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Transcript of Jan01 Seismic Inversion
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FEATURE ARTICLE
SEISMIC INVERSION THE BEST TOOL
FOR RESERVOIR CHARACTERIZATION
By John Pendrel, Jason Geosystems Canada
Introduction
The principle objective of seismic inversion is to transform
seismic reflection data into a quantitative rock property, descrip-
tive of the reservoir. In its most simple form, acoustic impedance
logs are computed at each CMP. In other words, if we had drilled
and logged wells at the CMPs, what would the impedance logs
have looked like? Compared to working with seismic ampli-
tudes, inversion results show higher resolution and support
more accurate interpretations. This in turn facilitates better esti-
mations of reservoir properties such as porosity and net pay. An
additional benefit is that interpretation efficiency is greatly
improved, more than offsetting the time spent in the inversion
process. It will also be demonstrated below that inversions make
possible the formal estimation of uncertainty and risk.
In various forms, seismic inversion has been around as a
viable exploration tool for about 25 years. It was in common use
in 1977 when the writer joined Gulf Oil Companys research lab
in Pittsburgh. During this time, it has suffered through a severe
identity crisis, having been alternately praised and vilified. Is it
just coloured seismic with a 90 deg phase rotation or a unique
window into the reservoir? Should we use well logs as a priori
information in the inversion process or would that be telling us
the answer? When should we use inversion and when should we
not? And what type: blocky, model-based, sparse spike? At
Jason Geosystems we offer all these algorithms - and a few more.
In the following, I will briefly discuss them in a somewhat quali-tative manner, keeping the equations to a minimum. After all, it
is drilling results which count and against which any algorithm
will ultimately be measured.
The Inversion Method
The modern era of seismic inversion started in the early 80s
when algorithms which accounted for both wavelet amplitude
and phase spectra started to appear. Previously, it had been
assumed that each and every sample in a seismic trace represent-
ed a unique reflection coefficient, unrelated to any other. This
was the so-called recursive method. The trace integration method
was a popular approximation. At the heart of any of the newer
generation algorithms is some sort of mathematics - usually in
the form of an objective function to be minimized. Here I will
write that objective function, in words, rather than symbols and
claim that it is valid for all modern inversion algorithms - blocky,
model-based or whatever.
Obj = Keep it Simple + Match the Seismic + Match the Logs (1)
Lets look at each of these terms, starting with the seismic. It
says that the inversion impedances should imply synthetics
which match the seismic. This is usually (but not always) done
in a least squares sense. Invoking this term also implies a knowledge of the seismic wavelet. Otherwise, synthetics could not be
made. At this point, life would be good except for one thing. The
wavelet is band-limited and any broadband impedances tha
would be obtained using the seismic term only would be non-
unique. Said another way, there is more than one inversion
impedance solution which, when converted to reflection coeffi
cients and convolved with the wavelet would match the seismic
In fact there are an unlimited number of such inversions. Worse
such one-term objective functions could even become unstable as
the algorithm relentlessly crunches on, its sole mission in life
being to match the seismic, noise and all, to the last decima
place.
Enter the simple term. Every algorithm has one. It could no
care less about matching the seismic data, preferring instead to
create an inversion impedance log with as few reflection coeffi-
cients as possible. Different algorithms invoke simplicity in dif
ferent ways. Some do it entirely outside of the objective function
by an a priori blocky assumption. In our most-used algorithm
it is placed inside the objective function in the form of an L-1
norm on the reflection coefficients themselves. This is advanta
geous since it locates all the important terms together where thei
interactions can easily be controlled. How much simplicity i
best? The answer is project-dependent and will be different fo
example, for hard-contrast carbonates and soft-contrast sandand shales. We exercise control by multiplying the seismic term
by a constant. When the constant is high, complexity rule
When it becomes smaller, the inversion becomes more simple
with a sparser set of reflection coefficients. I will now attach the
name sparse spike to these types of inversions.
