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GEOSTATISTICS
INTRODUCTION
Professor Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 1/45
Regionalized phenomenon
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Many natural phenomena exhibit variations
in time and space, for example, rainfall,
temperature, elevation, hydraulic
conductivity, soil moisture content, etc.
When a phenomenon spreads in space and
exhibits a certain spatial variation structure,
it is said to be regionalized.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 2/45
A regionalized variable generally possesses two
characteristics:
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a local and random aspect which can be characterized by
random variable (random feature)
a spatial dependence structure characterizing correlation
relationship between pairs of random variables
(structural feature)
A probabilistic interpretation of random function
or random field (RF) is required to take into
consideration both characteristics.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 3/45
Deterministic vs stochastic
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A phenomenon which can be completely
described by mathematical expressions
without error and uncertainty is
deterministic.
In contrast, a stochastic phenomenon can
not be fully described by mathematical
expressions due to embedded random
properties that are associated with
uncertainties.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 4/45
Stochastic Approach vs. Deterministic
Approach
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and Spatial Modeling
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Dept of Bioenvironmental Systems Engineering
National Taiwan University 5/45
Stochastic Approach vs. Deterministic
Approach Y  2 .8 X 1.3   ,  ~ iid N (0,    60 )
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y = 3.5218x1.2335
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R2 = 0.8852
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P(Y>460|X=50)=?
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(50, 438.9794) by deterministic
model.
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and Spatial Modeling
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Dept of Bioenvironmental Systems Engineering
National Taiwan University 6/45
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Probability plays a key role in modeling
stochastic phenomena.
In reality, many practices of stochastic
modeling arise from our inability of
developing deterministic models for
complex and complicated phenomena.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 7/45
Random variables
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A random variable is a mapping function
which assigns outcomes of a random
experiment to real numbers. Occurrence of
the outcome follows certain probability
distribution. Therefore, a random variable
is completely characterized by its
probability density function (PDF).
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 8/45
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 9/45
Random processes and random
Fields
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If random variables are time- (1-D) or
space-dependent (2-D or higher), they
jointly form random processes or random
fields.
In addition to the probability distributions,
correlations between random variables are
also needed to characterize random
processes and random fields.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 10/45
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 11/45
Completely stationary
Lab for Remote Sensing Hydrology
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Dept of Bioenvironmental Systems Engineering
National Taiwan University 12/45
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 13/45
Stationary up to order m
Recall the definition of moment.
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Second-order stationarity
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The second-order stationarity assumes the
existence of a covariance and a finite a
priori variance, Var[ Z ( x)]  C (0) .
However, there are many physical
phenomena and random functions which
have an infinite capacity for dispersion, i.e.,
which have neither an a priori variance nor
a covariance.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 16/45
The Brownian motion
(one-dimensional, also known as random walk)
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Consider a particle randomly moves on a real
line.
Suppose at small time intervals  the particle
jumps a small distance , randomly and equally
likely to the left or to the right.
Let X  (t ) be the position of the particle on the
real line at time t.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 17/45
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Assume the initial position of the particle
is at the origin, i.e. X  (0)  0
Position of the particle at time t can be
expressed as X  (t )   Y1  Y2    Y[t /  ] 
where Y1,Y2 , are independent random
variables, each having probability 1/2 of
equating 1 and 1.
( t /   represents the largest integer not
exceeding t /  .)
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 18/45
Distribution of X(t)
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Let the step length  equal  , then
X (t )   Y1  Y2    Y[t /  ] 
For fixed t, if  is small then the distribution
of X  (t ) is approximately normal with mean
0 and variance t, i.e., X (t ) ~ N 0, t  .
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 19/45
Graphical illustration of
Distribution of X(t)
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 20/45
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If t and h are fixed and  is sufficiently small
then
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X  (t  h)  X  (t )   Y1  Y2    Y[(t  h ) /  ]   Y1  Y2    Y[t /  ] 
  Y[t /  ]1  Y[t /  ] 2    Y[( t  h ) /  ] 
   Yt    Yt  2     Yt  h  


 

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and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 21/45
Distribution of the
displacement X (t  h)  X (t )
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The random variable X  (t  h)  X  (t ) is
normally distributed with mean 0 and
variance h, i.e.
2

