►
Description
Thibaut Astic presents the preliminary version of his Ph.D. defence: "A framework for joint petrophysically and geologically guided geophysical inversion".
Presentation's slides: http://bit.ly/astic_phd_defence; Dissertation: http://bit.ly/astic_phd_thesis
A
B
B
So,
and
so
this
research
was
done
on
the
superversion
of
douglas
oldenburg
with
a
jeep
group
and
we
are
implemented
within
the
simplex
package
on
in
python.
So
my
objective
today
is
to
present
you
the
work
that
I
did
for
to
tie
together:
geophysical,
petrophysical
and
geological
information
in
a
single
conventional
in
the
inversion
framework
and
by
petrophysical
here
I
mean
physical
property
information
about
the
different
different
work
units.
B
So
the
overview
for
this
presentation
is
gonna
mostly
follow
the
structures
of
the
thesis
and
publication.
So
I'm
gonna
start
by
an
overview
of
the
state
and
the
motivation
and
and
then
lay
out
the
foundation
of
the
framework
of
develop,
which
was
published
in
august
2019
and
then
we're
gonna
move
on
to
the
generalization
of
that
framework
to
multiphysic
inversion
and
we're
gonna
finish
then,
with
a
case
study
utilizing
this
framework
on
a
kimbo
light.
Pipe
in
the
northwestern
territories
in
canada-
and
this
case
today
was
published
just
recently
this
summer
too.
B
B
So
this
is
the
geological
model
from
bohol
of
the
do
27
kimballite
pipe
in
the
northwest
territories
in
canada.
So
so
this
this,
this
symbolic
pipe
is
dimenti
ferrous
and
so
the
way
and
the
way
it's
it's
composed.
So
we
have
it's
an
and
it's
embedded
in
a
granitic
background
and
overlaid
by
some
some
teal,
and
so
we
have
the
main
dimentiferous
unit
here.
B
The
pk
unit,
which
has
a
very
very
low
density,
compared
to
the
background,
and
also
so
pkf
stands
for
pure
classic
kimbo
light,
and
here
we
have
a
volcanic
volcano,
clastic
kimbo
light
we
named
vk,
which
has
also
kind
of
the
same
characteristics,
and
then
we
have
a
unit.
That's
strongly
magnetic
ibaby
soul,
kimbo
light.
B
So
it's
a
geophysical
problem
in
a
nutshell,
so
geophysics
its
goal
is
to
probe
the
subsurface
to
image
geological
structures
and
based
on
physical
property
contrast
in
the
background.
So,
for
example,
here
if
we
have
like
our
pipe
and
we
have
like
a
unit
with
low
densities
or
high
magnetic,
we
can,
for
example,
measure
the
variation
in
the
gravity,
field
or
variation
in
the
magnetic
field
of
the
earth.
B
B
So
the
way
we
do
that
to
go
from
the
data
to
an
image
is
through
the
minimization
of
an
objective
function
and
with
m
being
basically
the
physical
property
at
each
cell
here.
So
that's
our
variable
and
this
objective
function
has
two
parts:
one
is
the
data
misfit,
so
we
want
to.
We
want
that
the
physical
property
distribution
we
recover
in
the
image
reproduce
the
data
we
have.
B
So
that's
what
this
is
this
this
measures,
but
it's
also
a
highly
non-unique
problem
like
we
might
have
here
in
this
example,
a
thousand
data
point
in
each
data
set
where
we
might
have
like
close
to
a
million
pixels
in
the
image.
So,
for
for
that,
we
need
a
regularization
term,
which
basically
adds
prior
information
to
the
problem,
so
it
we
can
solve
it,
and
one
of
the
most
common
inversion
that
is,
that
is
done,
is
a
least
square
or
ticking
off
inversion.
B
And
so
so
it's
characterized
by
a
regularization
in
in
term
of
a
least
square
difference
between
a
reference
model
and
our
current
model,
and
it
usually
has
two
parts:
it
has
a
smallness
which
measures
the
distance
between
our
recovered
model
and
some
reference
models
that
we
divide.
We
we
design,
for
example,
from
geological
modeling
or
just
the
house,
half
space
so
like
a
constant
model.
B
If
we
don't
have
any
more
information-
and
we
also
have
a
smoothness
term,
meaning
that
we
want
the
variation
in
in
space
to
be
gradual
and
fairly
and
fairly
smooth
and,
for
example,
with
that
type
of
inversion.
This
is
what
we
would
recover
from
the
gravity
and
magnetic
data
we
have
so
as
we
see
here,
for
we
recover
a
density
model
for
gravity
and
the
magnetic
sensitivity
model
for
the
magnetic
data.
B
B
So
if
we
want
to
go
if
we
want
to
go
back
to
a
some
sort
of
a
geology
model
after
that,
we
can
do
a
post,
a
post,
inversion
classification.
So
here
I'm
plotting,
for
example,
the
the
scatter
plot
of
the
two
model
and
as
we
see
here,
we
have
like
a
continuous
range
of
values
as
we
as
we
expect
from
that
type
of
inversion,
and
we-
and
I
also
put
on
that
scatter
plots
a
real
physic
physical
properties,
signatures
of
each
work
unit.
