Jean-Laurent Mallet. Geomodeling
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538
CHAPTER
10.
DISCRETE SMOOTH PARTITION
Figure
10.3
The
cumulative
Membership
Function
$(a;
s)
associated
with
a
Membership
Function
(f>
and a
given
sorted
list
{SQ,SI,
..
.,s
n
}
of
real
values.
This
function
can be
used
to
build
a
simulator
S(u,u)
from
a
given
P-field
P(u,
o>).
10.2
The
probabilistic
approach
10.2.1
A
partitioning
method
based
on DSI
Membership
Function
(f>
Let
us
assume
that
the
number
n of
possible facies
{F
1
,...,
F
n
}
is
known.
As
suggested
in figure
(10.1),
to
each node
a G
O
we
propose attaching
a
vector
(p(ot)
with
n
components
{(p
l
(a),...,
(p
n
(a}}
such
that
represents
the
probability
for the
node
a to be
located
in the
facies
F
v
:
In
other words,
(f
defines
on
O
a
vectorial
function
called
the
"Membership
Function,"
which must
be
interpolated over
fi
while honoring
all the
available
data.
This suggests introducing
the
discrete model
.M
n
(f2,
A/",
<£>,
C^]
where
dp
corresponds
to a set of DSI
constraints
to be
defined
so as to
honor
all the
available
data.
Cumulative
Membership
Function
&(a;
s)
=
3>(a;
s\ip;
SQ,
si,...,
s
n
)
Let
{$„}
be a
sorted list
of (n + 1)
given
real
numbers
7
such
that
7
For
the
sake
of
generality,
the
meaning
and the
practical interest
of
this sorted list
{«i/}
is
not
detailed
at the
present time. Sections
(10.2.2)
and
(10.2.3)
give
examples
of how
this
list
can
actually
be
chosen.
10.2.
THE
PROBABILISTIC
APPROACH
539
and let
{$(0;;
$„)}
be the
associated list
of
real numbers
defined
as
follows:
By
definition,
one
calls "Cumulative Membership Function" associated with
(f>
and
{sjx}
and one
note
as
$(a;
s\(f>;
SQ,
si,
• •
-,
«n)
or
5
more simply,
$(a;
s)
the
function,
as
defined
by
In
other words,
as
shown
in figure
(10.3),
$(a;s)
linearly interpolates
the
values
{3>(a;s
v
)}
as
defined
by
equation (10.2).
It
should
be
noted
that,
if
$(a;
s) and the
associated series
{s
v
}
are
given,
then
it is
easy
to
retrieve
</?"(a:)
as
follows:
Simulator
S(u,
o>)
Let us
assume
that
the
graph
Q(£l,
AT)
has a
topological dimension equal
to
p. In
this case, according
to
section
(4.9.1),
it is
possible
to
build
a
global
parameterization
u
from
0
to a
p-dimensional
parametric domain
D
such
that
This parameterization
can
then
be
used
for
continuously interpolating
on
D any
discrete function
defined
on
Q,
in
particular
the
functions
(p(ot)
and
&(a;s):
It is
clear
that,
for any a G
O,
the
function
<!>(
u(a);
s ) has all the
properties
of
a cdf
function:
8
if a
global interpolator preserving these properties
on D
is
chosen
for
$,
then
^(u;
s) can be
considered
as a cdf
defined
on D. As in
the
case
of the
simulator introduced
in
section
(9.6.2)
and as
shown
in figure
(10.3),
to any
P-field
P(u,
w)
defined
9
on (D x
13),
this suggests associating
a
random
function
5(u,o;),
as
defined
by
According
to
equation (9.103),
the cdf
-Fs(u;
s) of the
simulator
so
defined
is
identical
to
<&(u;
s):
8
See
sections
(9.2.2)
and
(9.3.5).
9
According
to the
notations
introduced
in
section
(9.2.1),
15
represents
the set of all the
possible
elementary
statistical
events
u.
540
CHAPTER
10.
DISCRETE SMOOTH PARTITION
In
particular, according
to
equations (10.4)
and
(10.5),
this
implies
that
the
probability
that
S(u,uj)
belongs
to
js^-i,^]
is
equal
to
c/?"(u):
Comment
For the
sake
of
generality,
the
sorted list
{SQ,
si,...,
s
n
}
and the
associated
simulator
S(u,o;)
have been
formally
denned
without
any
explanation. Sec-
tions (10.2.2)
and
(10.2.3) show
how the
list
{SQ,SI,
• •
-,s
n
}
must
be
chosen
to
obtain
a
simulator
S(u,
a;)
that
allows
the
generation
of
equiprobable
in-
terpolations
for
•
either
the
partition
F
of fi,
• or a
scalar
function
2(11)
denned
on
O.
