
object-based data models (Egenhofer and Franzosa, 1995), and map algebra techniques
for the field-based data models (Tomlin, 1990).
The process of representing geographic reality goes through the stages of identifying the
landscape elements necessary to appropriately represent the phenomenon under study,
selecting the spatial data model that would be used to represent these spatial entities, and
specifying the consistent instructions the computer uses to reconstruct the landscape
albeit in a digital form (Peuquet, 1984). In general, GIS software applications use the
raster geospatial data model to represent reality from the field-based perspective and the
vector model for the object-based perspective (Frank, 1992). For example, the two-
dimensional network models would be in the object-based perspective while the
triangulated irregular network (TIN) models as well as the continuous surface of a digital
elevation model (DEM) are in the field-based perspective.
6
1.2.1. Raster Data Model
The raster data model is based on a cellular organization of space. It divides the entire
geographic area into a series of non-overlapping cells that are usually identical to each
other in size and shape but differ in attribute values (McCullagh, 1988). Thus, spatial
features are divided into cellular arrays and planar coordinates derived from a geographic
reference system are assigned to each cell. Elevation, slope and surface moisture are
examples of geographic themes whose attribute domains are typically represented using a
field-based perspective of geographic space.
1.2.2. Vector Data Model
In the vector data model, spatial representation is achieved with the aid of points, lines
and polygons (Burrough and McDonnell, 1998). Points are spatial objects with no area
but can have attached attributes, while lines are objects linking two or more connected
points. Polygons are areas within a closed circuit of two or more line segments and can
take on shape and size. For example, discrete objects for roads, forests, lakes, and
agricultural farms can be used to represent a rural landscape. This data model is
implemented on the assumption that all the elements in the landscape are discrete objects
that can be represented geometrically using points, lines, and polygons. As with the cells
in the raster data model, the latitude-longitude coordinate pairs are used to reference the
object’s location in the real landscape. In this instance, however, multiple coordinate
pairs are needed to define the spatial extent of the geometric object.
1.2.3. Unified Models
The field and object-based perspectives were originally seen as conflicting views of
geographic space and there was much discussion about their relative merits (Couclelis,
1992). In the intervening years perspectives have changed because the current research
efforts view the field and object-based perspectives as complementary and the need for a
model that conceptually unites them has been acknowledged for a long time (Peuquet,
7
1988; Worboys, 1994; Cova and Goodchild, 2002). However, the interest in adopting and
implementing such a unified model has been slow despite significant progress in the
research on modeling the third spatial dimension and the temporal component.
Geographic representation in GIS applications often takes the planimetric view.
1.2.4. The Geo-Atom Data Model
In recent years, an important proposal for a unified spatial data model, referred to as the
geo-atom data model, has been fully described (Goodchild, et al., 2007).This model links
both the object and the field-based perspectives and is capable of comprehensively
representing spatiotemporal processes because the temporal dimension is also modeled
explicitly. A geo-atom defines a spatial feature in terms of its three-dimensional location,
attribute domain and temporal component and it is theorized as an elemental form of
representing spatiotemporal data. Conceptually, a geo-field is an aggregation of geo-
atoms that represent a specific attribute type or geographic theme, while a geo-object is
an aggregation of geo-atoms based on the spatial objects’ similarity of identity.
Moreover, in between the geo-object and geo-field representations, field-objects can be
derived by aggregating geo-atoms exhibiting persistent object identity but with field like
characteristics internally. Similarly, geo-atoms can be aggregated to an object-field when
the geo-atoms map to objects rather than a field values. In this study, the geo-atom data
model is used to represent spatial features tessellated into uniforms voxels.
1.2.5. The Voxel-Based Data Model
The field-based and object-based views of the world are not mutually exclusive
ontological systems but are intimately interconnected (Galton, 2003). Although the raster
and vector data models may distinctively represent geographic phenomena as objects or
fields, some objects can behave as fields and some fields may have object characteristics.
A storm is an example of an object that behaves as field. It may have clearly defined
boundaries and an identity that persists in time; however, its internal structure is
characterised by variation in atmospheric pressure and the precipitation. A field that maps
8
viewshades in a topographic landscape is an example of a field of objects. In addition,
these data models in GIS software applications are atemporal and they represent
geographic entities as 2D objects, which make three-dimensional spatiotemporal analysis
difficult to perform. To address these limitations, a voxel-based representation capable of
storing both object and field data using the geo-atom concept can be used to model four-
dimensional spatiotemporal phenomena. The voxels, being three-dimensional volumetric
units, can be used to represent the solid forms of a variety of spatial entities (Qian, et al.,
1992; Marschallinger, 1996). In addition to incorporating the temporal component, a
single voxel or an aggregation of a group of voxels can represent a single object while a
field can be modeled if the study area is viewed as a collection of voxels. Furthermore,
temporary objects can be created out of fields and temporary fields out of objects. Also, it
is often necessary to model both objects and fields in a single simulation because of the
interrelationships between the attributes of the field and objects therein. Like an object-
based data model, the voxel-based representation can be defined with a set of attributes
which characterize the spatial entity being modeled. Likewise, the field-based data
modeling used to map a continuously varying spatial attribute can be defined in a voxel-
based representation using a singular attribute value to characterize a phenomenon. In
particular, the voxel-based data model is implemented by defining functions for storing,
retrieving and manipulating data depending on how the geographic phenomenon is
modeled as an object or a field.
1.3. Representing Dynamic Spatial Phenomena in four dimensions
1.3.1. An Overview of Temporal Dimensional Representation
In addition to the long running advocacy for a unified spatial data model, over the years
there has been research on extending the restrictive raster and vector geospatial data
models, for example, by improving the representation of the third dimension (Raper,
2000; Breunig and Zlatanova, 2011) and the temporal component. Notable among the
9
efforts for temporal modeling, is the event-based approach where for any change in the
data a corresponding time is recorded (Peuquet and Duan, 1995), the use of fuzzy logic to
interpolate between time instances (Dragicevic and Marceau, 2000), and the SNAP-
SPAN ontology where, depending on the phenomena, geographic entities can be
represented in terms of snapshot occurrents (incidental events), or in terms of continuant
spatiotemporal processes (Grenon and Smith, 2004).
Typically, the third spatial dimension in a GIS is handled as an attribute of the location in
the horizontal plane (a solution often referred to as “2.5D”) and has been used to display
elevation, depth or the intensity value of another variable. Similarly, the temporal
changes are modeled by representing time as an optional attribute of spatial location
(Langran, 1992; O'Sullivan, 2005). With this kind of data organization, it is difficult to
execute queries for multi-dimensional topological relationships that involve the temporal
component, as well as the third dimension of space in a GIS. Therefore, in the spatial
databases for information on dynamic geographic phenomena where the temporal
information is important the data are stored as a series of snapshots corresponding to
particular points in time. This multi-snapshot representation of the space-time domain
lends itself well to the use of movie-like animations to visualize spatial changes.
However, with an increasing number of tools used in spatial data collection, the new
computing platforms being developed, as well as the sophisticated data search and
visualization algorithms, the question of representing dynamic spatial processes is taking
on a more urgent role in GIScience discourse.
1.3.2. Complex System Theory for Three and Four-Dimensional Representations
In the last three decades, cellular automata (CA) modeling has emerged as an effective
methodology for simulating dynamic spatial phenomena using the theory of complex
systems. The CA method models a process by defining local relationships and exchanges
for a collection of heterogeneous entities whose geographic interactions define the
dynamics of the system. CA have their origins in the fields of mathematics and computer
10
science through the pioneering efforts of John von Neumann (1966) as he studied self-
reproducing systems soon after World War II. It was Conway's Game of Life (McIntosh,
2010), however, that popularised them into the mainstream scientific community
resulting in various applications in mechanical physics, biology, chemistry (Wolfram,
1983, 2002) and geography (Tobler, 1979). Since Carter Bays (1987) made the initial
application of three-dimensional CA to Conway's Game of Life, their application in
three-dimensional modelling in still largely limited to the field of computational studies
in the physical sciences.
In the CA modeling approach, the temporal component is represented by a series of
discrete time-steps to approximate the temporal changes in the phenomenon under study.
This temporal representation is well suited to the four-dimensional modeling of dynamic
processes because data associated with a spatial phenomenon can be captured at one or
multiple instances in time during the model's run. Indeed, there are studies that have
taken the principles of two-dimensional cellular automata and extended them to the three
dimensions of space so as to aid in the four-dimensional simulation of environmental
phenomena. Aitkenhead et al. (1999) demonstrated their usefulness in modeling water
absorption in volumes of different soil types, while Barpi (2007) used them to model the
paths of snow avalanches. Other works in which four-dimensional cellular automata have
been applied include the simulation of seismic activity in an earth-quake zone (Jimenez,
et al., 2005), modeling fluid dynamics using a lattice gas method (Kobori and Maruyama,
2003), the simulation of disease spread (Dibble and Feldman, 2004), fire evacuation
dynamics (Thorp, et al., 2006), while Sarkar and Abbasi (2006) have used them to
simulate the hazards of fire spread and the dispersion of toxic pollutants. Moreover,
Wolf-Gladrow (2000) has emphasised their usefulness in the simulation of gaseous
processes particularly using the principles of the Lattice-Gas automata (Rothman and
Zaleski, 2004) and the Lattice Boltzmann (Chen and Doolen, 1998) methods. In addition,
PCRaster is a useful application for the four-dimensional simulation of dynamic
processes using automata based methods (Burrough, et al., 2005; Schmitz, et al., 2009).
However, most of the models published to date represent four-dimensional phenomena
11
using a two-dimensional spatial framework because the third dimension is ignored in the
raster datasets used. Although the benefits of using four-dimensional simulations in order
to analyse dynamic processes have been outlined there are limited efforts in applying
them for geographical study and analysis in comparison to the applications developed
using the classical two-dimensional space-time formalism. Moreover, the approaches
employed in many of the studies do not attempt to address the underlying limitation of
contemporary geospatial data models, that is the representation of time and elevation as
an attribute of the lateral coordinates. They are, instead, focused on the methodologies of
dynamic process simulation and visualisation.
1.4. Spatiotemporal Queries
Spatiotemporal queries investigate a dataset’s space and time topological relationships.
Using Allen’s (1983) temporal logic, for example, one can investigate equivalence,
adjacency and overlaps in much the same way that spatial queries are done. With that
same logic, queries that investigate space and time relationships can be used to find out
the rate at which a phenomena occupies space and the duration of particular processes;
explore concurrent events or processes (as an example of temporal proximity), moments
of transition (for example, when the attribute values at a specific location or associated
with a particular object) begin to change (Yuan, 2001).
Currently, the queries that can be executed on the existing vector and raster data models
are limited by the planar conceptualization of space and two-dimensional topological
relationships. The need for wide ranging queries that encompass time and the three-
dimensions of space as highlighted over the years (Hazelton, et al., 1992; Peuquet, 1994;
Yuan, 1999; Yuan and McIntosh, 2002) ought to be addressed in GIS software
applications. Moreover, there is a need to move beyond the existing three-dimensional
models that are suited only to dynamic visualization and content display and advance
towards four-dimensional spatiotemporal models with which complex three-dimensional
12
relationships are expressed and where the time component is not just an attribute of space
but a fundamental dimension along which data can be interrogated.
1.5. Research Problem and Objectives
The comprehensive modeling of both space and time is essential for the realistic
simulation of dynamic geographic phenomena that evolves in the three-dimensions of
space over extended time periods. Cellular automata models are widely used to simulate
and study dynamic phenomena. However, they rely on the two-dimensional geospatial
data models even in situations where a four-dimensional representation of a process
would be more beneficial. Besides being two-dimensional in their representation of
geographic space, the raster and vector-based data models often used in automata models,
are restrictive because they also conceptualise the real world as either a field or a
collection of spatial objects. Despite their limitations, two-dimensional data are
predominantly used in GIS. However, with the costs of data collection becoming cheaper
and the availability of a variety of sensors to collect 3D data, it is imperative to move
towards a three-dimensional representation and analysis of geographic phenomena in
GIS. Moreover, spatial analysis procedures routinely generate objects from within field-
based data; and field data are often generated from object-based data.
In order to address the above limitations and the gap in the GIScience literature related to
the methods and concepts about representing four-dimensional spatiotemporal
phenomena, the following research problems have been developed:
1. How can three-dimensional space and time be conceptualised in GIScience? 2. How can the geographic automata theory be used to represent four-dimensional
spatiotemporal phenomena using a voxel-based simulation? 3. How can the geo-atom data model be implemented to manage four-dimensional
spatiotemporal representation and analysis? 4. What are some of the indices that can be developed to characterise the
spatiotemporal patterns that emerge from a four-dimensional process? 5. What are some of the queries that can be developed to understand spatiotemporal
processes generated from the geo-atom based geographic automata model?
13
These problems provide the focus for the research work to improve on the representation
of geographic entities and processes in GIS software applications. More importantly,
however, it seeks to develop and demonstrate how a data model based on the geo-atom
theory can be coupled with the complex systems theory of geographic automata to
develop simulation models that can be used to study the spatiotemporal aspects of a
phenomenon. The work advances GIScience and spatial modeling by taking advantage of
the usefulness and effectiveness of the geo-atom theory of representing dynamic spatial
phenomena by developing several prototype automata models of increasing complexity
and varied spatial representation that demonstrates how different complex dynamic
processes can be simulated. In order to test the 4D modelling concepts, the developed
prototype models are rooted in complex systems theory using the automata method.
The main objectives of this research are:
1. To design and develop novel voxel-based automata models for the representation of dynamic spatial phenomena in the four dimensions of the space-time continuum,
2. To implement the geo-atom spatial data model to enhance the representation of the dynamic spatial phenomena using a four-dimensional space-time paradigm,
3. To develop three dimensional indices to measure and evaluate the generated multi-dimensional spatiotemporal patterns, and
4. To develop specific queries that allow for an exploration of the three and four-dimensional aspects of dynamic spatial phenomena.
The impact of this study is in the application of a geospatial data model to a dynamic
simulation approach which allows for the four-dimensional modeling of phenomena and
the querying of spatiotemporal relationships. In addition it explores the characterization
of landscape structure using a voxel-based discretization of space. These aspects,
although important to our modeling and analysis of geographic phenomena, are not
possible in contemporary GIS.
14
1.6. Datasets and Software Tools
Spatiotemporal relationships of geographic processes can be explored and analysed with
the aid of a data model that inherently supports the representation of time and the three
dimensions of geographic space. In this dissertation, the recently proposed theoretical
conceptualisation of geographic space using the geo-atom theory is employed in the
modelling of dynamic geographic phenomena. The study area is discretized into uniform
voxels, each of which is described by a geo-atom. The association of this unified
geospatial data model with a volumetric spatial unit allows for the representation of a
broad range of phenomena. Also, since it is a four-dimensional geospatial data model, the
phenomena are correspondingly modeled in four dimensions of the space-time
continuum. In addition, the simulation of dynamic spatial phenomena lends itself well to
the use of the geo-atom concepts because it is possible to model the spatial entities
representing the phenomena with a single geo-atom or a collection of geo-atoms.
Moreover, the theory of the CA modeling paradigm that is widely used for extrapolative
simulation and the construction of what-if scenarios can be adapted to serve as an
appropriate tool for building scenarios of spatial processes and for generating four-
dimensional data that can be queried a posteriori.
The MATLAB software (MathWorks, 2012) was used to develop and implement the
modeling procedures. In all the implementations, geographic phenomena are modeled as
a collection of voxels that are encoded as geo-atoms in MATLAB. A complete four-
dimensional dataset for a spatiotemporal phenomenon was unavailable at the time of
conducting this research. Thus, it was necessary to use hypothetical data to demonstrate
the conceptualisation of a topographic landscape and to show how data for a dynamic
geographic phenomenon that operates therein can be managed and analysed. In addition,
the assembly of geographic datasets was beyond the scope of this research whose goal is
to develop a theoretical structure as a first step towards four-dimensional modeling. The
models developed in this dissertation are simple and, thus, their usefulness in providing
insights about dynamic phenomena is limited to the area of the conceptual representation
of dynamic phenomena and illustrations for few analytical operations. Nonetheless, for
15
representing more complex modeling procedures a digital elevation model (DEM) of
Metro Vancouver was used to generate topographic landscapes whose voxel units have
the same horizontal dimensions as the DEM grid pixels. This dataset was used to
generate a landscape that is more or less realistic in which to simulate a hypothetical
phenomenon but it also indicates that spatial data in the raster format can be imported and
applied to the geo-atom data model.
1.7. Structure of the Thesis
This introductory chapter sets the stage for the four-dimensional modeling work using the
geo-atom concepts in GIS and it provides the research questions and objectives of this
dissertation. The subsequent chapters are focused on how the research objectives have
been addressed. Chapter 2 presents a voxel-based automata modeling approach that is
used to simulate dynamic processes as they evolve over time in three-dimensional space.
The voxel automata are further integrated with the geo-atom data model which enables
the analysis of three-dimensional topology which has been presented in Chapter 3. The
modeling approach proposed here was applied to simulate the propagation of airborne
particles from the perspective of three-dimensional space. In addition to managing the
data within the automaton, the GIS-based geo-atom theory was used to provide linkages
between the object and field duality in spatial representation, which allows advanced
spatiotemporal analysis.
In Chapter 4, landscape indices were developed to describe the surface characteristics. It
provides fundamental approaches to include relief properties into large-scale landscape
analyses, including the calculation of standard landscape indices on the basis of “true”
surface geometries and the application of roughness parameters derived from the field of
surface metrology. The methods are tested for their explanatory power using simulated
landscape and they can be used in spatiotemporal models. The methods provide a
valuable extension of the existing set of indices that can be used mainly in “rough”
landscape sections, allowing for a more realistic assessment of the spatial structure.
16
Chapter 5 presents the application of the geo-atom spatial data model in the
representation and analysis of a dynamic field like process. The digital elevation model
(DEM) was introduced in the landscape to characterise surface configuration and to
demonstrate the adaptation of the widely applied raster model to the geo-atom format.
The geo-atom concepts were employed to spatially and temporally define the voxels used
in representing a phenomenon in space and time. Specific spatiotemporal queries were
developed to demonstrate the usefulness of this model in GIS analysis. Finally, the thesis
concludes with Chapter 6 which presents the overall summary of the findings, limitations
and some suggestions for the possible future directions for this research.
References
Aitkenhead, M. J., Foster, A. R., Fitzpatrick, E. A., and Townend, J. (1999). Modelling water release and absorption in soils using cellular automata. Journal of Hydrology, 220 (1-2), 104-112.
Allen, J. F. (1983). Maintaining knowledge about temporal intervals. Communications of the Association for Computing Machinery, 26 (11), 832-843.
Barpi, F., Borri-Brunetto, M., and Veveri Delli, L. (2007). Cellular-automata model for dense-snow avalanches. Journal of Cold Regions Engineering, 21 (4), 121-140.
Batty, M. (2005). Cities and Complexity: Understanding Cities with Cellular Automata, Agent-Based Models, and Fractals. MIT Press, Cambridge, Massachusetts.
Bays, C. (1987). Candidates for the game of life in three dimensions. Complex Systems, 1, 373-400.
Breunig, M., and Zlatanova, S. (2011). 3D geo-database research: retrospective and future directions. Computers and Geosciences, 37 (7), 791-803.
Burrough, P. A., Karssenberg, D., and Van Deursen, W. P. A. (2005). Environmental modelling with PCRaster. In D. J. Maguire, M. F. Goodchild and M. Batty (Eds.), GIS, Spatial Analysis and Modeling, Redlands, California.
Burrough, P. A., and McDonnell, R. (1998). Principles of geographical information systems. Oxford University Press, Oxford ; New York.
17
Chen, S., and Doolen, G. D. (1998). Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 30 (1), 329-364.
Couclelis, H. (1992). People manipulate objects (but cultivate fields): Beyond the raster-vector debate in GIS. In A. Frank, I. Campari and U. Formentini (Eds.), Theories and Methods of Spatio-Temporal Reasoning in Geographic Space (pp. 65-77). Springer, Berlin.
Couclelis, H. (2010). Ontologies of geographic information. International Journal of Geographical Information Science, 24 (12), 1785-1809.
Cova, T. J., and Goodchild, M. F. (2002). Extending geographical representation to include fields of spatial objects. International Journal of Geographical Information Science, 16 (6), 509-532.
Dibble, C., and Feldman, P. G. (2004). The GeoGraph 3D computational laboratory: Network and terrain landscapes for RePast. Journal of Artificial Societies and Social Simulation, 7 (1).
Dragicevic, S., and Marceau, D. J. (2000). A fuzzy set approach for modelling time in GIS. International Journal of Geographical Information Science, 14 (3), 225-245.
Egenhofer, M. J., and Franzosa, R. D. (1995). On the equivalence of topological relations. International Journal of Geographical Information Systems, 9 (2), 133-152.
Forrest, S. (1990). Emergent computation: self-organizing, collective, and cooperative phenomena in natural and artificial computing networks: Introduction to the proceedings of the ninth annual CNLS conference. Physica D: Nonlinear Phenomena, 42 (1–3), 1-11.
Frank, A. U. (1992). Spatial concepts, geometric data models, and geometric data structures. Computers & Geosciences, 18 (4), 409-417.
Galton, A. (2001). A formal theory of objects and fields. In D. R. Montello (Ed.), Spatial Information Theory (pp. 458-473). Springer, Berlin, Heidelberg.
Galton, A. (2003). Desiderata for a spatio-temporal geo-ontology. In W. Kuhn, M. Worboys and S. Timpf (Eds.), Spatial Information Theory. Foundations of Geographic Information Science (Vol. 2825, pp. 1-12). Springer Berlin Heidelberg.
Goodchild, M. F. (1989). Modelling error in objects and fields. In M. Goodchild and S. Gopal (Eds.), The accuracy of spatial databases. Taylor & Francis, London.
18
Goodchild, M. F. (1992). Geographical data modeling. Computers and Geosciences, 18 (4), 401-408.
Goodchild, M. F. (2010). Twenty years of progress: GIScience in 2010. Journal of Spatial Information Science (1), 3-20.
Goodchild, M. F., Yuan, M., and Cova, T. J. (2007). Towards a general theory of geographic representation in GIS. International Journal of Geographical Information Science, 21 (3), 239-260.
Grenon, P., and Smith, B. (2004). SNAP and SPAN: towards dynamic spatial ontology. Spatial Cognition and Computation, 4 (1), 69-104.
Hazelton, N., Leahy, F. J., and Williamson, I. P. (1992). Integrating dynamic modeling and geographic information systems. Journal of the Urban and Regional Information Systems Association, 4 (2), 47-58.
Holland, J. H. (1995). Hidden Order: How Adaptation Builds Complexity. Addison-Wesley, Reading, Mass.
Holland, J. H. (2006). Studying complex adaptive systems. Journal of Systems Science and Complexity, 19 (1), 1-8.
Jakle, J. A. (1971). Time, space, and the geographic past: a prospectus for historical geography. The American Historical Review, 76 (4), 1084-1103.
Jimenez, A., Posadas, A. M., and Marfil, J. M. (2005). A probabilistic seismic hazard model based on cellular automata and information theory. Nonlinear Processes in Geophysics, 12 (3), 381-396.
Keon, D., Steinberg, B., Yeh, H., Pancake, C. M., and Wright, D. (2014). Web-based spatiotemporal simulation modeling and visualization of tsunami inundation and potential human response. International Journal of Geographical Information Science, 28 (5), 987-1009.
Kobori, T., and Maruyama, T. (2003). A high speed computation system for 3D FCHC lattice gas model with FPGA. In Field-Programmable Logic and Applications, Proceedings (pp. 755-765).
Langran, G. (1992). Time in geographic information systems. Taylor and Francis, New York.
Langran, G., and Chrisman, N. R. (1988). A framework for temporal geographic information. Cartographica, 25 (4), 1-14.
19
Manson, S. M. (2001). Simplifying complexity: A review of complexity theory. Geoforum, 32 (3), 405-414.
Manson, S. M., and O'Sullivan, D. (2006). Complexity theory in the study of space and place. Environment and Planning A, 38 (4), 677-692.
Marschallinger, R. (1996). A voxel visualization and analysis system based on AutoCAD. Computers & Geosciences, 22 (4), 379-386.
MathWorks. (2012). MATLAB and Statistics Toolbox. In (Release 2012b ed.). The MathWorks, Inc., Natick, Massachusetts, United States.
McCullagh, M. J. (1988). Terrain and Surface Modelling Systems: Theory and Practice. The Photogrammetric Record, 12 (72), 747-779.
McIntosh, H. V. (2010). Conway’s life. In A. Adamatzky (Ed.), Game of Life Cellular Automata (pp. 17-33). Springer London.
O'Sullivan, D. (2004). Complexity science and human geography. Transactions of the Institute of British Geographers, 29 (3), 282-295.
O'Sullivan, D. (2005). Geographical information science: time changes everything. Progress in Human Geography, 29 (6), 749-756.
Parker, D. C., Manson, S. M., Janssen, M., Hoffmann, M., and Deadman, P. (2003). Multi-agent systems for the simulation of land-use and land-cover change: a review. Annals of the Association of American Geographers, 93 (2), 314-337.
Peuquet, D. J. (1984). A conceptual framework and comparison of spatial data models. Cartographica: The International Journal for Geographic Information and Geovisualization, 21 (4), 66-113.
Peuquet, D. J. (1988). Representations of geographic space: toward a conceptual synthesis. Annals - Association of American Geographers, 78 (3), 375-394.
Peuquet, D. J. (1994). It's About Time: a conceptual framework for the representation of temporal dynamics in geographic information systems. Annals of the Association of American Geographers, 84 (3), 441-461.
Peuquet, D. J., and Duan, N. (1995). An event-based spatiotemporal data model (ESTDM) for temporal analysis of geographical data. International Journal of Geographical Information Systems, 9 (1), 7-24.
20
Peuquet, D. J., Smith, B., and Brogaard, B. (1998). The ontology of fields. In Panel on Computational Implementations of Geographic Concepts. Report of a Specialist Meeting Held under the Auspices of the Varenius Project, Bar Harbor, Maine.
Qian, K., Lu, X., and Bhattacharya, P. (1992). A 3D object recognition system using voxel representation. Pattern Recognition Letters, 13 (10), 725-733.
Raper, J. (2000). Multidimensional Geographic Information Science (1 edition ed.). Taylor & Francis, London.
Rothman, D. H., and Zaleski, S. (2004). Lattice-gas cellular automata: simple models of complex hydrodynamics. Cambridge University Press, London.
Sarkar, C., and Abbasi, S. A. (2006). Cellular automata-based forecasting of the impact of accidental fire and toxic dispersion in process industries. Journal of Hazardous Materials, 137 (1), 8-30.
Schmitz, O., Karssenberg, D., van Deursen, W. P. A., and Wesseling, C. G. (2009). Linking external components to a spatio-temporal modelling framework: Coupling MODFLOW and PCRaster. Environmental Modelling & Software, 24 (9), 1088-1099.
Sinton, D. (1978). The inherent structure of information as a constraint to analysis: mapped thematic data as a case study. In First International Advanced Study Symposium on Topological Data Structures for Geographic Information Systems (Vol. 7, pp. 1-17). Laboratory for Computer Graphics and Spatial Analysis.
Snodgrass, R. T., and Ahn, I. (1986). Temporal databases. IEEE Computer, 19 (9), 35-42.
Thorp, J., Guerin, S., Wimberly, F., Rossbach, M., Densmore, O., Agar, M., and Roberts, D. (2006). Santa Fe on fire: Agent-based modeling of wildfire evacuation dynamics. In D. Sallach, C. M. Macal and M. J. North (Eds.). University of Chicago and Argonne National Laboratory, Chicago, IL.
