Geology Reference
In-Depth Information
kV
P
. In this case
elastic inversion was joint or simultaneous inversion
of near and far angle stacks (e.g. Rasmussen et al.,
2004
) in which additional constraints were applied,
such as the co-dependency of P and S, in order to
refine the inversion.
by a Gardner relation of the type
ρ ¼
R
s
is linearly related to R
p
and G:
R
S
¼
aR
p
+bG,
ð
9
:
4
Þ
2n
1+n
1+
ε
1
where a
¼
and b
¼
2
:
It is thus possible to create trace volumes of the
P and S wave reflection coefficients from AVO inter-
cept and gradient and invert them individually to P and
S wave impedance using exactly the same methods as
discussed previously for acoustic impedance inversion.
As with the Fatti approach a smoothed background
model of V
s
/V
p
is required as an additional input to
the calculation of R
s
. The equations above are a salient
reminder that the shear component in elastic inversion
is directly related to the AVO gradient. Noise in the
gradient (
Chapters 5
and
6
) will translate into errors in
the impedance estimation. This is why data condition-
ing is considered to be so important for AVO inversion.
8
γ
2
8
γ
9.2.7.2 Elastic inversion
-
the Fatti approach
The
(after Fatti et al.,
1994
)isbasedon
the extraction of fitting coefficients from pre-stack data
using two- or three-term Aki
'
Fatti approach
'
-
Richards approxima-
tions. A re-write of the Aki
Richards three term equa-
tions is shown below. By introducing constraints based
on a P wave velocity cube (i.e. offset to angle calcula-
tions and invoking a V
p
/V
s
transform (such as Castag-
na
-
s mudrock line (
Chapter 8
)) the Zoeppritz
approximation can be sufficiently constrained to obtain
P and S reflectvities (two-term) or P, S and density
reflectivities (three-term) from least-squares fitting:
'
R
PP
ðÞ¼
c
1
R
P0
+c
2
R
S0
+c
3
R
D
,
ð
9
:
2
Þ
9.2.7.4 Pre-stack simultaneous inversion
The two-step process of reflectivity estimation followed
bymodel based inversion is now commonly replaced by
one-step pre-stack simultaneous inversion algorithms
deriving Z
p
and Z
s
directly (e.g. Ma,
2002
;Hampson
et al.,
2005
;Russellet al.
2006
;
Fig. 9.23
). Clearly a good
quality control is the match of the gathers input to the
inversion with the synthetic gathers generated from the
inversion result (
Fig. 9.24
). Once Z
p
and Z
s
volumes
have been created, they can be easily manipulated to
create other useful volumes such as V
p
/V
s
(
V
p
2
sin
2
tan
2
8
V
s
where
c
1
¼
1
+
θ
,
c
2
¼
θ
,
V
p
2
h
i
,
2
2
Δ
V
P
V
P
+
Δ
ρ
tan
2
2
V
s
sin
2
c
3
¼
θ
θ
,
R
P0
¼
h
i
and R
D
¼
Δ
ρ
2
Δ
V
S
V
S
+
Δ
ρ
R
S0
¼
.
Reflectivities extracted from the gathers are subse-
quently inverted to acoustic impedance and shear
impedance, for example using the model based inver-
sion technique. The subsequent development was for
simultaneous inversion of Z
p
(acoustic impedance)
and Z
s
(shear impedance) from R
p
and R
s
(e.g. Pendrel
et al.,
2000
).
¼
Z
p
/Z
s
). It is
also possible to create impedance volumes
μρ
which are the product of density and the Lamé elastic
constants
λρ
and
Z
S
.These
might be regarded as fluid and lithology volumes
respectively (Goodway et al.,
1999
). As with AVO pro-
jections (
Chapter 5
) various adaptive combinations of
acoustic and shear impedance can be created to high-
light lithology and fluid (e.g. Espersen et al.,
2000
;
Russell et al.,
2006
).
To stabilise the inversion process, it is usual to
supply background models of the relation between
Z
p
and Z
s
and between Z
p
and density; the inversion
calculates differences from this background trend
(
Fig. 9.25
). This may be problematic if several differ-
ent lithologies are present in the zone of interest, with
very different V
p
/V
s
ratios. Pre-stack simultaneous
inversion is commercially available in a number of
software packages and is widely used. A typical result
is shown in
Fig. 9.26
.
Z
P
2Z
S
and
λ
and
μ
:
λρ ¼
μρ ¼
9.2.7.3 Elastic inversion
-
intercept and
gradient approach
Elastic inversion can also be approached from the
perspective of the AVO intercept and gradient
(White,
2000
). The AVO intercept, which is the
normal incidence P wave reflection coefficient R
p
,
and gradient G can be related to the normal incidence
S wave reflection coefficient R
s
as follows:
ε
Δ
ρ
ρ
2
R
S
+
G
¼
R
P
8
γ
,
ð
9
3
Þ
:
where V
p
, V
s
and
are the average values of the
P wave velocity, S wave velocity and density at the
interface,
ρ
V
s
γ ¼
V
p
,
Δ
ρ
is the density contrast across
2
1
the interface, and
is small
and the density term can be adequately approximated
ε ¼
2
γ
2
. In most cases
ε
211