Intrinsic
probability
constraints
Whatever
the
data,
the
Membership Function
ip
must always honor
the
fol-
lowing
"intrinsic" probability constraints:
These constraints
are
linear relative
to
ip
and can
easily
be
turned into three
hard
DSI
constraints belonging
to
C
v
:
• The
constraint
{^(a)
>
0} can be
written according
to the
canonic
form
of
a DSI
constraint
c =
c(a,
v}
that depends
on (a,
v}
and
belongs
to
C^:
• The
constraint
{^(a)
<
1} can be
written according
to the
canonic
form
of
a DSI
constraint
c =
c(a,
v)
that
depends
on (a,
v}
and
belongs
to
C^:
• The
constraint
(5^1™
^(CK)
= 1} can be
written according
to the
canonic
form
of a DSI
constraint
c =
c(a)
that depends
on a and
belongs
to
C^:
It
should
be
noted
that
this constraint
can
also
be
taken into account,
in a
least square sense,
as
belonging
to
C^
. In
such
a
case, according
to the
local
10.2.
THE
PROBABILISTIC
APPROACH
541
DSI
equation
at
node
a, the
terms
7^(a)
and
T"(a\(p)
associated
with
this
constraint
take
the
following
form:
One
can
observe
that
the
notion
of
"Membership Function"
is
close
to the
notion
of
"indicator" random
function
used
in
geostatistics [117, 118,
119].
Another possible approach
to our
problem would consist
of
using
the
indicator
Kriging
method
for
estimating
(p
over
Q.
Contrary
to
DSI, however, indicator
Kriging
equations
do not
allow
the
intrinsic probability constraints presented
above
to be
taken into account directly
and may
generate values
<p
v
(ot)
out
of
the
range
[0,1].
Fuzzy
Control-Point
constraints
In
practical applications,
the
available
"data
points" generally consist
of a
given
set of
points
P —
{p
1
,...,p
m
}
where
a
Membership Function
0
is
given:
For
each
data
point
p E P, two
cases have
to be
considered
to
take into
account
the
associated piece
of
information
0(p):
• If p is
close
to a
node
loftl,
then
it can be
added
to the set
L
of
Control-Nodes
(see
definition
(10.1))
where
9? is
assumed
to be
known:
• If p is a
long
way
from
any
node
of
Q,
then
it
should
be
considered
as a
fuzzy
Control-Point
(see
page
182);
such
a
fuzzy
Control-Point
will
be
relative
to the
set
f2*
=
f2*(p)
corresponding
to the
vertices
of the
cell
of the
graph
(?(f2,
TV)
containing
the
point
p.
According
to
section
(4.7),
this
fuzzy
Control-Point
generates
the
following
DSI
constraint
c —
c(r2*(p))
belonging
to
C~
where
the
coefficients
{14,3(p)}
represent
a set of
barycentric
coordinates
of p
relative
to the
location
of the
nodes
of fi*(p).
In
the
case where
the
data
points
p G P are
strongly clustered relative
to the
distribution
of the
nodes
a
e
f2,
it is far
better
to use
fuzzy
Control-Points
than Control-Nodes.
To
understand
the
superiority
of the
fuzzy
Control-
Points
over Control-Nodes,
let us
consider
two
data
points,
p^
and
p
j?
be-
longing
to P and
very close
to
each other:
542
CHAPTER
10.
DISCRETE SMOOTH PARTITION
• It is
likely
that
pj
and
p^
will
be
simultaneously
close
to the
same
node
i
G
fi;
since
it is
impossible
to
attach
two
distinct
Control-Node
values
</>(Pi)
and
</>(PJ)
to the
same
node
£,
considering
£ as a
Control-Node
will
result
in
one
of the two
data
0(pj)
or
0(p^)
being
discarded.
•
Even
in the
case
where
p^
and
p^-
belong
to the
same
cell
£7*,
it is
possible
to
take
into
account both
of the
pieces
of
data
(/>(PJ)
and
</>(p,-)
as two
distinct
fuzzy
Control-Point
DSI
constraints.
