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Sumário
O primeiro objectivo da presente dissertação corresponde à redução de erros numéricos em
formulações de advecção-difusão e é efectuado através da apresentação dos métodos DisPar. Estes
métodos são uma classe de formulações numéricas de advecção-difusão, baseada em distribuições
do deslocamento de partículas para processos de Markov. Estão incluídos os desenvolvimentos,
análises formais e testes de métodos DisPar explícitos e implícitos aplicados em malhas uniformes
uni-dimensionais e bi-dimensionais. O primeiro método, DisPar-1, é baseado no desenvolvimento da
distribuição de probabilidade discreta do movimento de uma partícula, cujos valores são inferidos a
partir da média e variância do deslocamento. Estes dois parâmetros estatísticos dependem das
condições físicas (velocidade, coeficientes de dispersão e fluxos). O Segundo método explícito,
DisPar-k, desenvolvido para uma e duas dimensões, é uma extensão do anterior. Para além da média
e da variância, a distribuição do deslocamento de uma partícula baseia-se num número específico de
momentos. Os momentos são obtidos através da relação entre as equações de advecção-difusão e
Fokker-Planck, assumindo uma distribuição de Gauss para o movimento das partículas. O número de
momentos afecta de uma forma directamente proporcional a precisão espacial do método, sendo
possível obter bons resultados em situações de advecção pura. Nestas situações, a comparação com
outros métodos demonstrou que a principal desvantagem do DisPar, em 1-D e 2-D, é a presença de
oscilações nas vizinhanças de perfis de concentração descontínuos. No entanto, os métodos que
evitam estas oscilações, apresentam piores resultados que o DisPar-k no transporte de perfis mais
alisados. A aplicação do DisPar 2-D ao estuário do Tejo demonstrou a capacidade do método de
representar o transporte de massa em escoamentos complexos. Finalmente, uma versão 1-D
implícita do DisPar é igualmente apresentada, obtendo-se uma relação semelhante entre os erros de
truncatura e os momentos de deslocamento das partículas.
O contributo para a redução do custo de modelação, segundo objectivo de dissertação, é obtido
através da apresentação da TangiTable, uma interface tangível para a simulação da dispersão de
poluentes, composta por uma computador pessoal, uma câmara, um projector de video e uma mesa.
Neste sistema, um ambiente virtual é projectado sobre uma mesa, na qual utilizadores colocam
objectos representando infra-estruturas que afectam a água de um rio e a qualidade do ar. O
ambiente e a dispersão da poluição são dinamicamente projectados na mesa. A usabilidade da
TangiTable é testada com resultados bastante positivos numa exposição aberta ao público e usos
potenciais incluem participação pública e trabalho colaborativo.
1 Introduction
1.1 Problem Definition
Environmental quality became one of the main society concerns during the 20th century.
Pollution caused by human activities, such as industry and agriculture, plays a harmful role in human
health and quality of life. There is, therefore, increasing interest in the understanding of environmental
processes to improve its planning and management. The transport of substances in surface waters,
such as rivers and estuaries, and in groundwater and atmosphere is one of the most important
processes that affect the quality of those natural systems. For instance, the impacts of industrial
discharge in a specific place of a river can have damaging consequences downstream, depending on
the local hydrodynamic conditions. Simulation can be a valuable tool to evaluate the impacts of
existing infrastructures and predict the consequences of different scenarios. Substance dispersion
simulation, in particular pollutant dispersion simulation, is the topic of the present thesis.
Pollutant dispersion simulation, as other models, is seen in engineering perspective as a tool to
solve problems and in scientific and mathematical fields as the problem to be solved. Goldberg (2002)
describes a theory towards an economy of modelling (Figure 1.1), whose concept is based on a trade-
off between model accuracy and cost of modelling1. For example, a high-accuracy model with high
costs could not generate a comparable marginal benefit in an engineering application, where lower
accurate models can be used. On the other hand, the aim of theoretical work will always be to
minimize the associated errors, leaving costs in the background. Goldberg thus built a modelling
spectrum that starts high cost, high fidelity models such as detailed equations of motion, goes past
facet-wise models, dimensional models and articulated qualitative models and ends at low cost, low
fidelity models, such as unarticulated wisdom.
