School of Computer Science and Software Engineering
Monash University
I declare that the work presented in this thesis is my own, and has not
been submitted in any form for another degree, diploma or award at any
other university or institution. Any information taken from other sources,
either published or unpublished, has been acknowledged in the text, and
a list of references accompanies this thesis.
Suzannah Molloy
November, 2002
The effects of the processes of erosion and deposition are extrememly varied, as can be seen from the wide variety of features in a natural landscape. This large variety is a result of many different variables. This thesis proposes a model for creating some of these characteristic features, with the focus being primarily on sand particles. The model is a very simplified approximation to the processes that occur in the natural world. The modeling technique chosen is particle systems. Particle systems lend themselves very well to this problem, as erosion and deposition are themselves processes involving particles. The rules describing the behaviour of the particles in the model are a mixture of real, physical laws, and simple approximations designed to reflect actual behaviour. The results from this model are compared with natural sand formations found in the natural world. A number of features are produced by the model which are reflections of features found in nature.
The natural landscape is a complex and varied structure, formed over a great length of time by countless physical processes that include the erosion and deposition of particles. Landscape models are used for a variety of purposes such as for generating images of terrain that may be used in a flight simulator, or perhaps for studying the effect of some force acting on the landscape. Depending on what the model is to be used for, the intention may be to produce a model that is capable of generating images that look as realistic as possible, or the intention may be to produce a model that is physically accurate, with less emphasis placed on the generation of visually pleasing and believable images. Regardless of the purpose of the model, most current methods of generating terrain do not take into account the features produced by erosion and deposition, and as a result, lack a great many features that are characteristic of these processes.
While still very realistic in appearance, methods not including the effect of erosion and deposition processes tend to produce terrain that is statistically similar over the whole surface. In a natural landscape, such features as depressions in the ground tend not to display quite the same characteristics as mountains and hills. The collection of debris at the bottom of slopes is one very characteristic feature of deposition, which is not modelled by most current landscape modelling techniques.
Erosion and deposition are dynamic processes, and this project aims to model the effects of these processes on terrain by using a dynamic modelling process, specifically, particle systems. Due to the deep complexity of the processes of erosion and deposition, extremely simplified models will need to be designed which are still capable of capturing the basic processes involved. The investigation will also concentrate primarily on sand particles, as these are arguably the most simple particles to model. Actual examples of sand dunes which are one of the more common results of erosion and deposition involving sand, are readily available. Results from the particle system models will be compared with true examples to gain some indication of the accuracy of the features obtained from the models. It is hoped that by the end of this investigation, a model will have been developed which has some degree of physical accuracy, and is capable of producing simple landscapes which display some of the features characteristic of the processes of erosion and deposition.
The remainder of this thesis is organised in the following way. Section 4 will begin by looking at erosion and deposition processes in the natural world, followed by a discussion of previous work in related areas, including the use of particle systems to date, and how they may be used in this investigation. Section 5 follows on from this with a description of the design and implementation of a particle system intended to model the erosion and deposition processes. The resulting images produced by the model are given in section 6 along with a comparison of these images with examples from the natural world. Discussion of results follows in section 7 , with conclusions and a discussion of possible future work is given in section 8 .
Erosion and deposition of terrain is a naturally occuring phenomenon, therefore this section will begin by looking at these processes in the natural world, with examples of characteristic landscape features, and a discussion of the causes and contributing factors. As the intention is to produce a model of these phenomena, terrain modeling techniques, in particular the popular fractal technique will be discussed along with drawbacks associated with these methods. Finally, this section will look at particle systems, and how a solution to the aforementioned drawbacks may be possible by developing a model for erosion and deposition based on these systems.
In the context of this investigation, the term erosion refers to the removal of particles from the landscape, while deposition refers to the dropping of particles. In both cases, some eroding medium provides the force required to pick up, transport or deposit the particles. This eroding medium may be flowing water, wind, or a creeping glacier, among others.
The effects of these processes are governed by a great many factors[ 1 ][ 3 ][ 10 ][ 4 ]. On the macroscopic scale, the current shape of the landscape, weather conditions and ground coverage will have a great effect on the erosion and deposition process. On a much smaller scale, the size and shape of the individual particles, their cohesion and adhesion, the angle of shearing resistance[ 3 ] and many other factors all combine to make the process of erosion and deposition a very complex one. It is not easy to describe exactly how the landscape will be effected by forces acting upon it.
Flowing water is perhaps one of the most common causes of erosion and deposition, with effects ranging from small drainage ditches and silt deposits, to deep canyons and ravines. Water is greatly restricted by gravity, and so flows downhill, unlike air which may flow in many more directions, including uphill. As a result, water carries sediment downhill, and tends to run in ditches and gullies carved out by previous water erosion. The more water that runs through these pathways, the more pronounced they become, and given sufficient time, they can become very deep. A classic example of water erosion on a large scale is the Grand Canyon in the US, (figure 1 ). What is clearly visible is the intricate network of gullies carved out of the canyon walls by water flowing from the rim to the bottom.
The Grand Canyon was formed over many thousands of years, almost entirely by the erosive power of water. What may have started out as a relatively flat plain, has been carved into an intricate system of gulleys and ravines, that stretches for some 217 miles. At it's widest point the canyon is about 13 miles wide, and reaches depths of over 6000 feet. Water running down the canyon walls picks up sand and dirt particles and carries them down to the Colorado river that runs through the middle of the canyon. The Colorado river carries this sediment away, where it is deposited somewhere further downstream. Water does not carry sediment for an indefinite length of time. Eventually, even sediment caught in fast flowing currents will be deposited. Ripples in the silt of a stream bed are a feature of particle deposition by water. In general, water is only capable of eroding small particles by rolling them along the bed of the stream, or by literally picking the particles up and carrying them some distance.
Like water, wind is usually only capable of moving fairly small particles, but is less restricted in the direction it can flow. This type of erosion and deposition caused by air flowing over the land is referred to as aeolian [ 25 ], and occurs mainly in dryer areas such as deserts. The force required to transport particles is much less if the particles are dry and free to move around, like sand and dry snow.
Aeolian transport is carried out in the following three ways[
25
]:
1) Suspension: the particles are picked up and carried
by the wind, before being deposited. Only small particles are capable
of being transported by suspension.
2) Saltation: the particles bounce along the surface, often
knocked loose in the first place by falling particles being transported
by suspension.
3) Surface creep: the particles roll along the surface.
The type of aeolian transport which moves a particle will depend on the mass of the particle. Lighter particles can be transported by suspension, and as the mass of the particles increases, the mode of transport changes from suspension, to saltation, then to surface creep, until the particles are too heavy for the wind to move.
