Geoscience Reference
In-Depth Information
2. Calculate and model the variogram of the uniform trans-
form of the original variable,
U(Z(
u
)
. Assume that the
probability field
P(
u
)
follows a uniform distribution, and
that
The weights (λ
α
(u), α = 1,…,
n
) are determined by kriging.
Simple kriging is the exact solution to the least-squares op-
timal estimate. If local departure from stationarity is a con-
cern, an estimate of the local mean can be used, but usually
with an inflated variance as cost.
As before, a random number is drawn to determine which
class
k
to assign to the node. Since the conditional probabili-
ties were estimated by kriging with a given variogram, the
original histogram and variograms of the data will be repro-
duced by the simulated values.
The steps in SIS are identical to those used for SGS. All
the relevant decisions regarding implementation are the same,
and the processes used to check the models are also the same.
However, in practice, SIS of a continuous variable has proven
to be difficult to calibrate. Variance inflation is common, as
is the difficulties in controlling the tails of the distribution,
particularly the highs on a positively skewed distribution.
SIS is mostly used with variables that exhibit high vari-
ability, as for example epithermal gold grades. Dealing with
variables characterized by a high coefficient of variation is
always difficult, and in these cases SIS can present signifi-
cant challenges.
While the indicator variograms are reproduced as ex-
pected (up to ergodic fluctuations), the reproduction of the
original
z
variogram is not assured. For this, a full indicator
co-kriging should be performed, which in practice is never
done. It can be shown that the SIS simulation reproduces
the madogram,
Ch C
≈
.
3. Generate a non-conditional simulation of
P(
u
)
honoring
the uniform distribution and the covariance
()
()
P
U
Ch
.
4. At each node, draw a simulated value
z
s
(
u
)
from the
local cdf
F(
u,
z)
using the probability value
p(
u
)
:
1
()
=
5. Repeat th
e
above two steps until a
sufficient number of
realizations have been obtained.
z u
()
F
−
(, ())
u pu
The main advantage of P-field simulation is its speed, and
that the distributions of uncertainty can be constructed to
honor all data and checked before any realizations are drawn.
Also, the simulations are consistent with the distributions of
uncertainty. Another interesting aspect of P-field is that is
fairly easy to integrate secondary data into the simulation,
without increasing significantly the time and effort it takes
to obtain a realization. Some of the potential disadvantages
of the method are that the local conditioning data have a
tendency to be reproduced as a local discontinuity and the
spatial correlation may be unrealistically increased. Also, the
spatial continuity features of the uniform transform of the
data and the probability field may not be similar.
10.3
Continuous Variables: Indicator-Based
Simulation
{
}
γ = −+
, since this
function is the integral of all indicator variograms (Alabert
1987a
; Goovaerts
1997
). But the practical consequence of
this is minor, since there is no particular reason to prefer the
z
variogram to the
z
madogram reproduction.
The reward for those who take on the challenges of using
SIS on continuous variable is a simulation model that does not
rely on Gaussian assumption, and thus do not have that under-
lying maximum entropy property. SIS of a continuous vari-
able when it is necessary to reproduce well the connectivity of
extreme values, and thus Gaussian features may be a concern.
2
()
h
E Zu
()
Zu h
(
)
M
Sequential Indicator Simulation (SIS) is in essence the same
as the sequential Gaussian simulation, except that instead of
simulating a Gaussian variable, the indicator transform of
the original variable is simulated. Aside from providing a
method that is not dependent on Gaussian assumptions, SIS
does not require any back-transformation, drawing directly
the simulated value in the original space from the local indi-
cator kriging-derived conditional distributions. Details of the
indicator formalism were presented in Chap. 9.
Recall that the average of the indicator transform is the
global proportion of the stationary domain. The variogram
of an indicator variable measures spatial correlation, as op-
posed to variability:
10.4
Simulated Annealing
Simulated annealing is a minimization/maximization tech-
nique that has attracted significant attention in recent years.
It is based on a thermodynamics analogy, specifically with
the way liquids freeze and crystallize, or metals cool and an-
neal. The basic principles are that: (a) molecules move freely
at high temperature; (b) thermal mobility is lost if cooled
slowly; (c) atoms can often line up over a distance billions
their size in all directions; and (d) such crystal structures are
minimum energy states. This state can always be found if
cooled slowly to allow time for redistribution of the atoms.
The annealing method mimicks nature's minimization al-
gorithm. It is different than conventional minimization algo-
1
γ = −+
()
h
E Iuk
{[(; )
Iu hk
(
; )]}
2
I
2
Inference of cumulative indicator variograms is easier than
class indicators because more conditioning data is always used,
and cumulative indicators carry information across cutoffs.
The ccdf is calculated using a linear combination of the
nearby indicator data:
n
n
∑
∑
*
p u
()
=
λ
() ( ; ) [1
u Iu k
⋅ +−
λ
()]
u
⋅
m
k
α
α
α
k
α
=
1
α
=
1