Tobler, W. R. (1979). Cellular geography. In S. Gale and G. Olsson (Eds.), Philosophy in Geography (pp. 379-386). Reidel Publishing Company, Dordrecht, Holland.
Tomlin, C. D. (1990). Geographic information systems and cartographic modeling. Prentice Hall, Englewood Cliffs, N.J.
von Neumann, J. (1966). Theory of Self-Reproducing Automata. University of Illinois Press, Urbana.
21
Wolf-Gladrow, D. A. (2000). Lattice-gas cellular automata and lattice Boltzmann models : an introduction. Springer, Berlin.
Wolfram, S. (1983). Statistical mechanics of cellular automata. Reviews of Modern Physics, 55 (3), 601-644.
Wolfram, S. (2002). A New Kind of Science. Wolfram Media, Champaign, IL.
Worboys, M. (1994). A unified model for spatial and temporal information. The Computer Journal, 37 (1), 26-34.
Worboys, M. (1995). GIS, a computing perspective. Taylor & Francis, London.
Worboys, M., and Duckham, M. (2004). GIS: a computing perspective (Vol. 2nd). CRC Press, Boca Raton, Fla.
Yi, J., Du, Y., Liang, F., Zhou, C., Wu, D., and Mo, Y. (2014). A representation framework for studying spatiotemporal changes and interactions of dynamic geographic phenomena. International Journal of Geographical Information Science, 28 (5), 1010-1027.
Yuan, M. (1999). Use of a three-domain representation to enhance GIS Support for complex spatiotemporal queries. Transactions in GIS, 3 (2), 137-159.
Yuan, M. (2001). Representing complex geographic phenomena in GIS. Cartography and Geographic Information Science, 28 (2), 83-96.
Yuan, M. (2007). Adding time into geographic information system databases. In J. P. Wilson and A. S. Fotheringham (Eds.), The Handbook of Geographic Information Science (pp. 169-184). Blackwell Publishing Ltd.
Yuan, M., and Hornsby, K. (2008). Computation and Visualization for Understanding Dynamics in Geographic Domains: A Research Agenda. CRC Press, Boca Raton, FL.
Yuan, M., and McIntosh, J. (2002). A typology of spatiotemporal information queries. In R. Ladner, K. Shaw and M. Abdelguerfi (Eds.), Mining Spatiotemporal Information Systems (pp. 63-82). Kuwer Academic Publishers, Norwell, MA.
22
Chapter 2. Voxel-based Geographic Automata for the Simulation of Geospatial Processes1
2.1. Abstract
Most geographic processes evolve in a three dimensional space and time continuum.
However, when they are represented with the aid of geographic information systems
(GIS) or geosimulation models, they are modelled in a framework of two-dimensional
space with an added temporal component. The objective of this study is to propose the
design and implementation of voxel-based automata as a methodological approach for
representing spatial processes evolving in the four-dimensional (4D) space-time domain.
Similar to geographic automata models which are developed to capture and forecast
geographical processes that change in a two-dimensional spatial framework using cells
(raster geospatial data), voxel automata rely on the automata theory and use a three
dimensional volumetric unit, a voxel. Transition rules were developed to represent
various spatial processes which range from the movement of an object in 3D to the
diffusion of airborne particles and landslide simulation. In addition, the proposed 4D
models demonstrate that complex processes can be readily reproduced from simple
transition functions without complex methodological approaches. The voxel-based
automata approach provides a unique basis to model geospatial processes in 4D for the
purpose of improving representation, analysis and understanding their spatiotemporal
dynamics. This study advances the concepts and framework of 4D GIS.
1 A version of this chapter has been co-authored with Suzana Dragićević and has been submitted
for review to the ISPRS Journal of Photogrammetry and Remote Sensing, special issue on Multi-dimensional Modeling, Analysis and Visualization
23
2.2. Introduction
Geographic automata (Torrens and Benenson, 2005) provide a flexible and adaptable
method for simulating a wide range of geospatial phenomena whose spatiotemporal
dynamics can be represented using the theory of complex systems. However, they have
been limited to the two-dimensional perspective of representing geographic processes
using the theory of cellular automata (Torrens and O'Sullivan, 2001). This is partly due to
the simplicity of the cellular automata (CA) formalism in representing complex
geospatial processes as well as the ease of integrating them with geographic information
systems (GIS), remotely sensed data and other geospatial information. Besides the
limited number of multi-dimensional modelling tools available in the field of Geographic
Information Science (GIScience), the geospatial data used in contemporary GIS
applications are also formatted with the raster or vector models, which are both
planimetric in their real world perspective. In these data models, the three-dimensional
nature of the geographic space is abstracted so that properties such as the elevation or the
temporal dimension are modelled as an attribute of a location represented by two
geographic coordinates (Chrisman, 1997). This study extends geographic automata
systems that are based on complex systems theory to represent multidimensional spatial
processes evolving in the four-dimensional space-time continuum.
First conceptualised by Stanislaw Ulam, the scientific study of CA models was initiated
by John von Neumann when he applied them in the investigation of the nature of self-
reproducing biological systems (von Neumann, 1966). Later on they were popularized by
the publication and eventual widespread use of Conway’s game of life (McIntosh, 2010).
They are well suited to the study of complex systems on the basis that from the
application of simple rules of interaction at a local level can emerge order and patterns
that are vastly complex and often unpredictable (Ilachinski, 2001; Wolfram, 2002;
Cotsaftis, 2009). Well specified CA models have been used to generate and study
complex patterns and behaviour and over the last two decades they have been applied in
various ways to simulate the complexity of dynamic geographic processes (Couclelis,
1985; Batty, 1997; White, et al., 1997; Clarke and Gaydos, 1998; Berjak and Hearne,
24
2002; Benenson and Torrens, 2004; Rothman and Zaleski, 2004). They use discrete
spatial elements (cells or rasters) which are undergoing either deterministic or stochastic
(non-deterministic) local interactions (Rabin, 1963; Grassberger, 1984). For deterministic
automata, the local interactions are actualized with the aid of transition functions that rely
on the existing conditions of the neighborhood to cause a specific outcome at the next
iteration (White and Engelen, 1993). Typically, the transition functions operate in such a
way that for a given local neighbourhood configuration, a specific change in state may be
applied to the cell at the centre of that neighbourhood. On the other hand, stochastic
automata are less restrictive because the state of the central cell at the next iteration is
selected from a range of possible outcomes given a specific neighbourhood configuration
(Hopcroft, 2007). Moreover, the transition functions that are used to define these local
interactions fall into three broad categories, namely, totalistic, semi-totalistic and non-
totalistic automata functions. In totalistic cellular automata, the transition functions
depend on the simple summation or average of the cell values in the neighborhood; in
semi-totalistic automata the transitions depend on the state of the central cell as well as
the summation of neighbors; while in the non-totalistic cellular automata the cell values
change based on the positioning of the other cell values in the neighbourhood (Wolfram,
1983). Various transition rules belonging to one of these categories have been described
in detail and used to mimic different processes that evolve over space and time.
With simple rules to govern the local interactions at specific time intervals, it has been
shown that as the states of the spatial elements change, the system as a whole gives rise to
emerging patterns, evolution and self-organizing behavior which are all general
characteristics of complex systems (Lansing, 2003; Holland, 2006). In particular, the
application of geographic automata is well suited to complex systems modelling because
the phenomena are themselves inherently spatial in character, time dependent, and exhibit
a complexity of pattern formation resulting from the various interactions of the systems'
components. Although the key concepts of the classical CA formalism have been
extended to develop geographic automata models to represent geographic processes and
to study complexity, most of these automata models are limited to the two-dimensional
25
spatial perspective (Benenson and Torrens, 2004). In reality, however, geographic
processes occur in a four-dimensional (4D) continuum and many of these are best
represented with the aid of the three dimensions of space with time being the fourth one.
For example, urban land use change has been modelled and analysed using two-
dimensional spatial data; however, other growth processes like fire spread, particle
diffusion and 3D landscape surface change would be more insightful when modelled
using a 3D spatial framework.
The main objective of this study is to develop a voxel-based automata modelling
approach, as an extension of the theory of geographic automata systems, capable of
simulating spatiotemporal processes in the four dimensions of the space-time domain.
The voxel, as a volumetric element, is used to represent the smallest 3D spatial unit that
changes its state based on the functioning of the automaton’s transition rules. By
designing and applying local transition rules, the voxel automata approach is used to
generate 3D spatial patterns that closely resemble those of geographic phenomena as they
evolve in the four dimensions of the space-time domain.
2.3. Literature Overview
In geography, CA were proposed by Tobler (1979) as a suitable modelling approach to
represent spatiotemporal change and have since found widespread use in modelling
processes such as land use change. For the most part, the two-dimensional CA models for
geographic processes operate on grid spaces with uniform tessellations due to the
prevalence and complementarities of the raster geospatial data format with geographic
information systems (Clarke and Gaydos, 1998; Stocks and Wise, 2000). However, this is
not a strict requirement and work has been done using irregular spatial tessellations
(Semboloni, 1997; Shi and Pang, 2000; Stevens and Dragicevic, 2007) or vector-based
CA (Moreno, et al., 2008).
26
The two-dimensional CA models have been used extensively to simulate land use and
regional change, for example to name a few studies by Batty and Xie (1994), White et
al.(1997), Clarke and Gaydos (1998), Lau and Kam (2005) and Kocabas and Dragicevic
(2007). In addition, they have been used to represent various environmental processes
such as pollutant diffusion (Guariso and Maniezzo, 1992), physical processes such as
landslides (Di Gregorio, et al., 1999; Lai and Dragicevic, 2011) and snow avalanches
(Barpi, et al., 2007), ecological processes such as insect forest infestations (Bone, et al.,
2006), the spread of forest fires (Berjak and Hearne, 2002; Yassemi, et al., 2008), and the
dynamics of marine animals (Vabø and Nøttestad, 1997). All of these environmental
processes operate in 3D spaces over extended temporal durations and should be simulated
with models designed to study and analyze the 4D space-time domain.
3D GIS-based representation and analysis requires the geographic space to be discretized
into volumetric units referred to as voxels (Hernandez-Pajares, et al., 2003; Popescu and
Zhao, 2008). Uniformly tessellated voxels are straightforward to generate using
computational algorithms and their analysis is adaptable to the many available analytical
methods of their cellular analogues. The subdivision of the 4D space-time continuum of
the real world into voxels and discrete time intervals lends itself well to the use of the
voxel-automata methodology. The methodology is rooted in the theoretical framework of
cellular automata and complex systems modelling and can be used to describe the local
interactions of a geographic process. In this context, voxel automata, like their cellular
counterparts, provide a mechanism with which dynamic geographic phenomena can be
simulated and analysed. Three-dimensional automata models of complex geographic
processes provide useful 4D representations of the change in spatial phenomena over
time. The developed 4D modelling approach can be particularly powerful in the analysis
and visualization and a contribution to the process of developing a framework for 4D GIS
(Raper, 2000).
The concepts of three-dimensional automata have only been used in a few applications
and even fewer of these are for the simulation of geographic processes. For example,
Vanderhoef and Frenkel (1991) have used 3D lattice gas automata formulation to
27
simulate the process of gaseous diffusion. Using a two-step process, gaseous particles
propagate with the aid of twenty possible velocities from their initial position to a new
node on the grid. Particles at the nodes undergo a collision that conserves momentum and
the number of particles in the system. Sarkar and Abbasi (2006), on the other hand, use
3D cellular automata to simulate the propagation of self-evolving phenomena. In
particular, they simulate the outward diffusion of gaseous particles, energy and
momentum from the site of an accident in order to determine the most vulnerable
locations in the landscape. Furthermore, Narteau et al. (2009) developed a 3D automata
model to simulate the process of dune formation by simulating the processes of particle
erosion, transport and deposition. They use the theoretical elements from both the 3D
lattice gas automata and numerical methods to simulate the dune fields. Subsequently, the
fields are used to analyze the sensitivity of a sand bed disturbed by small perturbations
caused by ground movements. Similarly, Gandin et al. (1996) as well as Eshraghi et al.
(2012) use numerical formulations in their different algorithms while at the same time
drawing on the gas lattice automata to implement a model that reproduces the growth
dendritic grain structures during the solidification process. Other studies have employed
the automata to simulate water lily growth (Lin, et al., 2011), rock formation (Lalonde, et
al., 2010), snow avalanche (Barpi, et al., 2007). In the life sciences, Hunt et al. (2005)
use a three-dimensional cellular automata model to simulate the dynamics of
antimicrobial agents in a biofilm and other applications have focused on the study of
tissue formation (Hotzendorfer, et al., 2009; Cai, et al., 2013; Ben Youssef, 2014).
The outputs from these three-dimensional models, however, are for the most part still in
two-dimensional representation. Differential equations, or other established mathematical
methods, are often used to represent the phenomena without considering 3D space in an
explicit manner. However, the voxel automata approach used here relies on deterministic
transition functions that describe the complex spatiotemporal processes on a 3D grid
space for the purpose of modelling spatial processes. The extension of the cellular
automata method to a voxel grid provides more versatility with which the 3D space in
which a geographic process unfolds can be conceptualized.
28
2.4. Methods
2.4.1. The Formalism for Voxel Automata
Using the existing geographic automata theory (White and Engelen, 2000; Torrens and
Benenson, 2005) as a foundation, the five key elements that can completely specify a
voxel automaton model are: the voxel grid space, the voxel states, the time-steps, the 3D
neighbourhood and the transition rules. The mathematical conceptualization of the voxel
automaton (VA) can be represented with a 4-tuple as:
),,,( tfNSVA ≡
(2.1)
where { }psssS ,...,, 21= , p, number of possible states s; neighborhood, { }qvvvN ,..., 21= ,q, number of voxels v, in the neighbourhood; the
function of transition rules SSf →×ω: , ω are the other input variables for the process, t = [1,2,..n], input values are the time-steps.
The state S of any voxel at time t+1 is a function of the states of voxels and their relative
positioning x, y, z within the three-dimensional neighbourhood V at time t formulated as
[ ]ttzyx SfS =+1
,,
(2.2)
The key elements of the voxel automaton in 3D space are described below:
29
2.4.2. Voxel Space
The voxel space refers to the representation of the geographic study area in which the
process under simulation unfolds. It is modelled using a uniform 3D lattice of volumetric
elements because similar to the raster data model, the subdivision algorithm is relatively
straightforward, the voxels are contiguously arranged with a uniform regularity, and they
form closely packed neighborhoods. The voxels are conceived as distinct volumetric
elements that represent portions of the study area and the spatial elements that are
contained therein. They have been used to illustrate the computational concepts of three-
dimensional problems in fields as diverse as tomography and computational biology
(Thaler, et al., 1978; Siddon, 1985; Ridgway, et al., 2012); mathematics and computer
science (Frieder, et al., 1985; Patterson, et al., 2012); as well as remote sensing
(Cucurull, et al., 1998; Hosoi and Omasa, 2009; Schilling, et al., 2012; Hosoi, et al.,
2013; Schneider, et al., 2014). Besides the use of cubic voxels, other approaches can be
used to decompose 3D space into smaller parts; it is important, however, that these
volumetric components are meaningful, logical and intuitively discernible parts of the
study area (Reniers and Telea, 2008). Analogous to the two-dimensional constructs used
in the discretization of space for the raster spatial data model, in 3D space the voxels can
also take on regular and irregular hexahedra forms from adjoining cubic, hexahedron,
triangular prism and rhombohedron voxels that fit compactly next to each other. Like
their 2D counterparts, the voxel space in 3D automata provides the ability to capture the
changes and to form spatial patterns over time at varying spatial and temporal resolution.
In this study, cubic voxels have been considered because they present the most logical
and intuitive tessellation of the spatial continuum however, an appropriate spatial
resolution can be defined in terms of the voxel size to suit the type of phenomenon to be
modelled. Thus, extending the traditional raster grid to a 3D continuum of voxels is a
logical extension of a GIS framework for 3D modelling of spatiotemporal phenomena.
30
2.4.3. The 3D Neighbourhood
The geographic automata methodology is a bottom-up modelling approach that mimics
the local dynamics of a voxel and its surrounding neighbourhood, which in turn generates
complex patterns at a larger spatial scale. The most commonly used neighbourhood types
around a central voxel are the von Neumann and Moore neighbourhood formed by the
adjacent voxels (Figure 2.1). However, in 3D voxel space the way that the
neighbourhoods can be conceptualised allows for many more configurations. The von
Neumann neighbourhood has six neighbouring voxels Figure 2.1(a) while the Moore
neighborhood comprises of twenty six voxels Figure 2.1(b) surrounding the voxel at the
center of the neighbourhood structure and whose state is to be changed at the next
iteration. Alternative neighbourhood configurations, for example, those shown in Figure
2.1(c-f) can also be used in simulations by voxel automata. However, one of the
implications of having the additional flexibility in characterising a neighbourhood is that
more work has to be devoted to the voxel automata calibration process to determine the
type of neighbourhood configuration that would be most appropriate for a particular
process under study.
31
Figure 2.1. Various configurations for 3D neighborhoods
2.4.4. Voxel State
During the simulation, each voxel in the 3D grid space has a state from a set of possible
alternatives. The voxel states are updated based on the transition rules which operate in
the local neighbourhood surrounding the voxel. The voxels are updated once in each
iteration depending on how the transition rules are driving the simulation. Binary classes
(black or white, 0 or 1) or multiple classes can be used to represent the state of the voxel
3D geographic objects. For example, the states of the voxels can describe a phenomenon
using one attribute to symbolise the presence and absence of a terrestrial or marine
species (Figure 2.2a). Multiple voxel states can be used to represent more complex
phenomena that ought to be characterised with multiple attribute values, for example, the
representation of the varied ages in a colony of marine organisms using in a multistate
automaton (Figure 2.2b). In a 3D continuum, the states of the voxels at any given time-
32
step will change at the subsequent iteration based on the transition thereby enabling the
generations of various pattern formalisation in 3D.
Figure 2.2. Types of voxel states (a) binary and (b) multistate automata
2.4.5. Transition Rules
The transition rules are an important element in specifying how the evolution of the
automaton unfolds with the explicit aim of correctly characterizing the process being
studied. The rules enable the voxel automaton to represent, in discrete terms, a dynamic
geographical process that operates in the four space-time dimensions. Using a totalistic
33
specification of the transition rule, the state S of any voxel at time t+1 is a function of the
states of voxel in the neighborhood at time t and can be formulated as follows:
(2.3)
where 1+ts is the state of a voxel i at time t+1; tiS is the state of a voxel i in
the neighbourhood at time, t; and n is the number of voxels in the neighbourhood.
Transition rules are functions that operationalize the representation of the modelled
process, which in this context is in a four dimension space-time continuum. They pull
together and analyse the local dynamics surrounding each voxel and consequently assign
a value for the voxel state to the centre voxel.
A process described using both the totalistic and the non-totalistic transition rules relies
on a standardized reference mechanism to each of the voxels in the neighborhood. The
referencing mechanism allows for the consistent application of change to particular
voxels whenever some conditions in the neighborhood have been attained as shown in
Figure 2.3. Each of the voxels in the 3D Moore neighbourhood has distinct set of
coordinates relative to the central voxel. Thus, the state S of the central voxel at time t+1
can be modeled as a function of the relative positions of the other voxels in the
neighborhood at time t and is formulated as:
[ ]tn
tn
ti
ti
ti
t sssssfs ,,...,,, 1211
−+++ =
34
[ ]tzyx
tzyx
tzyx
tzyx
tzyx
tzyx sssssfs 1,1,11,,11,1,11,,11,1,1
1,, ,,...,,, +−+++−−−−−−+−
+ =
(2.4)
where 1,,
+tzyxs is the state of a cell at time t+1 and t
iS is the states of the voxel positioned at index i in the neighborhood at time, t.
Figure 2.3. The relative position of the voxels in a 3D Moore neighborhood
The transition rules for voxel-automata are functions that permit simulations of four-
dimensional geospatial processes because they operate on the basis of discretizing space
and time followed by the modelling of local interactions and the cross-transfer of
information between neighbouring spatial entities at regularly specified time intervals. It
is a bottom-up approach to system modelling that differs from the application of top-
down deferential equations. Equation based approaches are too complex to manipulate,
do not provide means for explicitly analysing local drivers and spatial interactions in
addition to the construction of 'what-if' scenarios (Parunak, et al., 1998; Batty and
Torrens, 2005). In addition, numerical modelling of the geospatial processes presented
here may not be justified because of the difficulty in accessing complete and accurate
35
datasets that would allow for multiple runs of the model in the model training and testing
stages (Oreskes, et al., 1994; Neofytou, et al., 2006).
2.4.6. Time-steps
The voxel automaton models the temporal continuum as a sequence of discrete time
intervals. At each time-step transition functions that determine how the state of a given
voxel changes are applied to the neighbourhood. The updated state of the voxel comes
into effect at the subsequent time-step. Generally, the number of time-steps required is
the temporal extent needed to meet the objectives of the investigation and it depends on
the characteristics of the system under study. The temporal resolution refers to the time
period in the real world that each time-step represents within the model. Its duration
depends of the 4D phenomenon under study as it can be minutes for a particle dispersal
process or centuries for a geomorphological process. In addition, the model usually
requires calibration to fit the temporal resolution and temporal extent of the geographic
process that it is representing.
2.5. Simulation of Spatial Processes in 4D
In this study, several voxel-based automata models were developed to simulate the
evolution of various geospatial processes in a four-dimensional space-time domain using
binary and multistate voxel automata. MATLAB software (MathWorks, 2012) was used
to implement the proposed voxel automata models and to generate the simulation results.
2.5.1. Binary Voxel Automata
Sandpile Model
The sandpile model, introduced by Bak et al. (1987) has been used to study the frequency
of occurrence (as a function of the size of the affected area) as well as the fractal
behaviour of some geospatial processes such as landslides and forest-fires (S. Hergarten,
36
2003). Such processes tend to evolve in small increments towards critical points at which
the system gets out of equilibrium (Bak, 1999; Turcotte, 1999). In the sandpile
illustration, grains of sand, one at a time, are dropped in the centre of the grid and over
time a heap of sand begins to form. During this process the cascading reactions affecting
multiple sites as a result of the addition of a single grain of sand at the point of criticality
occur as avalanches, from time to time. The sandpile model has been used to study
natural hazards such as landslides (Stefan Hergarten and Neugebauer, 1998) and forest
fire propagation (Drossel and Schwabl, 1992).
Using a semi-totalistic voxel automaton, the model is used here to demonstrate 3D
pattern formation using a voxel automaton constructed on a 40 x 40 x 40 voxel lattice.
The sandpile model represents a multi stage spatial process. Firstly, a voxel is dropped
from the top and it makes its way down to the base of the grid or its sits on the already
existing voxels at the location. The next stage is the determination of whether there is a
stack of four voxels that have no neighbours on either of their sides corresponding to the
four cardinal directions. If this condition is fulfilled then a collapse of these four voxels
occurs. The collapse of the voxel stack may in turn lead to a cascade of collapses at other
locations within the grid space. It is assumed that these stages are occurring in a single
timestep. The automaton's neighbourhood is configured to be an extension of the von
Neumann neighbourhood. The presence of voxels in the three positions just below the
central voxel, coupled with the absence of any other voxels in the neighbourhood,
triggers the automaton's transition rule responsible for the avalanche effect. Overall, the
accumulation of voxels into a pile continues to be stable and predictable if the height
value of the column stack is less than four; however, at the critical value of four, an
avalanche unpredictability sets in. The collapsed voxels are randomly distributed to the
empty neighbouring grid locations along the four cardinal directions and the stack height
limit has been set to four in this study but other heights can be used. The pile generated
by a sandpile model cumulatively increases in volume (Figure 2.4a) but critical points are
reached for example at time-steps 47, 707 and 822 as shown in Figure 2.4b where
37
significant and unpredictable changes between two time-steps are observed due to the
effect of avalanches.
Figure 2.4. Voxel automata simulation outcomes for the sandpile model (a) for a run up to 3000 iterations and (b) for a series of successive critical moments when the form of the sandpile significantly changes its shape in one iteration.
Object Movement in Space
The voxel automata model can be used to simulate the movement of geographic objects
in space considering, for example, the case of objects gravitating to a central area. In the
real world this model can be used to represent suspended organisms, such as fish, moving
towards a feeding patch. To simulate this process, occupied voxels are randomly seeded
in the three-dimensional landscape. Within the landscape is a field which also acts as a
region of attraction for the seeded voxels. At every time-step, some of the seeded voxels
are randomly selected to move in a stepwise fashion, one voxel at a time closer, towards
the centre of this field. Several iterations of the simulation outcomes showing the
movement of suspended organisms towards a given area are presented in Figure 2.5.
38
Figure 2.5. Objects moving towards a central region of attraction, simulation outcomes for fish moving towards a feed patch as an example
In the model's implementation, the voxels' position coordinates are updated whenever the
movement occurs while the voxel states are not updated. The persistence of the voxel
identity allows for the 3D voxel automata to become a 3D agent-based model simulating
spatial dynamics.
Dispersal of Objects in Space
A binary automaton is used to implement an Eden growth model to simulate a species
cellular reproduction and migration of an underwater colony such as algae organisms. In
the model’s design, a single voxel is seeded in the study region which is then subdivides
into two or four other voxels at the next interaction. At the subsequent timestep one of the
previously created voxels is removed from the simulation to mimic the death of the
parent cell. Moreover, the voxels are given a directional movement to simulate their
migration to locations with nutrients under the influence of water currents. Figure 2.6
presents the simulation outcomes showing the dispersal of underwater species.
39
Figure 2.6. Simulation outcomes of a reproduction and dispersal of an organism in a fluid medium.
Landslide Processes
Important natural processes such as landslides and snow avalanches are initiated at
locations where there is instability in the layers of slope material caused by the
combination of gravitational forces, the angle of the slope and the weight of the material
itself (Wang, et al., 2006; Lai and Dragicevic, 2011). These processes start when the
shearing stress from gravitation pull exceeds the shearing resistance of the material
resulting in the flow of debris down the slope. In this study, a generic three-dimensional
simulation based on the sandpile model is used to approximate the removal of material
debris from a higher ground to lower ground.
Assuming that the simulation starts at the point when the instability of the ground is just
giving way for the start of the downward flow of the unstable material, a non-totalistic
voxel automata model is developed that approximates the movement of the debris. The
sequential flow of material starts from a single voxel location (a selected central voxel)
and moves to another location at a lower elevation but within the Moore neighbourhood
of the central voxel. The transition rules implementing the local dynamics are used to
determine the location to which the debris moves in such a way that the steeper the slope
the higher the likelihood of being selected while at same time limiting this movement to
down slope voxel locations only. The assumption is that the weight of the debris at the
new location increases the instability at that location and in turn leads to further
40
progression of the landslide. Although landslides last for short durations, the
implementation in this model represents the movement of debris between only two
neighbouring voxel locations at each time-step. The simulation results for a voxel
automata model representing the landslide process is presented in Figure 2.7.
41
Figure 2.7. Simulation of a landslide scenario using voxel automata for time T=50, T=500 and T=1800 as well as T=250 and T=500 for a portion of the study area at a larger scale.
42
2.5.2. Multiclass Voxel Automata
Advection with Multiple Classes
Many spatial processes are characterized by their gradual spatial spread starting from
boundary lines between regions or from individual point sources. Each local site in a
study region is defined by a value that represents the theme of the spatial process, which
is itself modelled on the basis of diffusive and advective flows between local sites.
The extent of the spread of airborne particles from a point source is greatly influenced by
the surrounding air flow among other factors (Lee Jr, et al., 1974), but a model for
particle advection can be simplified by assuming that the movement of the airborne
molecules is principally directed by the magnitude of a three-dimensional wind field. The
multistate model implemented in this study simulates a simple and hypothetical system of
gaseous particle movement using a totalistic description of the transition rules. It is
described with a four-state voxel automaton to represent particle concentration values in
each voxel and in terms of four ordinal categories, namely, negligible, low, medium and
high.