Sand dunes are a common example of erosion and deposition caused by the wind. The shape of the dune depends on a number of factors, such as the direction, regularity and strength of the dominant and cross winds, the size, density and consistency of the sand, and the presence of any obstacles such as trees or rocks[ 4 ][ 26 ]. The dune in figure is a seif dune, displaying the characteristic sharp ridge across the top. Seif dunes require a strong dominant wind, and strong periodic cross-winds to form, and run parallel to the direction of the dominant wind. The sharp ridge running the length of the dune is carved by the cross-winds[ 4 ].
Under different conditions, a very different sand dune is produced, such as the barchan dune in figure 3 . Barchan dunes are the result of a strong, steady dominant wind, and are shaped similar to a horseshoe. The length of the dune runs perpendicular to the direction of the dominant wind rather than parallel to it as was the case with the seif dune.
In mountainous regions where snow is able to remain unmelted even during
summer months, it is possible for glaciers to form. Snow builds up, and
as the depth increases, the pressure builds up causing the snow at the
bottom of the heap to compact. With enough pressure, the snow will compact
to form ice. As the pressure increases, the ice at the very bottom is
forced to thaw and become liquid water. This water begins to flow, carrying
the ice with it, and so the glacier is born. This is known as a warm
glacier , and these tend to occur in slightly warmer regions, that
allow the ice at the bottom to thaw. In colder regions, where the ice does
not thaw, but instead remains frozen to the bedrock, the glacier can still
begin to flow given a steep enough slope, and enough pressure from above.
In this case, the ice flows by deformation under its own weight. These
are known as cold glaciers[ 11
].
Compared to wind and water, glaciers move much more slowly, perhaps
only a few centimeters a day, but are capable of carrying anything from
fine rock flour (like the sediment in a stream), to rocks the size of
a house. Moraine is the term used to describe debris carried by
a glacier, whether it be rock flour or huge boulders.
Glaciers erode the landscape in a number of different ways:
1) Loose material from the underlying bedrock is picked up and
carried with the glacier. This becomes moraine.
2) Moraine carried at the interface between the glacier and the
bedrock abraids the bedrock surface, eroding particles, which then become
moraine themselves. This process is known as abrasion.
3) A reduction in pressure allows the liquid water at the bottom
of a warm glacier to refreeze, attaching the moving ice to the bedrock.
The ice continues to move and may pull large chunks of rock free.
4) Rock pieces may fall from the surrounding walls due to frost
shattering, or freeze-thaw activity. Although not technically eroded
by the glacier, the glacier provides the medium by which these rock pieces
are then transported.
Many features caused by glaciers are a result of erosion rather than the deposition of moraine, such as the arete in figure 4 . Aretes are formed when two small glaciers in hollows next to each other (corrie glaciers ), erode the side walls away to the extent that all that is left is a sharp ridge.
Moraine carried by the glacier eventually reaches either the mouth of the glacier or one side or another. It may be deposited, or if small enough, meltwater may carry the particles away to be deposited further downstream. This deposition of moraine creates large piles of debris which build up around the sides and snout of the glacier, as in figure 5 .
Glaciers are not the only way in which snow and ice participate in the process of erosion and deposition. Snow that builds up on a slope may become unstable, the result of which is an avalanche. In this situation, a potentially large amount of snow may be very quickly removed from one part of the landscape and deposited further downslope. The underlying terrain is relatively unchanged, however the visible surface greatly differs after the occurance of an avalanche. Where the slab of snow actually fails depends on many factors, as was mentioned earlier in regards to erosion and deposition in general. Earth slopes may also fail in a similar way to snow, with the result being a landslide or rockfall. For further discussion of slope instability, the reader is refered to Bromhead[ 1 ] and Chowdhury[ 3 ], and for avalanches, to Fredston and Fesler[ 10 ].
Some factors controlling the effects of erosion and deposition have been mentioned in previous sections. There are however, a great many more factors which effect the outcome of the erosion and deposition process, helping to create a great variety of natural features.
As previously mentioned, different types of particles behave in different ways. Damp snow blown by the wind produces different ripple patterns compared to the patterns produced by dry sand under the same wind conditions. The snow is more likely to stick to other particles, making it more difficult to transport. The gradient of the slope also effects what happens to particles that land there. If the slope is too steep, the particles will slide off, perhaps dislodging in large clumps in the case of an avalanche. Sticky particles, such as damp snow, are more able to stick to steeper slopes than dry particles, as the list of angles of shearing resistance shows in Chowdhury[ 3 ].
The internal properties of the particles which the terrain consists of also play a large part in determining how the erosion/deposition process proceeds. The cohesion and adhesion of materials helps determine the angle of shearing resistance, as do the masses of the particles. The hardness of the terrain is also a factor. Softer material will erode faster than hard material, and so differing layers of hardness in the landscape can help to produce an interesting array of features. Figure 6 shows a tall, thin spike rising up out of an assortment of other rocks. This feature is mainly a product of wind erosion, with the tall spire being composed of a softer rock than that which makes up the base, hence the top has eroded faster than the base.
With so many factors involved in determining how the erosion and deposition processes are to proceed, the task of modeling these processes may very easily become quite complex. It becomes apparent that a model will need to be a greatly simplified model of erosion and deposition to make it feasible to use.
Terrain is generally modeled using surface modeling techniques, rather than volume modeling methods. Volume modeling methods, such as particle systems, are very well suited to the generation of amorphous substances such as fire, and will be discussed shortly. As terrain is a solid, well-defined structure, it is reasonable to model it with surface modeling techniques. Surface modeling techniques aim to describe the surface of an object with polygons. The fractal technique is perhaps the most well-known surface modeling technique for landscapes, and is discussed in the next section.
One of the major problems faced when using a computer to generate landscapes,
is how to produce something that exhibits the irregularity of a natural
landscape. A solution to this problem began to emerge with the introduction
of fractional Brownian motion by Mandelbrot and Van Ness[
18
], and Mandelbrot's subsequent observation of the similarity between mountain
peaks on a skyline and a record of Brownian motion over time[
17
]. The idea that the irregularity of terrain could be modeled using Brownian
motion was expanded upon by Mandelbrot who produced some images generated
using the fractional Brownian motion method. Fournier et al [
9
] later used an approximation to Mandelbrot's fractional Brownian motion
as a very effective model for representing landscapes. These fractal landscapes
are now arguably the most popular method for generating terrain. They
are relatively simple to implement, and produce quite realistic and visually
pleasing results. The mountain in figure
was produced using the midpoint version of the fractal technique[
9
]. Simply speaking, the fractal technique is carried out in the following
way:
1) Beginnning with a single triangle, the midpoint of each edge
is found
2) All midpoints are perturbed upwards by some distance proportional
to the side length
3) The resulting perturbed points, and the original vertices are
joined together to form a new set of triangles
4) For each triangle, the process is repeated
Using this method, detail can be calculated down to an arbitrary
level.