What about the Match the Logs term? When turned on, i
makes the inversion somewhat model-based. Sounds reasonable
- the inversion impedances should agree or a least be consistent
with an impedance model constructed from the well logs. And i
is reasonable, as long as it is not overdone. The primary use o
the model term should be to help control those frequencies below
the seismic band. When it is used to add high frequency information above the seismic band, great care should be exercised
(more on this below). We control the model contribution in a
variety of ways both inside and outside of the objective function
In the initial iteration of a sparse spike inversion, the model term
is set firmly off. We need to very clearly isolate the unique and
separate contributions of the seismic and the model. In the fina
inversion, we may turn it on to achieve the best power matching
between the model and seismic at the transition frequency.
By now you might be thinking that no simple label can be
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FEATURE ARTICLE Contd
applied to any modern algorithm. You would be right, and as we
will see below, the distinctions can become even more blurred.
The Details - Low Frequencies, Constraints, QC, Annealing,
Global Inversion
It is important how the low frequency part of the inversion
spectrum is computed. By low, we mean the frequencies belowthe seismic band. They are important since they are present in
the impedance logs which we seek to emulate. Commonly, the
low frequencies are obtained from an impedance model derived
from the geologic interpolation of well control. They can be
added to the inversion afterwards as an end step or within the
objective function itself. In the latter case only very low frequen-
cies need to come from the model. At Jason, we can do both and
each has its advantages. Whichever method is used, you might
now be saying that all inversions must, to some degree, be
model-based. The writer would not dispute this assertion. The
important point though, is that the model contribution is essen-
tially in a different band than that of the seismic.
There are other strategies in seismic inversion which control
the way in which the output impedances are obtained. It is com-
mon to define high and low limits on the output impedances.
These are supposed to keep the inversions physical and consis-
tent with known analogues and theories. Our implementation is
unique in that these constraints are not a fixed percentage of the
log impedances but can be freely defined at each horizon, vari-
able with time. The constraints are interpolated along the hori-
zons throughout the project, riding them like a roller coaster.
Could the inversions be then critically dependent upon inaccu-
rate horizons interpreted from seismic data? We address this
potential problem in two ways. In the first pass of inversion, theconstraints are relaxed to allow for inaccuracies. The horizons
are then re-evaluated against the initial inversion before a final
pass with tighter constraints. Second, the re-evaluation can be
done on an inversion without the model-based low frequencies
added in at all. We call this the relative inversion and it is by def-
inition, free from any inaccuracies in the input model.
Quality control of the relative inversion comes from perhaps
an improbable source. In our sparse spike approach, when we
invert at the well locations, the impedance log is not made avail-
able to the algorithm. Only the user constraints and the objective
function settings control the output, leaving it free to disagree
with the logs. It then follows that comparing the relative inver-sion and the band-limited impedance logs is a very powerful
quality control tool. The corollary is that the addition of more
logs to the inversion project increases confidence in the result
rather than copying the answer into it.
Some algorithms incorporate a process called simulated
annealing. This method deliberately takes a round-about path to
the solution of the objective function in recognition of the fact the
solution space may not contain a single simple minimum - the
inversion solution. If the solution space is a bumpy place, then
simulated annealing is indeed needed to avoid being hung up on
local minima. So, is solution space bumpy or not? It depend
upon how the inversion problem is parameterized. In the Jason
sparse spike algorithm family, the parameterization is by
impedance and grid point, resulting in a solution space withou
local minima. Some algorithms parameterize by impedance andtime. That is a more problematic parameterization due to the fac
that the time parameter can be strongly non-linear,. The result i
a much more complicated solution space which may necessitate
the use of a simulated annealing algorithm to come to a mean
ingful solution.
Global is another term which has recently been used in con-
junction with inversion. In Global mode, more than one trace i
inverted at the same time within a common objective function
The idea is that seismic noise induced variations that are not con
sistent over a user-specified number of traces will tend to be sup
pressed in the output impedances. The result is a smoothe
looking inversion, which, if one has been careful, does not compromise resolution. The number of traces inverted at once varie
with algorithms. We typically simultaneously invert for large
slightly overlapping blocks of traces, the only limit being
imposed by CPU memory.