Configured for eastward flow from a continuous point source, the particles are simulated
to move from the central voxel by assigning directional magnitudes greater than zero to
the easterly voxels in the neighbourhood and zero for the westerly voxels. The transition
rules are applied to both a Moore neighbourhood configuration (Figure 2.8a) and a von
Neumann neighbourhood (Figure 2.8b) which creates a tunnel-like particle flow effect.
This example also indicates that similar to the cellular automata, the outcomes of voxel
automata simulations are also sensitive to the 3D neighbourhood configuration.
43
Figure 2.8. Simulation outcomes of multistate voxel automata of particle dispersion using (a) Moore and (b) von Neumann 3D neighborhoods
Particle Segmentation with Multiple Classes
Sediment-laden flows are characterized by the density differences of the particles within
the ambient fluid and are a key process for sediment transport downstream (O’Laughlin,
et al., 2014). Turbidity currents are important in the evolution of channel morphology and
the characterization of deposit structure within flow path of the ambient fluid. In the
model, the voxels are subspace of the fluid continuum each of which represents a mass of
particles.
The voxels are of a uniform size and they approximate a distribution of suspended
particulate in the fluid. At the initial stage, each of the voxels is randomly assigned a
value for density to be high, low or medium (equivalent to the density of the fluid).
During the course of the simulation, each voxel is acted upon by the forces of gravity,
buoyancy and the advection due to the flow of the ambient fluid. For simplicity, the
velocity vector of the ambient fluid is constant and moves the voxels East and likewise
44
buoyancy and gravity are constant. The effective resultant effects are sinking for high
density voxels, flotation for less dense voxels and advection to all in proportion to their
respective mass values. The simulation outcomes from the currents are shown in Figure
2.9.
Figure 2.9. Voxel automata simulation outcomes for segmentation of particles in a moving fluid
The processes simulated above demonstrate that the voxel-based automata method can
model spatiotemporal processes in a four-dimensional space-time continuum. In this
research, the factors responsible for the processes' dynamics are simplified. However,
with a complete four dimensional dataset, a voxel-based automata model can be specified
and calibrated for the study the evolution of a geographic process.
2.6. Conclusions
Similar to the typical geographic automata methods that are widely used to simulate
environmental processes two-dimensionally using the raster-based two-dimensional
representation of space, the voxel-based automata approach can be used to model
dynamic spatial processes. By discretizing the geographic environment into regularly
sized cubic units, voxel automata extend the representation to the four dimensions of the
space–time domain. The full integration of voxel automata with geospatial dataset in a
45
GIS environment is the next logical stage of research. The approach provides an
enhanced means with which to simulate geographic processes that if operationalized in a
GIS will improve spatial analysis and visualization in three and four dimensional
frameworks. Voxel automata have the potential to dynamically represent the evolution of
a complex geographical process.
In order to comprehensively test, calibrate and validate the voxel-automata models, the
voxels representing the spatial units in the process must be geo-referenced in order to
locate the automaton in space. This would necessitate the use of a geospatial data model
with a structure capable of representing and analysing three-dimensional space and time.
In addition, the scarcity of complete three-dimensional datasets for geospatial processes
implies that the testing and validation of voxel automata models is an area that will
require further research but the methods can be similar to those used in geographic
automata. However, as the financial costs of deploying geospatial sensors continue to fall
the necessary data may become more readily available, which will also accelerate the
efforts being put into the use four dimensional space-time geospatial data models. Using
partially indexed four-dimensional data within the sound theoretical framework of
complex systems modelling, the results from voxel automata models can be validated
against primary data.
The evolution of geographic processes is continuous in time and, barring obstacles in a
landscape or marine environment, the phenomenon itself spreads out continuously in the
volumetric space. As demonstrated in this study, the mechanisms transforming an
otherwise continuous geospatial process into discrete spatial and temporal units for
modelling purposes has a direct influence on the resultant patterns of the model. This
study further confirms that the use of different neighborhood configurations will result in
different spatial patterns as other researchers have demonstrated for 2D CA. The problem
manifests itself in the way the shape of a three dimensional pattern changes whenever a
different classification of the continuous variable is employed. The implemented models
serve as the initial step towards the development of more realistic models for the
simulation of environmental processes. In addition, the methodologies explored in this
46
study provide the initial efforts in the application of deterministic voxel automata that
could be used to model the evolution of geographic processes in 4D. Future work will
focus on applying the principles presented here to real-world problem situations by
translating the voxel automata approach into a geographical context and using real three-
dimensional geospatial datasets that are integrated into a GIS framework.
References
Bak, P. (1999). How Nature Works: The Science of Self-Organized Criticality (1 edition ed.). Copernicus.
Bak, P., Tang, C., and Wiesenfeld, K. (1987). Self-organized criticality: An explanation of the 1/f noise. Physical Review Letters, 59 (4), 381-384.
Barpi, F., Borri-Brunetto, M., and Veveri Delli, L. (2007). Cellular-automata model for dense-snow avalanches. Journal of Cold Regions Engineering, 21 (4), 121-140.
Batty, M. (1997). Cellular automata and urban form: a primer. Journal - American Planning Association, 63 (2), 266-274.
Batty, M., and Torrens, P. M. (2005). Modelling and prediction in a complex world. Futures, 37 (2005), 745-766.
Batty, M., and Xie, Y. (1994). From cells to cities. Environment and Planning B: Planning and Design, 21, 531-548.
Ben Youssef, B. (2014). A visualization tool of 3-D time-varying data for the simulation of tissue growth. Multimedia Tools and Applications, 73 (3), 1795-1817.
Benenson, I., and Torrens, P. M. (2004). Geosimulation: Automata-based modelling of urban phenomena. John Wiley & Sons, Hoboken, NJ.
Berjak, S. G., and Hearne, J. W. (2002). An improved cellular automaton model for simulating fire in a spatially heterogeneous Savanna system. Ecological Modelling, 148 (2), 133-151.
Bone, C., Dragicevic, S., and Roberts, A. (2006). A fuzzy-constrained cellular automata model of forest insect infestations. Ecological Modelling, 192 (1-2), 107-125.
47
Cai, Y., Wu, J., Xu, S., and Li, Z. (2013). A Hybrid Cellular Automata Model of Multicellular Tumour Spheroid Growth in Hypoxic Microenvironment. Journal of Applied Mathematics.
Chrisman, N. R. (1997). Exploring geographic information systems. John Wiley & Sons, New York.
Clarke, K. C., and Gaydos, L. J. (1998). Loose-coupling a cellular automaton model and GIS: long-term urban growth prediction for San Francisco and Washington/Baltimore. International Journal of Geographical Information Systems, 12 (7), 699-714.
Cotsaftis, M. (2009). A Passage to Complex Systems. In C. Bertelle, G. H. E. Duchamp and H. Kadri-Dahmani (Eds.), Complex systems and self-organization modelling. Springer, Berlin.
Couclelis, H. (1985). Cellular worlds: a framework for modeling micro-macro dynamics. Environment and Planning A, 17 (5), 585-596.
Cucurull, L., Ruffini, G., Flores, A., and Rius, A. (1998). Ingestion of GPS data into a parameterized ionospheric model for tomography of the electron distribution of the ionosphere. In J. E. Russell (Ed.), (Vol. 3495, pp. 397-404). SPIE - The International Society for Optics and Photonics.
Di Gregorio, S., Rongo, R., Siciliano, C., Sorriso-Valvo, M., and Spataro, W. (1999). Mount Ontake landslide simulation by the cellular automata model SCIDDICA-3. Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, 24 (2), 131-137.
Drossel, B., and Schwabl, F. (1992). Self-organized critical forest-fire model. Phys. Rev. Lett., 69 (11), 1629-1632.
Eshraghi, M., Felicelli, S. D., and Jelinek, B. (2012). Three dimensional simulation of solutal dendrite growth using lattice Boltzmann and cellular automaton methods. Journal of Crystal Growth, 354 (1), 129-134.
Frieder, G., Gordon, D., and Reynolds, R. (1985). Back-to-Front Display of Voxel-Based Objects. Ieee Computer Graphics and Applications, 5 (1), 52-60.
Gandin, C. A., Rappaz, M., Desbiolles, J. L., Lopez, E., Swierkosz, M., and Thevoz, P. (1996). 3D modeling of dendritic grain structures in turbine blade investment cast parts. Minerals, Metals & Materials Soc, Warrendale.
48
Grassberger, P. (1984). Chaos and diffusion in deterministic cellular automata. Physica D: Nonlinear Phenomena, 10 (1–2), 52-58.
Guariso, G., and Maniezzo, V. (1992). Air quality simulation through cellular automata. Environmental Software, 7 (3), 131-141.
Hergarten, S. (2003). Landslides, sandpiles, and self-organized criticality. Natural Hazards and Earth System Science, 3 (6), 505-514.
Hergarten, S., and Neugebauer, H. J. (1998). Self-organized criticality in a landslide model. Geophysical Research Letters, 25 (6), 801-804.
Hernandez-Pajares, M., Zornoza, J. M. J., Subirana, J. S., and Colombo, O. L. (2003). Feasibility of wide-area subdecimeter navigation with GALILEO and modernized GPS. IEEE Transactions on Geoscience and Remote Sensing, 41 (9), 2128-2131.
Holland, J. H. (2006). Studying complex adaptive systems. Journal of Systems Science and Complexity, 19 (1), 1-8.
Hopcroft, J. E. (2007). Introduction to automata theory, languages, and computation (3rd ed ed.). Pearson/Addison Wesley, Boston.
Hosoi, F., Nakai, Y., and Omasa, K. (2013). 3-D voxel-based solid modeling of a broad-leaved tree for accurate volume estimation using portable scanning lidar. ISPRS Journal of Photogrammetry and Remote Sensing, 82, 41-48.
Hosoi, F., and Omasa, K. (2009). Detecting seasonal change of broad-leaved woody canopy leaf area density profile using 3D portable LIDAR imaging. Functional Plant Biology, 36 (10-11), 998-1005.
Hotzendorfer, H., Estelberger, W., Breitenecker, F., and Wassertheurer, S. (2009). Three-dimensional cellular automaton simulation of tumour growth in inhomogeneous oxygen environment. Mathematical and Computer Modelling of Dynamical Systems, 15 (2), 177-189.
Hunt, S. M., Hamilton, M. A., and Stewart, P. S. (2005). A 3D model of antimicrobial action on biofilms. Water Science and Technology, 52 (7), 143-148.
Ilachinski, A. (2001). Cellular Automata: A Discrete Universe. World Scientific, Singapore; River Edge, NJ.
Kocabas, V., and Dragicevic, S. (2007). Enhancing a GIS cellular automata model of land use change: Bayesian networks, influence diagrams and causality. Transactions in GIS, 11 (5), 681-702.
49
Lai, T., and Dragicevic, S. (2011). Development of an urban landslide cellular automata model: A case study of North Vancouver, Canada. Earth Science Informatics, 4 (2), 69-80.
Lalonde, M., Tremblay, G., and Jebrak, M. (2010). A Cellular Automata Breccia Simulator (CABS) and its application to rounding in hydrothermal breccias. Computers and Geosciences, 36 (7), 827-838.
Lansing, J. S. (2003). Complex adaptive systems. Annual Review of Anthropology, 32, 183-204.
Lau, K. H., and Kam, B. H. (2005). A cellular automata model for urban land-use simulation. Environment and Planning B: Planning and Design, 32 (2), 247-263.
Lee Jr, R. E., Caldwell, J., Akland, G. G., and Fankhauser, R. (1974). The distribution and transport of airborne particulate matter and inorganic components in Great Britain. Atmospheric Environment (1967), 8 (11), 1095-1109.
Lin, Y., Mynett, A., and Li, H. (2011). Unstructured cellular automata for modelling macrophyte dynamics. International Journal of River Basin Management, 9 (3-4), 205-210.
MathWorks. (2012). MATLAB and Statistics Toolbox. In (Release 2012b ed.). The MathWorks, Inc., Natick, Massachusetts, United States.
McIntosh, H. V. (2010). Conway’s life. In A. Adamatzky (Ed.), Game of Life Cellular Automata (pp. 17-33). Springer London.
Moreno, N., Menard, A., and Marceau, D. J. (2008). VecGCA: A vector-based geographic cellular automata model allowing geometric transformations of objects. Environment and Planning B: Planning and Design, 35 (4), 647-665.
Narteau, C., Zhang, D., Rozier, O., and Claudin, P. (2009). Setting the length and time scales of a cellular automaton dune model from the analysis of superimposed bed forms. Journal of Geophysical Research - Part F - Earth Surfaces, 114 (F3), F03006-(03018 pp.).
Neofytou, P., Venetsanos, A. G., Rafailidis, S., and Bartzis, J. G. (2006). Numerical investigation of the pollution dispersion in an urban street canyon. Environmental Modelling & Software, 21 (4), 525-531.
O’Laughlin, C., van Proosdij, D., and Milligan, T. G. (2014). Flocculation and sediment deposition in a hypertidal creek. Continental Shelf Research, 82 (0), 72-84.
50
Oreskes, N., Shrader-Frechette, K., and Belitz, K. (1994). Verification, validation, and confirmation of numerical models in the Earth sciences. Science, 263 (5147), 641-646.
Parunak, H. V. D., Savit, R., and Riolo, R. L. (1998). Agent-Based Modeling vs. Equation-Based Modeling: A Case Study and Users’ Guide. In J. S. Sichman, R. Conte and N. G. Gilbert (Eds.). Springer.
Patterson, T., Mitchell, N., and Sifakis, E. (2012). Simulation of Complex Nonlinear Elastic Bodies using Lattice Deformers. Acm Transactions on Graphics, 31 (6).
Popescu, S. C., and Zhao, K. (2008). A voxel-based lidar method for estimating crown base height for deciduous and pine trees. Remote Sensing of Environment, 112 (3), 767-781.
Rabin, M. O. (1963). Probabilistic automata. Information and Control, 6 (3), 230-245.
Raper, J. (2000). Multidimensional Geographic Information Science (1 edition ed.). Taylor & Francis, London.
Reniers, D., and Telea, A. (2008). Part-type Segmentation of Articulated Voxel-Shapes using the Junction Rule. Computer Graphics Forum, 27 (7), 1845-1852.
Ridgway, G. R., Litvak, V., Flandin, G., Friston, K. J., and Penny, W. D. (2012). The problem of low variance voxels in statistical parametric mapping; a new hat avoids a 'haircut'. Neuroimage, 59 (3), 2131-2141.
Rothman, D. H., and Zaleski, S. (2004). Lattice-gas cellular automata: simple models of complex hydrodynamics. Cambridge University Press, London.
Sarkar, C., and Abbasi, S. A. (2006). Cellular automata-based forecasting of the impact of accidental fire and toxic dispersion in process industries. Journal of Hazardous Materials, 137 (1), 8-30.
Schilling, A., Schmidt, A., and Maas, H.-G. (2012). Tree topology representation from TLS point clouds using depth-first search in voxel space. Photogrammetric Engineering And Remote Sensing, 78 (4), 383-392.
Schneider, F. D., Letterer, R., Morsdorf, F., Gastellu-Etchegorry, J.-P., Lauret, N., Pfeifer, N., and Schaepman, M. E. (2014). Simulating imaging spectrometer data: 3D forest modeling based on LiDAR and in situ data. Remote Sensing of Environment, 152, 235-250.
51
Semboloni, F. (1997). An urban and regional model based on cellular automata. Environment and Planning B: Planning and Design, 24 (4), 589-612.
Shi, W., and Pang, M. Y. C. (2000). Development of Voronoi-based cellular automata-an integrated dynamic model for geographical information systems. International Journal of Geographical Information Science, 14 (5), 455-474.
Siddon, R. (1985). Fast Calculation of the Exact Radiological Path for a 3-Dimensional Ct Array. Medical Physics, 12 (2), 252-255.
Stevens, D., and Dragicevic, S. (2007). A GIS-based irregular cellular automata model of land-use change. Environment and Planning B: Planning and Design, 34 (4), 708-724.
Stocks, C. E., and Wise, S. (2000). The Role of GIS in Environmental Modelling. Geographical and Environmental Modelling, 4 (2), 219-235.
Thaler, H. T., Ferber, P. W., and Rottenberg, D. A. (1978). A statistical method for determining the proportions of gray matter, white matter, and CSF using computed tomography. Neuroradiology, 16 (1), 133-135.
Tobler, W. R. (1979). Cellular geography. In S. Gale and G. Olsson (Eds.), Philosophy in Geography (pp. 379-386). Reidel Publishing Company, Dordrecht, Holland.
Torrens, P. M., and Benenson, I. (2005). Geographic automata systems. International Journal of Geographical Information Science, 19 (4), 385-412.
Torrens, P. M., and O'Sullivan, D. (2001). Cellular automata and urban simulation: where do we go from here? Environment and Planning B: Planning and Design, 28 (2), 163-168.
Turcotte, D. L. (1999). Self-organized criticality. Reports on Progress in Physics, 62 (10), 1377-1429.
Vabø, R., and Nøttestad, L. (1997). An individual based model of fish school reactions: predicting antipredator behaviour as observed in nature. Fisheries Oceanography, 6 (3), 155-171.
Vanderhoef, M., and Frenkel, D. (1991). Tagged Particle Diffusion in 3d Lattice Gas Cellular Automata. Physica D, 47 (1-2), 191-197.
von Neumann, J. (1966). Theory of Self-Reproducing Automata. University of Illinois Press, Urbana.
52
Wang, C., Esaki, T., Xie, M., and Qiu, C. (2006). Landslide and debris-flow hazard analysis and prediction using GIS in Minamata–Hougawachi area, Japan. Environmental Geology, 51 (1), 91-102.
White, R., and Engelen, G. (1993). Cellular automata and fractal urban form: a cellular modelling approach to the evolution of urban land-use patterns. Environment & Planning A, 25 (8), 1175-1199.
White, R., and Engelen, G. (2000). High-resolution integrated modeling of spatial dynamics of urban and regional systems. Computers, Environment and Urban Systems, 24 (5), 383-400.
White, R., Engelen, G., and Uljee, I. (1997). The use of constrained cellular automata for high-resolution modelling of urban land-use dynamics. Environment and Planning B: Planning and Design, 24 (3), 323-343.
Wolfram, S. (1983). Statistical mechanics of cellular automata. Reviews of Modern Physics, 55 (3), 601-644.
Wolfram, S. (2002). The crucial experiment. In A new kind of science (pp. 23-50). Wolfram Media, Champaign, IL.
Yassemi, S., Dragicevic, S., and Schmidt, M. (2008). Design and implementation of an integrated GIS-based cellular automata model to characterize forest fire behaviour. Ecological Modelling, 210 (1-2), 71-84.
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Chapter 3. Integrating GIS-Based Geo-Atom Theory and Voxel Automata to Simulate the Dispersal of Airborne Pollutants1
3.1. Abstract
Environmental processes are usually conceptualized as complex systems whose dynamics
are best understood by examining the relationships and interactions of their constituent
parts. The cellular automata paradigm, as a bottom-up modeling approach, has been
widely used to study the macroscopic behavior of these complex natural processes.
However, cellular automata models are largely restricted to the two-dimensional spatial
perspective even though the process dynamics they represent evolve in the three spatial
dimensions. The objective of this study is to develop a voxel-based automata approach
for modeling the propagation of airborne pollutants in three-dimensional space over time.
The GIS-based geo-atom theory was used to manage the data within the automaton. The
simulation results indicate the model has the capability to generate effective four-
dimensional (4D) simulations from simple transition rules that describe the processes of
particle advection and diffusion. The application of voxel-based automata and the geo-
atom concepts allows for a detailed 4D analysis and tracking of the changes in the voxel
space at every time-step. The proposed modeling approach provides new a means to
examine the relationships between pattern and process in 4D.
1This chapter has been accepted for publication: Jjumba, A., and Dragićević, S., 2014, Integrating
GIS-Based Geo-Atom Theory and Voxel Automata to Simulate the Dispersal of Airborne Pollutants: Transactions in GIS (Reproduced with the permission of the publisher).
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3.2. Introduction
When modeling complex environmental systems, simulation models are particularly
valuable because they can be used to describe, analyze and reliably forecast the
trajectories of environmental processes using trends from historical data (Andretta, et al.,
2006; Heung, et al., 2013). Among the methods used to develop these simulation models
is cellular automata (CA), which is also a commonly employed dynamic modeling
method because of its straightforward computational theory (Camazine, et al., 2001;
Bertelle, et al., 2009; Rasmussen and Hamilton, 2012). CA models are developed
following a bottom-up approach where the system is described in terms of its
microscopic components. In fact, CA theory has been adopted to define the principles of
geographic automata systems (Benenson and Torrens, 2004), which can be used to
represent complex dynamic geographic processes. CA models can generate complex
global patterns from simple local interactions and as such they are used extensively to
study the macroscopic behavior of complex geographic processes (Batty, 2005b; Holland,
2006; Manson, 2007). However, most of the automata models of environmental processes
have been limited to the two-dimensional (2D) spatial perspective. Extending these 2D
models to a more realistic three-dimensional (3D) spatial context provides the means by
which the dynamics of the environmental processes can be analyzed intuitively and more
comprehensively. Even in the few cases where 3D CA results have been generated and
used in further environmental analysis with geographic information systems (GIS)
software applications (Ciolli, et al., 2004), they are limited to static spatial data queries
since modeling the time dimension in GIS is still a significant research challenge (Pultar,
et al., 2010).
The realistic simulation and analysis of the dynamics of many environmental phenomena,
such as the dispersal process of gaseous matter in the atmosphere, can be significantly
improved by representing them as four-dimensional (4D) space-time models. In general,
environmental phenomena are dynamic processes driven by systems composed of
multiple component parts and are sometimes organized into different sub-systems
characterized by geographic complexity (Manson, 2007). These systems evolve in 3D
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space and the interactions between the component parts occur over geographic space for
extended periods of time (Manson and O'Sullivan, 2006). The nonlinear local
relationships between the component parts tend to produce global characteristics such as
self-organization, adaptation, feedback loops, emergent behavior and bifurcations
(Holland, 1995; Epstein and Axtell, 1996; Lansing, 2003; J. H. Miller and Page, 2007;
Messina, et al., 2008). It has been shown that these properties of complex systems can be
reproduced by designing dynamic models based on the idea that the emergent
characteristics of a system are best understood by examining the relationships of its
component parts (Auyang, 1998; Macal and North, 2005; Tang and Bennett, 2010).
However, the spatial data models typically used in GIS applications rely on the raster and
the vector spatial representations which are not capable of adequately representing the
temporally dynamic changes of geographical entities. Instead, they describe spatial
objects in terms of their respective singular location and attribute values (Chrisman,
1997). Coupling CA and GIS applications has therefore enabled dynamic model outputs
to be represented as a series of temporal snapshots of the simulated system at the macro
scale (Couclelis, 1985; Clarke and Gaydos, 1998; Batty, et al., 1999; Schwarz, et al.,
2010).
Although the research on 3D CA has been undertaken in computational studies (Bays,
1987; Imai, et al., 2002) as well as in the physical sciences, these studies have been
focused on theoretical concepts that are not particularly related to geographical
representation. 3D CA have been used to simulate gas transport (Frisch, et al., 1986;
Vanderhoef and Frenkel, 1991; Weimar, 2002; Leonardi, et al., 2012) and in the
biological sciences to simulate tissue growth (Kansal, et al., 2000; Luebeck and deGunst,
2001) and to study behavioral dynamics (Jimenez-Morales, 2000). In the geosciences,
Narteau et al. (2009) have used the 3D automata to simulate dune formation, however,
the simplified concepts of 2.5D cellular automata were used in other efforts to simulate
phenomena that propagate in 4D in order to model, for example, snow avalanches
(Avolio, et al., 2010; Fonseca, et al., 2011) or geological formations (Baas, 2007) by
making the elevation an attribute in a two-dimensional automata system. While these
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applications contribute to the understanding of possible 3D patterns and formations over
space and time, these studies are still at the level of initial research and they do not yet
contribute towards the development of a spatial data model that enables the
conceptualization of GIS for 4D querying and analysis.
Given the gaps in the way spatial phenomena are conceptualized and how their dynamics
are represented in a GIS, the main objective of this study is to develop a proof-of-concept
voxel-based automata model that can simulate environmental processes in 4D. Concepts
from the GIS based geo-atom theory (Goodchild, et al., 2007) were used to conceptually
and practically handle 3D entities in voxel space. In addition, a case study of pollutant
dispersion from a single point source is used to demonstrate the mechanics of the
proposed model. The next section provides the theoretical basis for the developed model
and following that the practical implementation of the dispersal model is presented and
discussed.
3.3. Theoretical Background
The CA technique is a computational approach originally proposed by von Neumann
(1966) that has been widely used to develop models that generate complex spatial
patterns from locally interacting system elements. Formally, a cellular automaton can be
described by a mathematical function that represents the following key elements: a grid
space (also termed a lattice of cells), cell states, the neighborhood, transition rules to
guide cell changes from one state to another at each time-step, and the number of time-
steps model completion (Wolfram, 1983; Ilachinski, 2001). The automaton operates such
that each cell in the grid space computes its own state based on the current state of its
neighboring cells. The CA model formalism is:
57
C = (G, S, N, f)
(3.1)
where C is the automaton representing the CA model, G is the grid space of one or more spatial dimensions; S is the finite set of discrete states; N represents the cells in the neighborhood; f is the transition function.
The transition function that determines how the present state of a cell will change at the
next time-step can be represented as:
( ) 1,: +→ ttt TTT SNSf
(3.2)
where and are the internal states of the cell at the current time Ti and at the subsequent time Ti+1, respectively; is a neighborhood around a particular cell.
Even though the preceding formulation was designed for CA operating in a 2D cellular
space, it can be adapted for an automaton in 3D volumetric space because both 2D and
3D models use the same underlying principle of relying on local neighborhood-based
interactions to generate complex global spatial patterns.
Observable changes in natural phenomena evolve in a temporally and spatially
continuous 4D space-time domain. However, for the purposes of measurement,
simplification and eventual representation in GIS applications, both the temporal and
spatial dimensions are discretized into meaningful and well-defined units (Lo and Yeung,
2007). Moreover, the representation of the spatial and the temporal dimensions in discrete
terms, as they are conceptualized in the CA paradigm, is attractive for spatiotemporal
modeling in GIScience because it enables the dynamic representation of a geographic
process by using spatial data that would otherwise provide static representations of
iTS 1+iTSiTN
58
spatiotemporal processes (Benenson and Torrens, 2004; Batty, 2005a). Also as Bailey
(1990) noted, the usefulness of the spatial data models is rooted in their ability to provide
a GIS with a consistent discretization mechanism that can be applied to the geographic
objects of interest.
Temporal changes in the geospatial datasets which are structured around the vector or the
raster geospatial data formats are presented as a series of snapshot data layers in order to
simulate changes over time (Peuquet, 1994). For the purpose of mimicking the dynamic
nature of a phenomenon such as water flow, spatial interpolation between the snapshots
of the vector data objects has been used to describe change between successive points on
the temporal axis (Raper and Livingstone, 1995). In order to simulate changes in natural
processes using automata models, the temporal dimension is likewise discretized into a
series of distinct temporal intervals. Thus, due to the nature of the automata paradigm, the
visualization of complex geographical processes is made possible in CA models because
the outputs at every time-step can be displayed as raster-based GIS outputs. For 3D
modeling, the study area is represented in terms of the three dimensions of space and is
decomposed into logical and consistent volumetric elements referred to as voxels. Voxel
as volumetric tessellation has been used in the physical sciences literature when
analyzing volumetric shapes (Arata, et al., 1999; Fischer, et al., 2005) and within the
field of GIScience (Marschallinger, 1996; E. J. Miller, 1997; Grunwald and Barak, 2003;
Ledoux and Gold, 2008).