A different approach to terrain modeling was taken by Kelley et al[ 15 ]. Their idea was based on the idea that terrain is shaped primarily by the path of running water, and erosion and weathering restrict the amount of self-similarity in natural terrain. Fractal methods on the other hand are based on self-similarity, where the statistical properties of the surface are very similar over the whole area. Kelley et al's model is based around real-world hydrology data, which is used to describe a network of streams that flow over the landscape. These streams determine what the profiles of the valleys will be. This technique is also a type of fractal method, as the level of detail in the image can be increased by increasing the level of detail in the stream network. Resulting images of terrain generated using this method are also quite convincing. They display a large amount of randomness which is typical of a natural landscape.
While the very popular fractal method generates quite realistic terrain, it lacks a great many features that are typical of natural terrain. Current landscape modeling techniques do not capture features that are characteristic of the process of erosion and deposition. While terrain generated using a fractal method is quite convincing, a greater level of physical accuracy may be gained by modeling these effects. The fractal technique requires some simple piece of terrain as a seed, so to some degree, these features can still be modeled. By using a seed which already demonstrates some of these features, terrain can be generated that may appear to have been partially affected by erosion and/or deposition. However, characteristics such as the intricate system of gullies and ravines in the Grand Canyon, or ripples across a sand dune are still uncommon features in the average computer generated landscape.
As mentioned, the results of the fractal method are very good, and convincing images of nature can be produced relatively easily. However, the shape of terrain covered by substances such as snow can differ significantly to the shape of that same terrain without covering. Snow has the ability to cling to surfaces, and build up around objects, as does dust, although perhaps not to the same degree as snow. The basic fractal method does not provide a particularly realistic approximation to covered terrain.
Methods for handling covered terrain and surfaces has been investigated by a number of people. Nishita et al[ 21 ] proposed a volumetric method for modeling snow covered objects and snow piled up around the sides of objects, using metaballs. More realistic effects were gained by describing a method to calculate the scattering of light by snow particles. The same method was also used to model clouds. The resulting images show rather unrealistically large clumps of snow, but the method produces quite satisfactory results for clouds.
A more visually convincing method for generating snow covered objects was proposed by Fearing[ 8 ]. The model in fact consists of two separate models, one for snow accumulation to determine how much snow a surface receives, and the other for stability. The stability model moves material away from unstable areas by avalanching the material down to lower surfaces. The model is able to produce a much more complex image than that demonstrated by Nishita et al.
Dust accumulation, as modeled by Hsu and Wong[ 13 ], calculated the amount of dust that a surface was expected to receive based on the surface properties and geometry of the object. This calculated amount was then used to determine how the light would interact with the surface. The dust that was calculated to fall did not have a volume, and was merely used in conjunction with the light reflection technique to give the surface the appearance of being covered with dust, without actually increasing the thickness of the surface to account for the dust layer.
Researchers have made a number of investigations into methods for modeling erosion and deposition. Some earlier attempts to model the appearance of weathered, naturally worn surfaces, involved the use of texture maps, such as in the Pixar movie ``Toy Story''[ 5 ]. Difficulties with texture maps arise when trying to match the patterns across boundaries. The work involved in trying to produce a natural, and seamless image is difficult and can become quite laborious.
Chen and Fu[ 2 ] attempted to model the behaviour of dust as a vehicle passes over the landscape using amongst other things, particle systems. Although the method is able to determine the behaviour of the dust that is kicked up by the wheels, there is no modeling of any effects on the actual surface of the landscape. Regardless of how much dust is kicked up, the underlying shape of the landscape never changes.
Dorsey et al, while investigating a method for simulating the effects of environmental weathering on surfaces, in particular the patterns and stains left by flowing water, developed a model for deposition of sediment[ 7 ]. A flow model controlled where the water was able to flow to, and hence where the sediment may be deposited. This method also incorporated the use of particle systems, which were used to model the flow of water. The method developed was not expanded to include the erosion and deposition of particles in terrain models, although it may lend itself to that application very well. Dorsey and Hanrahan continued on in a similar manner, to model metallic patinas [ 6 ]. A patina is a film or encrustation of a surface by the removal or deposition of material, or the chemical alteration of the surface. Some examples are painting, where a film is being added to the surface, and oxidation, which is a chemical reaction altering the surface of a metal. The patina grows in layers, and the thickness increses with age. Similarly, dust, snow and sand may be deposited on a surface, with the thickness of the layer increasing with age. Dorsey and Hanrahan simulate the thickening of the layers over time by the use of one of several models of deposition. The models include steady, uniform thickening with time, random deposition, where particles are randomly deposited on the surface, and a ballistic deposition which is similar to the random deposition model, but includes lateral growth. The random deposition model incorporates a stabilizing technique where particles tend to move towards the lower regions.
As with the model in Dorsey et al[ 7 ], the method of Dorsey and Hanrahan[ 6 ] could potentially be used for a model of deposition of particles onto terrain. In particular, the random deposition may resemble the deposition of dry sand particles, which fall, or are blown along, and land in a random position, but tend to settle into a stable configuration. The ballistic model may be suitable for snow deposition, as the particles are able to stick to each other without actually coming to rest at the surface, hence the lateral growth. As snow particles are able to stick to each other in a similar way, there is the potential for lateral growth of snow particles also.
An erosion model was also part of the method described by Sumner et al[ 27 ] for the animation of sand, mud and snow when impacted by an object such as a falling runner, or the feet of a walking person. It was necessary to deal with the compression or removal of particles at the point of impact. The surface material, i.e. sand, mud or snow, was able to compress to some degree, while any material that could not be compressed was removed by way of the erosion model. Heightfields were used to represent vertical columns of ground/terrain material. The slope between the tops of neighbouring columns was compared, and if greater than the threshold, then material would be moved from the higher column to the lower column. In comparison to the physical world, the threshold slope would correspond to the angle of shear resistance[ 3 ]. This model, by Sumner et al produces very satisfactory images of footprints in sand, mud or snow when compared to images of real footprints.
A strongly physically based model was introduced by Li et al[ 16 ] to model the interaction of soil with the blade of a bulldozer. The model was designed to run in real time, where some range of actions would be taking place, such as excavations, piling up dirt and other such activities to do with the movement of soil. The model takes into account a number of factors mentioned earlier, which play a part in the effect of erosion and deposition. Namely, cohesion, adhesion, unit weight and internal friction in particular are used to calculate the angle of shear resistance. The information is used to calculate the stability of a slope, and if slides are to occur, where they will occur. The volume of soil is conserved in this model, unlike that of Chen and Fu[ 2 ], and is strongly related to the work presented in the paper by Kass et al[ 14 ], on modeling the flow of fluids.