In the preceding, our discussion has been centred on inversion
methods which emphasize sparsity as a way to invoke the keep
it simple term. In the following, we present some field example
of this and other inversion methods, some with potentially even
more powerful strategies.
EXAMPLES
Constrained Sparse Spike Inversion
This is the most common inversion algorithm in use by our
clients. Most importantly, they need to understand precisely
what statement the seismic data, itself is making abou
impedance. Sparse Spike is ideally suited for this since, among
its other virtues (noted above), it produces both a relative (no low
frequencies information from the logs added) and an absolut
(low frequencies added) version. The separate contributions o
the logs and the seismic are then clear.
Figure 1 shows such an inversion over a Nisku reef. On struc
ture, the blue colours represent the low impedances of the porou
reef which has both a west and an east build-up. Farther to theeast are low impedance shales. The plot is in an un-smoothed
format so that the contribution of each voxel (small rectangle) can
be more easily seen. Meaningful lateral variations in impedanc
of 1 or 2 samples have been achieved and variations in
impedance within the reef can be identified. The inversion vol
ume was converted to depth using a client-provided depth
datum, its associated time datum from the inversion and veloci
ties from the geologic model. The depth inversion was then con
verted to porosity using the observed relation in the logs. Map
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FEATURE ARTICLE Contd
of average porosity were made for several depth intervals and
one of these is shown in Figure 2. The best porosity is concen-
trated in the south-east reef but in a non-homogeneous way.
Wells were drilled from these maps and confirmed the inversion
predictions.
In Figure 3 we show a southeast Asia clastic example from
Latimer et al., 1999. The facies are an alternating sequence of
sands and shales. Interpretation is problematic due to the closevertical positioning of contrasting layers within half of a wavelet
length. The result is severe interference (tuning) and a general
complication of the seismic section. The interpretation of the yel-
low reservoir event is particularly difficult. Figure 4 shows the
inversion result. It is generally more simple and the interpreta-
tion of the yellow event is obvious. It is now interpreted as a
sequence boundary which is overlain by an incised valley sand.
Model-Based Inversion
We reserve the term model-based inversion for methods that
attempt to achieve resolution within and beyond the seismic
band by a stronger use of the a priori impedance logs in the model
term of equation 1. In one of the model-driven methods we sup-port, the simplicity term is implemented by limiting the solution
space to that spanned by (perturbed) well logs. The model term
is free to act both within the seismic band and outside of it.
Obviously we need to have confidence in our logs and in the rela-
tion between them and the seismic data if we are going to use
them in such a way. This confidence is generally provided byrunning a sparse spike inversion first. As in any model-based
approach we need to ensure that all possible facies are represent-
ed in the logs. The Jason inversion works by adapting the initial
model made from a simple interpolation of the input logs along
structure to a more detailed one. At any given CMP to be invert-
ed, an optimal weighting of the input wells is determined to min-
imize the residual between the input seismic and the inversion
synthetic. The whole operation is performed in a transformed
principle components space to remove non-uniqueness resulting
from the similarity of input wells. The final detailed model i
also the output inversion. Since high resolution logs are input
high resolution in the inversion is possible.
This approach has been used successfully by Helland-Hansen
et al., 1997 in the North Sea to highlight regions where sands
might occur with increased probability. Figure 5 shows the inpu
density model constructed from selected wells in principle com
ponents space. Figure 6 shows an enlarged section of the fina
model where the green colours represent densities indicative o
increased sand probability. Prior to doing the inversion, this are
had not been considered to be sand prone. A well was drilled on
this anomaly and the inferences from the inversion were confirmed. This resulted in the development of a new productive
Figure 1: A section through a 3D constrained sparse spike inversion of a Nisku reef. There aretwo main build-ups. Porosity is indicated by the lowest impedances and can be seen to be non-homogeneously distributed. In this un-smoothed display, each small colour square representsone voxel. There are apparently meaningful, consistent variations of 1 or 2 voxels.
Figure 2: The impedance volume in Figure 1 has been converted to depth using a client-supplied datum and velocities from the logs. It was then transformed to porosity using the observerelation between impedance and porosity in the logs. The figure shows the average porosity inone of a series of depth intervals. Note that the porosity is not distributed uniformly withineither of the build-ups.