Although this study presents a uniform voxel-based automata model, the theoretical and
practical GIS concepts for modeling in 4D using cubes of different sizes have been
presented by Karssenberg and de Jong (Karssenberg and De Jong, 2005) using the
modeling language of PCRaster software (Karssenberg, et al., 2010), which was specially
designed to accommodate a range of environmental dynamic models (Wesselung, et al.,
1996; Burrough, et al., 2005). The data inputs are of various formats including DEMs,
raster data and tabular representation of the multifaceted attributes of the environmental
phenomena. More particularly PCR-GLOBWB has been developed and used to model
large scale hydrology (Karssenberg, et al., 2010; Trambauer, et al., 2014), soil erosion
59
(Kamphorst, et al., 2005) and various global processes related to climate change
modeling. The PCR-GLOBWB has been designed using an enhanced ‘leaky bucket’
model developed by Bergström (1995) and combined with process equations and applied
on a cell-by-cell basis. In recent years, PCRaster BlockAnalyst has been in development
for 3D voxel-based modeling but it is yet to be documented in the literature. Moreover,
the effects of earthquakes on dynamics of fracture and collapse of the discrete building
objects represented in an urban environment has been simulated in 4D by using three
dimensional irregular object tessellations (Torrens, 2014).
The geographic automata modeling approach provides a mechanism for gathering
insights into the relationship between the micro-level interactions of the system’s
components and the macro-level manifestation of the geographically complex
phenomenon. For this to be achieved, CA models and GIS applications are coupled
together when simulating environmental phenomena. However, the existing GIS
applications are known for being ill-suited for temporal model querying (Langran, 1992;
Wachowicz and Healey, 1994; Dragicevic and Marceau, 2000; Peuquet, 2001; Worboys
and Duckham, 2004) since the technology was originally envisioned as a tool for
reproducing the static spatial contents of paper maps in a digital environment (Goodchild,
1992). Nevertheless, over the years researchers have proposed various ways to handle
time in GIS applications and the snapshot method has been used to represent the
spatiotemporal dynamics of geographic phenomena under the assumption that the
temporal interval in between snapshots is sufficiently small to represent the continuity of
the process. For automata modeling, the framework proposed by Grenon and Smith
(2004) lends itself more readily. They suggested a unified framework where events
taking place instantaneously can be represented as discrete marks on a temporal scale,
while those spatial objects whose identity is persistent through time are represented by
continuous intervals.
In recent times, it has been proposed that spatial features as well as their persistence in
time can be represented in terms of the geo-atom GIS-based spatial data model
(Goodchild, et al., 2007). Theoretically, a geo-atom describes a spatial feature as an
60
atomic element with a specific attribute value in a space-time domain. Using this
fundamental concept, raster representations of a geographic landscape can be described
as geo-fields or collections of geo-atoms, while geo-objects can be used to describe the
vector-based models of spatial features. In principle, the location and attribute value of a
geo-atom can only describe a snapshot of a geographic feature as it is measured at a
particular point in time. However, since time is explicitly recorded as a field in the tuple
that describes the structure of the geo-atom, it can be extended temporally to dynamically
show and query the spatial distribution of attribute values at different times. Thus, in the
context of a dynamic model output, records can be kept of all the changes to the objects
in the system along with their associated time of incidence, a feature that allows for
inferred explanations about the cause of the changes in the system with more clarity and
certainty as compared to the existing dynamic models that use raster and vector datasets.
Besides this study, Pultar et al. (2010) have implemented the concepts of the geo-atom
theory to develop a data structure which they use to simulate wildfire propagation and
pedestrian movement limited to a two-dimensional geographic space.
3.3.1. Modelling the Space-Time Domain
The structure of a geo-atom is described in terms of the location, attribute domain and the
temporal component of the spatial feature it represents. It is formalised as a tuple:
g = (p, A, a(p))
(3.3)
where g is a geo-atom that represents a spatial object being modeled, p is a vector that describes the object in space-time coordinates (x, y, z, t), A is the attribute or variable of interest such as carbon monoxide for example, and a(p) is the specific attribute value of A at that location in space-time.
For the purpose of illustration, let’s assume that the carbon monoxide concentration at
(123W, 49N) and 20m above mean sea level on 1st June 2010 at 13:00 hours was 50ppm.
61
From equation (3.3), the spatial data point could be expressed in the form of a geo-atom
as:
p ≡ [123, 49, 20, 2010/6/1/13] and a(p)≡50
For 3D automata, a voxel, being the smallest spatial unit in 3D space, can be
conceptualised as a geo-object having a state S. Using function (3.3) then
g = (p, A, S’)
(3.4)
where S’ = a(p) and p = (x,y,z,t); x,y,z are the coordinates for the centroid of the voxel and t is the time stamp. From equation (3.1), the finite set S with m possible voxel states can be represented as
''2
'1 ..., mSSSS =
(3.5)
The Geo-Atom Structure for a 3D Voxel Automaton
The development of a voxel automata model for the simulation of air pollution dynamics
is proposed based on the geo-atom data model and a bottom-up modeling approach. The
automata approach is used because CA models have been used effectively to simulate
complex and non-equilibrium physical processes such as pollutant dispersal.
A key aspect of the geo-atom theory is its ability to provide a description of a
geographical entity in terms of the three dimensions of space and time. In the case of 3D
CA, each voxel is a spatial object that would be represented by a geo-atom structure. It is
possible that at the beginning of the simulation, the state of each voxel is recorded and
whenever there is a change in the state of the voxel as a result of the application of the
transition rules, the value describing its state is recorded and is later used to analyse the
simulation outcomes. With this approach the automaton has memory encoded for each
62
voxel that indicates when the changes in each voxel took place. This approach differs
from the one by Alonso-Sanz and Martin (2003) that is used to remember the cumulative
average value of past states, and is distinctly different from reversible 2D CA. Although
it may be possible to model each voxel as a geo-dipole (Goodchild, et al., 2007) linked in
space and time to the other voxels in its neighborhood, the representation in this study is
limited to the elementary structure of the geo-atom for simplicity. Besides the difficulty
of designing a database structure that maps each voxel to its twenty-six neighbors, these
relationships and interactions are sufficiently defined and implemented using the
transition rules of the automata.
3.3.2. Simulating the Dispersal of Airborne Pollutants in 4D
The dispersal of airborne gases and particles is a one of the major causes of concern for
public health in addition to climate change and global warming as a result of fossil fuel
emissions. In the case of an air pollutant, the buoyancy effects within the plume coupled
with the transport mechanisms due to the surrounding airflow system influence the
dispersion of the particles from a given point source (Degrazia, et al., 2009). Gases that
are either neutrally or positively buoyant will stay afloat above the ground surface and
become mixed with the neighbourhood's airflow system (Britter, 1979). Dense negatively
buoyant gases, on the other hand, are greatly influenced by the gravitational force and
tend to move towards the ground surface with less influence from the wind speed in the
neighbourhood. The prevailing wind direction plays an important role in the transport
process because the pollutant’s movement is downwind and the greater the speed the
greater is the rate of dilution in the pollutant's concentration. More specifically, the
transportation of the pollutant in the air is directed by the processes of advection and
diffusion (Wark, et al., 1997). Advection is the movement of the pollutant's particles with
the flow of the fluid in which they are contained. The effective speed of this movement,
also termed the advective speed, depends on the time-varying 3D wind field and is
responsible for the advective transportation of the pollutant. Molecular diffusion on the
other hand is the spread of the pollutant from spaces of higher concentration to those of
63
lower concentration (Zang, 1991). In addition, turbulence causes additional diffusion as a
result of high transport velocities and the dramatic concentration differences between the
various points in the neighborhood.
Existing 3D models are typically based on complex mathematical formulations or they
use differential equations to represent the propagation of airborne particles (Yeh and
Huang, 1975; Tirabassi, 1989). This study proposes a voxel automata model derived from
complex systems theory which takes a ‘bottom up’ approach by looking at processes in
terms of locally defined discrete 3D spatial units and discrete temporal increments.
Taking a central voxel and its surrounding neighbors as the primary spatial units at a
local 3D scale, the processes of advection and diffusion are modeled to mimic the
relocation of airborne particles moving from one voxel to another, thereby reproducing
the process of propagation in 4D at a larger spatial scale.
3.4. Methods
In developing the 3D voxel automata model for this study, several generalizations and
assumptions regarding the spread of airborne pollutants were made. The effects of
turbulence on the movement of the pollutant’s molecular particles are not explicitly
considered. Instead, it is assumed the advective and diffusion processes are the major
contributors to the movement of the particles. Furthermore, the pollutants are transported
under minimally fluctuating environmental conditions with only the wind velocity
changing slightly. Other environmental factors such as temperature, relative humidity and
the chemical transformation of pollutants are not considered. The molecules of the
gaseous pollutants are assumed neutrally buoyant. Although excluded from consideration
here, it is also acknowledged that other processes involved in particle transport, such as
microturbulence, may operate at fine-grained resolutions in space and time and can
influence the dynamics of the environmental process over a large spatial extent and over
long periods of time (Perry and Enright, 2006). Moreover, for the purposes of improving
computational efficiency, this simulation study was designed to use a relatively coarse
64
spatial resolution over a small study area and the simulations run for short computational
durations.
The study area was divided into uniform volumetric cubes (voxels) in which the
pollutant’s particles (molecules) are in motion due to the wind action. Each voxel is
surrounded by twenty-six other contiguous neighbors to form a 3D Moore neighborhood.
In the model, the pollutant’s concentration in a voxel is used to infer the amount of
pollutant molecular particles per unit volume. A concentration value of 0 refers to
volumetric space that is devoid of pollutants while a value of 1 is assigned to a voxel that
is saturated, in other words the voxel volume is filled with the molecules of the
pollutants. To simulate the process of pollutant dispersal, the mass of the pollutant
particles in the system is conserved in the system. The magnitude of the wind speed and
its direction will influence the spread of the pollutants due to advection, while the
differences in pollutant concentration for any two voxels and diffusivity of the molecules
will influence the spread due to the diffusion process.
3.5. Transport due to Particle Advection
Advective transport is the movement of pollutants due to the magnitude of the wind
velocity. In general, as the wind speed increases, the advective flow of the pollutants
from any given voxel to one or more of its neighboring voxels also increases. This
process can be modeled by using equation (3.6) which defines the advective flux when
particles are carried by fluid motion (Logan, 2001).
vCkCa =
(3.6)
where Ca is the pollutant concentration transported either out or into a voxel due to advection, v is the magnitude of the wind speed in the voxel, k is a porosity constant that defines the amount that goes through the voxel’s boundaries.
65
Effectively, the pollutants are propagated from a neighboring voxel, located upwind, to
the central voxel in the neighborhood. During the same time-step, pollutants are moved
from the central voxel to the neighboring voxel located downwind relative to the central
voxel. The voxels are conceptualized as distinct entities adjoining one another via
varying surface area extents. For those that are connected to one another via the face
sides, more pollutants will flow between them due to the large surface area that is
exposed between them. However, those that are connected by an edge (diagonal
positions) will have lesser pollutant quantities due to the limited surface area exposed.
Likewise, even lesser pollutants will be exchanged between those connected via the
vertex positions (double diagonal) in the 3D neighborhood.
3.5.1. Transport due to Particle Diffusion
Diffusion is the process by which airborne pollutants spread from a space with a higher
molecular concentration to a space with a lesser concentration. It differs from advection
in that the pollutant’s particles spread from one voxel to another without requiring a force
for this movement. The flux into any given voxel as a result of diffusion can be described
as:
(3.7)
where Cd is the concentration lost or gained as a result of diffusion between a voxel and its neighbor, r is the dispersion coefficient, ∆C is the difference in concentration between a voxel and its neighbor, ∆d is the distance between centroids of the neighboring voxels.
If the voxel at the center of the neighborhood has a higher pollutant concentration than
that in a neighboring voxel, it will result in a negative concentration difference and,
consequently, pollutant particles are lost from the voxel due to diffusion. Conversely,
when the difference in the concentration gradient is positive, that is to say, less
dCrCd ∆
∆−=
66
concentration in the central voxel than its neighbor, then the pollutant particles are added
to the central voxel. Each of the twenty-six neighboring voxels is positioned in a specific
direction from the one at the center, and any one voxel can potentially gain and lose
pollutants to the central voxel due to diffusion. In addition, it is expected that face
neighbors will lose and gain more pollutants from the central voxel than those that are
diagonally positioned in the 3D neighborhood.
3.5.2. The Elements of the 3D Voxel Automaton
These local dynamics of pollutant movement are primarily modeling the pollutant’s
influx to and outflux from the central voxel due to diffusion and advective transport. By
implication, the transition rules governing this automata model replicate the process at
the local level. The concepts of the geo-atom theory are used to associate the identity of a
voxel to its location and the ongoing changes in its state as result of the application of the
neighborhood transition rules.
The key voxel automata elements are described as follows:
Voxel Space: The approach in this study differs from the classical formalism that uses a
2D cellular grid, in that it uses a 3D lattice of voxels. These voxels are encoded with a
description for the spatial and temporal dimensions of the study site. Although the voxel
space matches what would be expected in a real study area situation, it is representative
and not based on an existing dataset.
Voxel State: The state of a voxel is considered to be a function of the pollutant's
concentration characterised by the extent of dilution and the dispersion characteristics of
the gas. The pollutant itself is assumed to be uniformly distributed within the volumetric
space.
Voxel Neighbourhood: A 3D Moore neighborhood is used to describe the processes of
pollutant advection and diffusion from any given voxel to its neighbors. It is comprised
of 26 neighbors – 6 face neighbours, 12 diagonal neighbors (edge connected) and 8
67
double diagonal neighbors (vertex connected) as notionally shown in the Figure 3.1
below. All the voxels in the neighborhood are contiguous to the central voxel and so they
are all situated in unique positions and orientation with respect to the central one.
Figure 3.1. Representation of the 3D voxel automata neighborhood: (a) 3D Moore neighborhood, (b) face neighbors within this neighborhood, (c) diagonal neighbors and (d) double-diagonal neighbors
Transition Rules: The concentration of the pollutant in a voxel at the subsequent time
interval t + 1 depends upon:
• The level of the pollutant concentration in the voxel at the time t
• The quantity of the pollutant particles drawn from some or all of the 26 voxels in the neighborhood
• The quantity of the pollutant transported to some or all of the 26 voxels in the neighborhood
• The magnitude of the prevailing wind speed in a specified direction
Even though all the voxels are of the same size, the extent of their surface areas exposed
to the central voxel depends on their respective positions in the neighborhood. As a
result, three porosity constants were used to represent the extent of the exposed voxel
membranes for the face neighbours, diagonal neighbors and the double diagonal
68
neighbours, respectively. By combining the processes of advection and dispersal of
particles in a voxel, the pollutant concentration Ct+1 at time t+1 in is given by:
(3.8)
where Ct is the pollutant concentration that exists in the voxel at time t; Cd is the pollutant that is either lost or gained through diffusion; Ca is for the pollutants drawn into or from the central voxel as result of the action of advection.
Assuming the gaseous pollutants are neutrally buoyant and under steady-conditions, it is
further assumed that the effective transport of the pollutant’s particles is solely a result of
advection and diffusion. Using a 3D Moore neighborhood structure, there are 26
directions towards which the wind can blow from the central voxel. For every voxel in
the neighborhood, there is a direction point and all the directions point away from the
central voxel as notionally presented in Figure 3.2. Furthermore, a unique identifier is
assigned to each voxel in the neighborhood because each relates differently to the central
voxel.
adtt CCCC −+=+1
69
Figure 3.2. A two dimensional representation of the wind directions in (a) the horizontal plane and (b) the vertical plane
Transition Rules for Modeling the Dispersal of Pollutants
Advective transport, due to the force of the wind action, causes the pollutants to move
from one voxel to another. Thus, for any given voxel at any given time-step, the pollutant
molecules are entering and exiting through its boundaries as a result of advection. While
conserving the mass of the pollutant particles, equation (3.6) is used to model the flux
of pollutants into and out of the central voxel due to advective transport as follows:
))((26
1
26
1∑∑=
=
=
=
−=i
i
j
jjjjia CvCvkC
(3.9)
where ki is the porosity constant representing the exposed surface area between the two voxels; C is the concentration of the central voxel; Cj and vj respectively represent the magnitude of the pollutant concentration of a voxel and velocity in a given direction j. Thus, vjCj represents the influx into the central voxel while vjC models the outflux from the central voxel.
aC
in flux out flux
70
Diffusion occurs when there are differences in pollutant concentrations between any two
neighboring voxels. This concentration gradient will result in the flow of the pollutants
from the voxel with a higher concentration to one that has a lower concentration. The
function in equation (3.7) describing a sigmoid curve that relates the pollutants lost or
gained from a central voxel due to the differences in the concentration with any one
neighboring voxel can be rewritten as:
∆
∆−=
j
jd d
CrC
(3.10)
where Cd is the pollutants concentration lost or gained due to diffusion, r is the appropriate diffusion coefficient, ∆Cj is the difference in the concentration between the voxels, ∆dj constant representing the exposed surface area of neighboring voxel, j, to the central voxel.
The time-stepping scheme that is proposed to simulate advection and diffusion is
monotonic in character and conserves the mass of the particles during the propagation.
While advanced methods of numerical analysis typically apply upwinding methods to the
diffusion and propagation processes, they are not considered here for simplicity reasons.
3.5.3. Model Implementation
The model presented in this study was developed in the MATLAB software (MathWorks,
2012) using an object-oriented design paradigm. The use of classes and objects ensured
accurate tractability and reliable integration of the model components.
3.6. Simulation Results and Discussion
The developed model operated on a small and abstract landscape of 40 x 40 x 40 voxels.
This strategy provided computational processing advantages and also allowed for a
simplified model design and a less rigorous parameter calibration and estimation
71
procedure. The model was conceptualized as a five state automaton. The pollutant
particles are, firstly, categorized into three distinct states to mark the level of pollutant
concentration in them. Setting the maximum concentration in a voxel at 1 and the
minimum to be just above zero the Low state (L, marked with light gray) ranges from
0.001 to 0.3, the Medium state (M, marked with a darker gray) ranges from 0.3 to 0.6 and
the High state (H, marked with black) ranges from 0.6 to 1. The fourth state of the
automaton, which is consistently marked as empty space (translucent) in the model, is
used to represent the space that contains none or negligible pollutant concentration
values. Lastly, the fifth state is one that represents the physical characteristics of the
landscape and is itself consistently marked with a brown color. In addition, the values of
1, , are used as the estimates for the porosity constants for the voxels
membranes that are between face, diagonal and double diagonal neighbors respectively.
The manual adjustment of the parameter values allowed for the checking of the model’s
software code to ensure that the automaton is logically consistent with the mathematical
descriptions underpinning it. In particular, 0.009 and 4.5 are used as the reasonable
parameter values for the diffusion coefficient and the westerly wind speed respectively.
21
31
72
Figure 3.3. Voxel automata simulation of the propagation of pollutants from a single non-continuous point source in 3D space over time
73
Figure 3.4. Measuring pollutant concentration values:(a) the positions, located at 10, 20, 24 units from the point source, at which voxel values are recorded; (b) the line graph of the concentration values as recorded at the three positions for the scenario depicting the propagation of pollutants from a single non- continuous point source
The simulation results in Figure 3.3 were generated to represent the progression of the
pollutant dispersion from a non-continuous point source as influenced by westerly winds.
At every time-step, the values of the pollutant concentration at 10, 20 and 24 voxel units,
downwind from the point source, are recorded. Figure 3.4a shows the locations of the
voxel positions for which the concentration values are recorded while Figure 3.4b depicts
the concentration at the three different locations. There is no change in concentration at
the location that is 24 voxel units from the point source because the pollutants dissipate
before the propagation reaches that location. From these results, it is clear that the higher
pollutant concentration values are closer to the point source and that at these locations the
values quickly rise and dissipate as the simulation runs. These conclusions match what
would be expected in reality where, as the distance from the point source increases, the
effects of dispersed pollutants are only felt minimally.
The results showing the propagation of pollutants were generated on a level landscape
but in this scenario the pollutants are emanating from a continuous point source (Figure
3.5). The graph of the concentration values of the pollutants at 10, 20, 24 voxel units
74
from the point source is presented in Figure 3.6. In comparison to the results in Figure
3.4(b), the concentration values in Figure 3.6 increase steadily up to a maximum value.
The locations closer to the point source have higher concentration values than those
further away. After making modifications in the study area to create an uneven landscape,
simulation results present the variation in concentration values for both non-continuous
and continuous point sources.
75
Figure 3.5. Voxel automata simulation of the propagation of particles from a single continuous point source over 3D space and time
76
Figure 3.6. Concentration values measured at 10, 20, and 24 voxel units downwind from the point source for the scenario depicting the simulation from a single continuous point source
Figure 3.7(a) presents the results of the model’s simulation of pollutant dispersion from a
non-continuous point source. The landscape has uniformly rising elevations on the end
that is downwind from the source of the pollutants. The pollutants propagate following a
progression similar to that shown in Figure 3.3. In this case, however, the particles take
longer to dissipate in the voxels that are right at the base of the elevating ground.
Similarly, in the scenario (Figure 3.7(b)) that shows the propagation of particles from a
continuous point source, the pollutants accumulate on the sides of the rise as the force of
the wind pushes them upwards. The concentration values at 10, 20, 24 voxel units from
the point source are shown in Figure 3.8. The line plots in Figure 3.8(a) correspond to the
values recorded from the scenario where the pollutants emanate from a non-continuous
point source scenario (Figure 3.7(a)) while those plotted in Figure 3.8(b) correspond to
the simulation of pollutants from a continuous point source.
77
Figure 3.7. Voxel automata simulation of the propagation of particles in an uneven landscape from (a) a single non-continuous point source (b) a continuous point source
78
Figure 3.8. Graphical representations of the concentration values measured at 10, 20, and 24 voxel units for (a) a non-continuous point source and (b) a continuous point source scenario
Furthermore, conditions that represent a non-homogenous wind field in the study area
were designed by adding stochasticity in the values assigned to the wind speed in the
study area. The semblance to randomly fluctuating wind speeds was implemented by
assuming that the wind fluctuates in the study area at every time-step but is consistent in
all the voxels at that time-step. The heterogeneity in the wind field is achieved by
assigning a velocity magnitude in the range between 4.0 and 5.0 at every time-step
throughout the course of the simulation. The results from the simulations in which the
wind speed changes at every time-step but is uniform throughout the landscape are shown
in Figure 3.9. For these simulations, the measurement values at select distances from the
source are plotted in the graph shown in Figure 3.10.
79
Figure 3.9. Voxel automata simulation of the propagation of pollutants from (a) a single continuous point source under randomly fluctuating westerly wind speed magnitude and (b) multiple continuous point sources under randomly fluctuating northerly, southerly and westerly wind speed magnitudes
80
Figure 3.10. Graphical representations of the concentration values measured at 10, 20, and 24 voxel units for the scenario with random fluctuations of the wind speed magnitudes
The results indicate that there are fluctuations in the values at the measured points but are
not significantly different from the scenarios that use constant wind speed. As in the other
cases the values rise and plateau around a constant value. The simulation scenarios are
presented to depict the rise of pollutants from a source on the ground, for example, a
factory, but the point source of the pollutants can be located anywhere in the three-
dimensional space.
The obtained results demonstrate that although only the fundamental functions are used
to model the process of advection and particle diffusion, the prototype simulations mimic
the process of pollutant propagation. For example, the plots in Figure 3.8a and Figure
3.8b are similar to those reported elsewhere for analytical models of pollutant dispersion
(Tirabassi, 1989; Costa, et al., 2012) and this comparison represents a degree of internal
model validation.
Furthermore, the model was applied to a more heterogeneous landscape representing a
hypothetical urban setting. The simulation outcomes from this scenario are shown in
Figure 3.11. As the pollutants propagate eastwards from a raised continuous point source,
they envelop objects such as buildings while at the same time the concentration around
81
the point source increases steadily. This behavior is consistent with expectations from
real world situations.
Figure 3.11. Pollutant propagation from a continuous point source in a non-homogenous landscape that represents a hypothetical urban setting
3.7. Conclusions
The use of the voxel-based automata approach in modeling geographic processes
provided a means where these processes can be presented and analyzed in a four-
dimensional spatiotemporal perspective. Furthermore, the voxel automata draw on the
elements of geo-atom data model to record, for historical analysis, the changes that occur
in each of the voxels in the system. This is particularly advantageous when modeling
complex systems because it allows for further investigation of the attributes of the
system’s components. This is unlike the models that use raster data with which it is not
possible to record the attribute values of the systems components at every time-step. As it
is more generally the case, the raster and the vector spatial representations describe the
objects in terms of their respective singular location and attribute values without the
ability to explicitly record the temporal context. The model presented in this study can
record the value representing the state of each of voxel as the simulation runs.
The developed model has been used to generate results for various scenarios in 4D, all of
which are grounded in hypothetical landscapes with the main objective being to
demonstrate the potential of using the geo-automata theory in modeling environmental
processes and with real three-dimensional datasets. The results indicate that the model is
82
sensitive to the values of the constants used in defining the underlying functions, thus it
would be useful to apply these results to an actual geographic landscape for the purpose
of analyzing its potential to give reliable results. Furthermore, the addition of stochastic
elements to the wind magnitudes in the model also influences the shape of the resulting
plume although it is expected not to be adversely different from one simulation to the
next.
The next steps in further developing this model would be to undertake a thorough and
rigorous model testing, along with a validation procedure by using real geospatial
datasets. Given the broad assumptions taken in this study, and the fact that the data used
were not taken from an actual field sample, it is not yet possible to adequately validate
the model. The model has been internally verified for programming errors and logical
inconsistencies and calibrated to check if it represents the process of 4D dispersion of
pollution with a reasonable level of realism, but there is a need to validate it against real
data to make sure that its outputs match the real world. With the goal of showing that the
model correctly reproduces a process in the real world, data from similar modeling
studies of three-dimensional pollutant spread could be used and the corresponding
models and results could be compared to the results generated from this study. It would
be important to use a finer grained voxel resolution and to carefully consider the effects
of turbulence and crosswind interaction within the study site. However, the model has
been applied to various scenarios in order to verify its logic and the reliability of its
outputs with respect to the input parameters. The efforts to prepare the GIS data that will
be used in the model for model testing is a topic of a new research project.
Although not yet explored in this study, the proposed voxel automata model is a good
candidate for examining the dynamics of the voxel-states in order to describe pollutant
concentrations in 4D around the raised objects in a non-homogenous landscape.
However, the simplifications held in this study, including the consideration of vertical
transport due to convection, must be addressed as part of those efforts. Furthermore, work
needs to be done to translate geographic data from the raster and vector GIS data formats
to the format specified in the geo-atom theory. This change in data formats would enable
83
the application of 4D models to be linked to real datasets of environmental processes.
The work presented in this study complements the early efforts in the development of
spatial dynamics of voxel and object based representation in 3D. Its unique contribution
in is the implementation of the theoretical concepts related to the 4D geo-atom data
structure linked to geographical automata systems. Nevertheless, this developed model
provides a proof-of-concept and platform to extend theoretically and practically the use
of voxel automata in the analysis of more complex scenarios.
References
Alonso-Sanz, R., and Martín, M. (2003). Cellular automata with accumulative memory: legal rules starting from a single site seed. International Journal of Modern Physics C, 14 (05), 695-719.
Andretta, M., Serra, R., and Villani, M. (2006). A new model for polluted soil risk assessment. Computer Simulation of Naturel Phenomena for Hazard Assessment, 32 (7), 890-896.