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This cost includes time consumed, financial resources and all other kinds of costs required by the
modelling process
2
Error
Cost of modelling
Engineer/Inventor
Scientist/Mathematician
Figure 1.1 - Goldberg’s economy of modelling theory. A hypothetical engineer-inventor will
prefer lower cost, higher error models whereas a mathematician-scientist will choose the opposite.
Source: Goldberg, 2002.
Goldberg´s economy of modelling theory can be applied to pollutant dispersion simulation. In
Figure 1.2 an adaptation of that theory to pollutant dispersion simulation is presented, which includes
a classification of different modelling objectives:
Error
Cost of modelling
Public participation/political decision
Particle movement laws study
Infrastructure location planning
Real environment model implementation
Advection-diffusion numerical method development
Figure 1.2 - Example of economy of modelling theory applied to pollutant transport simulation
The objective of the higher fidelity/higher cost models, particle movement laws study,
corresponds to the developments of the theoretical assumptions, which have to be considered in any
research field. In the described example, it is considered that pollution is made up of particles whose
movements follow well-known statistical physics principles and advection-diffusion differential equation
describes a wide range of the substance transport in a fluid. Those assumptions result from extensive
and highly accurate work in mathematics and physics. Following these principles, the scope is then to
develop stable and convergent advection-diffusion or particle displacement numerical methods that
have the minimal numerical or truncation error. A substance transport model has to be parameterised
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with water velocities, water elevations and turbulence coefficients before being applied in real
environment. This process, known as calibration or parameter estimation, is performed to reduce the
differences between model results and available field observations, independently of the numerical or
physical nature of the errors.
In both situations, numerical method development and model application, the cost of modelling
and the importance of error minimization are still high. However, in the second situation, the numerical
error is not considered as the main motivation for choosing a specific numerical method. The choice
will consequently be mainly based on the availability of different numerical methods, since other
concerns affect the model calibration and validation. For example, it can be more efficient to use a
graphic user interface for a simulation then to access or to write the model source code, even if that
results in a decrease of the model user control.
The next stage of the modelling spectrum can be the pollution source location, which is
integrated in engineering or environmental impact assessment studies. Due to time constraints, they
usually require a model previously validated. Therefore, the cost of modelling has to be low, even if the
associated error is higher due to model assumptions and simplifications or due to lack of real data.
The modelling spectrum defined by Goldberg goes from mathematician/scientist (or theoretical)
to engineer/inventor (or practical) purposes. A pollutant dispersion model spectrum can, however, be
extended towards social objectives such as public information and political decision objectives, since
pollution level is an important indicator of quality of life. An analogy can be established with the
weather forecast, where public communication is supported on two spatial dimension simulations of
the most relevant climatic variables. This information is widely spread out by the media, including
websites, whereas visualization of environmental quality variables, such as air pollution and surface
water quality, is generally restricted to scientific and technical websites.
This dissertation aims at presenting two new methodologies that target on reducing errors or
costs associated with pollutant transport simulation. The first methodology is about advection-diffusion
numerical methods, which govern most of substance (and also pollutant) dispersion processes in
fluids. The goal is to increase the numerical accuracy of simulations, by reducing numerical errors.
The second methodology is an attempt to reduce the cost of modelling by introducing alternative user
interfaces to pollutant dispersion simulation. Next, a research context of these two areas will be given,
followed by the description of the principal objectives of this thesis.
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1.2 Numerical Formulations for Advection-Diffusion Transport
1.2.1 Research Context
Besides the problems resulting from background data insufficiencies, there are also numerical
errors associated with advection-diffusion transport simulations. Those errors do not appear due to
incorrect use of data, but are generated by the numerical method employed.
Advection-diffusion transport simulation can be numerically solved by analytical or by numerical
methods. The first type provides an exact solution of the problem, but can only be employed in
restricted physical conditions. Therefore, in common environmental conditions, such as complex flows
or boundaries, numerical models have to be used. The broad numerical method classes are Eulerian -
EMs, Eulerian-Lagrangian - ELMs and particle methods - PMs, and it is possible to find out
advantages and shortcomings in every type of scheme. Eulerian models, for instance, balanced
between stability problems and significant accuracy problems, whereas Eulerian-Lagrangian models
can present mass conversation errors. No grid is employed in particle models and thus spatial errors
are avoided. However, the large amount of particles required to simulate complex situations can lead
to unsustainable computational costs.