Figure 3: This seismic section is from southeast Asia and represents a sequence of sands anshales. The Yellow horizon interpretation has not been completed. It is made difficult by thclose proximity of events and the structure.
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FEATURE ARTICLE Contd
horizon. The model-based algorithm has two flavours. The sec-ond, not illustrated here, associates character changes in (multi-
ple) seismic data set(s) with well log attributes and obviates the
need for a wavelet.
Geostatistical (Stochastic) Inversion
Geostatistical simulation differs from all of the other methods
in one respect. There is no objective function and hence no need
for a simplicity term to stabilize it. Rather, property solutions
(impedance, porosity, etc.) are drawn from a probability density
function (pdf) of possible outcomes. The pdf is defined at each
grid point in space and time. A priori information comes from
well logs and spatial statistical property and lithology distribu-
tions. As in the other model-based methods, the logs are assumed
to represent the correct solution at the well locations. It is again
useful to run a sparse spike inversion first, to establish this.
Historically, away from wells, geostatistics has had problems. It
is the inversion aspect of geostatistics which has finally guaran-
teed its use as a modern inversion tool. The geostatistical (or
stochastic) inversion algorithm simply accepts or discards simu-
lations at individual grid points depending upon whether they
imply synthetics which agree with the input seismic. The deci-
sion to accept or reject simulations can optionally be controlled
by a simulated annealing strategy. The inversion option results
in a tighter set of simulations, the variation of which can be used
to estimate risk or make probability maps. The simulations canbe done at arbitrary sample intervals. Close to wells, resolution
beyond the seismic band can reasonably be inferred. Away from
wells, the absence of a simplicity term in the simulation and the
statistical conditioning still hold the possibility of resolution
beyond that of traditional inversion methods.
In geostatistical modelling, property and indicator (facies)
simulations can be combined to produce both property (eg.
impedance) and facies volumes. This is illustrated in Figure 7
from Torres-Verdin et al., 1999, which shows such an estimate
from Argentinean data. The green patches are sand bodies from
a single simulation. Favourable locations for new wells were
determined by integrating the sand volume at each CMP for a se
of simulations. The results of this development programme
showed a definite improvement in sand detection. Accumulated
production has been up to three times the field average in some
instances, more than justifying the effort and expense of the
inversion.
Important end results of 3D Geostatistical modelling areproperty probability volumes. A set of volume simulations o
porosity, for example, can be modelled as a Normal probability
density function at each grid point in time and space. From
these, volumes can be constructed giving the probability that the
porosity lies within a specified range. Figure 8 shows an exampl
of this for simulations of porosity over a Western Canadian
Devonian reef. Twenty simulations were used to generate a
probability volume for the occurrence of porosity above 10%
Figure 4: This shows the constrained sparse spike inversion of the data in Figure 3. It is a muchsimpler section due to the attenuation of wavelet sidelobes. The completion of the Yellow hori-zon is now easy. The low impedance region above the Yellow, in the middle of the figure hasbeen interpreted to be a valley fill.
Figure 5: This section is from a volume representing the initial density model of a model-basedinversion. It was constructed from selected well logs which had previously been transformed tprinciple components space.
Figure 6: The density model in Figure 5 (enlarged section shown) has been modified by modelbased inversion. This final model is in fact the inversion output. It is a weighted average of thwell logs, optimized to produce a synthetic which agrees with the seismic. The low density(green) area indicates increased probability of sand. A well was subsequently drilled on thianomaly and confirmed the predictions of the inversion.
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FEATURE ARTICLE Contd
This volume was then viewed in 3D perspective and probabilities
less than 80% were set to be transparent. The tops and bottoms
of the viewable remainders were picked automatically. It is the
thickness of one of these high-probability bodies which is
mapped in Figure 8. The colours represent the thickness, within
which, the probability of 10% or greater porosity exceeds 80%. In
this way, uncertainty can be formally measured and input direct-ly into risk management analyses.
