Stochastic Modelling of Fractures in Rock Masses, 2002
2
1. Introduction
The purpose of the program is to generate spatial points in R
2
as realisations of user-
defined Poisson processes. The type of processes implemented by the program are:
homogeneous Poisson process, non-homogeneous Poisson process, Poisson cluster
process, simple Cox process and simple Boolean model.
The realisation is to be constructed in a rectangular area, A=[x
1
,x
2
]×[y
1
,y
2
], where x
1
,
x
2
, y
1
and y
2
can be specified. A is divided into quadrate cells with each individual
cell being treated as an all bounded Borel set. The number of cells in X and Y can also
be specified by the user of the program. For this arrangement, all cells are disjoint
and therefore the simulation of Poisson points for each cell can be done
independently.
The commands to be used are all listed under the master menu of “SpatialPattern”.
Select the corresponding command will bring up the related parameter specification
window. Closing the window will activate the simulation and the generated points
will be displayed once the process finishes.
2. Homogeneous Poisson process
Homogeneous process is the simplest possible stochastic mechanism representing the
spatial point patterns. It generates realisations of stationary and isotropic
characteristics which resemble complete spatial randomness (CSR) in applications.
The process defines a spatial points pattern which bears the following properties
(characteristics):
• For any bounded Borel set A in the region of consideration ℜ, the number of
points falling inside A, N(A), follows a Poisson distribution with mean λ⋅ν(A),
where ν(A) is the measure of A. λ is referred to as the density of the
distribution. In two-dimensional case, A is a planar sub-region of ℜ and ν(A)
is the area of the sub-region.
• Conditioned on N(A) = n, the n event points are uniformly distributed inside
the sub-region A. In two-dimensional case, the events follow bi-variate
uniform distribution.
• For any disjoint sub-regions B
1
, B
2
, …, B
k
within the region ℜ, N(B
1
), N(B
2
),
…, N(B
k
) are independent random variables.
Two distinct ways can be used to generate realisations of homogeneous Poisson
process. One is to use the property that the spatial differences (areas in 2D case)
between successive point events follow an exponential distribution with parameter λ
(the density of the Poisson process). Different implementations can be derived to
explore this property in 2D and 3D cases. The other way is to simulate the Poisson
variable N(A) directly. Simulation of N(A) may sometimes be time-consuming, the
implementation nevertheless is simpler and less error prone. In the current exercise,
direct Poisson variable simulation is used.
The procedures for the realisation of homogeneous Poisson process in a two
dimensional region ℜ are given as:
• 1). Sub-divide the region ℜ into m sub-regions, A
1
, A
2
, …, A
m
. In most of the
cases, region ℜ is a rectangle and A
1
, A
2
, …, A
m
are disjoint rectangular
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