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4    Background

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.

4.1   Erosion and Deposition of Natural Terrain

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.

4.1.1   Water erosion and deposition

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.

4.1.2   Wind erosion and deposition

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.

4.1.3   Erosion and deposition by snow and ice

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 ].

4.1.4   Factors contributing to the effects of erosion and deposition

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.

4.2   Modelling Terrain

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.

4.2.1   Development of landscape modeling techniques

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.

4.2.2   Drawbacks Of Current Modeling Techniques

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.

4.2.3   Covered terrain

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.

4.2.4   Erosion and Deposition Models to Date

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.

4.3   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.

4.3.1   Using a particle system model

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.

4.4   Using Particle Systems to Model Erosion And Deposition

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.

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