Perhaps the best results so far, at least in a visually pleasing sense, are those obtained by Musgrave et al[ 20 ], who use the fractional Brownian motion technique to generate detailed terrain, and then run an erosion simulation on this terrain. Included is a model for hydraulic erosion, that being erosion caused by flowing water, and a model for thermal weathering, which covers any other type of erosion where material is worn away from slopes and deposited further downhill.
There have been a variety of methods put forward to deal with the problem of modeling terrain that exhibits the characteristic features of erosion and deposition. Other methods which have been used to address different problems may prove to be useful in this context also, such as the work mentioned previously by Dorsey et al[ 7 ] on environmental weathering of surfaces, and Dorsey and Hanrahan[ 6 ] on metallic patinas. Some of these methods may be worth investigating further. For this project however, the intention is to model the processes of erosion and deposition using particle systems.
Prior to the appearance of particle systems in 1983[ 22 ], most objects, both solid and amorphous, were modelled with surface modelling techniques. As described earlier, these techniques attempt to approximate the shape of an object with flat polygons or solid primitive shapes. For objects with well defined surfaces, this was quite adequate. However, for objects without well defined surfaces, such as clouds, fire and explosions, surface modelling techniques proved somewhat inadequate. These are dynamic objects without a rigid structure, and surface modelling techniques are not suited to modelling these.
In 1983, W.T. Reeves proposed the particle systems modelling technique[ 22 ]. Particle systems model the volume of an object rather than the shape of the surface by using a cloud of primitive particles. The particles themselves are not static. They are introduced to the system as they are needed, survive for some length of time, and then are allowed to die. Particle systems are able to model non-deterministic objects, such as clouds and explosions[ 22 ][ 19 ] far more effectively than current surface modelling techniques.
In addition, particle systems are not restricted to the modelling of ill-defined objects. Solid objects can also be modelled by their volume rather than surface shape. Reeves included an image in his '83 paper[ 22 ], of a clump of grass generated using particle systems by Alvy Ray Smith of Lucasfilm. Trees and forests were also modelled using structured particle systems by Reeves and Blau [ 23 ]. In an unstructured particle system, each individual particle is independant of all the other particles. The initial values of each attribute are assigned based on random distributions from when the particle is first generated. Structured particle systems differ, in that they are no longer independant of all the other particles. With structured particle systems, the intent may be to produce a solid, three-dimensional object such as a tree, which needs to exhibit a cohesive structure. Structured particle systems have a greater degree of complexity, and are able to model very complex objects such as forests.
As mentioned in the previous section, normal particle systems are made up of simple point light sources, that act as the modeling primitives rather than simple polygons as in surface modeling techniques. For this project, only basic particle systems will be used, rather than the structured variety. Each particle has a number of different attributes, such as colour, radius, velocity and position. When a particle is injected, or born, into the system, these attributes are set, and the behaviour of the particle over time is governed by these attributes. If given a lifetime, the particle will be allowed to exist for the given length of time, and then removed from the system.
For a model of an explosion, such as the one in figure 7 , the particles may start with a velocity, acceleration, position, colour and radius. As time passes, the particle may burn up, causing the radius to decrease, and the colour to darken. If the force of gravity is included in the model, the particle's velocity may increase in the downwards direction, and the position of the particle at any one time may be calculated from the previous position, velocity and acceleration. In this way, the behaviour of a particle can be controlled with only a few parameters.
As discussed in earlier sections, the modelling of erosion and deposition has to some degree been carried out, using a variety of techniques. Some techniques, such as Musgrave et al's[ 20 ] are not physically accurate, although the resulting images appear realistic and believable. Particle systems are made up of very simple primitives, that can be controlled by very few rules. These rules can be designed to model true physical laws, such as basic laws of force, velocity and acceleration. Due to this, it is possible that a more physically accurate model of erosion and deposition processes may be developed. Existing models of erosion and deposition, such as those used by Dorsey and Hanrahan[ 6 ] or Sumner et al[ 27 ] may be extended to the current research project.
As fractal techniques are already known to produce high quality images of terrain, it is hoped that a method can be developed which will model the basic structure of the terrain using the fractal technique and retain the high quality of detail which this method produces. As with Musgrave et al[ 20 ], the simulation of the erosion and deposition process is expected to follow the generation of the basic terrain. What needs to be developed is possibly a single process, possibly separate processes, for the determination of erosion and deposition effects. The use of particle systems should allow for a greater degree of physical correctness in the model compared to some of the models researched previously.
Designing a model to implement the processes of erosion and deposition was carried out in a number of steps. Firstly, as erosion and deposition is such a complex process, the problem needed to be simplified. Hence, the problem was separated into three parts. Design a model to simulate deposition, then design a model to simulate the erosion process, and a third model to stabilize the landscape. The models were designed and implemented, after which the three were combined to produce the final model. The deposition model involved injecting airborne particles into the system, and tracing their progress, until they collided with the landscape and came to rest. The erosion model involved removing particles from the landscape according to a number of rules, and determining where they eventually came to rest. Particles which became airborne as a result of the erosion process were controlled by the deposition model. The stabilization model involved checking all slope angles, and adjustin any that were considered unstable. This section details the design of the individual models, and the process by which they were combined.
As discussed earler, there are a great many variables involved in the process of erosion and deposition, and as a result, the variety of landscape features is vast. It was therefore not feasible to develop a complete model of the process, but only to concentrate on a small selection of variables and features. The basic shape of sand particles is extremely varied, and the shape, size and density of the particles affects their behaviour, and hence the effects of sand erosion and deposition. For this project, the emphasis was on sand particles, as they can be easily generalized to simplify the problem. The particles were generalized such that all particles were uniform in size, shape and density. This simplified the rules which would control the way in which sand was eroded and deposited on sand dunes. The physics controlling the behaviour of the particles is also a complex process, so simplifying the particles was not enough. In most cases, approximations were used to describe the behaviour of the particles under various conditions. These approximations will be discussed in further detail in later sections were appropriate.
The final representation of the terrain structure was a two-dimensional landscape. All particles were of a uniform shape and size, so all particles had an identical diameter. The terrain was divided into sections, with each section being the same width as the diameter of a particle. A single column of particles could be held in each section. An array was used to store the information on each section, this information being the number of particles in the column, and a pointer to the first member of a linked list.This list contained each particle in the column, and each particle held a pointer to the particle directly above it. The last particle, at the top of the column, held a null pointer. Particles could be added and removed from anywhere in the list, and the counter, recording the number of particles in the column, was updated to reflect these changes. This method of representing the terrain is similar to the method used by Sumner et al for representing terrain material while animating sand, mud and snow [ 27 ].