An important difference between the two first presented classes (EMs and ELMs) and PMs is
that random walk theory, whose foundations come from statistical physics concepts, serves a basis for
its development. Indeed, advection-diffusion is a stochastic process, which can be considered as a
Markov process, since particle movement does not depend on the presence of other particles (Van
Kampen, 1992). On the other hand, EMs and ELMs do not make explicit use of stochastic concepts,
which can be seen as disadvantage in the comprehension of physical processes involving
randomness, such as particle transport in turbulent fluids.
All the numerical methods described in the literature have advantages and shortcomings
associated in terms of accuracy and stability. Thus, it is possible to state that there is still some
research to be done in terms of error reduction in numerical simulation of advection-diffusion
problems, as it will be now described.
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1.2.2 Research Objectives
As it was previously mentioned, the first main objective of this thesis is to reduce numerical
errors in advection-diffusion modelling. This is accomplished by presenting the DisPar methods, which
are a class of numerical schemes of advection-diffusion or transport problems, based on a particle
displacement distribution for Markov processes.
A summary of the DisPar schemes developed and tested is presented in table I:
Table 1.I – DisPar Schemes
One-dimension Two-dimensions Three-dimensions
Discretization
Space
Time
Uniform Regular Uniform Regular Unstructured Uniform Regular Unstructured
Explicit a) a) a)
Implicit a)
a) Presented in this dissertation; - developed and tested; - developed, not tested; -
not developed and not tested.
The DisPar methods were developed for different combinations of time and spatial
discretizations. Therefore, there are explicit and implicit methods applied to uniform and regular grids.
DisPar was also developed and tested for one and two dimensional situations and it was
conceptualised for three dimensions. The present dissertation includes the development and analyses
of explicit and implicit DisPar formulations applied to one and two dimensional uniform grids. The
concept of explicit three-dimensional model is also presented in appendix 11.1 but not tested. The
models are tested in different theoretical situations and compared with other formulations in order to
point out the advantages and shortcomings of these methods.
1.3 User Interaction with Pollutant Dispersion Simulation
1.3.1 Research Context
Environmental simulation in general and pollutant transport (or dispersion) simulation in
particular are generally restricted to engineers and scientists, who are often the model developers.
Indeed, those simulation interfaces are used and understood only by one or two experts, even in multi-
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disciplinary studies embracing a whole range of collaborators. This can thus be the main reason for
considering simulation interaction, and in particular pollutant dispersion simulation, as a highly
specialized task. Therefore, a huge gap is created, which prevents this tool from being regarded as a
potential instrument for educational proposes and for public participation. Such application could be
attractive since air and water pollution is a very important quality of life and public health indicator. To
better understand these issues, a brief history of user interaction with pollutant dispersion models is
now introduced.
Before the advent of computational simulation, physical mock-ups were built and applied in
many fields, such as the simulation of hydrodynamic and transport processes in natural aquatic
systems. Estuarine scale models were built to study changes in tidal prisms, circulation patterns,
salinity concentration changes and pollution transport, among other issues. An example is the Tagus
estuary physical model, which reproduced a real environment area that extends from 15 km away in
the ocean to the head of tidal propagation, which distance 80 km from the estuary mouth. The model
was entirely housed in a building with a maximum width of 70 m and a length of 180 m (Elias, 1982).
The simulation set up was, however, expensive and time consuming and when the computer capacity
allowed the reproduction of these systems, numerical methods started replacing physical models in
almost all the situations.
Over the past two or three decades, numerical simulation interfaces evolved in a similar way as
general computational software and now they are based on Graphic User Interfaces (GUI). A personal
computer with GUI considerably enlarged the number of software end users and opened computation
to a wide range of non specialized public. Nevertheless, and as the name indicates, the personal
computer is for personal use and its standard interface, known as WIMP (windows, icons, menus,
pointers) style, restricts interaction at various levels (Gentner, D. & Nielsen J., 1996). Rosson & Caroll
(2002) discuss some themes that are already having significant impact on the design of new activities
and new user interaction techniques. One of them is collaborative systems and another one is
ubiquitous computing, which are also contextualized in terms of environmental applications in Camara
(2002).