Incidentally,
in the
case corresponding
to figure
(10.1) where dense
data
are
sampled
on
well-paths
and
where
the
graph
£/(Q,
N)
consists
of a
coarse
strati-
graphic grid,
it
should
be
noted
that
using
fuzzy
Control-Points
elegantly
re-
alizes
a
so-called
"upscaling"
(e.g.,
[45])
from
the
dense
well
data
to the
coarse
grid.
Estimating
the
partition
F
As
soon
as an
approximation
of the
Membership
Function
</?
is
known every-
where
on fi, a
deterministic estimation
J-(y>]
of
J-
can be
made
as
follows:
The
quality
of
this estimation
can be
measured
at
each node
a of
12
thanks
to the
following
function
£
lp
(a)
defined
on fi,
which
we
propose calling
the
"Likelihood
function":
It is
easy
to
check
that
the
function
£p(a)
so
defined
has the
following
inter-
esting properties
that
make
it a
good
candidate
for
measuring
the
quality
of
the
estimation
of
F
at
node
a:
The
Likelihood
function
could
be
used
in
answering
the
following
question:
Assuming
that
the set
of
data points
L is
given, where
is the
best
location
in
il
for
observing
a new
data point
(.Q
?
A
straightforward answer would
be to
select
IQ
in the set
/
=
(f2
—
L) in
such
a way
that
IQ
corresponds
to the
minimum value
of
£
lf
(a)
on /:
The
next section shows
that
it is
possible
to
build
stochastic
estimations
of
T
and the
deterministic solution presented above
has
only
to be
considered
as the
"most probable" one.
10.2.
THE
PROBABILISTIC APPROACH
543
10.2.2
Simulation
versus
estimation
Need
for
simulations
The set of
Control-Nodes
and
fuzzy
Control-Points
is
often
very small com-
pared
to the
local complexity
of the
partition
T
to be
built.
As a
consequence,
in
regions
a
long
way
from
these Control-Nodes
and
fuzzy
Control-Points,
the
DSI
method, very like
any
other estimation method, will tend
to
produce
a
smooth Membership Function
(p.
This implies that,
in
regions
a
long
way
from
the
Control-Nodes
and
fuzzy
Control-Points,
the
"most probable" parti-
tion presented
in the
previous section
can
only have
a
smooth geometry very
different
from
the
actual geometry
of the
unknown partition
F.
Therefore,
as
explained
in
section (9.1),
any
decision-making strategy
based
on
F
is
relevant only
if it is
possible
to
produce equiprobable solu-
tions
JF(ijj\(p)
called
simulations,
honoring
the two
following
fundamental
constraints:
1)
Each
simulation
F((jj
(p)
is
assumed
to
interpolate
the
Control-Nodes
and
approximate
as
closely
as
possible
the
fuzzy
Control-Points
and
other
constraints,
if
any.
2)
Each
simulation
F(u)\(p)
is
assumed
to
have
a
local
geometry
compatible
with
an a
priori
knowledge
of the
"style"
of the
local
variations
of
F.
The
next section presents
a
very
efficient
method based
on the
notion
of
"P-
field"
(see
section
(9.5))
specially designed
to
produce such solutions.
The
P-field
simulation
technique
First,
consider
the
cumulative Membership Function
$(a;
s)
defined
by
equa-
tion (10.3)
and
associated with
the
sorted list
{0,1,...,
n\:
The
random
function
5(u,
w),
defined
by
equation
(10.6)
and
associated with
a
given P-field
P
—
P(u,
a;),
can be
used
for
defining
a new
random function
S
D
(U,UJ)
derived
as
follows
from
S(u,uj):
Note
that
the
random
function
S
a
(u, u) so
defined
can
take only integer values
between
1 and
n:
Moreover, taking
into
account equation (10.7),
it can be
deduced
that
^"(a)
is
equal
to the
probability
that
5
D
(u,
uj]
is
equal
to
v:
544
CHAPTER
10.
DISCRETE SMOOTH PARTITION
This
suggests
that
the
following
technique
may be
used
for
associating
an
estimation
F(uj\(f>}
of the
partition
JF
with each elementary statistical event
u;:
It
should
be
noted
that
the
constraint
(10.9)-1
is
automatically honored
for
any
Control-Node
I
G L
because
where
v(£)
is the
index
of the
facies
F
v
^
containing
the
node
t
(see definition
(10.1)).
In
practice,
the
only
difficulty
lies
in
building
the
P-field
P(u,
u;).