As mentioned earlier in section 4.1 , the term deposition refers to the dropping of particles onto the landscape. The deposition model therefore deals with sand particles undergoing suspended aeolian transport which are then deposited on the landscape to form sand dunes. A method of dealing with airborne particles was required. This method covered the motion of the particles through the air, a collision detection algorithm to determine if the particle had collided with the landscape at any point. If the particle did collide with the landscape, the method needed to calculate where the final resting place would be. Sand dunes also offer some shelter to airborne particles, and a rule for calculating the sheltering effects of a dune was also incorporated. The simulations were run by injecting airborne particles into the system at pseudo-randomly determined initial heights. These heights were not purely random, as a bias was given against height. The probability of a particle being injected at a certain heigth decreased as the height increased. In a natural situation, there are generally fewer airborne particles as vertical distance from the landscape surface increases.
Particles suspended in the air have two main forces acting on them. The forces of gravity and wind resistance. Gravity is quite straightforward as it can be considered constant near the surface of the Earth. So the force of gravity acting on any airborne particles was taken to be 9.8 m / s2 downwards. Wind resistance on the other hand becomes a little more complicated. The force of the wind pushes the particle along through the air, and wind resistance also acts against the particle's motion through the air. In the downwards direction also, wind resistance effects the downwards motion of a particle. The addition of wind resistance, while adding to the degree of physical accuracy present in the model, would be outweighed by the increased complexity. The effect of wind resistance was not included in the model as a working, simplified model was not completed in time. Therefore the only force taken to be acting on airborne particles was gravity. Airborne particles still required a horizontal velocity however, so the velocity given was equivalent to the speed of the wind.
Given these simplifications, the behaviour of an airborne particle can
be descibed by the basic physical laws governing motion:
|
|
Equations 3 and 5 were used to describe the uninterrupted motion of airborne particles. However, the main purpose of the model was to deposit particles on the landscape, so a collision detection algorithm was developed to determine when a particle collided with the landscape. It was not sufficient to merely test if a collision had occured. It was also necessary to know where the collision had taken place, in order to deposit the particle in the correct place. The collision detection algorithm is discussed in section .
One component of equations 3 and 5 is time. A standard time increment and minimum time increment were chosen for use in the equations. The values of these increments needed to satisfy two conditions. The standard time increment was used to calculate the basic motion of the particles travelling in the air. Given the initial position of the particle, the new position was calculated. The possibility existed however, that the particle may pass through the terrain without the collision detection algoritm picking it up. The standard time increment needed to be sufficiently small to minimize this possibility. However, the smaller the increment, the greater the time required to deal with any particular particle. This tradeoff between accuracy and time was taken into consideration when choosing a value for the standard time increment.
The minimum time increment was used in conjunction with the collision detection algorithm. When determining where a collision took place, the new position of the particle was calculated with smaller and smaller time increments. The minimum time increment represented the smallest time increment that would be used to calculate the new position of the particle. The time required to deal with a particular particle increased as the time increment decreased, so the particle's position could not be calculated with arbitrarily small time increments. This method provided an approximation only to the point at which the particle collided with the terrain. Collision detection is discussed further in the following section.
The path of an airborne particle was traced, as described above, in time increments. Given the current position of the particle, the next position was calculated, but the particle was not automatically moved to the new position after calculation. The new position was first tested to ensure the particle was not going to collide with the terrain by moving it there. For the new position to be valid, the height of the new position must be greater than the height of the terrain at that point. The basic algorithm for collision detection proceeds as follows:
while( particle still airborne )
{
calculate new_X position with current time increment
calculate new_Y position with current time increment
while( new_Y < terrain height at new_X &&
current time increment >= minimum time increment )
{
calculate new_X position with current time increment
calculate new_Y position with current time increment
current time increment = current time increment / 2
}
if( no collision about to occur at new_X and new_Y )
{
place the particle at new_X and new_Y
}
else if( no collision at new_X and old_Y )
{
place the particle at new_X and old_Y
}
else
{
drop particle vertically, depositing on terrain at old_X
}
}
There are three possible outcomes with this algorithm. Firstly, the new X and Y coordinates are valid, and the particle is placed at this position. Secondly, the particle is found to be sliding along a surface, in which case the particle either comes to rest against another particle, or is blown off the other end. Lastly, the particle is blocked by another particle, and can travel no further in the horizontal X direction, and so falls straight down and comes to rest on the surface at the current X position.
A sand dune offers some shelter from the wind in the lee of the dune. Particles in this sheltered zone are affected less by the wind than those on the windward side. Any airborne particle which drops down into the shelter of the dune will lose some of its horizontal velocity. This will cause airborne particles to be deposited earlier than they might otherwise have been. The amount of shelter offered to a particular particle in the deposition model was approximated as follows:
|
In equation 6 , the gap is the amount of space between the airborne particle and any sheltering terrain on the windward side. For the terrain to offer any shelter, it must be at least as high as the particle. The sheltering ratio was the minimum gap size between the particle and sheltering terrain, such that the particle would no longer be effected by the dune. These values were arrived at experimentally by observing the effects of airborne particles falling into the leeward side of the dune, and are not based on actual physical values. A greater degree of physical accuracy could be acheived by determining these values with the help of an air viscosity model, similar to the model proposed by Wejchert and Haumann [ 28 ] in their paper on animation aerodynamics. However, the simple rule to approximate the sheltering effects of a dune given by equation 6 provided a satisfactory result. Again, a more accurate model of the sheltering effects of a dune would be obtained by looking at the flow of air around the dune, and how the shape of individual dunes affects the air currents. This would however, have added a great deal of complexity to the overall model, and it was decided that the benefits of the increased degree of physical accuracy would not outweigh the drawbacks of the added complexity.
The erosion model was designed to deal with the removal of particles from the landscape. Section 4.1.2 described the three types of aeolian transport which may erode particles from the landscape. These were suspension, saltation and surface creep. The erosion model concentrated on suspension and surface creep in particular, as saltation is essentially a mixture of these two.
There were three main steps involved in modeling the erosion process. The first step was to determine whether or not a particle could actually move given its current position. The next step was to define rules to decide if the particle did move, assuming it was able to do so, and the last step was to determine where the particle moved to given that it actually did move.
For a particle to be eroded from the landscape, the wind must be strong enough to uproot the particle from its current position. Particles which are sheltered by other parts of the dune may be affected very little by the flow of air. In particular, particles in the lee side of the dune may experience no air flow at all, and hence be unable to be eroded by the wind. Particles which are effected by the wind may be rolled along a flat surface, rolled up or down hill, or be picked up completely by the wind and undergo suspended aeolian transport as discussed in section 4.1.2 . For this model, only particles on the top of columns would be allowed to be eroded. The justification for this is that the particles on top of the column are far more likely to be blown away than any particle further down the column, as there is much less force required to shift them.
