8.1.
GEOMETRY
OF
FAULTS
AND
HORIZONS
• For
each node
a € fi,
compute
the
vector
N*(a)
defined
by
(8.8)
and
then
—
add a
fuzzy
Control-Normal-Vector
DSI
constraint
c'(a)
to
Cx
specifying
that
T(J-}
must
be
orthogonal
to
N*(a)
at
location
x(a)
(see page
265);
—
add a
fuzzy
Control-Straightline
DSI
constraint
c"(oi)
to
Cx
specifying
that
a can
move
only
on the
straight
line
passing
by the
point
x(a)
and
parallel
to the
direction
N*(a)
(see
page
264).
•
RunDSIonM
3
(ft,7V,x,Cx).
Figure (8.3) shows
an
example
of how filtered
fault-throw directions
can be
used
for
improving
the
geometry
of a
fault
surface
T(^):
after
applying
the
above algorithm,
the
normal vectors
to
T(F)
tend
to
become consistent with
the
computed
fault
striae.
It
should
be
noted
that
a
technique
to
align
the
minimum absolute normal
curvature directions with
specified
directions
will
be
presented
in the
next
sec-
tion.
If
need
be,
this technique
can be
combined with
the
procedure proposed
above.
8.1.2
Modeling horizons
In
sedimentary geology,
at the
time
of
deposition, horizons
are
assumed
to be
approximately horizontal
and are
thus approximately developable (see
defi-
nition
on
page 221).
Due to
tectonic events occurring throughout geological
time,
such horizons
are
deformed,
and
their shapes,
as
they
can be
observed
today, appear
as
folded
and
faulted versions
of
their initial horizontal devel-
opable shapes.
If
plastic deformations occurred during
the
tectonic events,
the
resulting
surface
for a
given horizon
<S
is no
longer "strictly" developable.
However,
in
practice,
folded
horizons remain
approximately
developable
and
their direction
of
minimum curvature
is
coincident with their (curvilinear)
axis.
For
example,
if we
consider
the two
surfaces represented
in figures
(8.4)-
A
and
(8.4)-B
respectively
and
having similar boundaries,
it can be
observed
that
the
saddle shaped surface
in figure
(8.4)-A
is far
from
being developable
while
the
anticline represented
in figure
(8.4)-B
can be
easily unfolded (devel-
oped)
on a
plane.
The
"basic"
DSI
harmonic weighting
coefficients
{v(a,fl}}
introduced
on
page
149 and
used
for
modeling most
of the
surfaces
of
this book
are
generally
quite
sufficient
for
interpolating
the
geometry
of
geological horizon. However,
as
shown
in
this section, when
the
(curvilinear) axis
of a
fold
is
known,
it is
possible
to use
anisotropic weighting
coefficients
{v(a,0}},
which take into
account such structural information.
Taking
into
account
the
axis
of a
fold
Consider
a
folded
horizon
<S
approximated
by a
triangulated surface
T(«S)
represented
by a
discrete model
,M
3
(0,
TV,
x,C
x
)-
Let us
assume
that
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