A
naive
solution would
be to
choose
for
P(u,u;)
a
simple
random
variable
10
not de-
pendent
on
u
and
uniformly
distributed
on
[0,1]:
However,
such
a
choice would generate solutions
J-(uj\(p)
strongly dependent
on
the
numbering
of the
facies:
• if
P(<jj)
is
close
to
zero,
then
F
l
(uj\(p)
will
be
overestimated relative
to the
other
facies,
and
• if
P(UJ)
is
close
to 1,
then
F
n
(uj\<£>)
will
be
overestimated relative
to the
other
facies.
To
remedy
this
bias,
a
random function whose realizations
depend
on u and
have
the
following
properties must
be
chosen
for
P(u,u;):
• if a £
O
is fixed,
then
the
values
of the
random variable
_P(u(a),u;)
are
uniformly
distributed
on
[0,1],
and
• if the
statistical event
u is fixed,
then
the
variations
of
P(u(a),w)
when
a
scans
11
must mimic
the
local variations
of the
unknown
partition
J-
to
honor
the
constraint
(10.9)-2.
In
practice,
a
P-field
similar
to the one
introduced
in
section (9.5)
will
be
used
for
P(u,
LJ)
whose associated covariance
function
{Cp(h)}
should
be
chosen
to
specify
the
style
of the
variations
of the
unknown partition
f'.
Practical
implementation
of
simulations
The
simulation technique presented above
is no
more
than
a
"downscaling"
method able
to
generate
high-frequency
solutions
from
low-frequency Control-
Points
and
Control-Nodes:
• the
DSI
method generates
a
smooth Membership Function
and is
responsible
for
interpolating
the
data
while
10
See
section (9.2.2).
10.2.
THE
PROBABILISTIC APPROACH
545
• the
P-field
realizations
are
responsible
for
generating
the
high
frequency
vari-
ations
between
the
data.
This
has a
consequence
for the
optimal implementation
of the
simulation
technique:
*-
The
Membership
Function
can be
represented
by a
discrete
model
M.
n
(£l,
N,
<p,
C
v
)
corresponding
to a
coarse
graph
Q(£l,
N).
• The
simulations
must
be
stored
in the
nodes
of a fine
graph
G(&,
N
f
)
whose
cells
have
a
size
11
compatible
with
the
high-frequency
variations
induced
by
the
P-field.
Thus,
it may be
wise
not to use the
same graph
for
storing
the
Membership
Function
and the
simulations.
For
example,
in the
case
of the
tutorial example
introduced
in
section
(10.1.2),
an
optimal implementation would consist
of
• a
coarse
graph
£/(£!,
N)
having
large
steps
in the
(u,
v)
directions,
while
re-
taining
small
steps
in the
w
direction
to
preserve
the
high
density
of
data
along
the
well-paths
which
are
more
or
less
parallel
to the w
axis,
and
• a fine
grid
$(&,
N*}
having
small
steps
in the (u,
v,
w)
directions
compatible
with
the
ranges
of the
covariance
functions
used
for
generating
the
P-field
(see
section
(9.5)).
An
efficient
technique
is to use a
local interpolation
0
to
compute
the
global
parameter
u(o/)
at the
nodes
a/
6
$V
of the fine
graph
£($V,
N?)
from
the
values
of
u(a)
stored
in the
nodes
of the
coarse graph
(/(£},
AT).
In
practice,
a
barycentric interpolant
0
similar
to the one
defined
by
equation (3.16),
can
be
chosen such
that
where
fi*(o/)
represents
the
subset
of
nodes
12
of
0
corresponding
to the
cell
of
G(tl,N)
containing
the
node
a*
G fiA The
simulator
5(u,
a;)
can
then
be
computed
at the
nodes
of the fine
grid
as
follows:
10.2.3
Simulating
a
bounded
scalar function
The
problem
to be
solved
Let
us
assume
that
an
estimate
is
needed
of a
scalar
function
z(a)
defined
on
the set fi
corresponding
to the
nodes
of a
graph
£(f2,
N) and
bounded
by
two
given values
ZQ
and
z
n
:
A
straightforward solution
to
solve this problem lies
in
applying
the
DSI
method
to the
discrete model
Ai
n
(O,
N,
Z,C
Z
)
where
C
contains
at
least
the
two
following
hard constraints belonging
to
C>:
11
In
practice,
the
cells
of the
fine
graph
G(ftf
,Nf)
are
built
as
regular
subdivisions
of
the
cells
of the
coarse graph
G(Q,
N).