In the current model, two main parameters were used to determine if a particle would be eroded. The strength of the wind, and the height that the particle must scale in order to roll from one column to another. This height was particularly important for any particles which might roll uphill. If the difference in height between the particle's current position, and the height of the next column was too great, the particle would not be blown up the slope. The wind needed to be sufficiently strong to overcome the difference in heights. Particles were assigned a probability of moving based on the strength, or speed of the wind which was acting on the particle, the height difference between the particle's current position and the particle's potential position, and the maximum allowable strength of the wind. The maximum wind strength was an upper limit on the velocity of the wind allowed within the model. The following equations were used to determine if a particular particle was to be eroded:
|
The new wind strength is calculated from
equation 6
.
The height difference is the height of the column the particle
is rolling to minus the current height of the particle.
Maximum wind strength is a value selected by trial and error.
Various values were used during the course of implementing and testing
the model.
Epsilon is a small constant added on to the maximum wind strength.
This ensures that regardless of the wind strength, there is always some
small probability that a particle will remain where it is. This is to
account for particles in a natural landscape which may be
P(moving) is the calculated probability that the particle will
move. This value is used to determine whether or not the particle will
move.
Once a particle is eroded from the landscape, the next task is to determine where it stops. For a particle which is rolling along the surface, this is a fairly simple task. The particle is moved to the next column, and from there the probability of the particle moving is recalculated for the new position, and the process continues as described in the previous section. In the natural world, particles which are already moving require less force to continue moving, compared with the force required to move particles which are stationary. This is directly related to Newton's laws of motion. Stationary particles require enough force to overcome static friction, before they can begin to move. The force of friction is being disregarded in the course of this project, so stationary particles have the same probability of moving as particles which are currently in motion. For particles which become airborne, this is obviously not an appropriate method for determining where the particle eventually comes to rest. Eroded particles which become airborne are controlled by the same processes outlined in section 5.3.1 . The particle is given a velocity equivalent to the velocity of the wind at that position, and the path of the particle is traced until either it is blown out of the system, or it comes to rest somewhere else.
During testing of the erosion and deposition models it became apparent that a new model was required to settle the resulting terrain structures into stable configurations. While running the two models, the particles tended to build up in a manner that did not appear to be physically possible. The angles of some slopes were much too steep than what could be expected from sand. A stabilization model was required to check the angle of all slopes, and adjust any that were too steep. The resulting model that was developed bears some similarity with the method by which Sumner et al produced more realistic deposition patterns for the purpose of modeling sand, mud and snow [ 27 ]. This section begins with an explaination of the angle of shearing resistance which is the maximum angle at which a particular slope will be stable, followed by a description of the design of the stabilization routine.
A soil slope has a particular angle at which it is stable, and any steeper than this causes the slope to become unstable, and slides to occur. This angle is known as the angle of shearing resistance. The value of this angle depends on a number of factors, including the size, shape and smoothness of the particles, and how densely packed the soil is. The Great Sea of Sand in the Libyan Desert is made up of sand particles which are for the most part, silica with oval or round shapes. The average angle of shearing resistance in the Great Sea of Sand appears to be roughly 33 °[ 4 ]. As the focus for this project is mainly on sand dunes, this angle will be used as the angle of shearing resistance in the stabilization model. Using the angle of shearing resistance for stabilizing the model implicitly includes some factors such as cohesion, adhesion, particle size and shape, consistency and density of the sand particles, into the model. This allows some degree of increase in the physical accuracy of the model with only a small cost in design and computational complexity. For further discussion on the angle of shearing resistance, and values for the angle for a range of soil types, the reader is refered to [ 3 ]
As with the erosion model, the only particles which may move are those which have room to do so, ie. they lack a neighbouring particle to one side. A particle may move to either the column on the left or the right, depending on whether or not space is available for the particle to move there. Particles are given a probability for moving left and right, and this probability is determined by the direction of air flow, and the velocity of the flow. The greater the velocity of the wind, the higher the probability that particles will fall in that direction. All particles are given the same probability. No sheltering effects are taken into account here. The probabilities are designed only to give emphasis to one direction or the other.
The stabilization process continues until all slopes are less than or equal to the angle of shearing resistance. For particles which fall from the very top of a column, the only tasks are to determine which side the particle falls to, and where it lands.
Particles further down the column, which have other particles above them, are not so simple. If a particle falls from the column, any particles which are above it are also forced to move. These particles above may fall either left, right or straight down. The values of the probabilities in this case are influenced by the direction and velocity of the wind, as for the previous case with the topmost particle moving. Individual particles can be removed and added to columns as necessary through the information in the linked lists representing the columns.
For the sand dune to be stable, all slopes must be at an angle equal to or less than the angle of shearing resistance. In the natural world, the angles of slopes are continuous values, whereas in this model, the slope angles are continuous as a result of separating the particles into columns with discrete horizontal coordinates. The angle of the slope between any two neighbouring particles on the same level is 0 ° , see figure , whereas if one particle is one level higher, the angle of the slope between them is 45 ° , see figure . The angles were calculated using equation 8, where the rise refers to the height in particles, and the run refers to the number of particles in the horizontal.
|
With such a large difference between discrete values for the angle of shearing resistance, it became clear that the angle of the slope would need to be averaged over a number of columns. The number of particles required in the rise to produce the required angle of shearing resistance needed to be calculated. For an angle of 33° , over a run of five particles, the rise was calculated to be approximately three particles, see figure . A module was included in the implementation to calculate the number of particles required in the rise, given the number in the run.
The greater the number of particles in the run, the more accurate the value would be for the number of particles in the rise. As with some previous calculations, the greater the degree of accuracy, the more time is required to handle the problem. It was decided that using five particles in the run produced results that were visually accurate enough for the present model. With values for the number of particles in the rise and the run, the remaining task was to ensure that for any block of five columns the maximum height difference between columns in that block was no more than the number of particles allowed in the rise. To carry out this test, the maximum and minimum heights of every block of five consecutive particles was tested against the allowable values. Any block where these constraints were not met was adjusted according to the rules for controlling the movement of particles outlined in section 5.5.2 . The outcome was a section of terrain where all slopes approximated physically stable slopes.
Recall that the main aim of the project was to design a model capable of producing terrain which exhibited some characteristic features of erosion and deposition. Given this aim, testing of the models was predominantly visual. Simple terrain structures were built out of small collections of particles, and the behaviour of individual particles was observed under various conditions. Velocity of the wind, angle of shear resistance, and probabilities affecting the behaviour of the particles were all adjusted and the particles observed. These values were given to the system in input files, and once entered could not be adjusted until the next run. The particles were found to behave in the intended manner, which was to mimic the behaviour of real particles, albeit in simplified conditions.