1
u 
du
P X (t  h)  X (t )   x 
exp 

2h  
 2h 
x
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 22/45
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Variance of X  (t ) is dependent on t, while
variance of X (t  h)  X (t ) is not.
If 0  t1  t2    t2 m , then X  (t2 )  X  (t1 ) ,
X (t4 )  X (t3 ),, X  (t2 m )  X  (t2 m 1 )
are independent random variables.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 23/45
Covariance and Correlation
functions of X  (t )
CovX  (t ), X  (t  h)  t
t
Correl X  (t ), X  (t  h) 
t  t  h
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 24/45
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 25/45
Remarks

It is custom to use capital and lower case letters
respectively to represent random variables and
observations. For example, X(t) and Z(x) are
random variables and x(t) and z(x) are
measurements. Also, X(t) and Z(x) are often used
for random processes (time series) and random
fields and should be made clear from the context
of the manuscript.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 26/45
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 27/45
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 28/45
Ensemble space
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The set of all realizations of a random
process is called the ensemble space.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
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Ensemble space
t
Particular realizations of a random process.
PDF of a random variable at time t
Figure 3. Ensemble space is a collection of all realizations.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
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Examples
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f(t)=C1+C2t
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f (t )  25Cos(4t )
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E[ f (t )]  0
Lab for Remote Sensing Hydrology
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Cov[ f (t ), f (t  r )]  E[ f (t ) f (t  r )]
2
a 2

Cos(t   )Cos( (t  r )   )d

2 0
2
a 2


Cos[( 2t  r )  2 ]  Cosr d

4 0
2
a
 Cos(r )
2
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
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Support of a ReV
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Many random variables of physical
quantities are observed based on certain
time and/or spatial domains.
Such time/spatial domain from which
observations are made is called “support” of
the random variables.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 35/45
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A support is characterized by not only its time,
area or volumetric coverage but also the
geometrical shape and orientation.
Even for the same physical phenomenon,
measurements of different supports are
considered realizations of different random
variables.
Change of support will change the probability
distribution and spatial variation structure of the
phenomenon under study and therefore define a
new random variable.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 36/45
Stochastic Simulation
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Given a random variable X, there are situations
that we want to obtain a desired number of
random samples, each of size n.
Similarly, we may need to generate as many
realizations of a random process or random field
under investigation. The advances in computer
technology have made it possible to generate such
random samples and realizations using computers.
The work of this nature is termed “simulation”, or
more precisely “stochastic simulation”.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 37/45
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 38/45
Pseudo-random number
generation
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Pseudorandom number generation (PRNG)
is the technique of generating a sequence of
numbers that appears to be a random
sample of random variables uniformly
distributed over (0,1).
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 39/45
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A commonly applied approach of PRNG
starts with an initial seed and the following
recursive algorithm (Ross, 2002)
xn  axn 1 modulo m
where a and m are given positive integers,
and where the above means that axn 1 is
divided by m and the remainder is taken as
the value of xn .
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 40/45
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The quantity xn / m is then taken as an
approximation to the value of a uniform
(0,1) random variable.
Such algorithm will deterministically
generate a sequence of values and repeat
itself again and again. Consequently, the
constants a and m should be chosen to
satisfy the following criteria:
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 41/45
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For any initial seed, the resultant sequence has
the “appearance” of being a sequence of
independent uniform (0,1) random variables.
For any initial seed, the number of random
variables that can be generated before repetition
begins is large.
The values can be computed efficiently on a
digital computer.
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 42/45
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A guideline for selection of a and m is that m
be chosen to be a large prime number that
can be fitted to the computer word size. For
31
a 32-bit word computer, m = 2  1 and a = 75
result in desired properties (Ross, 2002).
Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 43/45
Simulating a continuous random
variable
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probability integral transformation
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Risk and Reliability
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Definition of risk and reliability in
hydrology
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Definition of risk in risk management
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Another example – target cancer risk
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Dept of Bioenvironmental Systems Engineering, NTU
Modeling MCSinorg – Log-normal
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Dept of Bioenvironmental Systems Engineering, NTU
Cumulative distribution of the target
cancer risk
There is no need for stochastic simulation since the risk is
completely dependent on only one random variable (MCS).
Once the parameters of MCS are determined, the
distribution of TR is completely specified.
5/4/2017
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Dept of Bioenvironmental Systems Engineering, NTU
An example of two-dimensional
random walk simulation
Lab for Remote Sensing Hydrology
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Lab for Remote Sensing Hydrology
and Spatial Modeling
Dept of Bioenvironmental Systems Engineering
National Taiwan University 51/45