jan01 seismic inversion

8/6/2019 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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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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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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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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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 transformed 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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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.

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