A working model of erosion and deposition required the three individual models to be combined. The final implementation involved cycling through the algorithm of each model in turn. Particles were deposited on the landscape, after which some amount of time was given for particles to be eroded. Finally, the entire landscape was tested for stability, and any unstable regions were adjusted as required. At this point the cycle began again with particles being deposited. Parameters for wind velocity, angle of shearing resistance and probabilities affecting the particles were adjusted and the resulting effects observed.
With the three models combined, one last addition was made. This addition was to cause the terrain to wrap-around, so that any particles blown out of the system would reappear again on the other side. Also, in the case of sand dune creep, as the dune shifted out of the visible viewing area, it would begin to reappear on the opposite side. The stabilization model was also wrapped around in the same manner.
This section details the results of simulations using each of the three models separately, followed by results from the combined model. Parameters were varied and details of these variations are given. The results for the erosion and deposition models run individually do not appear particularly accurate. This is to be expected as the results of one process effect the results of another in a natural environment. The individual models were each designed to focus on one process only, when in the natural world, these processes run in parallel with each other. Only in combination do the models begin to produce satisfactory results. Discussion of the results follows in section 7.
The deposition simulation starts with a simple piece of symmetrical terrain as shown in figure 11 . In all examples shown, airborne sand particles were injected into the system from the left hand side of the image. All airborne sand particles were given a velocity equal to the velocity of the wind. Variable wind speeds were used. Wind speed is not measured in meters per second or any other standard measure. The velocities used in this investigation can be regarded as dimensionless, and only relative to each other, hence the lack of units. Figure shows a snapshot from the deposition simulation using a wind velocity of 40, and figure shows a snapshot from a deposition simulation using a wind velocity of 20. For the simulation with wind velocity of 20, the simulation was run for significantly longer than for the simulation run with wind velocity of 40. Maximum allowable wind velocity is 50.
As with the deposition model, the erosion model begins with the simple piece of terrain shown in figure 11 . Variable wind velocities are used which simulate erosion under weak and strong wind conditions. Wind blows from a constant direction with constant force. Sand particles were blown up the slope, and in some cases particles became airborne. Airborne particles were controlled as described in section 5.3.1 . The probability of a particle becoming airborne could be adjusted to simulate larger, heavier particles. The sequence of erosion images in figures , and illustrate the progress of particles undergoing a steady wind which originates from the left hand side of the images. Part A (figure ) is early in the simulation, with part B (figure ) following some time later, and part C (figure ), later still.
For testing purposes, the stabilization model was given completely random terrain as a starting point. Each column was allocated a random number of particles, as seen in figure .
From this point, the terrain was stabilized according to the rules outlined in section 5.5 , until the angle of every slope was less than or equal to the angle of shearing resistance. Figure 18 shows the terrain from figure 17 after several passes of the stabilization routine. The final result of stabilizing figure 17 using an angle of shearing resistance of approximately 33 ° is shown in figure 19 .
For the sand particles used in this project, the angle of shearing resistance was assigned a value of 33° . Adjustments to this angle were made for the purposes of testing the model. In most cases, the number of particles in the run was set to five when calculating the maximum height difference allowed between any two particles. This value was varied for the purpose of testing the routine. Figure shows the initial random terrain which was then stabilized with angle of shearing resistance = 33 ° , and twenty particles in the run. For a run of twenty particles, the approximate maximum number of particles in the rise is eleven. Figure shows the results of stabilizing the terrain in figure .
As for the erosion and deposition simulations, the initial terrain for the combined model was represented as a simple, symmetrical pile of sand (see figure ). The simulation firstly concentrated on deposition. Particles were allowed to build up over some length of time, during which the stabilization model was constantly in effect. The results of deposition on the initial landscape in figure are shown in figure . After this, the emphasis was placed on the erosion model. The force of the wind was allowed to erode the terrain for some length of time, again with the stabilization model in effect.
The results of erosion in the combined model can be seen in figure 24 . Additionally, the results of all three processes running together can be downloaded from
http://www.csse.monash.edu.au/~molloy/thesis/
Each of the three individual models was designed to approximate a physical process. With the enormous level of complexity involved naturally in each of the processes, the models were completely physically accurate. Some level of physical accuracy was maintained where possible by the use of real physical laws and constants, such as for tracing the path of an airborne particle using equations of motion, or calculating the angle of a slope. This section discusses the results obtained from the three models of erosion, deposition and stabilization, as individual models, and as a combined process. Where appropriate, comparisons will be made with natural terrain.
Results from the deposition simulation appeared to be much the same regardless of the velocity of the wind. Figures 12 and 13 show very similar results despite having very dissimilar velocities. In figure 12 , the velocity of the wind was twice as fast as that in figure 13 . This appears to indicate that the basic result of depositing airborne sand particles is much the same redardless of the velocity of the wind. The only significant difference was the time required for the particles to build up. It was no surprise to find that the greater the air flow velocity, the more quickly the effects of deposition would be visible. This is due to the basic fact that particles are injected into the system more rapidly for higher velocities.
Giving a bias towards particles being injected at lower altitudes was intended to emulate the tendency for the density of airborne particles in natural situations to decrease as the altitude increases. This caused a greater build up of particles towards the base of the dune, which tapered off higher up the slope. Visually, the dune appeared to develop a longer, gentler slope on the windward face, compared with the slope in the lee side.
The results from the erosion model, in particular figures 15 and 16 , show very clearly the need for a stabilization model. Particles were blown up the slope, and toppled over the crest. Without a stability model, the particles tended to bunch up into tall columns, completely unlike any characteristic feature of loose sand. One important feature to note, however, is the overall movement of the sand pile. With particles constantly rolling up and over the sand pile in the direction of the wind, the entire pile creeps in the direction of the wind. This reflects the behaviour of natural sand dunes, which have a tendency to creep in the direction of the wind.
Testing of the stabilization algorithm was carried out on random terrain, as was shown in figure 17 . The terrain in this figure was not meant to represent characteristics of natural terrain. The intention was to provide an extremely unstable structure on which the stabilization routine could be tested. Given an angle of shearing resistance of 33° , the final output from the stabilization routine is a much smoother section of terrain, as shown in figure 19 . Some slope angles appear to be more than the required 33 °. This inaccuracy is due to the method by which the angle of the slope is tested. As described in section 5.5.3 , the angles which the slope may take are discrete, whereas in the natural world, slope angles take continuous values. This discretization is responsible for the inaccuracy of the slopes seen in figure 19 . In this case, the number of particles in the run was taken as five. This required that for the angle of the slope to be no more than the allowable angle of shearing resistance, there could be a height difference of no more than approximately 2.7 particles between any pair of particles which were separated horizontally by a maximum of five columns. As this height difference is measured in particles, and the particles are discrete, using 2.7 particles was not possible. Rounding this result to three particles had the effect of increasing the maximum observed slope angles to approximately 37 ° instead of the prefered 33 ° .