Elastic Impedance Inversion
AVO analysis, which has been with us for many years, has
taken a new and dramatic turn recently with the introduction of
the concept of Elastic Inversion. Briefly, the thought process is as
follows. We are familiar with the transformation of reflection
coefficients to acoustic (zero incidence) impedance. Zoeppritz
tells us that reflection coefficients are angle-dependent. What
then would a transformation of angle-dependent reflection coef-
ficients to impedance represent? We call this angle-dependent
quantity Elastic Impedance and claim that an inversion of an
angle or offset-limited stack would measure it. Such an approach
addresses a whole host of problems including offset-dependent
scaling, phase and tuning (Connolly, 1999). Both traditional AVO
methods and Elastic methods which determine P and S reflectiv-
ities from seismic directly, do not deal with these issues correctly.
Simultaneous Zp, Zs Inversion
The elastic impedance inversion process can in fact be carried
one step further with the simultaneous inversion of angle or off-
set-limited sub-stacks. This procedure brings all the benefits of
inversion to bear upon problems encountered in traditional seis-
mic weighted stacking approaches to AVO analysis. The out-
comes are inversion volumes of P-impedance, S-impedance and
density. Density is usually ill-defined and set to be constrained
to the relationships observed in the logs.
In Figure 9, we present an example from Pendrel et al., 2000
Detection of Cretaceous valley sands in the Blackfoot, Alberta
area is problematic due to the similarity of their P impedanc
with that of shales. In Figure 9, the near and far offset stacks have
been inverted separately. Note that the valley sands indicated by
the arrows are softer (lower impedance) at the far offset. Figur
10 shows the results of the simultaneous inversion for the Upper
Valley. The P-impedance is as much responsive to the off-valley
shales as it is to the valley sands. The S-impedance section, on
the other hand, highlights the highly-rigid valley sandsTransforming to the familiar Lame parameters LambdaRho and
MuRho (Goodway et al., 1997), and then forming their ratio
results in a good sand discriminator (Figure 11).
Conclusions / Discussion
I hope in the above that we have conveyed the wide range o
possibilities in modern seismic inversion and in particular those
available through Jason Geosystems. Interpreters need to con
sider seismic inversion whenever interpretation is complicated
by interference from nearby reflectors or the end result is to be a
quantitative reservoir property such as porosity. Output in the
format of geologic cross-sections of rock properties (as opposedto seismic reflection amplitudes) is endlessly useful as a means to
put geologists, geophysicists, petrophysicists and engineers on
the same page.
It is almost always useful to do sparse spike inversion first so
that we may be crystal clear about what the seismic is telling us
free from bias from logs. These ideas are best expressed in th
relative inversion. The absolute inversion brings in the logs as a
geologic context for the relative inversion and at the same time
broadens the inversion band. Only after this, when the relation
Figure 7: This is a perspective view showing the results of the simulaneous geostatisticalsimulation of density and lithotype (shale, tight sand, porous sand). The green bodies arelow density sands. A set of simulations were computed and integrated at each CMP toestablish the probability of occurrence of sand. Wells drilled on these analyses have shownsignificantly higher production rates.
Figure 8: Geostatistical simulation of porosity was done over a Western CanadianDevonian reef. At each point in time and space, the set of simulations was modelled by probability density function (pdf). Then a volume was created, the elements of which were probabilities that the porosity was in excess of 10%. Next, the probabilities themselves werviewed in 3D perspective, and values less than 80% made transparent. Finally, the remaining visible probability bodies were automatically interpreted. A small area of the final mapof probability body thickness is shown here. The colours are the thickness (ms) of bodiewhich should have greater than 10% porosity, 80% of the time.
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FEATURE ARTICLE Contd
between the logs and the seismic has been ascertained should we
consider methods, which make a stronger use of the logs. Model
based Inversion, Elastic Inversion, Simultaneous Zp, Z
Inversion and Geostatistical Inversion are all possibilities
depending upon the data and the goals of the project.