Collaborative activities can be classified according to whether they take place in the same (co-
located) or different (remote) locations and at the same (synchronous) or different (asynchronous)
points in time. The applications written to support the collaboration of several users are generally
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identified as groupware or as Computer Supported Collaborative Work (CSCW) systems (Dix et al,
1997), which can be useful for multi-user interaction with environmental simulations.
The term ubiquitous computing was first used by Weiser, M. (1991) to describe a vision of the
future in which computers are integrated in the real world, supporting everyday tasks. An important
element of the ubiquitous computing vision is to consider the physical objects and the environment as
input and output mechanisms interacting with digital information. Ishii & Ullmer (1997) systematize this
idea, paying special attention to the concept of tangible user interface in which the control of the digital
information is achieved, for instance, by graspable physical objects. These authors also refer that the
digital outputs can be displayed on interactive surfaces, such as walls, desktops and tables.
In order to contextualize the visualization of environmental simulation, the concept of mixed
reality introduced by Milgram & Kishino (1994) is applied. These authors defined a "virtuality
continuum" where classes of objects are mixed up in any particular visual display situation. At one end
of the continuum there are real environments and at the other end there are virtual environments.
Figure 1.3 illustrates the mixed reality concept applied to typical visualization of pollutant dispersion
simulation:
Mixed reality
Real Environment Augmented Reality Augmented Virtuality Virtual Environment
Physical scale model ? ? Virtual objects in GUI or
in Immersive virtual reality.
Examples of visualization in pollutant dispersion simulation
Figure 1.3 - Milgram & Kishino mixed reality concept applied to typical visualisation of pollutant
dispersion simulation
As can be seen, visualisation of virtual (i.e. not real) images in a typical desktop computer GUI
or in an immersive virtual environment, such as Camara et al (1998), are positioned as a virtual
environment. The physical mock-up of the Tagus Estuary previously mentioned is situated at the other
extreme of the "virtuality continuum", the real environment.
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Augmented Reality is a slice of mixed reality defined by Milgram & Kishino (1994) as any
situation where real environment is "augmented", in visual terms, by means of virtual objects. Another
part of mixed reality is augmented virtuality, which is defined by the same authors as any case where
virtual environment is "augmented" by means of real objects. In terms of the visualization of a spatial
simulation, augmented reality can be the superimposition of virtual elements, such as pollution, over
an aquatic environment. On the other hand augmented virtuality would be the visualization of virtual
landscape with real objects helping to understand the overall context of the digital information.
Augmented reality and augmented virtuality have concepts that can serve as a basis for new
approaches in user visualization and interaction with pollutant dispersion simulation, as it will be
demonstrated afterwards in the present thesis.
After presenting all these concepts, a question emerges: why not apply these new human-
computer interaction paradigms to improve understanding and usability of pollutant dispersion
simulation. These improvements include the increase of the range of potential users, by replacing
input mechanisms such as mouse by more intuitive ones. Furthermore, user interaction with pollutant
dispersion simulation requires new hardware and software schemes to support collaborative work,
since the popular personal computer is not designed to serve, for example, face-to-face collaborative
work. The study of all these issues may lead to a modelling cost reduction, which was defined as the
second main objective of the present thesis.
1.3.2 Research Objectives
The second main objective of this thesis, to contribute to modelling cost reduction, is
accomplished by presenting TangiTable, a tangible interface for pollutant dispersion simulation
composed by a personal computer, a camera, a video projector and a table. In this system, a virtual
environment is projected on the table, where the users place objects representing some infrastructures
that affect the water of an existent river and the air quality. The environment and the pollution
dispersion along the river are then projected on the table. TangiTable usability was tested in a public
exhibition visited by nearly 60,000 people and its future uses can be public participation or technical
meetings in collaborative environments.
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1.4 Outline of the Thesis
Chapter 1 corresponds to the present introduction and chapter 8 contains the main conclusions
of this dissertation.