12
This
subset
can
easily
be
retrieved
in the
case
where
the
cells
of the fine
graph
<7(JV,
N?)
are
built
as
regular
subdivisions
of the
cells
of the
coarse graph
G(Q,
N).
546
CHAPTER
10.
DISCRETE SMOOTH PARTITION
Such
an
approach provides
one
single
solution.
However,
this
is not
sufficient
to
assess uncertainties, which leads
to a
different
approach
to
this problem
proposed
in the
next section.
The
associated
transformed
problem
To
assess uncertainties
as
described
in
section
(10.2.2),
it is
proposed
to
trans-
form
the
initial problem, described above, into
the
problem
of
estimating
a
partition
T
of
f2
as
follows:
•
Choose
a
series
of (n + 1)
real
numbers
{z
v
}
such
that
•
Define
the
discrete model
.M
n
(il,
N,
<£>,
dp)
where
<p
is a
Membership Function
whose
n
components
are
denned
as follows for any
Control-Node
I
G
L:
Similarly,
if a
data
point
p
holds
a
value
z(p)
then,
as
explained
on
page
541,
an
associated
fuzzy
Control-Point
constraint
(j)
z
(p) can be
installed such
that
•
Interpolate
if
with
DSL
The
Membership Function
<p
so
denned
can
then
be
used
for
estimating (see
page 542)
a
partition
{F
1
,...,
F
u
}
of 17.
This partition can,
in
turn,
be
used
for
estimating
the
range
to
which
z(a)
belongs
to
If
the
P-field
simulation technique
is
used
for
generating equiprobable random
partitions, then
the
classes
{F"}
are
replaced
by
random classes
{F
v
(uj\(p)}
and
z(a)
must
be
replaced
by a
random
function
5(u,u;),
such
that
where
15
is the set of all the
possible elementary
statistical
events.
The
next
section shows
how to
build such
a
random
function
5(u,
w).
A
simulation
method
Consider
the
cumulative Membership Function
$(a;
s)
defined
by
equation
(10.2)
and
associated with
the
sorted list
{ZQ,
z\,...,
z
n
}
introduced
in the
above section:
The
random
function
5(11,
u;)
denned
by
equation
(10.6)
is
bounded
and can
take only values between
ZQ
and
z
n
whatever
the
P-field
P(u,
cj):
10.2.
THE
PROBABILISTIC APPROACH
547
Moreover,
taking into account equation
(10.7),
it can be
observed
that
^(a)
is
equal
to the
probability
that
5(u,
o>)
belongs
to the
interval
]z
v
-i,z
v
}:
This suggests using
5(u(ce),o;)
as a
simulator (see sections (9.1)
and
(9.6))
associating
a
simulation
Zj(a)
with each statistical event
MJ:
The
exact
interpolation
problem
Assuming
that
z(£)
is
known
for a
node
t
G
L,
note, however,
that
the
simu-
lation technique presented above
is not
precise because:
In
the
case where precision
is
important, proceeding
as
follows
will
correct
the
problem:
• for
each
simulation
Zj(a)
=
S(u(a),ujj)
of the
function
z(a)
built
as
indicated
above,
define
the
associated
discrete
model
M
n
(£l,
N,
Zj,
C
z
)
in
such
a way
that
C
reduces
to the
following
Control-Node
constraints:
•
apply
the
local
iterative
form
of the
DSI
method
(see
section
(4.4.3))
limiting
it to the
subset
L*,
which
consists
of
nodes
belonging
to
£1
and
located
in a
given
neighborhood
of the
nodes
of L.
Proceeding
in
this
way
ensures
that
the
simulations
Zj(a)
remain unchanged,
except
in the
neighborhood
L* of the
subset
L
where
the
values
{zj(l*}
:
t*
6
L*}
are
changed smoothly
to
interpolate
the
given values
{z(l}
:
I
E
L}.
Comment
The
"Gaussian-simulation" technique introduced
in
section
(9.6.2)
is a
very
efficient
alternative
to the
method presented above
for
simulating
a
scalar
function.
The
main advantage
of the
above method, based
on the
Discrete
Smooth
Partition
technique,
is
that
it can
easily take into account
the
struc-
tural constraints presented
in the
next
section:
13
About
the
Heisenberg
uncertainty
principle
As
quoted
on
page 512, within
the
framework
of
particle physics,
the
Heisen-
berg uncertainty principle tells
us
that
13
For
example,
see
page
553.