B
And
if
we
do
any
sort
of
interpretation,
as
you
see
with
the
coloring
here,
it
would
be
highly
dependent
of
any
threshold
we
choose
and
all
these
behaviors
are
actually
very,
very,
very
expected
because
in
what
what
the
least
square
inversion
does
in
the
end
is
by,
if
we
take
a
smallness,
that's
in
the
shape
of
a
of
a
list
where
it
actually
means
that
we're
assuming
a
normal
distribution
of
the
parameters
m.
So
that's
so
this
is
so
this
we
call
it
instead
of
thinking
of
it
as
a
regularization.
B
B
So
so
that's
kind
of
summarized
kind
of
the
state
of
the
field
as
it
is
for
inversion
and
how
it's
commonly
done-
and
there
is
two
problems
I
can
see
with.
That
is
that
so
we
don't
reproduce
the
petrophysical
characteristic
of
the
rocks.
Even
if
they
are
known
we,
we
we
don't
reproduce
them
in
the
input
version
and
because
of
that,
and
because
we
get
that
very
continuous
distribution
of
physical
property
parameters,
it's
very
hard
to
do
some
work
identification.
B
B
So
so
to
include
physical
properties,
we
first
need
to
to
model
it
and
like
one
very
common.
So
as
we
already
seen
like
one
very
common
probability,
distribution
is
the
gaussian
distribution
and
what
we
can,
and
so
what
we
can
do
is
try
to
characterize
each
work.
You
need
each
work
unit
and
its
physical
properties
by
your
gaussian
distribution.
B
So,
for
example,
here
we
can
have
the
background
and
we
can
characterize
it,
characterize
its
density
and
magnetic
susceptibility
with
gaussians
in
a
two-dimensional
space,
and
we
can
do
the
same
for
the
pk
unit
and
the
hk
unit.
So
that's
so
like
if
we
have
samp.
If
we
have
samples
for
each
rock
each
rock
unit,
we
can
compute
the
mean
and
covariance
of
that
gaussian
distribution
quite
easily,
and
the
way
I
can
put
that
all
together
is
actually
very
simple
is
by
a
weighted
summation
of
this
different
gaussian
distribution.
B
So
that's
what
we
call
a
gaussian
mixture
model.
So
it's
it's
just
the
sum
of
the
different
equations
and
that's
that's
going
to
be
my
representation
for
the
physical
property
information
of
the
model.
So,
for
example,
here
for
the
dkc
example-
that's
what
I
will
get
with
a
background
unit,
an
hk
unit
and
a
vk
unit.
So
each
gaussian
distribution
is
centered
on
the
mean
physical
properties
of
each
unit
and
with
some
variation
around
that
mean.
B
So
it's
quite
a
complex
problem
now
because,
like
I
have
geophysical
data
that
I
want
to
invert,
I
will
also
have
petrol
physical
data.
I
want
to
include-
and
I
might
also
have
some
geological
data
like,
for
example,
from
bohol
or
like
relationship
between
units
etc.
So
so
and
I
want
to
link
all
of
those
together
and
to
be
able
also
to
formulate
assumptions
on
the
number
of
units
or
the
physical
properties
if
they
are
not
known,
etc.
B
So
how
I,
how
I
design
these
priors
now
for
this
new
inverse
problem
is:
is
this
way?
So
it's
it's,
it's
it's
a
gma
model,
but
it
includes
different
type
of
information.
So
here
I
have
the
product
just
to
say
that
it's
all
over
the
whole
mesh
with
ends
the
number
of
cells.
This
summation
here
is
the
number
of
expected
units.
So
here
in
that
case,
that
would
be
three.
Here's,
like
the
the
proportion
in
in
front
of
each
each
unit,
is
actually
at
each
cell.
B
How
much
do
I
expect
each
unit
to
appear?
So
that's
where
you
can
include,
for
example,
geology
information
about.
If
you
expect
you
need
to
appear
more
at
one
place
than
others,
and
then
we
have
the
gaussian
distribution,
and
so
what's
the
what
it
does
here.
So
we
have
our
mean,
which
has
a
mean
of
the
physical
properties
of
each
unit
unit.
B
So
this
is
so.
This
is
quite
a
complex
priors,
but
I'm
gonna
use
now
like
an
approximation,
that's
more
detailed
in
the
thesis,
but
basically
if
the
units
are
distinct
enough,
like
kind
of
in
that
case.
So
if
there
is
not
too
much
overlap
between
units,
I
can
still
approximate
that
priors
with
the
least
square
inversion.
So
it
means
I
can
rely
on.
B
I
have
some
input
geophysical
data
and
I
have
some
prior
information
in
terms
of
reference
model
and
the
smallness
weight,
and
then
we
take
a
step
of
minimizing
the
objective
objective
function
with
that
information,
and
here
I'm
just
solving
it
as
a
nonlinear
problem.
So
I'm
just
taking
one
step
of
one
step
of
the
inverse
problem
and
I
get
a
new
model,
a
new
model
m
from
that
in
geophysical,
inversion
step.