The first part of the thesis, devoted to developments on advection-diffusion numerical modelling,
is composed by five chapters (chapter 2 to 6).
Chapter 2 outlines the advection-diffusion numerical methods, beginning with a brief overview of
the main advantages and shortcomings of the Eulerian and Eulerian-Lagragian methods. Particle
Methods are then described paying special attention to their stochastic conceptualization and
including some theoretical issues on statistical physics that will be applied in this thesis. Other less
common numerical method categories, such as cellular automata, are also referred.
Chapter 3 describes and analyses the first one-dimension DisPar method developed. The
method is based on the development of a discrete probability distribution for a particle displacement,
whose numerical values are evaluated by analysing average and variance. This DisPar formulation
does not completely follow other described modelling principles and new contributions are presented
in the following chapters.
Chapter 4 presents DisPar-k, an extension of the previous chapter work, which is also based on
the particle displacement moments obtained by the relation between the advection-diffusion and the
Fokker-Planck equation. It is assumed a Guassian distribution for the particle displacement
distribution. Therefore, the developed method consists of dividing the Gaussian distribution in a user
specified number of discrete probabilities, which are evaluated as function of the particle displacement
moments. These numerical probabilities are used as coefficients to calculate mass transfers between
domain nodes. Thus, DisPar version presented in chapter 3 corresponds to a particular situation of
DisPar-k, where the user specified number of probabilities is 3. However, DisPar-k is much more
flexible and attractive in terms of numerical error control. The relation between Gaussian moments and
numerical errors is studied in the truncation error analysis.
In chapter 5, the two-dimensional DisPar-k version is developed and tested. Thus, the 1D
probabilities for each dimension are evaluated following the 1-D DisPar-k (chapter 4). Then, the
product of the combined independent probabilities produces the 2-D displacement probability
distribution. The method is assessed in theoretical situations by comparing the numerical results with
known analytical solutions and in a practical situation in the Tagus estuary, Portugal.
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Chapter 6 presents the one-dimensional implicit version of DisPar, called Implicit DisPar, which
is based on the evaluation of particle displacement distribution for Markov processes, as the explicit
formulation. The model analyses show that this formulation has some stability restrictions that were
avoided in the explicit formulation. In high-diffusive situations this model can be an alternative. As it
happened in the explicit formulation, it is proved that there is a relation between errors associated with
numerical methods for advection-diffusion and the Markov particle displacement moments.
The two and three dimension models development in uniform grid follows the same principles.
Thus, the three dimension version is presented in Appendix 11.1.
The second part of the thesis, composed by chapters 7 and 8, includes the presentation of an
approach about user interaction with pollutant dispersion simulation based on tangible interfaces.
In chapter 7, an overview of user interfaces in environmental modelling is described. The focus
is the comparison between usability of current graphic interfaces based on personal computer and
new concepts such as ubiquitous computing and tangible user interfaces. Some references of spatial
simulation with interactive tabletop surfaces are presented.
Chapter 8 describes TangiTable, a tangible interface applied to pollutant dispersion, which was
installed in a public exhibition. A vivid landscape environment with a main river, its affluents and green
pastures is projected onto a table and users place physical objects representing infrastructures that
affect the water quality of the virtual river. These infrastructures can be pollution sources (factories and
pig-farms) or waste water treatment plants, which are identified by high contrast colours. A camera
suspended above the table allows the infrastructure position identification, which is then connected by
virtual sewage pipes to a river point where pollution is discharged. This discharge position depends on
proximity and topography. If a pollution source is within the treatment plant radius of action, wastes are
then conducted to them and only a percentage is discharged into the river. The factories also release
atmospheric pollution that will be dispersed due to wind effect. The pollutant simulation results are
continuously displayed by a video projector suspended near the camera and different users around
the table handle the infrastructures and visualize the overall effects in real time. New users start
interacting and others abandon the table while simulation keeps going on. The usability of TangiTable
has been tested with the visitors of the pubic exhibition, through the observation of participants,
general remarks and comments of engineering students and exhibition guides. Finally, chapter 8 ends
with some concluding remarks, focusing on possible applications of TangiTable in public participation
and collaborative work. Possible improvements of the system are also listed.
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