Geoscience Reference
In-Depth Information
Fig. 10.5
Step required obtaining a P-field simulation
The typical sequential simulation approach is based on
drawing from the local conditional distributions random
probabilities and then adding each simulated value to the
pool of conditioning data. P-field, by dissociating the two
steps, starts with the premise that the local cdf's are known,
and they become input parameters. Probability values are
used to draw from the local cdf's, constituting a probability
field, and which are interpreted as outcomes of a RF with
uniform distribution and a known covariance function.
Assume that a simulated value has been obtained for each
location
u
within the area of interest. The simulated value
z
s
(
u
)
corresponds to a specific
p(
u
)
probability of the local cdf:
recommends starting with two intuitive basic assumptions:
(a) the probability field
P(
u
)
follows a Uniform Distribution,
as one would expect; and (b) the covariance of the prob-
ability field
P(
u
)
and the uniform transform of the variable
U(Z(
u
))
are the same
,
such that:
C h
()
=
C h
(), with
Uu
()
=
FZu
( ())
P
U
In essence, it is assumed that the two-point continuity of the
uniform transform of the original variable is similar to that of
the probability field. These features include relative nugget
effect, anisotropy ratios, and direction and range of maxi-
mum continuity. This is a rather strong assumption, and dif-
ficult to verify. Typically, the more hard data exists, the less
similar would
Fu z u
(, ())
=
pu
()
s
Ch
and
U
Ch
be.
()
()
The local probabilities
p(
u
)
are interpreted as outcomes of
the RF
P(
u
)
. If the local cdf's are available and the univari-
ate and bivariate statistics of
P(
u
)
can be inferred, then the
P-field simulation can be completed in a straightforward
manner.
The local cdf's can be derived using hard data, soft, ex-
haustive data, or empirically using a geologist's subjective
opinion as to likely range of values for the local cdf's. Per-
haps more importantly, the inference of the probability field
parameters presents bigger challenges. Froidevaux (
1992
)
The implementation of P-field follows these basic steps,
see Fig.
10.5
for a schematic explanation:
1. After defining the grid spacing and nodes to be simulated,
the local cdf's of the variable being simulated must be
obtained. These cdf's can be derived from hard, local data
through a non-linear estimation method (Chap. 9), with or
without secondary information, and can even be defined
empirically. The local cdf's are independent of the simu-
lation processes.