B
The
next
step
is
a
petrol
physical
characterization.
I'm
going
to
show
you
an
application
example
soon,
but
basically,
what
I
want
to
do
here
in
the
here
is
to
be
able
to
include
a
prior
information,
for
example,
if
you
have
physical
property
measurement
from
a
lab,
etc.
So
that's
my
prior
information
I
have,
but
I
also
want
to
be
able
to
include
information
from
the
geophysical
inversion
into
characterizing,
the
petrophysical
characteristic.
B
So
I
want
to
be
able
to
learn
a
petrophysical
distribution
from
both
what
I
know
from
the
lab,
but
also
from
the
geophysical
data
itself.
So
that's
that's
done
through
what
I
what
I
call
the
mapping
algorithm.
So
what
it
does
here
basically,
is
that
if
I
have
my
geophysical
model
m
here,
that
has
this
distribution
with
histogram
in
blue
and
my
prior
distribution
of
physical
properties.
Here
is
a
gray
line
and
dashed
from,
for
example,
from
labs
or
ball
measurement.
B
And
then
we
update
that
new
gmm
to
give
us
new
parameters
for
the
petrophysical
distribution,
and
then
the
geophysical
step
is
basically
for
each
cell
to
identify
to
which
unit
it
belongs
to
based
on
its
physical
property
and
the
current
gmm
distribution.
And
from
that
we
can
update
the
from
this
assignment.
We
can
update
the
reference
model
and
the
smallness
weight
at
each
cell,
and
then
we
start
over
the
process
of
with
the
geophysical
inversion
with
the
new,
updated
values
so
to
show
it
in
a
bit
in
action.
B
Here
I
showed
a
very
simple
problem
so
with
here
I
show
the
ticon
of
inversion
so
like
the
black
line
is
the
true
model.
The
blue
dashed
line
is
the
reference
model
and
the
red
is
a
geophysical
model
at
each
iteration.
So
you
see
in
the
kind
of
inversion
we
usually
have
like
the
reference
model
that
stays
at
the
same
place
and
we
stop
when
we
feed
the
data
themselves.
B
In
my
pgi
inversion,
we
see
that
the
geophys,
the
reference
model
is
updated
at
each
iteration
and
and
we
stop
when
we
fit
both
the
petrophysical
and
the
geological
data.
So
I've
designed
target
misfits
for
both
5d
and
for
my
smallness
and
they
they
follow
kind
of
the
same
pattern
because
they're,
both
this
square
term.
B
And
so
I
talked
about
before
like
about
now,
but
why
that
I
want
to
learn
the
petrophysical
distribution
and
why
I
want
to
do.
That
is
because
I
want
to
be
able
to
work
with
partial,
incomplete
or
even
like
no
petrophysical
information,
so
here
I'll
show
you
an
example
with
a
dc
example.
So
that's
here
my
true
model
with
two
cylinders,
one
conductive
and
one
resistive,
and
here
I
show
you
what
you
get
with
a
tick
on
off
type
of
information
below.
I
show
you
the
result.
B
If
I
provide
the
algorithm
with
the
true
physical
properties,
values
and-
and
I
fix
them
so
like
I,
like
the
the
algorithm
knows
the
true
petrophysical
distribution
and
in
that
that
last
slide
is
where
it
gets.
Interesting
is
that
I
launched
my
pgi
algorithm,
but
I
didn't
provide
any
type
of
information
to
the
in
terms
of
physical
properties,
except
that
there
are
three
units
and
I
let
the
algorithm.
B
So
the
mapping
algorithm
define
a
new
petrophytical
distribution
at
each
iteration,
and
this
is
what
I
get
so
I
do
not
recover
the
true
the
true
mean
of
the
physical
property
of
the
physical
properties,
but
I'm
able
to
recover
sharp
bodies
that
have
somewhat
the
right,
the
right,
the
right
shape.
So
it
can
it
can
you
and
that's
that
is
all
just
from
making
the
assumptions
that
you
can
now
make
with
that
algorithm
that
there
are
three
units
find
me
a
model
with
three
distinct
units
and
that's
what
this
is
the
algorithm
does.
B
So
that's
summarize
the
the
first
papers.
So
that's
so.
We've
developed
the
the
pgi
framework
we
formulated
as
three
interlocked
inverse
problem
and
we
did
and
which
each
is
a
fitting
criteria,
and
it
provided
us
tools
that
we
didn't
have
before
about
making
assumption
about
either
the
number
of
units
in
the
model
or
the
petrol
physical
signature
of
the
unit
and
so
far
we've
applied.
B
I've
applied
it
to
a
single
physical
property
problem,
but,
like
the
real
power
of
that
approach,
is
that
I
can
now
use
the
physical
property
and
geology
information
to
to
make
a
joint
version,
and
so
that's
going
to
be
the
next
chapters.
So
this
one
was
published
just
this
summers
and
it
also
come
with
github
repositories
to
reproduce
all
the
examples
from
the
papers
so
back
to
the
tkc
example.
B
B
So
now
I
can
apply
what
I
just
developed
in
the
previous
chapters.
