Extended Elastic
Impedance using HRS-9
Brian Russell
Hampson-Russell, A CGGVeritas Company
Introduction
In a recent SEG Distinguished Lecture, Patrick Connolly outlined BP’s company-wide approach to fluid and lithology prediction using seismic data.
The cornerstones of this approach are the Coloured Inversion (CI) and Extended Elastic Impedance (EEI) methods.
This talk will first present a general framework for pre-stack and post-stack inversion methods.
I will then review the principles of EEI inversion within this framework.
Finally, I will show how the EEI method has been
implemented in HRS-9, using both model-based and coloured inversion
Inversion in general
The above flowchart shows the general approach to seismic trace inversion, which involves a geological model, a seismic volume and an inversion algorithm. We can apply the method to either pre- or post-stack seismic data.
Geological
Model Seismic Volume Inversion
Algorithm
Inverted Seismic Volume
Post-stack inversion
The earliest trace inversion approach involved building an AI (acoustic impedance, or rVP) model and inverting the stacked
volume to create an AI output.
AI Model
Volume Stacked Volume Inversion
Algorithm
AI
Inversion
Model building
Building a model volume involves the following steps:
Create the log property of interest at each well location and insert it into the model (in this case, we create
acoustic impedance by multiplying VP by density).
Make sure the well logs match the seismic data in time by performing correlation with an extracted wavelet. Interpolate the logs using an algorithm such as
inverse-distance weighting or kriging.
Insert the seismic picks to guide the interpolation structurally.
Apply a low pass filter (typically 0 – 15 Hz) so that the detail in the inversion will come from the seismic data.
6
Recursive: Bandlimited inversion, in which the seismic trace is integrated and added to the low frequency part of the model.
Model Based: Iteratively updates the initial model to find a best fit to the synthetic.
Sparse Spike: Constrained to produce as few events as
possible, with the low frequency model added in. Coloured: Spectrum of seismic data is shaped to the well log
spectrum and a 90 degree phase shift applied. In the standard implementation, no low frequencies are added back (relative impedance) but in our implementation we can add them back (absolute impedance).
These post-stack inversion methods are available in HRS-9:
Gas sand stack
For example, here is a stack over a gas sand from Alberta, showing a “bright-spot” anomaly.
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Post-stack inversion
AIAI
AI
=
R
AI2
Here is the acoustic impedance inversion of the previous stack. The gas sand is a low impedance event, which is ambiguous because the shales are also low impedance.
Note that the seismic reflectivity can be written as the AI difference divided by twice its average:
Pre-stack simultaneous inversion
A more recent inversion approach is to build AI (rVP), SI (rVS)
and density (r) models and invert the seismic angle gathers
simultaneously. The next two slides show the gathers and the pre-stack inversion result.
AI, SI and r
models gathers Angle Simultaneous
inversion
AI, SI and
density volumes
The seismic gathers
The seismic line is the “stack” of a series of CMP gathers, as shown here. Here is a portion of a 2D seismic line showing the gas sand “bright-spot”.The gas sand is a typical Class 3
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AI
Vp/Vs
Pre-stack simultaneous inversion
12
A very robust AVO method involves analyzing near and far angle stacks, as shown here.
Note the amplitude of the “bright-spot” event is stronger on the far-angle stack than it is on the
near-angle stack. But what does this mean?
Near and far trace stacks
Near angle (0-15o) stack
Elastic Impedance
The equivalent impedance method to near and far angle stacking is Elastic Impedance, or EI (Connolly,1999). To understand EI, recall the Aki-Richards equation:
.
sin
4
1
and
,
sin
8
,
tan
1
:
where
,
2
2
2
)
(
2 2 2
r
r
K
c
K
b
a
c
V
V
b
V
V
a
R
S S P P P
c b S a P EIEI
EI
V
V
EI
EI
R
r
ln
(
)
,
w here
(
)
2
1
)
(
)
(
2
1
)
(
Connolly postulated that associated with this equation is an underlying elastic impedance, written (where I have re-named the EI reflectivity to match the AI concept):
Elastic impedance inversion
The inversion approach for EI involves building an EI() model
and inverting an angle stack volume at an angle to create an
EI output.
EI() Model
Volume Angle Stack Volume Inversion
Algorithm
EI() Inversion
Here is the comparison between the EI inversions of the near-angle stack and far-angle stack. Notice the decrease in the elastic impedance value on the far-angle stack.
Gas sand case study
EI(7.5o)
The figures show the (a) crossplot between near and far EI logs, and (b) the zones on the logs. Notice the clear indication of the gas sand (yellow).
EI from logs
(a) (b)
EI_Near EI_Far
Gas sand case study
This figure shows a crossplot between
EI (7.5o) and EI (22.5o). The
background trend is the grey ellipse, and the anomaly is the yellow ellipse. As shown below, the yellow zone
corresponds to the known gas sand.
EI at 7.5o
EI
at 22
.5
18
Scaled Elastic Impedance
One of the problems with EI is the fact that the values do not scale correctly for different angles.
This is due to the variable dimensionality found by raising the velocity and density terms to increasing powers.
Whitcombe (2002), proposed solving this by scaling the EI equation as follows:
constants.
reference
are
and
,
,
where
,
)
(
0 0 0 sin 4 1 0 sin 8 0 tan 1 0 0 0 2 2 2r
r
r
r
S P K K S S P P PV
V
V
V
V
V
V
EI
This also leads us to extended elastic impedance (EEI). But first we need to review the intercept/gradient method.