Arata, H., Takai, Y., Takai, N. K., and Yamamoto, T. (1999). Free-form shape modeling by 3D cellular automata. In (pp. 242-247).
Auyang, S. Y. (1998). Foundations of complex-system theories: in economics, evolutionary biology, and statistical physics. Cambridge University Press, Cambridge, U.K. ; New York.
Avolio, M., Errera, A., Lupiano, V., Mazzanti, P., and Gregorio, S. (2010). Development and calibration of a preliminary cellular automata model for snow avalanches. In S. Bandini, S. Manzoni, H. Umeo and G. Vizzari (Eds.), Cellular Automata (Vol. 6350, pp. 83-94). Springer Berlin Heidelberg.
Baas, A. C. W. (2007). Complex systems in aeolian geomorphology. Geomorphology, 91 (3–4), 311-331.
Bailey, T. C. (1990). GIS and simple systems for visual, interactive, spatial analysis. The Cartographic Journal, 27 (2), 79-84.
Batty, M. (2005a). Agents, cells, and cities: New representational models for simulating multiscale urban dynamics. Environ.Plann.A, 37 (8), 1373-1394.
84
Batty, M. (2005b). Cities and Complexity: Understanding Cities with Cellular Automata, Agent-Based Models, and Fractals. MIT Press, Cambridge, Massachusetts.
Batty, M., Xie, Y., and Sun, Z. (1999). Modeling urban dynamics through GIS-based cellular automata. Computers, Environment and Urban Systems, 23 (3), 205-233.
Bays, C. (1987). Candidates for the game of life in three dimensions. Complex Systems, 1, 373-400.
Benenson, I., and Torrens, P. M. (2004). Geosimulation: Automata-based modelling of urban phenomena. John Wiley & Sons, Hoboken, NJ.
Bergström, S. (1995). The HBV model. In V. P. Singh (Ed.), Computer models of watershed hydrology (pp. 443-476). Water Resources Publications, Littleton, CO.
Bertelle, C., Duchamp, G. H. E., and Kadri-Dahmani, H. (2009). Complex systems and self-organization modelling. Springer, Berlin.
Britter, R. E. (1979). The spread of a negatively buoyant plume in a calm environment. Atmospheric Environment (1967), 13 (9), 1241-1247.
Burrough, P. A., Karssenberg, D., and Van Deursen, W. P. A. (2005). Environmental modelling with PCRaster. In D. J. Maguire, M. F. Goodchild and M. Batty (Eds.), GIS, Spatial Analysis and Modeling, Redlands, California.
Camazine, S., Jean-Louis, D., Nigel, R. F., james, S., Guy, T., and Eric, B. (2001). Self-organization in biological systems. Princeton University Press, Princeton, N.J.
Chrisman, N. R. (1997). Exploring geographic information systems. John Wiley & Sons, New York.
Ciolli, M., Rea, R., Zardi, D., De Franceschi, M., Vitti, A., and Zatelli, P. (2004). Development and application of 2D and 3D GRASS modules for simulation of thermally driven slope winds. Transactions in GIS, 8 (2), 191-210.
Clarke, K. C., and Gaydos, L. J. (1998). Loose-coupling a cellular automaton model and GIS: long-term urban growth prediction for San Francisco and Washington/Baltimore. International Journal of Geographical Information Systems, 12 (7), 699-714.
Costa, C. P., Tirabassi, T., Vilhena, M. T. M., and Moreira, D. M. (2012). A general formulation for pollutant dispersion in the atmosphere. Journal of Engineering Mathematics, 74 (1), 159-173.
85
Couclelis, H. (1985). Cellular worlds: a framework for modeling micro-macro dynamics. Environment and Planning A, 17 (5), 585-596.
Degrazia, G., Gledson, A., Goulart, O., and Roberti, D. (2009). Turbulence and dispersion of contaminants in the planetary boundary layer. In D. M. Moreira and M. Vilhena (Eds.), Air Pollution and Turbulence (pp. 33-68). CRC Press, Boca Raton, Florida.
Dragicevic, S., and Marceau, D. J. (2000). A fuzzy set approach for modelling time in GIS. International Journal of Geographical Information Science, 14 (3), 225-245.
Epstein, J. M., and Axtell, R. (1996). Growing artificial societies: social science from the bottom up. Brookings Institution Press, Washington, D.C.
Fischer, T., Burry, M., and Frazer, J. (2005). Triangulation of generative form for parametric design and rapid prototyping. Automation in Construction, 14 (2), 233-240.
Fonseca, P., Colls, M., and Casanovas, J. (2011). A novel model to predict a slab avalanche configuration using m:n-CAk cellular automata. Computers, Environment and Urban Systems, 35 (1), 12-24.
Frisch, U., Hasslacher, B., and Pomeau, Y. (1986). Lattice-gas automata for the Navier-Stokes equation. Physical Review Letters, 56 (14), 1505-1508.
Goodchild, M. F. (1992). Geographical data modeling. Computers and Geosciences, 18 (4), 401-408.
Goodchild, M. F., Yuan, M., and Cova, T. J. (2007). Towards a general theory of geographic representation in GIS. International Journal of Geographical Information Science, 21 (3), 239-260.
Grenon, P., and Smith, B. (2004). SNAP and SPAN: towards dynamic spatial ontology. Spatial Cognition and Computation, 4 (1), 69-104.
Grunwald, S., and Barak, P. (2003). 3D geographic reconstruction and visualization techniques applied to land resource management. Transactions in GIS, 7 (2), 231-241.
Heung, B., Bakker, L., Schmidt, M. G., and Dragićević, S. (2013). Modelling the dynamics of soil redistribution induced by sheet erosion using the Universal Soil Loss Equation and cellular automata. Geoderma, 202–203 (0), 112-125.
86
Holland, J. H. (1995). Hidden Order: How Adaptation Builds Complexity. Addison-Wesley, Reading, Mass.
Holland, J. H. (2006). Studying complex adaptive systems. Journal of Systems Science and Complexity, 19 (1), 1-8.
Ilachinski, A. (2001). Cellular Automata: A Discrete Universe. World Scientific, Singapore; River Edge, NJ.
Imai, K., Hori, T., and Morita, K. (2002). Self-reproduction in three-dimensional reversible cellular space. Artificial Life, 8 (2), 155-174.
Jimenez-Morales, F. (2000). The evolution of 3-d CA to perform a collective behavior task. In Evolvable Systems: From Biology to Hardware, Proceedings (pp. 90-102).
Kamphorst, E. C., Chadœuf, J., Jetten, V., and Guérif, J. (2005). Generating 3D soil surfaces from 2D height measurements to determine depression storage. CATENA, 62 (2–3), 189-205.
Kansal, A. R., Torquato, S., Harsh Iv, G. R., Chiocca, E. A., and Deisboeck, T. S. (2000). Simulated brain tumor growth dynamics using a three-dimensional cellular automaton. Journal of Theoretical Biology, 203 (4), 367-382.
Karssenberg, D., and De Jong, K. (2005). Dynamic environmental modelling in GIS: 1. Modelling in three spatial dimensions. International Journal of Geographical Information Science, 19 (5), 559-579.
Karssenberg, D., Schmitz, O., Salamon, P., de Jong, K., and Bierkens, M. F. P. (2010). A software framework for construction of process-based stochastic spatio-temporal models and data assimilation. Environmental Modelling & Software, 25 (4), 489-502.
Langran, G. (1992). Time in geographic information systems. Taylor and Francis, New York.
Lansing, J. S. (2003). Complex adaptive systems. Annual Review of Anthropology, 32, 183-204.
Ledoux, H., and Gold, C. M. (2008). Modelling three-dimensional geoscientific fields with the Voronoi diagram and its dual. International Journal of Geographical Information Science, 22 (4-5), 547-574.
87
Leonardi, C. R., Owen, D. R. J., and Feng, Y. T. (2012). Simulation of fines migration using a non-Newtonian lattice Boltzmann-discrete element model Part II: 3D extension and applications. Engineering Computations, 29 (3-4), 392-418.
Lo, C. P., and Yeung, A. K. W. (2007). Concepts and techniques of geographic information systems (Vol. 2nd). Pearson Prentice Hall, Upper Saddle River, NJ.
Logan, J. D. (2001). Reaction–Advection–Dispersion Equation. In Transport Modeling in Hydrogeochemical Systems. Springer, New York.
Luebeck, E. G., and deGunst, M. C. M. (2001). A stereological method for the analysis of cellular lesions in tissue sections using three-dimensional cellular automata. Mathematical and Computer Modelling, 33 (12-13), 1387-1400.
Macal, C. M., and North, M. J. (2005). Tutorial on Agent-Based Modeling and Simulation. In M. E. Kuhl, N. M. Steiger, F. B. Armstrong and J. A. Joines (Eds.), Proceedings of the 2005 Winter Simulation Conference.
Manson, S. M. (2007). Challenges in evaluating models of geographic complexity. Environment and Planning B: Planning and Design, 34 (2), 245-260.
Manson, S. M., and O'Sullivan, D. (2006). Complexity theory in the study of space and place. Environment and Planning A, 38 (4), 677-692.
Marschallinger, R. (1996). A voxel visualization and analysis system based on AutoCAD. Computers & Geosciences, 22 (4), 379-386.
MathWorks. (2012). MATLAB and Statistics Toolbox. In (Release 2012b ed.). The MathWorks, Inc., Natick, Massachusetts, United States.
Messina, J. P., Evans, T. P., Manson, S. M., Shortridge, A. M., Deadman, P. J., and Verburg, P. H. (2008). Complex systems models and the management of error and uncertainty. Journal of Land Use Science, 3 (1), 11-25.
Miller, E. J. (1997). Towards a 4D GIS: Four-dimensional interpolation utilizing kriging. In Z. Kemp (Ed.), Innovations In GIS. CRC Press.
Miller, J. H., and Page, S. E. (2007). Complex adaptive systems : an introduction to computational models of social life. Princeton University Press, Princeton, New Jersey.
Narteau, C., Zhang, D., Rozier, O., and Claudin, P. (2009). Setting the length and time scales of a cellular automaton dune model from the analysis of superimposed bed
88
forms. Journal of Geophysical Research - Part F - Earth Surfaces, 114 (F3), F03006-(03018 pp.).
Perry, G. L. W., and Enright, N. J. (2006). Spatial modelling of vegetation change in dynamic landscapes: a review of methods and applications. Progress in Physical Geography, 30 (1), 47-72.
Peuquet, D. J. (1994). It's About Time: a conceptual framework for the representation of temporal dynamics in geographic information systems. Annals of the Association of American Geographers, 84 (3), 441-461.
Peuquet, D. J. (2001). Making space for time: Issues in space-time data representation. GeoInformatica, 5 (1), 11-32.
Pultar, E., Cova, T. J., Yuan, M., and Goodchild, M. F. (2010). EDGIS: a dynamic GIS based on space time points. International Journal of Geographical Information Science, 24 (3), 329-346.
Raper, J., and Livingstone, D. (1995). Development of a geomorphological spatial model using object-oriented design. International Journal of Geographical Information Systems, 9 (4), 359-383.
Rasmussen, R., and Hamilton, G. (2012). An approximate bayesian computation approach for estimating parameters of complex environmental processes in a cellular automata. Environmental Modelling and Software, 29 (1), 1-10.
Schwarz, N., Haase, D., and Seppelt, R. (2010). Omnipresent sprawl? A review of urban simulation models with respect to urban shrinkage. Environment and Planning B: Planning and Design, 37 (2), 265-283.
Tang, W., and Bennett, D. A. (2010). Agent-based Modeling of Animal Movement: A Review. Geography Compass, 4 (7), 682-700.
Tirabassi, T. (1989). Analytical air pollution advection and diffusion models. Water, Air, and Soil Pollution, 47 (1-2), 19-24.
Torrens, P. M. (2014). High-resolution space–time processes for agents at the built–human interface of urban earthquakes. International Journal of Geographical Information Science, 28 (5), 964-986.
Trambauer, P., Maskey, S., Werner, M., Pappenberger, F., Van Beek, L. P. H., and Uhlenbrook, S. (2014). Identification and simulation of space-time variability of past hydrological drought events in the Limpopo river basin, Southern Africa. Hydrology and Earth System Sciences Discussions, 11 (3), 2639-2677.
89
Vanderhoef, M., and Frenkel, D. (1991). Tagged Particle Diffusion in 3d Lattice Gas Cellular Automata. Physica D, 47 (1-2), 191-197.
von Neumann, J. (1966). Theory of Self-Reproducing Automata. University of Illinois Press, Urbana.
Wachowicz, M., and Healey, R. G. (1994). Towards temporality in GIS. Innovations in GIS 1, 105-115.
Wark, K., Warner, C. F., and Davis, W. T. (1997). Air Pollution: Its Origin and Control (Vol. 3rd Edition). Prentice Hall, New Jersey.
Weimar, J. R. (2002). Three-dimensional cellular automata for reaction-diffusion systems. Fundamenta Informaticae, 52 (1-3), 277-284.
Wesselung, C. G., Karssenberg, D., Burrough, P. A., and Van Deursen, W. P. A. (1996). Integrating dynamic environmental models in GIS: The development of a Dynamic Modelling language. Transactions in GIS, 1 (1), 40-48.
Wolfram, S. (1983). Statistical mechanics of cellular automata. Reviews of Modern Physics, 55 (3), 601-644.
Worboys, M., and Duckham, M. (2004). GIS: a computing perspective (Vol. 2nd). CRC Press, Boca Raton, Fla.
Yeh, G.-T., and Huang, C.-H. (1975). Three-dimensional air pollutant modeling in the lower atmosphere. Boundary-Layer Meteorology, 9 (4), 381-390.
Zang, T. A. (1991). On the rotation and skew-symmetric forms for incompressible flow
simulations. Applied Numerical Mathematics, 7 (1), 27-40.
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Chapter 4. Spatial Indices for Measuring Three-Dimensional Patterns in Voxel-Based Space
4.1. Abstract
Spatial indices are used to quantitatively describe the spatial arrangements of the features
within a study region. However, most of the indices used are two-dimensional in their
representation of the surface characteristics and this is insufficient to quantify the three-
dimensional topographic properties of an area. With the increased availability of 3D data
from laser scanning and other collection methods, a voxel-based representation of space
is an important methodology that allows for an intuitive visualization of geospatial
features and their analysis with 3D GIS techniques. The objective of this study is to
conceptualise, develop and implement indices that can characterise three-dimensional
space and can be used to analyse the structure of spatial features in a landscape. The
indices for three-dimensional space that are implemented are, namely, surface area
volume, fractal dimension, lacunarity, and Moran's I which are all useful in the
quantification of spatial organisation found in ecological landscapes. In addition to
providing the quantitative descriptors, the results indicate that a voxel-based
representation provides a straightforward means of characterizing the form and
composition of the spatial features using 3D indices.
4.2. Introduction
The general effects of dynamic spatial processes arise from the multifaceted interactions
in a landscape are imprinted as spatial patterns in its topography. In order to quantify and
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understand the spatial dynamics in terrestrial and marine spaces, the spatial patterns are
assessed and evaluated using landscape measures (O'Neill, et al., 1988; Turner, 1990b;
Cushman, et al., 2008). In the research literature, the landscape measures are sometimes
referred to as landscape indices or metrics. For consistency in this study, a more general
term 'spatial indices' is used as some descriptors of landscape character are only
applicable to smaller geographic spaces that don’t quite fit the inference of a landscape
scale. In brief, spatial indices are quantitative descriptors used to characterise the
compositional and spatial aspects of a geographic space using relevant data about the
phenomenon under study (Gustafson, 1998). Although they are important in the
interpretation of spatial patterns, it has been often emphasised that their usefulness lies in
the broader linkage they provide between the evolution of the environmental processes
and the resultant structure of the geographic space (O'Neill, et al., 1997; Tischendorf,
2001; Kupfer, 2012).
The desire to understand the relationships between geographic patterns and their
connections with spatial functions and processes makes the measurement of geographic
composition and spatial configuration an important area in ecological studies (Hermy,
2012) and geographical research. These indices have been developed and applied to
various aspects of spatial composition, heterogeneity and pattern complexity at the
landscape level of scale (Turner, 1990a; Fuller, 2001; Stubblefield, et al., 2006).
However, it has been pointed out that application of the indices has inherent limitations
that are often unacknowledged and consequently become misused when using them to
perform spatial index analyses (Li and Wu, 2004).
One of the challenges related to the misapplication of the spatial indices is in the methods
used to characterise the spatial objects and the response variables that drive the geospatial
processes. For the most part, the indices have taken a planimetric perspective of space
while ignoring the vertical or the third dimension. In addition, the widely used patch
mosaic approach (Urban, et al., 1987) is an object-based view of space that discretizes
space regardless of the level of continuity within natural environment of the
phenomenon. This simplification in the representation of a landscape's structure has been
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pointed out and specific efforts have been made to quantify some aspects of topographic
variability. Stupariu et al. (2010) analysed landscape relief by using a triangular irregular
network to approximate the terrain; Jenness (2004) developed a methodology that
approximates a three dimensional surface area using DEM grids; and McGarigal and
Cushman (2005) as well as McGarigal et al. (2009) have contributed three-dimensional
indices developed on the basis that a landscape is a continuous and heterogeneous surface
that can be suitably described by using the varying intensity of a gradient surface. In
another study, the structure of the landscape was reconstructed from underwater
photogrammetry following which the three-dimensional surface area of the marine
environment was calculated (Courtney, et al., 2007). However, all these efforts have been
applied to raster datasets, which are inherently two-dimensional (2D) in their
conceptualization of space (Batista, et al., 2012). But newer data collection methods such
as multiple return Airborne Light Detection and Ranging (LiDAR) generate 3D point
cloud data (Sithole and Vosselman, 2004) that can be voxelised to model landscape
topography, anthropogenic artifacts as well as vegetation structure (Schilling, et al.,
2012). Moreover, voxels are computationally straightforward to implement and they have
an advantage of providing an intuitive 3D spatial awareness. In addition, the multitude of
sources and the frequency with which 3D data of the earth's surface are collected requires
researchers to move beyond the study of landscapes as planar, static and stable two-
dimensional surfaces over which processes evolve. For a more realistic representation,
landscapes must be viewed as dynamic, multilayered three-dimensional fabrics with
vegetation canopy, plant cover, geological forms and anthropogenic features that change
overtime (Mitasova, et al., 2012). Voxels lend themselves to representation and analysis
of 3D geographic space. Voxel data can be generated from LiDAR and photogrammetry
data, and where needed, 3D point data from other sensors can be interpolated to generate
3D continuous fields akin to those generated by the two-dimensional raster data model.
The main objective of this study is to conceptualize, develop and implement a variety of
indices that can be used to describe the 3D spatial patterns within a landscape using a
voxel-based representation of space. Discretization of space into 3D voxel units, which
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can be data-typed with a multidimensional geospatial data model, provides an effective
means whereby 3D spatial patterns can be quantified within the larger context of the
other aspects that may characterize the topography of the area. The voxel-based surface
characterization provides an alternative method for representing 3D objects and
landscape patterns and it allows for a computationally straightforward extraction of 3D
surface structures that are more realistic and meaningful in spatial analysis.
4.3. An Overview of Spatial Indices for 2D Space
Geographic areas contain complex spatial patterns that are outcomes of environmental
processes as well as the complex interactions of the biotic entities within them. The
patterns are conceptualized as mosaics of discrete patches and the objective of spatial
pattern analysis is to quantify the configuration, shape and composition of these patterns
(Batista, et al., 2012). Ecologists have generally adopted this patch mosaic model of
landscape representation as the standard paradigm for quantifying landscape structure
(Kupfer, 2012) and as a result a variety of software packages have been developed to
generate these indices using predominantly raster data. For example, SPAN, a raster-
based spatial analysis program for landscape structure was developed by Turner (1990b);
the r.le software program is coupled with the GRASS GIS system (Baker and Cai, 1992);
and the widely used FRAGSTATS application (McGarigal and Marks, 1995). ArcGIS
software by ESRI (ESRI, 2014) and the R-package, SDMTools (VanDerWal, et al.,
2014), are also equipped to provide indices that can characterise the structure of a
landscape.
In the patch mosaic model, an area is viewed as a complex heterogeneous collection of
contiguous areas in which processes of interest operate. The representative indices can be
defined at different levels of scale. Patch-level indices are used for individual patches to
characterize the spatial attributes and context of those specific patches (Magura, et al.,
2001); class level indices are integrated over patches of a given type or class (Neel, et al.,
2004) and landscape level patches are integrated over all the classes across the entire
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study area (Turner, 1990a). Depending on the type of data collected and the objectives of
the analysis, the indices can be used to investigate patterns of spatial points, linear
networks, continuous surfaces, and thematic maps (Forman, 1995). It is within the
context of the wide-spread use of raster-based representation and thematic mapping that
the patch mosaic paradigm has developed and become popular.
The indices that have been developed based on raster thematic maps can be grouped into
three general categories that quantify composition (Knutson, et al., 1999), connectivity
(Bélisle, 2005), and spatial configuration or structure of the map (Riitters, et al., 1995).
The composition of a map refers to the relative abundance and heterogeneity in the types
of patches within the landscape, the connectivity quantifies how patches are linked, and
spatial configuration refers to the arrangement, size, position and orientation of a given
class of patches or all for the patches within the landscape. Further indices that make
reference to particular ecological process are referred to as functional indices while those
that limit themselves to the physical composition or configuration of the patch are
referred to as structural indices (Noss, 1990). Functional indices are particularly
important because they explicitly measure landscape patterns in a manner that is relevant
to an underlying process under study.
Although there are incidences of a non-interactive relationship between pattern and
processes as well as cases in which the pattern and process operate at different spatial and
temporal scales (Li and Wu, 2004), the reciprocal linkage between them is an important
principle and objective in modeling and analysis (Turner, 1989). It is generally
emphasized that the exploration of the relationship between pattern and process gives
meaning to heterogeneity in the landscape, contextualizes the interactions between the
landscape elements, and assists in making inferences as to how the process dynamics
operate across spatial and temporal scales. Indeed, great strides have been achieved with
respect to the development of landscape indices as it is evident by the large volume of
research literature that lists their importance and accompanying limitations (Hargis, et al.,
1998; Neel, et al., 2004). However, these research and analytical efforts have been mostly
limited to the planimetric perspective or 2D representation of geographic space. In order
95
to address this limitation, specific indices were reported in the research literature to
quantify the 3D nature of the landscape. Using digital elevation models, algorithms that
determine 3D patch surface area and perimeter as well as other topographic aspects have
been developed (Jenness, 2004; Hoechstetter, et al., 2006; Stupariu, et al., 2010).
Similarly, the 3D surface area of a regional landscape has been estimated by overlaying
thematically mapped data onto a terrain surface defined by a triangulated irregular
network (Rogers, et al., 2012). In addition, a proposition for the 3D indices for spatial
patterns has been made on the basis of the “landscape gradient” concept (McGarigal and
Cushman, 2005; McGarigal, et al., 2009). With the gradient concept, the spatial data
represent a 3D surface where the measured value at any geographic location is treated as
the height of the surface. The spatial indices that have been derived this way such as
patch density, percentage of landscape, largest patch index, edge density, nearest
neighbor index and fractal dimension index have their equivalents in the area of surface
metrology which is within the field of molecular physics (Villarrubia, 1997). Efforts to
develop three-dimensional indices with this methodology are still in their early stages.
However, it has been argued that they can be useful in landscape ecology in the same
manner as they have proved useful in other fields (Pike, 2001). Parrott (2005) has
contributed directly towards the measurement of spatiotemporal complexity where the
indices quantify the dynamics of planimetric space along the temporal dimension. The
indices developed include space–time density, number of 3D patches, distribution
frequency, 3D patch shape complexity, fractal dimension, contagion and spatiotemporal
complexity (Parrott, et al., 2008).
Most of the methodologies used to date by the researchers to develop 3D indices are
based on the existing raster and vector GIS data models to calculate the relevant indices.
These are advantageous and pragmatic approaches that directly link theory to GIS
practice as the developed indices can be applied directly to existing datasets in response
to the lack of 3D indices for spatial dynamics. However, in this is research, it is argued
that a voxel-based discretization of space allows for the explicit utilization of the third
spatial dimension which has particular relevance in process analysis and visualizations
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within disciplines such as hydroecology, climatology and meteorology. Furthermore, for
raster based data, which is the data model widely used in the management of remotely
sensed spatial data, each pixel can only represent one value corresponding to an attribute
on the ground such as elevation, vegetation or temperature. This limitation in the data
model makes it complicated and cumbersome to analyze issues such as how land use
patterns change with elevation over time. It also forces a conflation of landform with
landscape surface characteristics even though sometimes it may be beneficial to analyze
them separately.
Comparatively speaking, topographic surfaces extend widely in the horizontal directions
with smaller irregularities along the elevation axis (Pike, 2001). This observation may in
part provide justification for representing a 3D landscape using a simpler 2D perspective,
and the generation of the topographical aspects using the digital elevation model (DEM)
and triangular irregular network (TIN) methods. The level of detail represented by the
DEMs and TINs is determined mainly by the density of the source data and the latest data
collection methods provide the opportunity for acquiring highly accurate and detailed
landscape data. Particular advances in remote sensing as well as air and terrestrial LiDAR
scanning provide detailed data using dense and accurate measurements (Bishop, et al.,
2012). Thus, beyond this general surface representation, it is now possible to identify
highly accurate and detailed three-dimensional objects such as plants, tree canopies,
urban feature, geological structures or other entities which may be on the landscape using
these advanced methods of data collection (Blaschkea, et al., 2004; Dupuy, et al., 2013).
The recent advancements in voxelization of point cloud data (Hinks, et al., 2013) which
are collected using laser scanning methods, as well as the extraction of volumetric data
from airborne LiDAR imaging satellites (Weishampel, et al., 2000; Olsoy, et al., 2014)
provide the means by which these objects can be extracted and processed for 3D spatial
analysis.
The issues related to data conversion and processing to generate voxels are beyond the
scope of this study. However, the main focus is on the conceptualization, development
and implementation of 3D indices that can be used to quantify objects represented with
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voxel space within a given natural landscape. The physical objects under examination are
abstract representations of spatial entities or patterns in 3D that have resulted from
geospatial processes.
4.4. Methods
The 3D features in the landscape could be static elements such as vegetation cover,
geological outcrops and anthropogenic structures, or representations of dynamic
phenomena like forest fires, landslides and snow avalanches. A 3D object in a landscape
is modeled using a collection of voxels, referred to as a bloxel in this study. The term
bloxel is derived from an amalgamation of the words 'blob' (for its connotation for 3D
mass form) and 'voxel'. Multiple groups of voxels will be referred to as bloxels. The
procedure relies on a simplification of an already voxelised landscape into a binary
composition by selecting the voxels of interest and assigning them a value of 1 while the
rest are treated as a background with a value 0. It is assumed that contiguous voxels are
connected along the edges, at vertices or on adjoining faces such that the surface area of
the bloxel is defined by full faces only. The 3D indices conceptualised in this study to
describe landscape features are: the number of bloxels, bloxel surface area, expected
surface area, volume, fractal dimension, lucanarity, and 3D Moran's I and are described
as follows:
4.4.1. Number of Bloxels
The number of bloxels, which corresponds to the number of distinct entities or objects in
3D the landscape, is determined by identifying and counting the distinct bloxels
unconnected to any other voxels in the study area. The number of separate bloxels can
also be counted by class when the input data is simplified to a binary representation.
98
4.4.2. Bloxel Surface Area
The surface area of a bloxel is equivalent to the total area of the exposed faces of the
constituent voxels. Assuming that each of the six faces of a uniform voxel has a surface
area, a, then the voxel will have total surface area of 6a. Similarly, if two voxels are
connected to each other at one of the faces then the total surface area will be 10a.