Single
physics
inversion
with
the
framework
so
for
the
gravity
and
the
magnetic-
and
one
thing
that
happens
here
is
actually,
if
I
consider
the
gravity
and
magnetic
data
alone,
I
can
explain
them
totally
by
assuming
a
single,
a
single
unit
like
a
for
the
gravity.
For
example,
I
don't
need
the
signature
of
the
hk
unit.
I
can
reproduce
all
the
data
with
only
pk
and
same
for
the
magnetic.
B
I
can
only
assume
h,
I
can
also
assume
hk
unit
and
it
will
reproduce
the
the
geophysical,
the
geophysical
data
and
I'm
not
going
to
develop
it
here.
But
if
you
don't
provide
any
other
information,
adding
more
units
to
the
info,
I'm
more
unique
to
the
information.
It's
not
gonna
do
do
any
good.
It's
gonna
be
mostly
like
doing
layers
and
like
halos
during
inversion,
because
there
is
some
sort
of
a
discrepancy
principle
here
that
you
can.
B
With
that
petrophytical
properties,
information
added
to
the
inversion,
I
still
have
a
problem
when
I
combine
the
model,
because
I
still
have
a
massive
volume
that
has
both
a
very
low
density
and
a
high
magnetic
susceptibility
and
which
corresponds
to
no
known
rock
unit.
So
I
I
still
have
a
lot
of
uncertainties
in
that
area,
for
example
from
the
inversion
so
to
to
counteract
that.
B
This
is
quite
a
big
improvement
now,
because
so
I
don't
have
this
overlapping
of
density
contrast
and
magnetic
susceptibility
as
before,
and
so
you
can
see
in
the
inversion
that
I
recov
I
recover
fairly
well
like
the
location
of
each
pk
and
hk
unit.
The
physical
property
distribution
are
well
recovered
and
in
the
geological
model,
what
we
can
see
here
is
that
we
have
an
information
we
didn't
have
before,
like
we
have
like
some
information
about
the
dip
of
hk,
which
was
appearing
just
as
some
things
we
recall
before.
B
So
that's
that's
the
result
using
the
full
picture
of
petrophysical
in
information
but,
as
I
said
before,
it's
like
it's
we're
unlikely
to
have
it
in
the
in
real
life.
So
what
do
I
do
if
I
don't
know
the
real
petrophysical
signatures?
Well,
I
can
make
assumptions
about
what's
happening
here
and
if
we
look
just
back
at
the
least
square
inversion,
we
saw
that
the
two
magnetic
and
gravity
anomaly
are
not
centered
at
the
same
place.
B
So
what
I
can
do,
for
example,
with
this
framework
is,
I
can
assume
that
there
are
three
units.
One
is
a
background.
I
can
assume
that
one
has
a
low
density,
but
no
magnetic
response
and
that
another
one
has
a
high
magnetic
response,
but
no
density
contrast.
So
it's
it's
not
true,
but
it's
a
good
approximation
enough
and
that
we
can
make
just
by
looking
at
the
least
square
inversion
and
so
making
the
using
that
assumption
within
the
pgi
algorithm.
This
is
what
we
will
recover.
B
So
that's
the
if
that's
the
end
of
the
methodology
papers
for
the
multi-physics
inversion,
so
we
generalized
the
pgi
framework
to
to
multiphysic
inversion.
They
are
linked
through
a
small,
a
single
smallness
term
that
include
all
the
petrophysical
information
and
we
also
generalize
the
learning
of
the
gmm
parameters
to
still
be
able
to
learn
a
proper
petrophical
characterization
with
joint
inversion
and
so
on.
Now
I'm
gonna
move
to
the
real
case
study
of
the
of
tkc,
and
that
was
published
just
this
summer
in
the
interpretation.
B
So
to
present
you
with
a
real
data,
set
that
I
worked
with
at
tkc.
They
still
like
magnetic
data
data
sets.
But,
as
you
see
here
like
it's,
not
well
centered
on
hk-
and
there
is
a
there
is
a
negative
magnetic
here
and
it
was
identified
in
previous
papers
that
there
is
a
remains.
So
now
we
are
going
to
work
with
a
magnetic
vector
inversion,
so
meaning
we're
not
gonna
invert
only
for
amplitudes,
but
for
the
full
magnetic
vectors,
because
now
the
hk
units
we
have
to
recover
its
own
orientation
too.
B
This
is
a
gravity
data
very
similar
to
what
we
got
in
the
synthetic
except
for
that
norton
extension
here.
That
is
not
explained
by
the
pipe
and
we're
going
to
see
how
that
affects
our
approach
here,
and
we
also
have
like
a
airborne
gravity,
graviometry
dataset
that
we
also
include
so
now.
I
have
like
three
the
three
geological
surveys
that
I'm
inverting
all
together.
B
So
how
I
how
I
include
that
information
in
the
inversion
is
actually
by
including
the
elevation
as
a
fixed
parameter
in
the
inversion
and
in
my
gmm,
so
that's
gonna
be
so
here
I
present
you,
the
2d
gmm,
that's
going
to
be
for
my
density,
so
here
you
see
that
I
have
density,
so
it
doesn't
like
ends
here
on
the
y-axis,
the
elevation,
so
you
see
elevation
doesn't
affect
at
all,
but
I
can
identify
as
background,
but
here
I
have
a
linear
trend
for
density
and
from
the
petrol
physical
measurement.