Adjusting the number of particles in the run produced some interesing results. Initially, it was thought that increasing the number of particles in the horizontal plane over which the slope was being calculated, may produce a result which more closely resembled the true angle of shearing resistance. The calculations were carried out with twenty particles in the horizontal plane, which produced the figure of approximately 10.9 particles being the maximum height difference allowed in the vertical plane, or rise. Again, this result was rounded off, with the final value set at eleven particles. Repeating the calculations using eleven particles in the vertical, and twenty in the horizontal, produced a much more accurate figure of 33.4 ° for the observed angle of shearing resistance.
Despite the increase in accuracy produced by increasing the number of particles in the run, the observed results in figure 21 were not as convincing as those seen in figure 19 . Using a shorter run length produced smoother slopes, with the drawback that some slopes still appeared to be too steep. The longer run length produced slopes which exhibited the occasional small, jagged peak. These sorts of peaks are not characteristic of sand dunes, whose slopes tend to resemble those seen in figure 19 . A possible solution to this problem is to restrict the maximum height difference between adjacent particles. If the maximum height difference between any two adjacent particles is restricted to one particle, and maintaining the previous restrictions on groups of particles, this problem may be solved. Values given to the probabilities of particles moving to the left and right did not seem to have any visible effect on the outcome. The only difference noticed was that it took perhaps a little longer for a stable configuration to be reached when these probabilities were reduced. It would seem then, that for this model, it matters very little whether or not particles are more likely to fall to one direction. It may give some small bias towards the direction and speed with which a sand dune creeps in a particular direction, but the evidence was not conclusive enough to know for certain.
Individually, each of the three models displayed many flaws. In particular, the erosion model produced extremely unconvincing images of terrain without the aid of a stabilization routine to control the angles of the slopes. In the natural world, the processes of erosion and deposition do not act alone, but rather as closely intertwined processes. The flaws in one process seemed to be mended by the other processes. For these reasons, the combined model produces much more satisfactory results.
There were a number of features characteristic of the processes of erosion and deposition which became noticeable in the combined model. In the individual implementations of these models, these features were also present, but not as clear. The conditions under which all the simulations were run are very similar to the conditions required for the formation of a barchan dune. Recalling section 4.1.2 , a barchan dune is the result of a strong, steady, dominant wind. The basic shape of a barchan dune was given in figure 3 . A cross-section of a barchan dune would show a long, gentle slope on the windward side of the dune. The leeward slope would settle at an angle approximately equal to the angle of shearing resistance. The observed value of this slope for dunes in the Great Sea of Sand, in the Libyan Desert, is approximately 33° [ 4 ]. Barchan dunes are also known to increase in size as sand builds up on the windward side, and creep across the desert floor in the direction of the wind. These are all features that the combined model was observed to have. The deposition model resulted in the formation of a long, gentle slope on the windward side, and steeper slope on the leeward side, approximately equal to the angle of shearing resistance. The erosion model was able to produce the effect of sand dune creep, and with the aid of the deposition model, the size of the dune increased. In particular, the sand dune became higher, and the breadth also increased. The effect of sand dune creep was small, and for the dune in figure , a significant amount of time was required for the sand dune to progress any distance in the direction of the wind. However, in the natural world, sand dunes do not creep particularly quickly either. Sand dune creep, and other features produced by the combined model can be seen on the accompanying CD.
The conclusions presented in this section are based on the discussion from section 7 . The thesis concludes with directions for further work in section .
Erosion and deposition are extremely complex processes, and it was decided early on that the focus of the project would be to model just a small selection of features, concentrating primarily on sand. Sand was considered to be a relatively simple particle, that was easily simplified further. The simplifications made in the model were necessary to make the problem feasible. Overall, the models are very cut down versions of natural processes, and individually, the models are not capable of producing convincing results. Each of the models is however, necessary to the combined process. Once combined the results obtained were much more satisfactory. A number of features of sand erosion and deposition were produced. These features included the basic shape of a cross-section of a barchan dune, in particular the gentle slope on the windward side, and the steeper face in the lee of the dune. Additionally, the sand dune was found to creep in the direction of the wind, and increase in size over time. The terrain structures were two dimensional, so it is not known if the model would be able to produce the characteristic 'arms' of a barchan dune.
Overall, the individual models, and the combined models are very limited. Compared with many other models for erosion and deposition, this model is probably much more limited in the effects it can produce. The model does bear more resemblence to what really happens in the physical world, compared with, for example Musgrave et al [ 20 ]. However, for the model to be particularly useful for producing images of terrain that appear to have undergone erosion and deposition, it would require some expansion. It is particularly limited in that the focus was primarily on sand dunes. The variable wind conditions which naturally produce a number of different types of sand dune, was limited to coming solely from one direction. This also caused to reduce the number of features the model was able to produce. However, it was expected from the beginning that many simplifications would need to be made. In conclusion, this model is capable of producing several characteristic features of sand dunes, which are formed by the processes of erosion and deposition.
One of the most important steps that could be taken to improve the current model, would be to implement it in three dimensions. This change would greatly increase the number of features that could potentially be modeled. In particular, the full shape of a barchan dune, with the characteristic 'arms' on either side, may be possible with very few adjustments.
One very important factor in the formation of sand dunes is the direction from which the wind blows. For barchan dunes, the wind tends to come only from one direction. However, the results are quite varied when crosswinds are added into the equation. Allowing variable winds in the model may produce some interesting effects.
The present model only deals with one particle at a time, regardless of which process is in effect. This tends to slow the process down considerably. Adapting the existing model so it can deal with more than one particle at a time may help considerably in speeding up the model. Increases in the speed of the model may also be acheived by generalizing the way in which masses of sand move. In the natural world, sand dunes tend to retain their shape as they creep across the desert floor. The particle model proposed in this thesis imitates sand dune creep on a particle by particle basis. In future, a particle by particle basis could be used to model sand dune creep only when the dune encounters an obstacle. At other times, when the dune is free to move, a generalization involving moving the crest of the dune in the direction of the wind may suffice. This movement of the crest may resemble a water wave, which bears some resemblance to the shape of a dune. Both share a long shallow slope on one face, and a steeper slope in the direction of travel.
A great deal of work could be carried out in expanding the existing model. There is very little limit to the number of variables which effect the landscape, and inclusion of other variables may produce a wide variety of effects. In particular, looking at different types of particles could produce an interesting range of results.
I would like to thank my supervisor Dr. Alan Dorin for his guidance over the course of the year. I would also like to thank Adam Koh for his help with finding a particularly frustrating bug in the code, and Tim Bate, for last minute help with burning the CD.