The Intercept/Gradient method
The Intercept/Gradient method is an approach to AVO which involves re-arranging the Aki-Richards equation to:
:
w here
,
tan
sin
sin
)
(
A
B
2
C
2
2
R
P
This is again a weighted reflectivity equation with weights of a = 1, b = sin2, c = sin2 tan2.
. and , 2 , 2 4 2 8 2 , 2 2 2 P S p P VP S S p P p P AI V V K V V R C K V V K V V B V V R A
r
r
r
r
20 Offset +A +B - B sin2 Time
The Aki-Richards equation predicts a linear relationship between these amplitudes and sin2θ. Regression curves are
calculated to give A and B values for each time sample. The amplitudes are extracted at all times, two of which are
shown:
-A
The result of this calculation is to produce 2 basic attribute volumes Intercept: A Gradient: B
Extended Elastic Impedance
As the next step from scaled EI, Whitcombe et al.
(2002) introduced Extended Elastic Impedance, or EEI. First, they replaced the sin2 term in the two-term
Aki-Richards equation with tanc, to give the following
expression for EEI reflectivity, REEI.
22
c
c
c
c
c
c
c
sin
cos
cos
)
(
)
(
tan
)
(
sin
)
(
2B
A
R
R
B
A
R
B
A
R
EEI P
Notice that EEI will equal acoustic impedance at c = 0o and gradient impedance (GI) at c = 90o. The limits of c
Extended Elastic Impedance
This leads to the extended elastic impedance expression, which involves substituting the EEI(c) reflectivity expression
into the scaled elastic impedance expression, to give:
Different values of c correlate with various rock properties, as shown on the next slide.
.
sin
4
cos
,
sin
8
,
sin
cos
where
,
)
(
0 0 0 0 0c
c
c
c
c
r
r
r
K
r
K
q
p
V
V
V
V
V
EEI
r q S S p P P P
Extended Elastic Impedance
Figure (a) shows EEI values at different angles and figure (b) compares elastic parameters to their equivalent EEI curves.
Whitcombe et al. (2002) (a) (b) 24 Vp/Vs EEI(45o) l EEI(19o) K EEI(10o) m EEI(-58o) rVs EEI(-45o)
EEI inversion
The EEI inversion approach involves building an EEI(c) model
and inverting the EEI(c) volume using an inversion algorithm
to create an EEI output.
EEI(c) Model
Volume Volume EEI(c) Inversion
Algorithm
EEI(c) Inversion
Implementation in HRS-9
Now that we have discussed the theory of EEI, let’s see how to implement the process in HRS-9.
The process involves four steps:
Choose a target log and find the optimum c angle. Build the log parameter model.
Compute the EEI(c) seismic volume from the intercept and gradient.
Perform the inversion.
We have recently built this functionality into HRS-9.
Since the process involves a number of steps, we will first build a Workflow.
Creating an EEI Workflow
We start by creating a new workflow Group Name and then start the Workflow Builder.
EEI Workflow
The workflow is built by moving processes from the Process to the Workflow list. The final Workflow is on the right. 28
Gas sand case study
We will now apply this Workflow to the dataset just described.
As our target log, we have chosen the Vp/Vs ratio. Note that this will produce the same output as
pre-stack simultaneous inversion.
However, the approach will be different since we are building an EEI model and inverting the rotated
intercept and gradient stacks.
In both cases, we are using the pre-stack data as input, rather than the post-stack data.
We will apply both model-based and coloured inversion.
First, use the AVO Attribute Volume option to create A and B:
30
Creating A and B volumes
This display is the product of A and B:
31
Correlation plot
We then find the maximum correlation value, which is at 39o with a correlation coefficient of close to 1.00. In this case,
almost any value between 30o and 60o would work reasonably well. However, in some cases there is a clear peak.
A good display option is the EEI Spectrum, which shows the EEI computation for every angle between -90o and +90o. 32
Next, we compute the EEI log at c = 39o. It closely resembles
the Vp/Vs ratio log but the units are impedance. 33
EEI log curve at
c
= 39
oVp/Vs Ratio EEI at 39o Vs Density Vp
EEI model
Next, we compute the EEI reflectivity section, as shown above (Note: REEI(39o) = A*cos(39o) + B*sin(39o))
EEI model
EEI wavelet
We then extract the statistical wavelet from the EEI reflectivity: 36EEI inversion analysis
Model-based EEI inversion
Model-based inversion with the EEI log is shown here:
38
Model-based EEI inversion
Here, we scaled to Vp/Vs units using a single scaler:
39
Coloured EEI inversion
Alternately, we apply coloured inversion, with relative scaling.
40
Conclusions
This presentation has been an overview of the extended elastic impedance (EEI) approach using HRS-9.
I first reviewed post and pre-stack inversion methods.
I then discussed EEI theory and how the general inversion method could be modified to implement EEI inversion.
I then showed how to find an optimum c angle to create the
EEI section. I chose Vp/Vs ratio as the target log.
I then showed how to create the EEI section in HRS-9. Next, I created the inverted Vp/Vs volume, using both
model-based and coloured inversion.
Both of the EEI inversion methods gave us excellent
definition of the gas sand zone, comparable to pre-stack inversion.