We
have
we.
B
B
With
that
linear
linear
trend
updates,
as
we
saw-
and
this
is
the
result
we
got
so
we
see-
the
outline
of
the
pipe-
is
well
recovered.
But
we
have
this
massive
extension
here
in
the
northern
area.
So
we
know
from
the
drilling
that,
as
you
see
here,
there
are
some
near
surface
kimbo
light,
but
this
seems
to
appear
only
as
a
smooth
features.
Here,
that's
still
classified
as
background
and
we're
going
to
come
back
to
that
a
bit
later
in
term
of
magnetic
representation.
B
We
only
had
four
sample
for
the
amplitude
of
hk
and
we,
but
we
measured
both
the
magnetic
sensitivity
and
the
koenigs
burger
ratio
which
measures
the
strength
of
the
remaining
field,
and
what
we
see
here
is
that
the
remaining
field
is
10
10
times
stronger
than
the
induced
field.
So
this
is
the
dominant
field
really
here
at
the
at
tkc
and
from
previous
studies,
that's
from
devries
and
al.
B
We
designed
this
3d
gmm
for
the
three
for
the
three
parameters
of
the
magnetic
vectors
in
cartesian
space
for
simplicity
so
like
we
here,
we
represent
the
amplitudes
in
x,
y
and
z
of
the
magnetic
field,
vectors
and
inputting
the
magnetic
data
with
that
information.
This.
This
is
a
magnetic
vector
model
that
we
recover.
So
we
see
that
we
recover
fairly
well
the
distribution.
B
But
same
as
in
the
in
the
synthetic,
when
I
combine
them
together,
I
have
like
a
have
a
big
area
that
has
both
gravity
and
magnetic
response
which
doesn't
correspond
to
anything.
We
know
about
the
petrol
physics
in
the
area,
so
next
step,
obviously
is
to
move
to
the
multiphysics
pgi
and
that's
what
we
get
with
the
same
kind
of
outcome
as
before.
So
we
we
remove
that
uncertain
zone
with
with
the
wrong
petrol.
B
Physical
signatures
and
magnetic
vectors
are
still
kind
of
well
recovered,
and
we-
you
see
here,
for
example,
in
the
magnetic
inversion,
that
the
orientation
of
the
magnetic
field
for
pk
and
hk
are
different,
because
pk
is
mostly
induced
field,
so
it's
oriented
as
the
earth's
magnetic
field
and
we
for
hk.
We
orient
it's
based
on
instrument
information
and
we're
able
to
have
this
different
orientation
within
the
same
inversion.
B
B
So
so
that's
so
now
we
can.
With
the
with
that
framework,
we
can
make
the
assumption
about
a
force
unit
and
we
can
make
assumptions
about
its
petrophysical
characteristics
and
what
we,
what
we
we
did
here
is
actually
we
run
the
pgi
inversion
with
many
different
parameters
for
that
first
unit.
So
here
I
show
you
the
new
here,
for
example,
a
new
gmm
distribution.
So
you
see
here
we
still
have
the
background
and
pick
a
unit
with
a
linear
trend.
B
But
here
now
we
add
another
unit
for
the
near
surface,
so
we
limit
it
in
term
of
elevation
to
quite
high
and
we
we
tested
it
for
various
mean
and
and
covariance
and
based
and
we
choose
what
we
choose.
Basically,
as
expert
knowledge
with
expert
knowledge,
what
we
thought
would
be
the
best
in
the
area
and
that's
then,
and
then
we
include
that
information
in
the
final
multiphysics
inversion
with
four
units
and
that's
the
final
result.
B
B
So,
and
that
concludes
our
presentation
of
this
new
framework
so
to
as
a
summaries,
I
consider
the
inverse
problem
now
as
a
string
interconnected
estimation
of
the
geophysical
model,
the
petrophysical
distribution
and
quasi
geology
model,
and
it's
this
framework
is
quite
general.
It
doesn't
depend
on
the
geophysical
method.
You
use.
B
We
solve
everything
as
a
nonlinear
problem,
and
so
we
can
couple
any
type
of
geophysical
data
together,
and
it
adds
quite
a
lot
of
tools
to
formulate
assumptions
that
are
directly
related
to
the
geological
problem
about
the
subsurface
and,
as
a
last
note,
I
also
want
to
thanks
all
of
the
thin
peg
and
open
source
community
that
made
this
work
possible
like
it
was
quite
a
great
experience
that
I
really
enjoyed.
So
thank
you.
So.
B
It's
it's
as
it's
as
you
wish.
If
you
have
questions,
I'm
very
happy
to
answer
them
and
if
you
have
comments
about
the
presentation
itself,
it's
it's
welcome
to
it.
I
I
don't
think
I
don't
think
it
needs
to
be
separated.
If
you
have
comments,
makes
comment,
if
you
have
questions,
ask
questions.
It's
it's!
It's
fine.