However, if they are connected together at the edges or vertices then the surface area will
be 12a. Using formula (4.1) below the surface area S of a bloxel can be algorithmically
determined by considering how the voxels are face-connected to each other and can be
formulated as:
(4.1)
where N is the total number of voxels in the bloxel, a is the area of a voxel face, i is the number of face-connections, Ni is the number of voxels with i face-connections.
If there are no face connected neighbors in a bloxel then the surface area would
automatically be the number of voxel N scaled by a factor of 6. For example, in the case
of two voxels connected at one face each with area a, the surface area S = (2 x 6a) - (2 x
1 x a) = 10a.
4.4.3. Expected Surface Area
The bloxel can take on numerous configurations. Three-dimensional features can be
characterized by compactly arranged voxels, in other cases the arrangement could be that
of a surface facade where the actual 3D entities themselves are hollow on the inside. In
addition, the manner in which the voxels are arranged in a bloxel is related to the number
of exposed faces and hence the exposed surface area. Consider the surface area of a
bloxel composed of 27 voxels: if the voxels are arranged in a single file, the surface area
( )∑ =
=××−×=
6
1)6( i
i i aiNaNS
99
would be different from the case where they are arranged in a 3-voxel sided cubic form.
Thus, for a methodology that uses the exposed voxel faces to determine the surface area,
a useful indicator is the surface area that would be expected if the bloxel's voxels were
tightly compacted. A comparison between the expected surface area and the one that is
actually calculated for a given bloxel is a useful indicator of the form of the bloxel. Given
two bloxels, both with the same number of voxels, it would be expected that the one
whose voxels are more compactly arranged will have a smaller surface area. The most
compact arrangement for cubic voxels is achieved when they are arranged in a
framework that is as close as possible to a cubic bloxel.
The expected surface area is a comparative index that illustrates the expansiveness of the
three-dimensional object. In general, the expected surface area is less than or equal to the
actual surface area. However, the greater the difference between these two, the higher the
spread and the possible complexity of the object. Of particular interest is the observation
that those bloxels which are compactly arranged tend to have a high volume to surface
area ratio. This measure, when combined with the fractal dimension, is also indicative of
the complexity of the feature being represented.
4.4.4. Volume
Besides the surface area, another fundamental attribute of a bloxel is its volume which
indicates the amount of 3D space that it occupies. Spatial analyses of the volumetric
properties provide the descriptive indicators of the space occupied by biotic and abiotic
entities in the study area. The volume of a bloxel is determined by counting the
constituent number of voxels equivalent to the number of non-zero elements. However,
given that a bloxel can have empty spaces concealed on the inside, it is useful to compare
the calculated volume to surface area ratio with the similar ratio that would be expected if
the same number of voxels were tightly packed.
100
4.4.5. Fractal Dimension
The patterns and the forms of irregular three-dimensional objects are difficult to describe
using Euclidean measures like diameter or length but can be quantitatively described
using measures that infer the complexity of their pattern and form. A feature that shows
self-similarity in its shape or form can be characterized by its fractal dimension, which is
arrived at by splitting the object into self-similar parts each of which is a reduced-size
copy of the original (O'Neill, et al., 1988). As a 3D index, the fractal dimension
characterizes the structure of the shape of the bloxel and is calculated using the box-
counting technique. Conceptually, the box counting technique involves covering the
selected feature with grids of different voxel sizes and estimating how many voxels of the
different sizes can cover the bloxel (Liebovitch and Toth, 1989). To estimate a value for
the three-dimensional fractal dimension, the slope of the regression line that has the
natural logarithm of the inverse of the size of the voxels on the x-axis and the y-axis
representing the natural logarithm of the number of voxels of that size covering the bloxel
as shown in formula (4.2).
(4.2)
where fd is the 3D fractal dimension, N is the number of voxels of with side size is r
The value estimating the fractal dimension corresponds to the slope of the best fit line
through the points.
4.4.6. Lacunarity
Lacunarity analysis has been used to quantify the overall textural characteristics of a
landscape (Plotnick, et al., 1993). Because it provides insights into scale-dependent
variation in the dips and crests (or patches and gaps for 2D images) that may exist on a
)/1ln()ln(r
Nfd =
101
3D spatial object, it has been used in ecology to study the heterogeneity in the physical
characteristics of landscapes (Hoechstetter, et al., 2011). There are several algorithms for
measuring surface lacunarity of a set but the most commonly used method is the "gliding
box" that was introduced by Allain and Cloitre (1991) and exhaustively scans the entire
landscape with a kernel. The gliding box method involves moving a cube of size r
throughout the whole bloxel, one unit at a time, to determine the number of voxels that it
encloses at that location. The lacunarity, )(rΛ , of a bloxel can be computed using
formula (4.3) (Hanen, et al., 2009; Soares, et al., 2013) as follows:
2
2
),(
),()(
=Λ
∑
∑
M
M
rMMQ
rMQMr
(4.3)
where M is the number of voxels in a bloxel of size r and Q(M,r) is the mass distribution probability.
The lacunarity of a perfectly smooth surface is 1 and this value increases infinitely as the
roughness of the surfaces increases.
4.4.7. 3D Moran's I
The Moran’s index I was introduced as a two-dimensional index for spatial
autocorrelation in ecological landscapes (Moran, 1948; Anselin, 1988). A common
example in landscape studies is the spatial autocorrelation of elevations data where
similar elevations are in close proximity. The application of the Moran's I statistic has
been extended to the 3D perspective in medicine in order to understand the variation in
bone tissue density (Marwan, et al., 2007). In this study, the same formulation is adopted
and applied for 3D spatial analysis as shown in formula (4.4) below:
102
(4.4)
where I3D is the 3D Moran's I index; N, total number of voxel elements in the dataset; So, total number of contiguous pairs in the voxel dataset; ∂ij, weighted value of distance, 1 for contiguous neighbors and 0 otherwise; d1,d2,d3 respectively the length, width, and height of the voxel-tessellated study area; and xi, xj the value of the variable at voxel i and voxel j respectively.
The Moran's I value is analogous to the coefficient of correlation between two variables
where the mean of a variable is subtracted from each sample value in the numerator. In
this instance however, it is used to measure the correlation of a variable with itself in
space. The values of Moran's I range from -1 for a strong negative spatial autocorrelation,
through 0 for a random pattern (no correlation) to +1 for strong positive spatial
autocorrelation. Figure 4.1 shows various configurations and their corresponding 3D
Moran's I values.
Figure 4.1. Moran's I values for the different 3D voxel state configurations depicting (a) negative spatial autocorrelation for a perfectly uniform pattern (b) no spatial autocorrelation for a random pattern and (c) a strong positive correlation for a clustered pattern.
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The 3D Moran's I would find ready application in fine scale 3D spatial distribution
studies. For example, Varner and Dearing (2014) argued for using fine scale temperature
and elevation data to predict habitat suitability as they found interstitial temperatures at
particular elevations differed significantly from those measured at ambient temperatures.
In such cases, the 3D autocorrelation of the variables characterizing the microclimates
would be useful in understanding how some species locally adapt to the changes in the
macroscopic climatic changes. Another application is in the extraction of structural
information related to spatial dependence from 3D point cloud data by identifying
homogeneous regions using the actual voxel values in the dataset.
4.5. Implementation of the Spatial Indices for 3D Space
In order to demonstrate their usability, the indices proposed in this study were applied to
static and dynamic 3D features. Although environmental processes are dynamic in nature,
it is often sufficient to analyze and quantify 3D spatial patterns as they exist at a single
point in time. Similarly, the analysis of time series data can be important in order to
understand spatiotemporal complexity of a phenomenon. Although the measures for the
complexity of dynamic phenomena can be categorized according to their spatial,
temporal and structural functions, this study focuses on the static representation of
objects because the temporal component is not an integral component of the indices
formulation; however the indices outcomes can be analyzed as time series. The proposed
3D indices were implemented in the MATLAB software application (MathWorks, 2012)
and several built-in MATLAB functions were programmed into the functions to compute
the 3D indices proposed in this study.
4.5.1. Hypothetical Results
Number of Bloxels
The number of distinct bloxels in the study area is determined using MATLAB's
bwconncomp function, which calculates the number of connected components in a binary
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array of multiple dimensions. The function, when applied to the 3D binary bloxel
representing the spatial objects, returns the number of connected objects. The number of
connected features is equivalent to the number of distinct bloxels in the dataset. For
example, given a dataset with N voxels that have the same attribute value, the
bwconncomp function automatically computes the distinct bloxels and provides the total
count as an output.
Bloxel Surface Area
Providing that the faces of the voxels are equivalent, the surface area of a 3D object is
directly related to the number of exposed faces it has. For cubic voxels, the sum of the
areas of the exposed faces that connect to each other equals the total surface area of the
bloxel. The number of connected faces were determined using a 3D kernel in the shape of
a von-Neumann neighborhood (six voxels, without the central voxel) applied to the 3D
voxel data using MATLAB's convn function. The convn function computes a convolution
of two matrices, which, in this case, are a matrix that represents the spatial data and
another for the von Neumann kernel. Convolution is used in image analysis to emphasize
specific values in the image while applying the opposite effect on the other values. In this
instance the convolution function is used to identify the face connected neighbors for
each voxel.
The formulation operates by first assuming that the faces of all the voxels in the bloxel
are uniform and exposed because this would be the case if the voxels are connected at the
vertices or along the edges. Next, the voxels connected at the faces are identified and the
face connections are subtracted in order to arrive at the actual number of exposed faces.
For example, a bloxel of three face-connected voxels (Figure 4.2) would have 2 voxels
with singular face-connections and one voxel with two face connections. Thus, using the
equation (4.1) and assuming each face has a unit square area, the surface area, S, would
be calculated as follows:
S = (3x6) – [(2x1) + (1x2)] = 14 square units
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Figure 4.2. Different patterns of three voxels arrangements in a bloxel: (a) in a row, and (b, c) at right angles
Therefore, the three configurations in Figure 4.2 would have the same surface area
irrespective of the fact that the 3D arrangements of the voxels are different thereby
representing three distinct bloxels as 3D objects in space. This becomes more
complicated, however, when a bloxel contains a higher number of voxels and the
configuration could take on a myriad of forms.
Expected Surface Area
In order to determine the expected surface area, the number of voxels equivalent to those
in the bloxel are compactly arranged programmatically. First, the cubic root r of the
number of voxels N in the bloxel is determined and rounded off to the nearest integer. If
[N – (r x r x r)] is greater than [r x r] then the N voxels are arranged in layered arrays of
r+1 by r+1, one on top of the other, until all the voxels are positioned. Otherwise, if [N –
(r x r x r)] is less than or equal to [r x r] then all the N voxels are arranged in layered
square arrays of side r until all the voxels are positioned in the 3D grid. After the voxels
are compactly arranged, the expected surface area of the new bloxel is calculated using
the formula outlined in the formula (4.1) above. A ratio of surface area to the expected
surface area is indicative of the bloxel's shape. The closer the ratio is to 1, the more
compactly arranged is the bloxel. Consider a bloxel with 21 voxels where r will be 3 and
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the bloxel formed will have two full layers of 3x3 voxels and an additional line of 3
voxels on top.
Fractal Dimension
The fractal dimension is determined by processing a binary three-dimensional matrix
representing the objects in the landscape. The box-counting method of determining the
fractal dimension was used. A three-dimensional array counts the number of cubes each
of a particular size r that are needed to cover the matrix. For this study, the
implementation of the box-counting algorithm is based on a script by Moisy (2006). It
determines the fractal dimension using cubes whose sizes are powers of two, i.e., r = 1, 2,
4, 8, 32... 2p, where p is the smallest integer such that 2p is equal to the largest side of the
input dataset.
The validity of the calculated fractal dimension was tested using a menger sponge
(Mandelbrot, 1977) of 243x243x243 voxels, which gave a result of 2.72. The known
fractal dimension of a menger sponge is 2.73 (Vita, et al., 2012). A cube comprising of
32x32x32 voxels has a fractal dimension of 3, while that with 3x3x3 voxels has a
dimension of 2.37 according to the implementation in this study. It should be noted that
the fractal dimension is influenced by the range of box sizes used in the analysis
(Foroutan-pour, et al., 1999) and this is a contributing factor in the differences between
the observed and the values that would otherwise be expected.
Listed in Figure 4.3 are the estimated values of the fractal dimension for 1D, 2D and 3D
bloxels where a single voxel is akin to a dimensionless point. In the implementation of
the box counting technique for this study, the size of the bloxel does influence the
computed fractal dimension because it is directly related to the number of iterations
(range of box sizes) it takes to come to the value. For example, there is a significant
difference in the outcomes for Figure 4.3(c), when the bloxel size changes from a six
sided bloxel to an eight sided bloxel that has a fractal dimension of 2.8 because the
repeating pattern of the bloxel's bounding box in Figure 4.3(c) is not as distinct as that in
Figure 4.3(d) whose sides are 8x8x8 and are exponential factors of 2. In principle,
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however, it is possible to generate fractal dimension values for complex 3D forms as
shown in Figure 4.4 were for four distinct bloxels.
Figure 4.3. The calculated fractal dimension values of various bloxels: a) voxels arranged as a single row are effectively a linear 1D object, (b) arranged contiguously in a single plane, whereas (c) and (d) are 3D bloxels.
Figure 4.4. Fractal dimension values obtain for various complex bloxels
108
Lacunarity
The lacunarity measure is used to quantify the surface characteristics of three-
dimensional structures. It can also be used to distinguish between structures with the
same fractal dimension but different surface characteristics. The quantification of the
surface structure is by its roughness, which is essentially a topographical property of
elevation variation. In this study, the lacunarity was implemented from the original
MATLAB script written by Vadakkan (2009). Figure 4.5 shows the lacunarity values of
various surface configurations. The lacunarity value of a bloxel with a smooth surface
(Figure 4.5a) varied slightly from the bloxel whose gaps are symmetrically arranged
(Figure 4.5b). However, as the surface complexity increases the roughness also increases
as shown in Figure 4.5c whose gaps are randomly arranged.
Figure 4.5. Lacunarity measure of a bloxel with a) smooth surface b) symmetrically arranged gaps (uniformly distributed), and c) randomly distributed gaps on the surface
The lacunarity and fractal dimension of an object provide complementary measures to
describe the complexity of the object shape. In Figure 4.5, the bloxel in (a) has a fractal
dimension of 3, the one in (b) has a fractal dimension value of 2.6 and that in (c) has a
fractal dimension estimated at 2.5. This indicates the form of the bloxel in (c) has less of
the repeating patterns compared to those in (a) and (b) which is further highlighted by the
high lacunarity value that point to the irregular nature of the surface. In the real world the
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lacunarity can be used to classify vegetation canopies based on the gaps in the leaves as
represented in the collected voxel-based data.
4.5.2. Complexity of Static Phenomena
A hypothetical landscape was generated in order to apply the indices on a voxel-based
representation of forest stands extracted from 3D point cloud data. The voxels
representing the hypothetical tree stands are overlaid on the freely available DEM of
Madison County, Idaho from the GeoCommunity data portal (GeoCommunity, 2015).
For visualization purposes, the DEM is sampled at lower resolution to smoothen the
details in the surface and to accentuate the distinction between the voxels of interest and
the underling topography (Figure 4.6). The developed functions for measuring the indices
are applied to the data that are represented as the voxels. The functions automatically
determined that there were three distinct regions in the study area and calculated the
corresponding indices accordingly (Table 4.1).
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Figure 4.6. A hypothetical landscape showing accentuated sample 3D bloxels
Region Expected Area (m2)
Volume (m3)
Surface Area (m2)
Fractal Dimension Lacunarity
1 15,400 12,600 17,600 2.3 1.1 2 23,800 24,200 33,000 2.7 1 3 20,800 19,000 39,800 2.1 2.1
Table 4.1. The calculated 3D indices for the bloxel representations in Figure 4.6
Similarly, Figure 4.7 represents the distribution of a plant species in a landscape where
the black voxels represent the species of interest while the white ones represent the other
vegetation cover in the area.
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Figure 4.7. A hypothetical binary distribution of voxels in the landscape: black voxels representing a plant species of interest and white the other vegetation cover
The Moran's I value for the bloxel in Figure 4.7 is 0.47, the fractal dimension is 1.71, and
the lacunarity is 1.39. Being a binary distribution, the voxels of interest (colored black) in
the bloxel can be agglomerated into 7 sub-bloxels with the indices listed in Table 4.2.
Expected Area (m2)
Volume (m3)
Surface Area(m2)
Fractal Dimension Lacunarity
3000 1000 3400 1.7 1.0 9600 6200 22600 1.9 2.8 4000 1600 6000 1.7 1.6 5800 2800 10000 2 2.4 1000 200 1000 1 0 1400 400 1600 2 1 1000 200 1000 1 0
Table 4.2. 3D Indices for the bloxels extracted from a binary vegetation dataset
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4.5.3. Complexity of Dynamic Phenomena
The focus of this study is in the indices and hence an in-depth treatment of
spatiotemporal analysis in 4D is beyond the scope of the work. However, the different
indices proposed can be arrayed as sets of time series that could be used in further
correlation analyses. These analyses would indicate the complexity and variations of
spatial patterns as the processes evolve along the temporal dimension. As an illustration,
a three-dimensional Eden growth model (Eden, 1961) intended to represent the formation
of bacterial clusters or deposition of particulates is used. In the model, the process of
cluster formation starts with a seed randomly placed in the 3D voxel grid. The
unoccupied sites in the von-Neumann neighborhood around the seed (the occupied
location) are referred to as the growth sites. At the subsequent time-step, a new site,
drawn randomly from the growth sites, is occupied. Without going into a detailed
treatment of the Eden growth process (Meakin, 1988; Cao and Wong, 1991), the clusters
formed by the Eden model have dense geometries that are compact on the inside with
tortuous surface character and change at each iteration (Gouyet, 1996).
Figure 4.8 shows the results of a three-dimensional Eden growth process at time-steps 1,
500, 1000, 2000 and 3000, with the focus being maintained on the formation of the
output patterns rather than the details and working of the system's dynamics. The
proposed 3D spatial indices were applied to the simulation outputs at every time-step.
The measures are, namely, the fractal dimension, lacunarity, surface area and volume - all
describing the spatial features on a landscape, which are represented with the aid of
uniformly tessellated cubic voxels. The graphical outputs in Figure 4.9 show the changes
in the various indices as applied to the output forms.
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Figure 4.8. Simulations results of a dynamic process at selected time-steps T representing iterations 1, 500, 1000, 2000 and 3000.
Figure 4.9. Graphical representation the 3D indices over time a) fractal dimension and lacunarity, and b) volume, surface area and expected surface area indices. On the y-axis is the scale for the index value and the number of iterations on the x-axis
In Figure 4.9a, the lacunarity varies widely in the first few time-steps before it starts a
gradual change while the fractal dimension on the other hand has gradual change over
time. The surface area, the expected surface area and volume change gradually as would
be expected for a form that is gradually enlarged (Figure 4.9b).
The indices are useful in analysing a landscape's three-dimensional characterises by
providing a structural description of its features. Moreover, the voxelised representation
of the landscape's 3D features lends itself well to point cloud data collection methods and
3D feature representation at high resolution. The approach differs from the other methods
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which use a digital elevation model of the entire landscape to estimate the surface's
characteristics. The indices can be used to describe natural features such as forest stands,
plant colonies or anthropogenic features.
4.6. Conclusions
A key objective of landscape ecology is to determine how spatial and temporal
heterogeneity come about and how they reciprocally influence processes. One aspect of
this objective involves the depiction and quantification of the spatial entities in a
landscape. This study develops a methodology that can be used to quantify the geological
formations, plant growth and other volumetric features in a landscape. The approach used
requires that the features need to be represented using uniformly tessellated voxels.
The indices presented in this study can be used to quantify the properties of volumetric
patches in a landscape. The indices are: volume, surface area, expected surface area (that
is if the voxels are compactly arranged), fractal dimension as well as the lacunarity. In
addition, these indices have been used to quantify the results from a three dimensional
Eden growth process. This work suggests that the quantitative assessment of voxel-based
features, as proposed here, can be translated into an alternative method for landscape
characterization and could potentially be integrated in a 3D GIS. It contributes towards
the development of algorithms and methods for analysing voxel-based data. The
voxelisation of the landscape would enable the use of an atomic data model capable of
representing field and object-based geographic data. In addition, further work is needed
to apply the indices in the characterization of an actual landscape and with use of
geospatial data in 3D.
References
Allain, C., and Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. Physical Review A, 44 (6), 3552-3558.
115
Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht ; Boston.
Baker, W. L., and Cai, Y. (1992). The r.le programs for multiscale analysis of landscape structure using the GRASS geographical information system. Landscape Ecology, 7 (4), 291-302.
Batista, T., Mendes, P., Carvalho, L., Vila-Vicosa, C., and Gomes, C. P. (2012). Suitable methods for landscape evaluation and valorization: the third dimension in landscape metrics. Acta Botanica Gallica, 159 (2), 161-168.
Bélisle, M. (2005). Measuring landscape connectivity: the challenge of behavioral landscape ecology. Ecology, 86 (8), 1988-1995.
Bishop, M. P., James, L. A., Shroder, J. F., and Walsh, S. J. (2012). Geospatial technologies and digital geomorphological mapping: Concepts, issues and research. Geomorphology, 137 (1), 5-26.
Blaschkea, T., Tiedea, D., and Heurichb, M. (2004). 3D landscape metrics to modelling forest structure and diversity based on laser scanning data. The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 36, 129-132.
Cao, Q.-z., and Wong, P.-z. (1991). Fractal interfaces in heterogeneous Eden-like growth. Physical Review Letters, 67 (1), 77-80.
Courtney, L. A., Fisher, W. S., Raimondo, S., Oliver, L. M., and Davis, W. P. (2007). Estimating 3-dimensional colony surface area of field corals. Journal of Experimental Marine Biology and Ecology, 351 (1–2), 234-242.
Cushman, S. A., McGarigal, K., and Neel, M. C. (2008). Parsimony in landscape metrics: strength, universality, and consistency. Ecological Indicators, 8 (5), 691-703.
Dupuy, S., Lainé, G., Tassin, J., and Sarrailh, J.-M. (2013). Characterization of the horizontal structure of the tropical forest canopy using object-based LiDAR and multispectral image analysis. International Journal of Applied Earth Observation and Geoinformation, 25 (0), 76-86.
Eden, M. (1961). A two-dimensional growth process. In Proceedings of Fourth Berkeley Symposium on Mathematics, Statistics, and Probability (Vol. 4, pp. 223-239). University of California Press, Berkeley.
ESRI. (2014). ArcMap 10.3. In. ESRI (Environmental Systems Resource Institute), Redlands, California. .
116
Forman, R. T. T. (1995). Land Mosaics: the ecology of landscapes and regions. Cambridge University Press, Cambridge ; New York.
Foroutan-pour, K., Dutilleul, P., and Smith, D. L. (1999). Advances in the implementation of the box-counting method of fractal dimension estimation. Applied Mathematics and Computation, 105 (2–3), 195-210.
Fuller, D. O. (2001). Forest fragmentation in Loudoun County, Virginia, USA evaluated with multitemporal Landsat imagery. Landscape Ecology, 16 (7), 627-642.
GeoCommunity. (2015). Free GIS data - GIS data depot. GeoCommunity, The Premier Portal for Geospatial Technology Professionals.
Gouyet, J.-F. (1996). Physics and fractal structures. Masson, New York; Paris.
Gustafson, E. J. (1998). Quantifying landscape spatial pattern: what is the state of the art? Ecosystems, 1 (2), 143-156.
Hanen, A., Imen, B., Asma, B. A., Patrick, D., and Hédi, B. M. (2009). Multifractal modelling and 3D lacunarity analysis. Physics Letters A, 373 (40), 3604-3609.
Hargis, C., Bissonette, J., and David, J. (1998). The behavior of landscape metrics commonly used in the study of habitat fragmentation. Landscape Ecology, 13 (3), 167-186.
Hermy, M. (2012). Landscape Ecology. Oxford University Press, New York.
Hinks, T., Carr, H., Truong-Hong, L., and Laefer, D. F. (2013). Point cloud data conversion into solid models via point-based voxelization. Journal of Surveying Engineering, 139 (2), 72-83.
Hoechstetter, S., Thinh, N. X., and Walz, U. (2006). 3D-indices for the analysis of spatial patterns of landscape structure. In H. Kremers and V. Tikunov (Eds.), InterCarto-InterGIS 12 (pp. 108-118).
Hoechstetter, S., Walz, U., and Thinh, N. X. (2011). Adapting lacunarity techniques for gradient-based analyses of landscape surfaces. Ecological Complexity, 8 (3), 229-238.
Jenness, J. S. (2004). Calculating landscape surface area from digital elevation models. Wildlife Society Bulletin, 32 (3), 829-839.
Knutson, M. G., Sauer, J. R., Olsen, D. A., Mossman, M. J., Hemesath, L. M., and Lannoo, M. J. (1999). Effects of landscape composition and wetland
117
fragmentation on frog and toad abundance and species richness in Iowa and Wisconsin, U.S.A. Conservation Biology, 13 (6), 1437-1446.
Kupfer, J. A. (2012). Landscape ecology and biogeography Rethinking landscape metrics in a post-FRAGSTATS landscape. Progress in Physical Geography, 36 (3), 400-420.
Li, H., and Wu, J. (2004). Use and misuse of landscape indices. Landscape Ecology, 19 (4), 389-399.
Liebovitch, L. S., and Toth, T. (1989). A fast algorithm to determine fractal dimensions by box counting. Physics Letters A, 141 (8–9), 386-390.
Magura, T., Ködöböcz, V., and Tóthmérész, B. (2001). Effects of habitat fragmentation on carabids in forest patches. Journal of Biogeography, 28 (1), 129-138.
Mandelbrot, B. B. (1977). Fractals: Form, Chance and Dimension. W. H. Freeman, San Francisco.
Marwan, N., Saparin, P., and Kurths, J. (2007). Measures of complexity for 3D image analysis of trabecular bone. The European Physical Journal Special Topics, 143 (1), 109-116.
MathWorks. (2012). MATLAB and Statistics Toolbox. In (Release 2012b ed.). The MathWorks, Inc., Natick, Massachusetts, United States.
McGarigal, K., and Cushman, S. A. (2005). The gradient concept of landscape structure. In J. A. Wiens and M. R. Moss (Eds.), Issues and perspectives in landscape ecology (pp. 112-119). Cambridge University Press, New York.
McGarigal, K., and Marks, B. J. (1995). FRAGSTATS: spatial pattern analysis program for quantifying landscape structure. In, Portland (OR): USDA Forest Service, Pacific Northwest Research Station.
McGarigal, K., Tagil, S., and Cushman, S. (2009). Surface metrics: an alternative to patch metrics for the quantification of landscape structure. Landscape Ecology, 24 (3), 433-450.
Meakin, P. (1988). Invasion percolation and invading Eden growth on multifractal lattices. Journal of Physics A: Mathematical and General, 21 (17).
Mitasova, H., Harmon, R. S., Weaver, K. J., Lyons, N. J., and Overton, M. F. (2012). Scientific visualization of landscapes and landforms. Geospatial Technologies
118
and Geomorphological Mapping Proceedings of the 41st Annual Binghamton Geomorphology Symposium, 137 (1), 122-137.
Moisy, F. (2006). Fractal dimension using the 'box-counting' method for 1D, 2D and 3D sets. In Online Resources: MATLAB Central, File Exchange. The MathWorks, Inc.
Moran, P. A. P. (1948). The interpretation of statistical maps. Journal of the Royal Statistical Society. Series B (Methodological), 10 (2), 243-251.
Neel, M. C., McGarigal, K., and Cushman, S. A. (2004). Behavior of class-level landscape metrics across gradients of class aggregation and area. Landscape Ecology, 19 (4), 435-455.
