110
with a list of values of z( )
u for a discrete set of u’s. To obtain the FT pair for one variable,
meaning forward and back transform Fourier equations, z ( )
u α must be sampled in uniform
steps (Yoo, 2001). Thus from a real data set available it is not trivial to perform fast Fourier
transform FFT algorithm. For that reason, we propose to estimate values that reproduce
samples spatial continuity property in a regular grid (BMEC).
3.2 Conditioning the covariance table
Given that covariance models must honor conditional negative definiteness to provide licit
values for any lag and direction, and the proposed method does not guarantee a conditional
negative definite CT, we need to overcome this issue. Figure 2 shows an example of a semivariogram model and its respective CT that does not honor conditional negative definiteness.
To solve it the Bochners theorem is used. It defines the general form of a continuous conditional negative definite function C(h), without taking the nugget effect into account, for h = 0:
C
cos
dS
d
( )
h =
( ) ( )
−∞
∞
∫ −
dS d d
) ( ,
(7)
under the constraints that dS ( )
ω > 0 and
dS
C( )< .S
∞
= C( )< .
( )
ω
( )
ω
−∞
∞
∫ −
is the spectral cumulative distribution function. In practice, the CT is corrected for conditional negative definiteness by transforming all negative real values to zero and then standardizing all spectrum
values to sum to the variance desired (Pyrcz and Deutsch, 2006).
4 METHODOLOGY
Once all the fundamental theory have been presented, it is possible to describe the proposed
work flow. The first step is to make all variables z 1 , z, ..., z N independent from each other
through PPMT method. This is the key step to overpass cross CT. What would make the
entire workflow more laborious. The second step is to apply the proposed CT method for
each independent variable. The third step is to simulate each independent variable using any
geostatistical simulation method based on covariance. The fourth step is to back transform
each realization to make the multiple variables as they were originally correlated. Figure 3
illustrates the entire step by step methodology for a multiple variable data set simulation case.
Figure 3. Step by step for the proposed methodology.
• z1
• Z2
• z.
-
- SGS Zl
• SGS Z2
• SGS Z3
• SGS z.
..
-
SGS Zl,Z2, ...
Independent SGS Zl, Z2, ...
with a list of values of z( )
u for a discrete set of u’s. To obtain the FT pair for one variable,
meaning forward and back transform Fourier equations, z ( )
u α must be sampled in uniform
steps (Yoo, 2001). Thus from a real data set available it is not trivial to perform fast Fourier
transform FFT algorithm. For that reason, we propose to estimate values that reproduce
samples spatial continuity property in a regular grid (BMEC).
3.2 Conditioning the covariance table
Given that covariance models must honor conditional negative definiteness to provide licit
values for any lag and direction, and the proposed method does not guarantee a conditional
negative definite CT, we need to overcome this issue. Figure 2 shows an example of a semivariogram model and its respective CT that does not honor conditional negative definiteness.
To solve it the Bochners theorem is used. It defines the general form of a continuous conditional negative definite function C(h), without taking the nugget effect into account, for h = 0:
C
cos
dS
d
( )
h =
( ) ( )
−∞
∞
∫ −
dS d d
) ( ,
(7)
under the constraints that dS ( )
ω > 0 and
dS
C( )< .S
∞
= C( )< .
( )
ω
( )
ω
−∞
∞
∫ −
is the spectral cumulative distribution function. In practice, the CT is corrected for conditional negative definiteness by transforming all negative real values to zero and then standardizing all spectrum
values to sum to the variance desired (Pyrcz and Deutsch, 2006).
4 METHODOLOGY
Once all the fundamental theory have been presented, it is possible to describe the proposed
work flow. The first step is to make all variables z 1 , z, ..., z N independent from each other
through PPMT method. This is the key step to overpass cross CT. What would make the
entire workflow more laborious. The second step is to apply the proposed CT method for
each independent variable. The third step is to simulate each independent variable using any
geostatistical simulation method based on covariance. The fourth step is to back transform
each realization to make the multiple variables as they were originally correlated. Figure 3
illustrates the entire step by step methodology for a multiple variable data set simulation case.
Figure 3. Step by step for the proposed methodology.
• z1
• Z2
• z.
-
- SGS Zl
• SGS Z2
• SGS Z3
• SGS z.
..
-
SGS Zl,Z2, ...
Independent SGS Zl, Z2, ...