D
B
So
I
like,
I
have
not
emphasized
this
here,
but
it
can
be
done
in
in
different
ways
and
that
might
work
well
or
not
of
the
problem
so,
for
example,
for
electric
connectivity,
it's
the
distribution
that
your
gaussian
has
a
log
space,
but
it's
it's
fine,
because
we
often
invert
with
the
in
the
logarithmic
space
for
the
inversion
too.
So
in
that
case,
it's
quite
it's
it's!
It's
quite
non-problematic
for
the
where
I
got.
Problems
was
more
for
the
magnetic
susceptibility
inversion,
because
magnetic
susceptibility
is
in
reality.
B
But
it
didn't
work
quite
well.
I
think
there
is
a
lot
of
things
to
do
on
that
side,
and
so
my
main
problem
was
that,
because
it's
a
linear
problem
using
the
log
of
systematic
sustainability,
my
sensitivities,
then,
where
we're
having
kind
of
like
an
exponential
range
there
was
and
because
we're
doing
inversion
based
on
the
sensitivity
we're
taking
a
step
very
quickly.
The
induction
was
getting
stuck
because
a
small
area
was
at
a
high
sensitivity
and
the
rest
was
not
that
sensitive
and
was
kind
of
flat.
B
So,
but
what
I
did
for
the
real
case
scenario
for
the
magnetic
for
the
magnetic
vectors
was,
I
I
define
and
and
also
have
I
didn't,
have
much
much
information
like
I
had.
I
said
I
had
four
samples
and
then
I
have
an
estimation
of
the
orientation
based
on
the
on
the
degrees
and
I'll
studies.
So
what
I
did
is
I
defined.
D
B
B
To
take
some
sort
of
like
a
middle
ground
like
a
and
so
that's
why,
if
I
go
back
to
the
distribution
here
like
this,
is
my
background
and
you
see
this
is
super
small
compared
to
the
extension
of
the
edge
of
the
hk
unit.
But
at
the
same
time
that's
also.
I
got
all
the
correlation
assuming
the
amplitude
and
the
angle
were
independent.
B
E
B
B
My
background
and
here's
already
zero
engine,
you
see
that
here
in
that
inversion,
most
of
my
mac
cells
are
included
within
that
very
small,
very
small
gaussian
distribution-
and
this
is
the
red,
so
actually
here
most
of
the
cells
are
are
contained
within
the
the
smallest
gaussian.
Is
that
what
you're
asking.
E
B
Are
you
right,
yeah,
it's
because
here
it's
just
so
small,
you
don't
see
it.
It's
that's!
That's
that's!
The
tricky
part
is
that
there's
I.
I
agree
that
the
area
of
influence
of
the
hk
unit
is
very
big
when
you
come
because
we
convert
it
from
a
logarithmic,
specs
logarithmic
space
to
a
linear
space.
So
so
what
what?
What
seems?
Okay
in
log
space
in
log
space,
now
linear,
like
there's
a
small
variation
for
the
background,
hk
app,
is
so
small,
but
they
still
contain
90
percent
of
the
mesh
or
more.
D
E
Yes,
so,
like
you,
my
next,
my
next
question
is
the
issue
that
I
always
see
is
that
the
the
influence
of
the
gaussian
mixtures
is
there's
nothing,
no
spatial
component
to
it
right,
so
they
kind
of
expand
throughout
the
entire
entire
match
right.
So
what's
your
next
step
to
to
be
able
to
be
a
little
bit
more
spatially
oriented.
B
B
How
likely
each
unit
is
like
how
like
what's
my
expectation
of
each
unit
and
so
basically
hey?
What
I
did
is
that
we
have
like
from
the
geological
like
what
are,
including
as
geological
information,
is
that
the
pipe
the
maximum
of
the
pipe
is
that
nothing
at
seven
one
133
680
meters
for
the
noting
so
so.
Basically,
here's
a
the
like
the
the
likelihood
of
the
pipe
you
need
to
appears
north
of
that
limit
is
zero
like
this,
and
you
need
the
pipe
you
need
cannot
appease
for
the
nose.
B
That's
what
we
mapped
with
the
with
the
bore
holes
and
that's.
Why
also
that's
why
we
kind
of
like
cut
it
here,
but
also,
but
it's
taken
over
by
this
by
this
near
surface
in
near
surface
unit,
so
we
can
actually,
we
can
actually
vary
the
the
properties
of
the
gmm,
especially
by
varying
the
the
proportion.
E
Yeah,
that's
great,
that's
great!
If
you
know
that's
right,
but
in
let's
say
you're
in
kind
of
more
of
a
green
field
setting
and
then
you're,
not
too
sure
you
can
you
foresee
I'm
not
asking
like
like
if
you
ever
have
done
it
already,
but
can
you
foresee
like
a
way
that
we
could
automate
that
kind
of
picking
of
you
know
or
have
more
like
a
spatial
variations
of
your
groups?
Basically
who
can
turn
off
groups
in
one
area
because
something
I
don't
know,
I
don't
know
if
you
thought
about
it.
E
B
It's
it's
like
there
is
possibilities
for
that.