Noss, R. F. (1990). Indicators for monitoring biodiversity: a hierarchical approach. Conservation Biology, 4 (4), 355-364.
O'Neill, R. V., Hunsaker, C. T., Jones, K. B., Riitters, K. H., Wickham, J. D., Schwartz, P. M., Goodman, I. A., Jackson, B. L., and Baillargeon, W. S. (1997). Monitoring environmental quality at the landscape scale. BioScience, 47 (8), 513-519.
O'Neill, R. V., Krummel, J. R., Gardner, R. H., Sugihara, G., Jackson, B., DeAngelis, D. L., Milne, B. T., Turner, M. G., Zygmunt, B., Christensen, S. W., Dale, V. H., and Graham, R. L. (1988). Indices of landscape pattern. Landscape Ecology, 1 (3), 153-162.
Olsoy, P. J., Glenn, N. F., Clark, P. E., and Derryberry, D. R. (2014). Aboveground total and green biomass of dryland shrub derived from terrestrial laser scanning. ISPRS Journal of Photogrammetry and Remote Sensing, 88 (0), 166-173.
Parrott, L. (2005). Quantifying the complexity of simulated spatiotemporal population dynamics. Ecological Complexity, 2 (2), 175-184.
Parrott, L., Proulx, R., and Thibert-Plante, X. (2008). Three-dimensional metrics for the analysis of spatiotemporal data in ecology. Ecological Informatics, 3 (6), 343-353.
Pike, R. J. (2001). Digital terrain modeling and industrial surface metrology: Converging realms. Professional Geographer, 53 (2), 263-274.
Plotnick, R. E., Gardner, R. H., and O'Neill, R. V. (1993). Lacunarity indices as measures of landscape texture. Landscape Ecology, 8 (3), 201-211.
119
Riitters, K. H., O'Neill, R. V., Hunsaker, C. T., Wickham, J. D., Yankee, D. H., Timmins, S. P., Jones, K. B., and Jackson, B. L. (1995). A factor analysis of landscape pattern and structure metrics. Landscape Ecology, 10 (1), 23-39.
Rogers, D., Cooper, A., McKenzie, P., and McCann, T. (2012). Assessing regional scale habitat area with a three dimensional measure. Ecological Informatics, 7 (1), 1-6.
Schilling, A., Schmidt, A., and Maas, H.-G. (2012). Tree topology representation from TLS point clouds using depth-first search in voxel space. Photogrammetric Engineering And Remote Sensing, 78 (4), 383-392.
Sithole, G., and Vosselman, G. (2004). Experimental comparison of filter algorithms for bare-Earth extraction from airborne laser scanning point clouds. Advanced Techniques for Analysis of Geo-spatial Data, 59 (1–2), 85-101.
Soares, F., Janela, F., Pereira, M., Seabra, J., and Freire, M. M. (2013). 3D lacunarity in multifractal analysis of breast tumor lesions in dynamic contrast-enhanced magnetic resonance imaging. IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 22 (11), 4422-4435.
Stubblefield, C. H., Vierling, K. T., and Rumble, M. A. (2006). Landscape-scale attributes of elk centers of activity in the central black hills of South Dakota. The Journal of Wildlife Management, 70 (4), 1060-1069.
Stupariu, M. S., Stupariu, I. P., and Cuculici, R. (2010). Geometric approaches to computing 3D-landscape metrics. Landscape Online, 24, 1-12.
Tischendorf, L. (2001). Can landscape indices predict ecological processes consistently? Landscape Ecology, 16 (3), 235-254.
Turner, M. G. (1989). Landscape ecology: the effect of pattern on process. Annual Review of Ecology and Systematics, 20 (1), 171-197.
Turner, M. G. (1990a). Landscape changes in nine rural counties in Georgia. PE & RS, Photogrammetric Engineering & Remote Sensing, 56 (3), 379-386.
Turner, M. G. (1990b). Spatial and temporal analysis of landscape patterns. Landscape Ecology, 4 (1), 21-30.
Urban, D. L., O'Neill, R. V., and Shugart, H. H., Jr. (1987). Landscape ecology: a hierarchical perspective can help scientists understand spatial patterns. BioScience, 37 (2), 119-127.
120
Vadakkan, T. (2009). 3D lacunarity. In Online Resources: MATLAB Central, File Exchange. The MathWorks, Inc.
VanDerWal, J., Falconi, L., Januchowski, S., Shoo, L., and Storlie, C. (2014). SDMTools: species distribution modelling tools. In.
Varner, J., and Dearing, M. D. (2014). The importance of biologically relevant microclimates in habitat suitability assessments. PLoS ONE, 9 (8).
Villarrubia, J. S. (1997). Algorithms for scanned probe microscope image simulation, surface reconstruction, and tip estimation. Journal of Research of the National Institute of Standards and Technology, 102 (4), 425-454.
Vita, M. C., De Bartolo, S., Fallico, C., and Veltri, M. (2012). Usage of infinitesimals in the Menger’s Sponge model of porosity. Special Issue dedicated to the international workshop "Infinite and Infinitesimal in Mathematics, Computing and Natural Sciences", 218 (16), 8187-8195.
Weishampel, J. F., Blair, J. B., Knox, R. G., Dubayah, R., and Clark, D. B. (2000). Volumetric lidar return patterns from an old-growth tropical rainforest canopy. International Journal of Remote Sensing, 21 (2), 409-415.
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Chapter 5. A Representation and Analysis of a Process' Spatiotemporal Dynamics Using a 4D Data Model
5.1. Abstract
The raster and vector spatial data models are the most commonly used in geographic
information systems (GIS) practice but are insufficient for the representation of dynamic
spatiotemporal phenomena. Though many improvements have been proposed to the
spatial data models and numerous prototype implementations have been developed in
order to address this limitation, the problem has persisted. One of these proposals is the
geo-atom which a general theory for geographic information representation in a four-
dimensional space-time continuum. The objective of this study is to implement the geo-
atom spatial data model in order to represent and analyse a dynamic four-dimensional
field-based process. The queries for spatiotemporal analysis are related to volume,
surface area, rate of change and temporal ordering. The data used are voxels units
generated from the simulation of the process and specific spatiotemporal queries have
been developed to demonstrate the usefulness of this model in GIS-based modeling.
5.2. Introduction
The representation of the varied and complex dynamics on the earth’s surface, in addition
to the particularities of the constituent elements in a given landscape, continues to be a
subject of significant interest in the discipline of geographic information science
(GIScience). In most geographic information system (GIS) software applications, it is not
possible to perform full three-dimensional spatial analysis because the third dimension of
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space is encoded as an additional attribute of the planimetric elements (Longley, et al.,
2011). In addition, the temporal dimension, which is a fourth addition to the three spatial
ones, is largely modeled with the aid of multiple data layers as snapshots of the temporal
changes (Peuquet, 1994; Worboys, 1994; Grenon and Smith, 2004). Although, the
significance of handling the third spatial dimension as well as temporal information in
GIS has long been recognized by the GIS research community, the raster and vector data
models that are predominantly used for geographic representation and analysis are
inadequate for this task.
To address this limitation, several proposals have been put forward by researchers over
the last three decades. One of these describes and suggests the adoption of the geo-atom
spatial data model to conceptualise the space-time domain (Goodchild, et al., 2007). This
geospatial data model provides the theoretical basis for a geographic abstraction capable
of representing the four-dimensional space-time continuum. It uses a space-time
primitive, referred to as a geo-atom, to model the spatial entities in a multidimensional
perspective and enhances GIS support for spatiotemporal analysis of geographic
processes. The integration of 3D space and time is important because dynamics are a
defining characteristic of processes and some of these processes (for example, ground
water and avalanche flows) evolve with the third dimension as a significant factor.
The dynamics of processes, for example, the spread or the spatial and temporal
interconnections between the constituent parts, can be simulated for study and analysis
with the aid of the geographic automata framework (Torrens and Benenson, 2005). With
their origins in complex systems theory, the geographic automata systems are an
important starting point for representing dynamic geographic phenomena (O'Sullivan,
2005). Although the framework has been predominantly used in two-dimensional
conceptualisations of space, it can be extended to 3D space in which case voxel units can
be used (Burrough, et al., 2005; Jjumba and Dragicevic, in press). In their formulation
voxel-based geographic automata simulations follow the same underlying principles as
their antecedents, the classical cellular automata models (von Neumann, 1966). The
automata theory requires the application of transition functions on a locally defined
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neighborhood so as to effect change to the attributes of the voxels there-in at a
subsequent iteration. The local neighborhood itself is a configuration is of contiguous
voxels around a central voxel. The data outputs from such a simulation would be four-
dimensional and could be interrogated for 3D spatial character of the process as well as
the spatiotemporal evolution.
Over the years there has been considerable interest in the research efforts to design GIS-
based frameworks that can represent and query dynamic phenomena. Erwig et al. (1999)
have proposed model for representing moving objects, similarly Lin and Su (2008)
highlight a methodology for determining the trajectory of an object's movement, and
McIntosh and Yuan (2005) demonstrate a method for query support in analyzing
spatiotemporal similarity for large datasets. In addition, Hubbard and Hornsby (2011)
propose a method of analyzing alternate trajectories of events while Yi et al. (2014) have
proposed a model for integrating space and time interactions of spatial objects. These
efforts highlight important consideration in the management and analysis of
spatiotemporal information using two-dimensional object-based and field-based
representations of geographic processes. However, there is still a need to explore
spatiotemporal analysis in three dimensional space plus time, which is particularly
important for phenomena with both object and field-like behavior. The objective of this
study is to analyse the evolution of a spatiotemporal process whose data are structured
using the geo-atom data model. Although geographic automata models are often
employed in the description, study and analysis of spatiotemporal phenomena, in this
study a voxel-based automata model is used to represent a dynamic process in four
dimensions and to generate an abstract spatiotemporal dataset that is structured with the
geo-atom data model for further analysis. A generalized four-dimensional process of
snow cover retreat was simulated as a proof of concept. Specific queries designed to
explore the four-dimensional nature of the spatiotemporal process were also developed
and applied to the simulation outcomes.
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5.3. Theoretical Background
The historical development of GIS is marked by the dichotomy of representing
geographic space with either the raster data model or the vector data model (Jones, 1997;
DeMers, 2009). However, these 2D data models are inadequate at representing
geographic processes as they evolve in the 4D space-time continuum because of the
limitations imposed by their respective data structures. Over the last three decades
researchers have explored and suggested ways for representing the vertical dimension of
space along with the two lateral ones, as well as the incorporation of time to capture the
dynamism of geographic processes. This section reviews the literature related to the
conceptualization of a multidimensional view of space followed by an overview of 4D
spatiotemporal modeling and the suitability of voxel-based automata for the simulation of
dynamic phenomena.
5.3.1. Two-Dimensional Representation of Geospatial Phenomena
Traditionally, GIScience has emphasized the two-dimensional spatial representation, and
the raster and vector geospatial data models continue to be the dominant formats for
representing geographic phenomena in GIS software applications based on an underlying
assumption that they are representable as either discrete entities (the object-based view)
or continuous surfaces (the field-based view)(Couclelis, 1992; Goodchild, 1992; Mennis,
2011). The vector-based representations of geographic space closely resemble the
fundamental graphical elements of a paper map and in a GIS, the symbolic geometries of
points, lines and polygons can maintain their distinct identities, as well as the attributes
and shape of the geographic entities they represent (Frank, 1992; Burrough and
McDonnell, 1998). On the other hand, the raster-based representations use a grid of
uniform cells whose attribute values represent the specific entities in the real world.
Efforts to represent the elevation in a GIS are often quasi 3D in that they treat the height
as an attribute of the two-dimensional coordinates of the location - a solution often
referred to as “2.5D”. In a truly three-dimensional GIS, the unique coordinates of a
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location would be defined by three independent variables, one for each spatial dimension;
otherwise such a location would not be displayed just as it is not possible to plot an
element without coordinate pairs in a Cartesian plane system (Longley, et al., 2011). In
addition, work has been done to extend the vector and raster data models in order to
incorporate the temporal component when representing geospatial phenomena (Langran,
1992; Dragicevic and Marceau, 2000; Peuquet, 2002; Yuan, et al., 2014) and to
accommodate the third spatial dimension (Raper, 2000; Breunig and Zlatanova, 2011).
However, the new ways with which spatiotemporal data are collected and the growing
need to analyse dynamic phenomena for the purpose of understanding the connections
between complex spatial patterns and the underlying geographic processes are pushing
the research community to develop and implement dynamic models of the space-time
continuum (Gately, et al., 2013; Lee, et al., 2013; Dias and Tchepel, 2014). Geographic
processes evolve in 3D space over extended temporal periods. They are characterised by
complex interactions between fixed and mobile entities and changes to the geometric
shape and internal structures of some of these entities (Gebbert and Pebesma, 2014).
Hence, it is important to represent, analyze, and model geographic dynamics in a 4D
space-time continuum.
5.3.2. Four-dimensional Representation of Geospatial Phenomena
An elemental representation of geographic phenomena in 4D has been formalised with
the geo-atom theory (Goodchild, et al., 2007). A geo-atom is a geographic primitive for a
space-time point described by the spatial coordinates, the temporal dimension, as well as
the attribute values of the geographic theme being represented. It is expressed as a tuple
in the following formulation:
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( )[ ]paApg ,,≈
(5.1)
where g is a geo-atom that represents a spatial object being modeled, p is a vector that describes the object in space-time coordinates (x, y, z, t), A is the variable of interest such as salt concentration in water, temperature variation in a lake or the distribution of particulate matter in the atmosphere, and a(p) is the specific attribute value of A at that location in space-time.
By altering the criteria for aggregating the geo-atoms, different types of spatial entities
can be represented as geo-objects, geo-fields, field-objects or object-fields based on the
identity of the geo-atoms or the values of the attributes describing the modeled
geographic theme. In order to model a spatial entity, a geo-atom structure can use
multiple values that represent the temporally varying attribute states at that location. The
use of multiple values for the attributes implies that in addition to being able to query a
GIS for a given value at all locations in the entire dataset, it is also possible to query for
multiple attribute values at a single location (Goodchild, 2010). Thus, the geo-atom data
structure is particularly important and useful because it broadens the flexibility with
which spatiotemporal queries on geographic data can be executed.
Beyond providing the means to identify spatiotemporal phenomena with semantic
descriptors (such as an attribute name and value), spatial properties (location coordinates
and spatial extents) as well as the temporal component (for example timestamps or the
start and end time marks), the ontology of the space-time model must provide a means for
defining the topological relationships of adjacency, connectivity and containment
(Langran, 1993; Thomsen, et al., 2008). It is these topological predicates, as well as the
geometric ones, that a GIS uses to answer queries about the relative locations of spatial
entities (Egenhofer, et al., 1994; Erwig and Schneider, 2002). Although specific
frameworks for modeling 3D spatial topology have been proposed, they are yet to be
sufficiently implemented in GIS practice (Billen and Zlatanova, 2003; Ellul and Haklay,
2006; Breunig and Zlatanova, 2011). Similarly, the integration of temporal topology in
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GIS databases is an area of ongoing research (Marceau, et al., 2001; Bittner, et al., 2009;
Galton, 2009; Couclelis, 2010; Nyerges, et al., 2014) that is yet to be included in both
commercial and open-source GIS software applications.
Although designed on the basis of 2D spatial data, the spatiotemporal data models that
have been proposed include the space-time composite (Langran and Chrisman, 1988), the
spatio-bitemporal model (Worboys, 1994), the three domain model (Yuan, 1999), the
OOgeomorph model (Raper and Livingstone, 1995), the event-based spatiotemporal
model (Peuquet and Duan, 1995) and a space-time object model for forestry data
management (Rasinmäki, 2003). Other efforts have been invested in developing 4D
spatiotemporal data models. Among these is a model for simulating a river based system
using numerical methods (Goodall and Maidment, 2009). It uses a Geographic Markup
Language (GML) spatial data structure but the volumetric units are modeled implicitly
using depth and height as attributes of the grid tessellations. Beni et al.(2011) have
developed a spatiotemporal model of movement point data using 3D voronoid objects.
Other data models include that proposed by van Oosterom and Stoter (2010), which in
addition to representing the four dimensions of the space-time continuum incorporates
scale as fifth dimension. The others are the Network Common Data Form (NetCDF) used
to store model outputs from numerical simulations in the geosciences (Nativi, et al.,
2005) and the Geographic Markup Language (GML) specification for dynamic features
(Lake, et al., 2004).
The 4D representation of geospatial phenomena allows for the application of
spatiotemporal queries to retrieve information about a process' patterns and the
interactions of the component parts in a multidimensional framework. Pultar et al. (2009)
applied the geo-atom concepts to represent and query emergency evacuation situations
from a two-dimensional perspective while Zhao et al. (2015) have adapted raster data to
create a space-time model for use in querying and analysing how vegetation was affected
by a cold weather. These models use related approaches to enhance raster data in the
representation of spatiotemporal phenomena modeled from a 2D perspective. However,
some spatiotemporal processes such as snow cover melt, landslides, avalanches and
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organism movement in marine environments, are truly four-dimensional. For this reason,
this study integrates the geo-atom data model with a voxel-based tessellation of space to
represent dynamic phenomena in the three dimensions of space plus time and allow for
specific queries. A hypothetical voxel automata model is developed and run to generate
the 4D data used for the spatiotemporal analysis. Specific queries pertinent to the
retrieval of information about changes in spatial attributes at a given location in time are
applied to the data.
5.4. Methods
5.4.1. Conceptualization of Space and Time
This study uses the geo-atom data model in the management of the spatiotemporal data
for a simulated phenomenon. The changes in the attribute values of the phenomenon
along with the time values corresponding to the time-steps of the voxel automata
simulation are all stored in the data structure. Although the time format recorded in the
model structure corresponds to the time-step at which the change in an attribute occurred,
it is possible to use a suitable temporal format such as the ISO 19108 GIS temporal
schema (Kresse, et al., 2012) when the automata model calibrated to the temporal
evolution of the process. Consider a study area that is tessellated into uniform and
contiguously arranged voxels where each voxel is defined by three characteristics: first,
the three geometrical dimensions as defined by the three geographic coordinates of
voxel’s center position; second, the attribute of the geographic phenomenon under study;
and third the temporal dimension modeled with the time-step value. These voxel
characteristics are formulated as the variables in the geo-atom tuple (5.1).
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Figure 5.1. Three snapshots of a field process in voxel space representing the attribute and temporal value updated at every time-step using the geo-atom model
Figure 5.1 shows three consecutive snapshots of a field process represented by voxels.
Each voxel is an element in a 3D field and is modeled with the geo-atom data structure
where the temporal dimension is associated with the phenomenon's attribute
characteristics by linking an attribute value directly with its corresponding timestamp at
which it is marked. The structure of the voxel also includes an identity field (ID) that is
useful in generating a unique identification for the field to object transition. The voxel's
attribute variable (A) can be represented with categorical or continuous data to model the
phenomenon's state or intensity at that location. The full record of the attribute values and
their corresponding timestamps is organized in two separate lists of equal length - one for
the temporal values (timestamps) and the other for the attribute values. Since the voxels
are immobile their respective coordinates do not need to be updated over time.
5.4.2. Spatiotemporal Queries
In a dynamic process, the measured values that represent the phenomenon under study
will vary through space and time. In order to analyse these data, spatiotemporal queries
can be constructed to investigate specific spatiotemporal aspects about the process. The
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procedure involves identifying a set of voxels that fall within the spatial limits, followed
by the attribute limits and lastly the temporal limits as provided in the query (Figure 5.2).
The program implementing the queries is structured such that a single time value,
attribute value or a triplet coordinate set of a voxel are also treated in a similar manner as
boundary limits.
Figure 5.2. The data query procedure narrows the search by sequentially limiting the spatial extents, attribute values and temporal bounds.
Using Langran's (1992) spatiotemporal query type categorization as well as Yuan's
(1999) extension of the same, the following spatiotemporal queries were developed.
Simple Spatiotemporal Queries
When applied on a collection of geo-atoms, the simple spatiotemporal queries extract the
attribute values of the voxels at an instant in time. The queries can provide results to
inquiries such as:
• Select all the voxels with attribute value a at time t, or
• Display all the voxels at time t on map.
For the first query, the procedure involves identifying the voxels whose attribute values
are equivalent to a at a corresponding time-step t. Given an array of attribute values A =
[a1,a2,...an] and a corresponding array of temporal values T = [t1, t2,...tn] to represent the
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attribute and temporal aspects of a geo-atom, the second query is constructed according
to a general structure that limits the selection of the voxels to both a defined attribute
value and a time value as follows:
SELECT voxels
FROM geo-atom dataset
WHERE T = ti AND A = ai
Spatiotemporal Range Queries
Spatiotemporal range queries extract the attribute values of the voxels as they pertain to a
specified period of time. These queries may answer questions such as:
• Display the voxels that had a value a between the specified time period
• What is the highest attribute value of the voxel (x,y,z) within a time period t1-t2?
• What is the average value at voxel (x,y,z) between time period t1-t2?
Similar to the case of the simple spatiotemporal queries, the steps for the first query
involve identifying the voxels whose attribute values are equivalent to a at a
corresponding time-step t. Given an array of attribute values A = [a1,a2,...an] and a
corresponding array of temporal values T = [t1, t2,...tn] to represent the attribute and
temporal aspects of a geo-atom. These queries are applied in a form that outputs the
voxels based on the attribute value and a time period such as:
SELECT voxels
FROM geo-atom dataset
WHERE ti >=T =< ti AND A = ai
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This query category and the simple spatiotemporal category above provide the basis for
the construction of the ones that follow which are more complex.
Spatiotemporal Behavior Queries
Rate of Change
The rate of change of a process can be measured using the change in the number of
voxels within a specified time interval using the formula below.
ij
ij
ttnn
−
−
(5.2)
where nj, ni, are the number of voxels at time-steps tj, ti, respectively, and i<j
This is accomplished by first applying a simple spatiotemporal query to identify the
voxels ni that fit a specified criteria at time ti. Those voxels are assigned an ID tag. At a
later time-step tj the same query is applied to the voxels selected in the earlier application
of the query to generate the value of nj. The difference in voxel count (nj-ni) is used in the
formula to determine the rate of change.
Location Descriptor Queries
The descriptions of a phenomenon at a particular portion of the study can be determined
by examining how the attribute values are apparent at those locations over time. The
queries proposed here build on the simple spatiotemporal queries above to determine
such descriptions as:
• Find the region with an average value µ at time t
The development of the proposed query requires the user to define a moving 3D cubic
window in whose boundaries the average value is determined. The cubic window is
moved throughout the study region and the mean values of these voxel attributes are
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calculated. If the arithmetic mean value is equivalent to µ then the central coordinates of
the window are stored in memory and the program continues to search. Although this
approach can be used to determine other descriptive statistics it is only the arithmetic
mean that is applied in this study.
Temporal Relationship Queries
The temporal relationship queries are used to investigate patterns in a sequential format
where the temporal equivalence and ordering (exact or relative) of the specified values
matters. They can be used to understand the temporal relationship between multiple
regions could be formulated as:
• What is the area that has these data values between t1 and t2?
• And, is there a sequential ordering to the existence of these areas?
The query is developed as an extension to the spatiotemporal range query by splitting a
time period into smaller sections for spatiotemporal segmentation. The method relies on
comparing temporal relations between time-steps that represent the interval time
specified by the query.
5.4.3. Case Study: A Model for Snow Cover Retreat
The process of snow cover retreat can be characterized with a spatially continuous
variable and is dependent on topographic factors such as elevation, slope, aspect and
shading (Dunn and Colohan, 1999; Jain, et al., 2009), which can be incorporated in the
structure of the model that simulates snow cover change. In general, snow cover change
models fall into two broad categories, namely, the energy-balance models and those
based on the temperature-index methods (Cazorzi and Fontana, 1996). Data for
temperature-index modeling are more readily available and the methodology assumes a
direct relationship between elevation and positive air temperature (Hock, 2003).
The changes in mountain snow cover are affected by the energy factors in the
surrounding climate in complex ways. At the most basic level, however, as the local
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temperature rises, the snow melt increases along with the rate of change in the snow
cover. The inverse relationship between temperature and altitude implies that in general
snow will recede from the lower elevations of the mountain towards the top.
A voxel automata model simulating the process of snow cover change was designed and
developed to generate the experimental data to which the spatiotemporal queries were
applied. This was necessitated by the absence of an appropriate dataset that could be
formatted and stored in the geo-atom data structure. This study uses the same approach
by relying on elevation as the principle factor but the model has sufficient flexibility to
add other factors pertinent to the process. In addition, stochasticity is added to the process
to mimic the unpredicability of snowstorms and to provide for the possibility more snow
may fall even though the snow cover continues to deplete due to increase in the general
ambient temperature. This is achieved with the addition of random localised snow fall in
parts of the study area. The rate of snow melt is directly related to the degrees of
atmospheric temperature above a critical temperature for snowmelt and can be formalized
as follows (Dunn and Colohan, 1999):
(5.3)
where, M is the melt rate, K the degree–day factor, T the mean temperature, and Tm the critical temperature for melt to occur. The degree–day factor is an empirical constant for proportionality that accounts for all the physical factors not included in the model.
As an example, the ground could be covered with snow of varying thickness across the
study area. It will generally melt from the lower altitude regions first and gradually to the
higher altitudes in a manner shown in Figure 5.3 below. Also, locations with thicker
snow cover will take longer to melt.
2/)( mTTKM −=
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Figure 5.3. A voxel automata simulation of snow cover change as a function of elevation for three selected time periods t1, t2 and t3, which could be days or weeks depending on the atmospheric temperature.
The transition rules enforce the disappearance of the snow in the inverse relationship to
the altitude and the thickness of the snow. At each time-step, the function in equation
(5.3) is applied to the voxels representing the snow cover field as the transition rule and
the changes are recorded as attribute values in their respective arrays (Figure 5.1).
5.5. Implementation and Results
Using MATLAB, the geo-atom spatial data model was organized as a structured array
with attributes of various data types including arrays of numeric and alphanumerical data
(Figure 5.1). The use of arrays allows for the recording of multiple values for a specific
aspect of the geo-atom. Whenever there is a change recorded in one of the aspects of the
voxel-based representation, the program simultaneously records the corresponding
temporal value. Thus, a voxel has a fixed position but its attribute values are changing
over time and are represented by an array of the changing attribute values as well as a
separate but corresponding array of temporal values indicating when these changes took
place.
The geographic landscape was derived from topographic data of Aggipah Mountain,
Idaho, USA, formatted according to the USGS 30 meter resolution digital elevation
model (DEM). The area measures about 6.3 km East-West and 5.7 km North-South with
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an elevation difference of about 1.4 km between the lowest and highest point. The lateral
sides of the voxel are 30 meters x 30 meters while the height is 1 meter. In MATLAB, the
USGS data file is organised separately into three two-dimensional arrays, one for the
latitude values, the longitude values and the elevation values. These coordinate arrays are
transferred to the structure of geo-atom data model by assigning the values for the
latitude, longitude and elevation coordinates from their respective two-dimensional
arrays. The general topographical structure of the landscape is modeled with a smooth
overlay while the snow cover is modeled explicitly with the voxels. This arrangement
was done to enhance the visual contrast between the fixed topography and the dynamics
of the changing phenomenon. In addition, the developed query algorithms use brute force
computation with nested loops to examine the geo-atom data in search of the voxels that
fit the parameters specified by the respective queries. Also, a projected coordinate system
would provide a more accurate measure of distances and areas, however, the built-in
function that reads the DEM data file into MATLAB projects the map in geographic
coordinates only. In the sections that follow, the proposed spatiotemporal queries are
applied to data from a model for snow cover retreat.