I do not expect that the confusion over inversion nomencla-
ture will disappear anytime soon. Perhaps, though, we hav
managed to shed some light on this topic. May I suggest that we
reserve the term, model-based for those algorithms which, in a
significant way, use logs to modify the inversion band within or
above seismic frequencies?
The days of viewing seismic inversion as an extra processing
step or subject of an isolated special study are long gone. Modern
inversions are intimately connected to detailed and quantitative
reservoir characterization and enhanced interpretation produc
tivity. The process requires and integrates input from all mem
bers of the asset team. Horizons should be re-assessed, model
re-built, log processing reviewed and inversion steps iterated
toward the best result. After drilling, new information should bused to create a living cube, always up-to-date with all avail-
able information. It is this partnership directed to the solution o
real reservoir characterization problems which leads to success.
References
Connolly, P., 1999, Elastic Inversion, The Leading Edge, 18 #4
p.440
Goodway, B., Chen, J.,Downton, J., 1997, AVO and Prestac
Inversion, CSEG Ann. Mtg. Abs. p.148
Helland-Hansen, D., Magnus, I., Edvardsen, A., Hansen, E., 1997Seismic Inversion for Reservoir Characterization and Well Planning in
the Snorre Field, The Leading Edge, 16 #3, p.269
Latimer, R.B., Davison, R., Van Riel, P., 2000,An Interpreters Guide
to Understanding and Working with Seismic-Derived Acoustic
Impedance Data, The Leading Edge, 19 #3, p.242
Pendrel, J., Debye, H. Pedersen-Tatalovic, R., Goodway, B.
Dufour, J., Bogaards, M., Stewart, R., 2000, Estimation and
Interpretation of P and S Impedance Volumes from the Simultaneous
Inversion of P-Wave Offset Data, CSEG Ann. Mtg. Abs. paper
AVO 2.5
Torres-Verdin, C., Victoria, M., Merletti, G., Pendrel, J., 1999
Trace-Based and Geostatistical Inversion of 3-D Seismic Data for Thin
Sand Delineation: An Application to San Jorge Basin, Argentina, The
Leading Edge, 18, #9, p.107
Figure 10: The near and far offset stacks from the Blackfoot seismic data were simultane-ously inverted for P-impedance and S-impedance. These figures are maps of the averages ofthese impedances in the Upper valley. Note that the S-impedance map shows the rigid sandsas relatively high impedances. The P-impedance map is sensitive to both sands and shales.
Figure 11: The P- and S- impedance volumes from simultaneous inversion were trans- formed to the Lame constants, ??and ??(multiplied by density). Then a ratio volume wasformed by dividing them. The result is a very sensitive valley sand detector.
Figure 9: The figures show the independent inversions of near and far offset stacks from theBlackfoot 3D survey. At Blackfoot, Glauconitic sands and shales have similar P-impedances. Note that the valley sands (arrows) appear softer at the far offsets. This is theAVO effect. The estimation of separate wavelets for each inversion means that offset-depen-
dent scaling, phase and tuning have been optimally addressed.
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FEATURE ARTICLE Contd
JOHN PENDREL
John Pendrel is Chief Geophysicist with Jason Geosystems Canada. In this positionhe is responsible for applying proprietary, leading edge technology to problems in reser-voir analysis and characterization. From 1981 to 1995, he was Sr. Geophysicist and thenManager, Geophysical Technology with Gulf Canada Resources in Calgary. He beganhis career in the oil industry in 1978 with Gulf Science and Technology Company inPittsburgh, PA, the research arm of the former Gulf Oil.
Johns academic career included a B.Sc. at The University of Saskatchewan (1968), and
an M.Sc. from The University of Saskatchewan, Saskatoon, (1972) where he did research
in auroral magnetic fields. John also holds a Saskatchewan Class A teachers certificate. He obtained a Ph.D.
(geophysics) from York University, Toronto in 1978 where his interests were in two-dimensional time series and
spectral analysis.
Johns early research was in the areas of pattern recognition and principal components analysis. More
recently he has done applied research and published papers in seismic inversion, geostatistical analysis and
AVO. Away from work he plays on an ice hockey team and volunteers with a local youth group, The Calgary
Stampede Showband.