I've
not
played
too
much
about
it
because,
as
I
say
like
for
this
mapping
algorithm
like
here,
I've
shown
mostly
what
I
can
do
by
varying
the
mean,
but
I
I've
defined
similar
problem
for
the
for
the
covariance
and
the
proportion,
and
I
have
an
example
in
the
thesis
for
the
covariance
with
our
groundwater
problem.
Where
we
don't
know
the
mean-
and
we
don't
know
too
much
about
the
covariance
either.
B
So
I
let
both
works
but
and
the
proportion
I
used
it
less
and
because
I
what
you
just
said,
I
consider
it
as
a
danger
at
some
point,
but
it
maybe
there
is
some
application.
Is
that
you
can?
You
can
also
vary
the
proportion
either
global,
either
globally
or
locally
the
the
problem.
That
is,
that
can
happen,
maybe
and
like.
I
turn
it
off
in
this
problem,
for
that
is
that
yeah
like,
if
is
that
some
clusters
might
like,
if
you
vary
the
proportion.
B
But
you
can
also
try
to
maybe
to
vary
the
proportions
locally,
but
you
have
the
risk
of
like
maybe
losing
some
units
and
and
I've
done
some
work,
but
it's
more
like
in
the
development
section
of
the
thesis
too.
I
have
some
picture
here
where
I'm
using
image
segmentation
to
varies
the
proportion
also
locally,
and
let
me
find
the
picture
for
that.
B
It's
not
a
geological
image,
but
so
here,
if
you
see
the
example,
it's
I'm
just
using
this
the
gmm
for
image
segmentation,
so
like
really
for
the
clustering.
What's
what
it's
been
used
for
before
and
you
see
if
we
do
just
a
normal
gmm
classification,
we
got
that
type
of
image
and
it
can
be
kind
of
noisy
or
pixelated
if
you
wish,
but
within
the
proportion
you
can
also
use.
B
So
there
is
definitely
maybe
some
application
to
do
on
the
geological
side
to
to
force
to
force
special
continuities,
but
maybe,
but
as
I
as
I
showed
nick's
at
some
point
also,
we
pointed
out
that,
for
example,
the
eyes
that
are
still
visible
in
the
inversion
in
the
gmm
here
are
not
visible
anymore
here.
So,
but
you
have
a
balance
between
special
continuities
and
small
features.
So
I
I
still
have
a
lot
of
work
to
do
on
that
side,
but
that's
that's
where
I'm
hoping
to
head
next
yeah
the
special
continuities.
F
F
B
Okay
yeah,
when
I
use
the
mvi
so
well.
We
we
have
the
so
like
it's
true
that
in
if
we
do
the
single
physical
inversion,
it's
not
quite
necessary
because
it's
all
in
the
same
direction-
and
I
agree
with
that-
like
you,
can
you
for
the
single
physics
pgi
with
a
single
unit?
With
that
only
hk,
I
could
have
fixed
the
orientation
and
say
well,
this
is
the
orientation
of
the
field
and
you
you
can
invert
for
that.
But
the
the
real
gain
is
really
for.
B
When
we
go
to
the
multiphysics,
because
here
we
have,
we
have
two
kimberlite
units
with
different
different
orientation,
and
I
don't
know-
and
I
don't
know
the
location
priors
so
I
need
the
I
needed
the
inversion
to
be
able
to
to
to
have
the
cells
in
different
directions
like
and
but
still
fall
within
bins
for
either
the
pka
induced
field
or
the
hk
remain
on
field.
Is
that.
F
F
F
B
Yeah,
so
right
now,
I've
mostly
experimented
with
kind
of
like
a
black
and
white
approach.
It's
is
it's
either.
I
I
like,
as
for
most
of
the
case
study
example
where
I
fixed
the
physical
property
values
to
what
I
measured
in
the
lab.
So
so,
basically,
there
is
no
waiting
here
like
so
so,
if
I
take
here
this
example,
my
prior
distribution
will
be
my
distribution
at
all
iterations,
so
that
black
line
will
be
the
same
as
the
gray
line.
B
That's
that
would
be
a
way
that
will
be
the
weight
where
I
put
all
the
weight
on
the
on
the
on
the
laboratory
prior
measurement
side,
if
I
put-
and
I've
also
exp
and
the
other
type
of
experiment
is,
if
I
put
all
the
weight
on
the
geophysical
side.
So,
for
example,
here
that
black
line
will
fit
the
blue
histogram
here
and
that's,
for
example,
what
I
what
what
I've
used
in
the
dc
example
here
there
is
no
petrophysical
information
except
the
number
of
units.
B
So
where
I
have
that
where,
where
I,
I
assume
that
one
clust
one
rock
unit
is
responsible
for
the
magnetic
response
and
one
unit
for
the
gravity
response.
So
on
those
two
on
those
two
panels
here
I
show
the
mean
of
the
three
units,
so
the
black
is
always
the
background.
So
the
background
is
fixed.
The
background
is
fixed
at
zero
contrast
for
both
physical
properties.
So
it's
all
fixed
from
our
price
of
our
priors
orange
here
is
the
hk
unit,
and
so
I
fixed
its
its
density
to
be
to
be
a
zero
contrast.