5.5.1. Queries for Snow Cover Retreat Model
A snow cover retreat model was developed and implemented to generate 4D datasets for
spatiotemporal analysis. At the start of the simulation, the simulation model is specified
with an average voxel attribute value of 20 to represent a general snow cover thickness of
20 cm throughout the study area. Because the program uses brute force procedures to
search the data, it was necessary to limit the amount of searchable data so as to have a
meaningful level of efficiency in the search for the voxels of interest per the query's
parameters. The average temperature, as well as the degree-day factor listed in formula
(5.3), were overadjusted such that simulation of the snow depletion in the entire area lasts
about 10 weeks, where each week corresponds to an iteration. The results from the
simulation are shown in Figure 5.4 below.
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Figure 5.4. Snapshots of the results from the snow cover retreat process where T1, T3, T6, T10 represent 1 week, 3 weeks, 6 weeks and 10 weeks, respectively, from the start of the simulation
5.5.2. Simple Spatiotemporal Queries
Provided that one has a complete dataset, a simple spatiotemporal query can be used to
find those areas in a study region that have a particulars set of attribute values at a
specified instance in time. In this study, this query was applied to identify the voxels that
have an attribute value that falls between 14.0 cm and 15.0 cm thickness of snow at week
5. The application of the query on the data resulted in an area of 9.9 square kilometer
being selected and the corresponding graphical output is shown in Figure 5.5a. The
output for another simple query that seeks to identify the voxels with an attribute value
greater than 0 (i.e. all active areas with snow) at week 5 are shown in Figure 5.5b and the
count of the voxels themselves is 25,741 which is equivalent to 23.2 square kilometers.
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Figure 5.5. Selected voxels for the simple and spatiotemporal range queries (a) query for 14.0 cm and 15.0 cm thickness of snow at time-step 5, and (b) all active voxels with snow at time-step 5
5.5.3. Spatiotemporal Range Queries
A spatiotemporal range query can be used to find areas that have a particular attribute
within a specific time period. It was applied to find voxels which have had attribute
values ranging between 19cm and 20.1cm during time period of 10 to 15 weeks. There
was a total of 274 voxels which correspond to the area of 0.2 square kilometers selected
as a result of the query and are shown in Figure 5.6. It should be noted, however, that this
query's outcome is a result of the stochastic snowstorm artifact of the model because
most of the snow setup at the beginning of the model is depleted at about time value
10cm.
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Figure 5.6. Results for spatiotemporal range queries
An additional temporal range query was applied to find the average amount of snow at a
voxel specified by a set of coordinates (-114.7o W, 45.2o N, 2285 m) during the course of
the simulation. This location was arbitrarily selected and the average attribute value was
found to be 13.9cm. In other words, at this particular location there was an average snow
cover of 13.9cm for the duration of the simulation run. Although, by extension, it would
be possible to determine the average value for a region, that component is not included in
the program at the moment. However, as indicated earlier in the methods section, it is
possible to identify a region fitting a certain criteria.
5.5.4. Spatiotemporal Behavioral Queries
Rate of Change Query
The rate of change can be determined by using time and the area or volumetric change in
the phenomena. However, in this study, the number of voxels, which can be used to infer
the volumetric change by an application of a scale factor, is used. The number of voxels
with an attribute value greater than 0 at time-step value 10 was computed to be 1449.
Similary, the number of voxels at time-step value 5 were computed to be 25741. With the
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application of formula (5.3) where nj=25741, ni=1449, tj=10 and ti=5, the rate of change
is computed as 4858.4 voxels per iteration from (25741-1441)/(10-5). With the voxel's
horizontal dimensions measuring 30 m x 30 m and the height set at 1 m the volumetric
rate of change is equivalent to about 4.4 x 106 cubic meters per time-step. The rate of
change computed applies to the entire study area; however, another measure of
spatiotemporal behavior would be to determine local rates of change by focusing the
analysis on specific regions. In simulation studies such a local measure could be used to
quantify how the variation in vegetation cover within the study region affects the snow
cover retreat.
Location Descriptor
A query was applied to the data to find the average of the voxels within a region that is
approximately East: -144.469 to -144.467 and North: 45.155 to 45.175 at time-step 6.
The computed average value of the attributes at that time-step is 5.7 and the region is
shown in Figure 5.7 below.
Figure 5.7. Output for a pre-defined boundary selection
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5.5.5. Temporal Relationship Queries
The implementation of this query is useful in trying to analyse the sequential ordering of
events or the progress of change around some clearly delineated boundaries. This is
achieved by selecting voxels that meet a given attribute value criteria within a defined
time period. The query is then applied at different time-steps. In the implementation the
procedure is manual and tedious but it can be used to analyse elements of concurrency
and the temporal ordering of events. The query was applied to determine the voxels that
had an attribute value of 20cm between time-steps 1 to 10 (Figure 5.8a) followed by a
segmentation for those that had the same value at time-step 3 (Figure 5.8c) and time-step
5(Figure 5.8b).
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Figure 5.8. Sequential occurrence of events: (a) all voxels with snow depth 20 cm at any point between weeks 1-10, (b) voxels with snow depth 20 cm at week 3, (c) voxels with snow depth of 20cm at week 5
a)
b)
c)
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5.6. Conclusion
This study uses the geo-atom spatial data model in the representation of dynamic spatial
process using four-dimensional data generated from voxel automata simulations and in
order to postulate various spatiotemporal queries. The geo-atom spatial data model
provides a means by which static and dynamic phenomena can be modeled and the ‘geo-
atom’ term is used in reference to a space-time primitive as a building block in describing
the spatial entities and is a data model that can represent the multiple perspectives of
geographic space.
The voxel automata model is used to demonstrate the dynamics of a spatial phenomenon
that is receding in its spatial coverage by modelling a simplified process of snow cover
retreat. During the course of the simulation, the attribute values of the voxels are updated
together with the time-step at which the change in value takes place. Specific queries
related to the evolution of a process such as the rate of movement and the frequency of
occupancy have been developed and demonstrated.
This research study contributes to the advancement in knowledge in the field of
GIScience about the conceptualization and representation of geographic data and
processes by implementing a 4D spatial data model in the simulation of dynamic
spatiotemporal phenomena. A computational geography approach provides a suitable
medium for validating the proposed theory through the development of prototype
implementations that can be further improved with real world data.
The methods for representing spatial processes have implications for the design of
spatiotemporal databases, the collection of geographic data and the development of new
algorithms for space-time analysis. Further study in all these aspects is needed and this
study serves to demonstrate the capability of the geo-atom spatial data model. For this
reason, a useable GIS software interface has not designed and the queries presented are
focused on just a few aspects of spatiotemporal analysis. GIS software being a numerical
and graphic tool, the aspects of graphic design and presentation outside of the MATLAB
144
computing environment, need to be considered as well in order to have an independent
4D GIS software product.
References
Batty, M. (2005). Network Geography: relations, interactions, scaling and spatial processes in GIS. In P. Fisher and D. Unwin (Eds.), Re-presenting GIS. John Wiley, Chichester.
Beni, L. H., Mostafavi, M. A., Pouliot, J., and Gavrilova, M. (2011). Toward 3D spatial dynamic field simulation within GIS using kinetic Voronoi diagram and Delaunay tetrahedralization. International Journal of Geographical Information Science, 25 (1), 25-50.
Billen, R., and Zlatanova, S. (2003). 3D spatial relationships model: a useful concept for 3D cadastre? 3D Cadastres, 27 (4), 411-425.
Bittner, T., Donnelly, M., and Smith, B. (2009). A spatio-temporal ontology for geographic information integration. International Journal of Geographical Information Science, 23 (6), 765-798.
Breunig, M., and Zlatanova, S. (2011). 3D geo-database research: retrospective and future directions. Computers and Geosciences, 37 (7), 791-803.
Burrough, P. A., Karssenberg, D., and Van Deursen, W. P. A. (2005). Environmental modelling with PCRaster. In D. J. Maguire, M. F. Goodchild and M. Batty (Eds.), GIS, Spatial Analysis and Modeling, Redlands, California.
Burrough, P. A., and McDonnell, R. (1998). Principles of geographical information systems. Oxford University Press, Oxford ; New York.
Cazorzi, F., and Fontana, G. D. (1996). Snowmelt modelling by combining air temperature and a distributed radiation index. Journal of Hydrology, 181 (1–4), 169-187.
Couclelis, H. (1992). People manipulate objects (but cultivate fields): Beyond the raster-vector debate in GIS. In A. Frank, I. Campari and U. Formentini (Eds.), Theories and Methods of Spatio-Temporal Reasoning in Geographic Space (pp. 65-77). Springer, Berlin.
145
Couclelis, H. (2010). Ontologies of geographic information. International Journal of Geographical Information Science, 24 (12), 1785-1809.
DeMers, M. N. (2009). Fundamentals of geographic information systems (4th ed.). Wiley, Hoboken, NJ.
Dias, D., and Tchepel, O. (2014). Modelling of human exposure to air pollution in the urban environment: A GPS-based approach. Environmental Science and Pollution Research, 21 (5), 3558-3571.
Dragicevic, S., and Marceau, D. J. (2000). A fuzzy set approach for modelling time in GIS. International Journal of Geographical Information Science, 14 (3), 225-245.
Dunn, S. M., and Colohan, R. J. E. (1999). Developing the snow component of a distributed hydrological model: a step-wise approach based on multi-objective analysis. Journal of Hydrology, 223 (1–2), 1-16.
Egenhofer, M. J., Clementini, E., and Di Felice, P. (1994). Topological relations between regions with holes. International Journal of Geographical Information Systems, 8 (2), 129-142.
Ellul, C., and Haklay, M. (2006). Requirements for topology in 3D GIS. Transactions in GIS, 10 (2), 157-175.
Erwig, M., Guting, R. H., Schneider, M., and Vazirgiannis, M. (1999). Spatio-temporal data types: an approach to modeling and querying moving objects in databases. GeoInformatica, 3 (3), 269-296.
Erwig, M., and Schneider, M. (2002). Spatio-temporal predicates. IEEE Transactions on Knowledge and Data Engineering, 14 (4), 881-901.
Frank, A. U. (1992). Spatial concepts, geometric data models, and geometric data structures. Computers & Geosciences, 18 (4), 409-417.
Galton, A. (2009). Spatial and temporal knowledge representation. Earth Science Informatics, 2 (3), 169-187.
Gately, C. K., Hutyra, L. R., Wing, I. S., and Brondfield, M. N. (2013). A bottom up approach to on-road CO2 emissions estimates: Improved spatial accuracy and applications for regional planning. Environmental Science and Technology, 47 (5), 2423-2430.
Gebbert, S., and Pebesma, E. (2014). A temporal GIS for field based environmental modeling. Environmental Modelling & Software, 53, 1-12.
146
Goodall, J. L., and Maidment, D. R. (2009). A spatiotemporal data model for river basin‐scale hydrologic systems. International Journal of Geographical Information Science, 23 (2), 233-247.
Goodchild, M. F. (1992). Geographical data modeling. Computers and Geosciences, 18 (4), 401-408.
Goodchild, M. F. (2010). Twenty years of progress: GIScience in 2010. Journal of Spatial Information Science (1), 3-20.
Goodchild, M. F., Yuan, M., and Cova, T. J. (2007). Towards a general theory of geographic representation in GIS. International Journal of Geographical Information Science, 21 (3), 239-260.
Grenon, P., and Smith, B. (2004). SNAP and SPAN: towards dynamic spatial ontology. Spatial Cognition and Computation, 4 (1), 69-104.
Hock, R. (2003). Temperature index melt modelling in mountain areas. Mountain Hydrology and Water Resources, 282 (1–4), 104-115.
Hubbard, S., and Hornsby, K. S. (2011). Modeling alternative sequences of events in dynamic geographic domains. Transactions in GIS, 15 (5), 557-575.
Jain, S., Goswami, A., and Saraf, A. K. (2009). Role of elevation and aspect in snow distribution in Western Himalaya. Water Resources Management, 23 (1), 71-83.
Jjumba, A., and Dragicevic, S. (in press). Integrating GIS-based geo-atom theory and voxel automata to simulate the dispersal of airborne pollutants. Transactions in GIS.
Jones, C. B. (1997). Geographical Information Systems and Computer Cartography. Longman, Harlow, England.
Kresse, W., Danko, D. M., and Fadaie, K. (2012). Standardization. In W. Kresse and D. M. Danko (Eds.), Springer Handbook of Geographic Information (pp. 393-566). Springer, New York.
Lake, R., Burggraf, D., Trninic, M., and Rae, L. (2004). Geography Mark-Up Language: Foundation for the Geo-Web. Wiley, Sussex.
Langran, G. (1992). Time in geographic information systems. Taylor and Francis, New York.
147
Langran, G. (1993). Issues of implementing a spatiotemporal system. International Journal of Geographical Information Systems, 7 (4), 305-314.
Langran, G., and Chrisman, N. R. (1988). A framework for temporal geographic information. Cartographica, 25 (4), 1-14.
Lee, G., Joo, S., Oh, C., and Choi, K. (2013). An evaluation framework for traffic calming measures in residential areas. Transportation Research Part D: Transport and Environment, 25, 68-76.
Lin, B., and Su, J. (2008). One way distance: for shape based similarity search of moving object trajectories. GeoInformatica, 12 (2), 117-142.
Longley, P., Goodchild, M. F., Maguire, D. J., and Rhind, D. W. (2011). Geographic information systems and science (Vol. 3rd). John Wiley and Sons, Hoboken, NJ.
Marceau, D. J., Guindon, L., Bruel, M., and Marois, C. (2001). Building temporal topology in a GIS database to study the land-use changes in a rural-urban environment. The Professional Geographer, 53 (4), 546-558.
McIntosh, J., and Yuan, M. (2005). Assessing similarity of geographic processes and events. Transactions in GIS, 9 (2), 223-245.
Mennis, J. (2011). Reflection essay: a conceptual framework and comparison of spatial data models. In Classics in Cartography (pp. 197-207). John Wiley & Sons, Ltd.
Nativi, S., Caron, J., Davis, E., and Domenico, B. (2005). Design and implementation of netCDF markup language (NcML) and its GML-based extension (NcML-GML). Computers and Geosciences, 31 (9), 1104-1118.
Nyerges, T., Roderick, M., Prager, S., Bennett, D., and Lam, N. (2014). Foundations of sustainability information representation theory: spatial–temporal dynamics of sustainable systems. International Journal of Geographical Information Science, 28 (5), 1165-1185.
O'Sullivan, D. (2005). Geographical information science: time changes everything. Progress in Human Geography, 29 (6), 749-756.
Peuquet, D. J. (1994). It's About Time: a conceptual framework for the representation of temporal dynamics in geographic information systems. Annals of the Association of American Geographers, 84 (3), 441-461.
Peuquet, D. J. (2002). Representations of space and time. Guilford Press, New York.
148
Peuquet, D. J., and Duan, N. (1995). An event-based spatiotemporal data model (ESTDM) for temporal analysis of geographical data. International Journal of Geographical Information Systems, 9 (1), 7-24.
Pultar, E., Raubal, M., Cova, T. J., and Goodchild, M. F. (2009). Dynamic GIS case studies: wildfire evacuation and volunteered geographic information. In (Vol. 13, pp. 85-104).
Raper, J. (2000). Multidimensional Geographic Information Science (1 edition ed.). Taylor & Francis, London.
Raper, J., and Livingstone, D. (1995). Development of a geomorphological spatial model using object-oriented design. International Journal of Geographical Information Systems, 9 (4), 359-383.
Rasinmäki, J. (2003). Modelling spatio-temporal environmental data. Integrating Environmental Modelling and GI-Technology, 18 (10), 877-886.
Thomsen, A., Breunig, M., Butwilowski, E., and Broscheit, B. (2008). Modelling and managing topology in 3D geoinformation systems. In P. van Oosterom, S. Zlatanova, F. Penninga and E. Fendel (Eds.), Advances in 3D Geoinformation Systems (pp. 229-246). Springer Berlin Heidelberg.
Torrens, P. M., and Benenson, I. (2005). Geographic automata systems. International Journal of Geographical Information Science, 19 (4), 385-412.
van Oosterom, P., and Stoter, J. (2010). 5D data modelling: full integration of 2D/3D space, time and scale dimensions. In (pp. 310-324). Springer-Verlag.
von Neumann, J. (1966). Theory of Self-Reproducing Automata. University of Illinois Press, Urbana.
Worboys, M. (1994). A unified model for spatial and temporal information. The Computer Journal, 37 (1), 26-34.
Yi, J., Du, Y., Liang, F., Zhou, C., Wu, D., and Mo, Y. (2014). A representation framework for studying spatiotemporal changes and interactions of dynamic geographic phenomena. International Journal of Geographical Information Science, 28 (5), 1010-1027.
Yuan, M. (1999). Use of a three-domain representation to enhance GIS Support for complex spatiotemporal queries. Transactions in GIS, 3 (2), 137-159.
149
Yuan, M., Nara, A., and Bothwell, J. (2014). Space–time representation and analytics. Annals of GIS, 20 (1), 1-9.
Zhao, Z., Shaw, S.-L., and Wang, D. (2015). A space-time raster GIS data model for spatiotemporal analysis of vegetation responses to a freeze event. Transactions in GIS, 19 (1), 151-168.
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Chapter 6. Conclusion
Geographic processes often behave like complex systems due to their nonlinear and
unpredictable dynamics. Although they evolve in three-dimensional space over extended
periods of time, their representation in a GIS is insufficient as the real world is often
collapsed to a two-dimensional plane. In addition, the temporal dimension is modeled by
organising an area’s spatial data as a series of snapshots each representing time along a
temporal continuum. Moreover, geographic representation is often limited to either an
object-based view of the world where space is modelled as a collection of geometric
objects, or to a field-based view where space is conceptualised as a grid of uniform cells
each having a unique attribute value representing the geographic theme at that location.
Yet, with the nature of geographic complexity and the need to analyse it, the internal
dynamics of some objects display field-like characteristics while some fields can be seen
as containers of independent spatial objects.
In this dissertation, a novel approach integrating voxel-based automata simulation and the
geo-atom data model for four-dimensional modeling is implemented to simulate several
spatial dynamic processes. The proposed suite of models represent a methodology well
suited to modelling dynamic phenomena and thus represent a foundation for exploring
further development of spatiotemporal analysis and visualisation of processes in four
dimensions. It is also proposed that the implementation of the voxel-based tessellation of
space and the geo-atom data model allows for the simultaneous modeling of the field and
object perspectives on phenomena without transforming data formats. These concepts
will assist with moving the field of GIScience towards a comprehensive representation
and analysis of spatiotemporal processes in the four-dimensions of the space-time
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continuum. In particular, the concepts of the geo-atom theory, which has been introduced
in recent years, are used to explain how geographic entities can be represented and
dynamic phenomena mimicked for four-dimensional analysis.
6.1. Summary of Findings
In Chapter 2, the voxel automata approach (cellular automata utilizing three-dimensional
volumetric units) was proposed, designed and implemented to model dynamic spatial
processes in the four dimensions of the space–time domain. The results indicate that the
developed voxel automata model, just like cellular automata - their two-dimensional
counterparts - are capable of generating three dimensional spatial patterns. Traditionally,
the automata paradigm forces the central voxel in a neighborhood to change depending
on the states of the other voxels in the neighborhood. This study proposed a relaxation of
this classical formulation when simulating geographic processes with the aid of voxel
automata. Simple deterministic rules were used to operate in a local neighborhood. The
model is capable of qualitatively reproducing the processes that are typically modeled
with differential equations or other non-spatial but stochastically-driven approaches. In
addition, this study demonstrates that the use of deterministic transition rules designed
intuitively using the non-totalistic approach are capable of generating dynamic behavior
that is analogous to the propagation of spatiotemporal phenomena in real life. Showing
the evolution of the phenomenon in the three spatial dimensions allows for examining the
process from multiple perspectives and viewpoints not possible otherwise. The study
further confirms that the use of different neighborhood configurations will result in
different spatial patterns as other researchers have demonstrated for two-dimensional
cellular automata. Topological queries were applied to the data that represents the three-
dimensional forms generated from the model simulation.
Similarly, voxel automata were used in Chapter 3 to demonstrate the transport of airborne
pollutant particles. The voxel automata draw on the elements of the geo-atom data model
to record, for spatiotemporal analysis, the changes that occur in each of the voxels in the
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system. This is particularly advantageous when modeling complex systems because it
allows for further investigation of the attributes of the system’s components. Although,
the model simplifies some of the elements related to the dynamics of particle spread, the
results indicate that it is sensitive to the parameters that define rate of diffusion as well as
the wind speed. The data generated from running the simulation are stored in a structure
of a MATLAB matrix and can be investigated further for spatiotemporal analysis.
In Chapter 4, the increasing interest in the indices that quantify the topographical aspects
of the landscape is emphasised. The described indices are for the number of volumetric
patches in a landscape. They provide measures for the volume, surface area, expected
surface area (if the same number of voxels is compacted), fractal dimension and
lacunarity for each volumetric patch. These indices provide the information necessary to
determine the points at which a phenomenon merges or agglomerates and when particular
volumes disappear and get formed. The indices presented in this chapter can be used to
quantify geological formations and other volumetric phenomena in the landscape. In
order to achieve this, it was necessary to represent the phenomena using the geo-atom
data.
In chapter 5, two proof-of-concept models that integrate the geo-atom concepts with the
complex systems theory were been developed to represent the real world characterises.
First, automata system was used to demonstrate the dynamics of a spatial phenomenon
that is spreading as if by diffusion in 3D geographic space. The second model simulates
spatial contraction to demonstrate a receding process such as the spatiotemporal change
of snow cover. Although the dynamics of these processes are more complex they were
used here in a reduced since form the objective was to demonstrate dynamic changes in a
field-like geographic space. During the course of the simulation, the attribute values of
the voxels are updated together with the time-step at which the change in value takes
place. Specific queries related to the evolution of a process such as the rate of movement
and the frequency of occupancy have been developed and demonstrated.
153
6.2. Perspectives on Further Research Work
The methodologies explored in this study extend previous work in the field and provide a
platform that can be used to simulate the evolution of spatial and geographic processes in
the four dimensions of the space-time domain. The models developed provide a
theoretical foundation for integrating voxel automata and the geo-atom theory.
Nonetheless, integration of empirical data, with improvements in specification and
testing, they could be used in the construction of ‘what-if’ scenarios that could be
integrated within spatial decision support systems. However, future work needs to focus
on applying the principles presented here to real-world situations by translating the voxel-
based automata approach into more real-world contexts. Unlike the subject area of
geography, GIScience research is limited by the lack of widely accepted and well tested
theoretical principles. The continued development and implementation of the geo-atom
concepts helps to address this limitation by defining a comprehensive geospatial data
model for geographic representation in GIS. For example, recent research efforts on
applied spatiotemporal representation (Fang and Lu, 2011; Nyerges, et al., 2014; Torrens,
2014) also provides a good basis for spatiotemporal investigation with the data model
because there is a need for four-dimensional geospatial datasets that can be integrated in
GIS settings.
In a wider perspective, this dissertation demonstrated the potential and the merits of using
the geo-atom data model, which is a unified field-object data model for four-dimensional
geographic representation. However, a useable interface as would be expected of a GIS
application interface was not developed and the queries presented are focused on selected
aspects of spatiotemporal analysis and four-dimensional data encoding and storage. The
development of a GIS software application that has both a numerical and graphical tool
as well as the modules for data storage, retrieval and processing needs to be considered.
Further integration with GIS will require an in-depth examination of how existing
databases with geospatial data in the raster and vector format can be restructured to
support the geo-atom model. This would make the development of a suite of fundamental
154
GIS analytical functions attractive and it would, in turn, make the adoption of the geo-
atom data model easier.
In GIS modeling, it is necessary to discretize the geographic space and the temporal
continuum in order to achieve effective spatiotemporal representation and to satisfy the
need for efficiency in data storage and management. In this study, uniform cubic units,
the simplest volumetric units with which space can be discretized are used to model
dynamic phenomena. However, this presents challenges for the spatial resolution along
the vertical axis where the geographical expanse tends to be much less than along the
horizontal plane. Attention to the appropriate voxel dimensions needs consideration in
further descriptions of voxel-based automata modelling.
The process of developing a simulation model involves an examination of its behavior
under various parameter settings and how its outputs compare to some other independent
observations of the dynamic process. Normally, rigorous procedure for calibration,
sensitivity analysis and validation would need to be applied. However, the prototype
models developed in this dissertation have undergone nominal calibration procedures to
ensure the model, qualitatively matches what would be expected in the real word.
Without a comprehensive set of geospatial data validation, the validation and sensitivity
analysis are not particularly relevant. Model testing procedures, the incorporation of real
datasets as well as the handling of uncertainty outputs are beyond the scope of this
research which focussed on the theoretical levels of simulation modeling four-
dimensional spatiotemporal processes and using hypothetical landscapes.
6.3. Thesis Contribution to the Body of Knowledge
This research contributes towards the previous efforts of GIS researchers who have been
examining and proposing ways to handle geographic processes, spatial changes and
movement for the last three decades. In GIScience, the methods for representing spatial
processes have implications for the design of spatiotemporal databases, the collection of
155
geographic data and the development of new algorithms for space-time analysis. The
study adds to the advancement in knowledge about the conceptualization and
representation of geographic data and processes by demonstrating the use of a four-
dimensional voxel-based automata model in simulation of spatiotemporal phenomena. It
is particularly focused on the application of the geo-atom data model, which is well
suited for four-dimensional spatiotemporal representation and analysis. Currently, the
prototypes described in the dissertation are among some of the first models using a
unified object-field geospatial data model for complex four-dimensional simulation
modelling. In addition, this data model allows for a three-dimensional representation of
spatial entities as well as the querying of topological relationships. The topological
relationships of adjacency, connectivity and containment of spatial objects are used as
examples that can be generated from the geo-atom data model. Furthermore, the indices
typically used to characterise a landscape such as three dimensional surface areas,
volumetric measures for the phenomena, fractal dimension and lacunarity have been
developed. In addition, spatiotemporal queries are also proposed which could help in
making inferences about cause and effect in spatial dynamics.
The work presented in this dissertation is of particular relevance in geospatial data
management and the modeling of spatiotemporal processes in four dimensions. The
contributions are of primary relevance to the development of geographic automata
systems in the field of GIScience. Besides the applications in GIScience and geography,
the proposed work can be applied to represent four-dimensional processes in the
application areas of geomorphology, hydrology, climatology, environmental science and
urban planning where spatiotemporal dynamics regularly occur and are investigated.
References
Fang, T. B., and Lu, Y. (2011). Constructing a near real-time space-time cube to depict urban ambient air pollution scenario. Transactions in GIS, 15 (5), 635-649.
Nyerges, T., Roderick, M., Prager, S., Bennett, D., and Lam, N. (2014). Foundations of sustainability information representation theory: spatial–temporal dynamics of
156
sustainable systems. International Journal of Geographical Information Science, 28 (5), 1165-1185.
Torrens, P. M. (2014). High-resolution space–time processes for agents at the built–human interface of urban earthquakes. International Journal of Geographical Information Science, 28 (5), 964-986.
157
Appendix A. Co-authorship Statement
Chapter 2: Voxel-based Geographic Automata for the Simulation of Geospatial Processes
Co-author: Suzana Dragićević
Role of co-author: Manuscript preparation
Chapter 3: Integrating GIS-Based Geo-Atom Theory and Voxel Automata to Simulate the Dispersal of Airborne Pollutants
Co-author: Suzana Dragićević
Role of co-author: Manuscript preparation
Chapter 4: Spatial Indices For Measuring Three-Dimensional Patterns in Voxel-Based Space
Co-author: Suzana Dragićević
Role of co-author: Manuscript preparation
Chapter 5: A Representation and Analysis of a Process' Spatiotemporal Dynamics Using a 4D Data Model
Co-author: Suzana Dragićević
Role of co-author: Manuscript preparation
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