B
B
The
background
is
still
held
fixed,
but
this
time
this
is
the
pka
magnetic
sensitivity
that
is
kept
fixed
at
zero,
and
I
leave
the
hk
magnetic
sensitivity
free
so
like
all
the
weight
now
is
on
the
geophysical
model
side
and
it
and
it
moves,
and
then
you
can
feel
like
the
things
I've
experimented
less
I've.
I've
shown
some
examples
in
the
thesis
where
you
put
like
a
half
weight
between,
for
example,
they
go
like
the
between
the
the
two.
B
So
that's
that's
what
I
show,
for
example,
in
this
example
here
between
the
geophysical
model
in
blue
and
the
prior
distribution
in
gray,
and
I
get
the
black
one
here.
This
is
basically
by
putting
equal
weight
for
both
magnet
the
geophysical
model
and
the
priors,
and
what
I
show
in
the
testis
basically,
is
that
we,
as
as
you
see
as
you
see
here
like
I
didn't
like
this-
is
the
result
I
get
with
a
zero
with
putting
like
zero
weight
for
the
for
the
true
physical
properties.
B
But
if
I
put
additional
additional
weights
on
for
with
a
for
mean
outside
of
the
inv
of
the
geophysical
inversion,
it's
gonna
slowly
push
it
to
the
to
the
through
the
through
distribution.
So
there
is
a
continuum
here
so
like
if
you
have
some
information,
but
you
don't
want
to
put,
if
you
don't
want
to
force
it
too
much,
you
might
put,
you
might
put
like
a
90
and
10
percent
weight
and
things
like
that.
B
I
don't
have
like
a
like
good
estimate
of
now
for
what
to
what
to
put
as
weight,
and
there
seems
to
be
a
threshold
too
like
if
you
put
a
50
50
weight
between
the
geophysical
model
and
the
petrophysical
distribution.
Priors
like
a
50
weight
is
actually
enough
that
it's
gonna
push
the
inversions
toward
reproducing
what
you
put
as
priors.
B
B
B
Yes,
so
I
have
a
slide
for
that
and
I
removed
it
because
it
was
taking
already
too
long,
but
that's
the
slide.
I
have
for
that.
So
to
explain
for
you
the
to
use
the
what's
what
the
approximation
come
from
is
that
here
I
put,
I
showed
two
gaussian
for
representing
inter
watch
a
rock
unit
in
blue
and
red
and
in
the
fine
black
line.
B
Here
is
the
resulting
gaussian
mixtures
and
use
like-
and
you
see
like
if,
if
I
have
like
good
separation
like
the
gaussian
mixtures,
can
be
well
approximated
locally
by
the
dominant
gaussian.
So
and
that's
how
I
get
back
to
early
square
to
a
least
square
formulation,
because,
for
example,
here
I
will
approximate
the
the
ocean
mixtures
by
the
blue,
the
blue
gaussian
and
then
because
it's
simply
a
gaussian,
it
translates
back
to
a
list
early
square.
But
then,
if
you
have
things
that
are
overlapping
like
that,
you
see
this.
This
approximation
is
not
as
good.
B
B
There
is
a
practical
reason
and
that's
something
I
didn't
really
develop
here,
but
I
had
the
idea
at
the
beginning
was
that
with
that
approximation,
as
I
said,
I
can
use
comp
file
code
and-
and
I
can
use,
for
example,
the
ubc
code
that
I
can
exit
at
each
iteration,
rewrite
a
new
reference
model
and
smallest
weight
and
send
it
back
to
the
ubc
code
without
modifying
it
modifying
it.
So
there
is
for
people
that
relies
on
legacy
codes.
It
could
be,
it
could
be
something
you
can
build
around
and
that's,
I
think,
that's
important.
B
The
second
one
was
more
like
a
educational
reason.
Is
that
imagine
that
so,
like
you
saw
the
big
priors
like
with
that
that
that
that
product
with
the
sum
of
different
of
that
things,
then
put
an
additional
minus
log
in
front
and
then
put
plug
it
into
an
objective
function
and
it's,
it
could
be
very
hard
at
first
glance
to
wrap
your
head
around
it.
B
B
And
and
I've
tested
both
formulation
in
my
code
so
with
the
approximation
and
without
the
approximation
and
in
most
cases
that
I
work
with
so
far
like
they
re,
like
I'm
at
least
at
three
sigma.
So
it's
the
approximation
works
quite
well
and
I
haven't
seen
in
those
cases
any
significant.
This
difference
in
the
inversion
result
with
or
without
the
approximation.
B
F
B
D
F
A
I
need
to
to
duck
out
and
head
to
my
next
meeting
but
wonderful
job
tebow
I'll
follow
up.
I
left
some
notes
in
the
google
doc
and
I
I
do
have
some
questions
in
mind,
but
I
can
follow
up
with
you
afterwards,
on
slack
so
I'll
hand
you
the
the
host,
so
that
you
can
continue
chatting
with
with
folks,
and
I
think
that'll
keep
the
recording
going
and
I
can
always
edit
it
afterwards
if
there's
stuff
that
you
don't
want
posted
so
just.