## Abstract

#### Introduction

The principle objective of seismic inversion is to transform seismic reflection data into a quantitative rock property, descriptive 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 all CMP's, what would the impedance logs have looked like? Compared to working with seismic amplitudes, inversion results show higher resolution and support more accurate interpretations. This, in turn facilitates better estimations 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. In addition, 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 30 years. It was in common use in 1977 when the writer joined Gulf Oil Company's 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? In the following, I will briefly discuss the most common methods in a somewhat qualitative manner, keeping the equations to a minimum.

#### The Post-Stack Inversion Method

The modern era of seismic inversion started in the early 80's 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 represented 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)

Let's look at each of these terms, starting with the seismic. It says that synthetics computed from the inversion impedances should match the input seismic. This is usually (but not always) done in a least squares sense. Invoking this term also implies 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 that 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 coefficients 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 decimal place.

Enter the simple term. Every algorithm has one. It could not care less about matching the seismic data, preferring instead to create an inversion impedance log with as few reflection coefficients as possible. Different algorithms invoke simplicity in different ways. Some do it entirely outside of the objective function by an a priori "blocky" assumption. It can also be placed inside the objective function in the form of an L-1 norm (sum of absolute values) on the reflection coefficients themselves. This is advantageous since it locates all the important terms together where their interactions can easily be controlled. How much simplicity is best? The answer is project-dependent and will be different for example, for hard-contrast carbonates and soft-contrast sands and shales. Control can be exercised by multiplying the seismic term by a constant. When the constant is high, complexity rules. When it becomes smaller, the inversion becomes simpler with a sparser set of reflection coefficients. We refer to these as mixed-norm types of inversions. The term, sparse-spike is used to describe algorithms wherein the simplicity term is outside the objective function.

What about the "Match the Logs" term? When turned on, it 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 it is reasonable, as long as it is not overdone. The primary use of 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. High frequencies from a model will be unaddressed and unchanged by the input seismic when they are above the seismic band. They can then appear in the output inversion, even though they are completely model driven.

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 band in which the model has influence.

#### The Details - Constraints, QC, Annealing, Global and Colored Inversion

There are other strategies in seismic inversion which control the way in which the output impedances are obtained. It is common to define high and low limits on the output impedances. These are supposed to keep the inversions physical and consistent with known analogues and theories. Some implementations offer a non-fixed percentage of the log impedances, freely defined at each horizon and variable with time. The constraints are interpolated along the horizons throughout the project, riding them like a roller coaster. Could the inversions be then critically dependent upon inaccurate horizons interpreted from seismic data? This potential problem can be addressed in two ways. In the first pass of inversion, the constraints 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 definition, free from any inaccuracies in the input model.

The relative inversion can play a vital role in quality control if the algorithm is constructed such that the impedance logs are not made available 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 inversion 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.

Another inversion strategy is the imprint of stratigraphy - should it influence the result and in which band? Including it in the low frequency mode is easy and does not affect the computation adversely. Algorithms which opt to constrain the seismic band to assumed stratigraphy require special solution techniques. This is because the solution space becomes more complex with many local minima beside the one representing the optimum result. Stochastic strategies such as simulated annealing are used to avoid local trapping.

Global is another term which has recently been used in conjunction with inversion. In Global mode, more than one trace is inverted at the same time within a common objective function. The idea is that seismic noise induced variations that are not consistent over a user-specified number of traces will tend to be suppressed in the output impedances. The result is a smoother looking inversion, which, if one has been careful, does not compromise resolution.

Another technique introduced recently is the so-called Colored Inversion. It trades computation speed for resolution. Phase is first assumed to be known. Then the spectrum of the reservoir impedances is assumed to be a straight line on a log frequency cross-plot. The slope of the line is determined from available logs. Then, an operator is designed which transforms the seismic spectrum to the desired log spectrum. This matching operator can be very ringy and some stabilization is usually required. However, once obtained, the inversion can be produced by a simple convolution of the operator with the input data.

Putting it all together, seismic inversion can play the central role in an improved understanding of the reservoir. In Figure 1, upper panel, is 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 close vertical 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 yellow reservoir event is particularly difficult. Figure 1, lower panel, shows the inversion result. It is generally simpler and the interpretation of the yellow event is obvious. It is now interpreted as a sequence boundary which is overlain by an incised valley sand.

The value of the inversion process is illustrated again in Figure 2 from Caulfield et al., 2005. The facies of interest are McLaren sandstones as indicated. The figure shows the original seismic, the inversion in colour with smoothed P Impedance logs overlain. Well cross-plot analyses showed that the best sandstones should be resolvable by P Impedance alone. The inversion method used here was blind to the logs in the seismic band, making the good agreement between the logs and the inversion a strong QC. As shown in the figure, there is a strong change in the inversion at well 121-16 which is indicative of a shale member. Shale had not been encountered at the nearby 141-16. The seismic reflection in the zone of interest (partially hidden by the overlying logs) does not suggest this change of reservoir property. At well 121-12, resolution also appears to be improved as there seems to be two separate levels of sandstone deposition. These are revealed upon converting to depth and preparing impedance slices (Figure 3). The probability of sandstone deposition can also be formalized for any inversion (post-stack or AVO), as demonstrated in Figure 4. In 3D probability space, is the likelihood of occurrence of a single McLaren sandstone. Figure 5 is a comparison of interpretations from the seismic and the inversion. There is better definition of channeling in the inversion and the authors judge that the accuracy of net pay estimates were improved by a factor of at least two.

#### AVO Inversion

It should come as no surprise that all of the above ideas transfer readily to the AVO World (see, for example, Pendrel et al., 2000). Instead of a single full-stack, we have a set of partial offset or angle stacks, each with their own wavelets. In addition to a P Impedance model for the low frequencies, we now need two more â€“ S Impedance and Density. We also want to include two more "keep it simple" terms for S Impedance and Density. After that, it is pretty much the same. The Zoeppritz equations dictate the range of allowable solutions. Alternate parameterizations are possible, P Impedance, Vp/Vs and Density being popular. Other modes can be inverted too, although PP is the most common. It is important, however, to ensure that the NMO is correct to sub-sample accuracy. Failure to observe this criterion will result in an S measure which will have too much dynamic range – too many strong lows and highs. Commonly, the S Impedance and any other reservoir parameter derived from it will exhibit a narrower bandwidth compared to the P Impedance inversion. This is natural and a consequence of the loss of frequency with offset.

The example in Figure 6 from the CREWES / EnCana Blackfoot data set illustrates the classic problem of separating sandstones from shales when discrimination is not possible from P Impedance alone. In Figure 6 are slices of P Impedance and Vp/Vs from a Simultaneous AVO Inversion. The two reservoir properties are indeed different. Regional and valley shales dominate the P Impedance slice. The major feature of the Vp/Vs slice is the sandstone valley itself, brighter in the south due to the presence of gas. Figure 7 is a 3D perspective of the Vp/Vs volume where it can bee seen that the valley development is essentially defined by Vp/Vs. The LambdaRho-MuRho (LMR) technology popularized by Goodway et al., 1997) can offer advantages to interpretation by optimally separating fluid and rock effects. LMR volumes are easily computed from any AVO Inversion.

Density has its own particular problems. The density contribution to AVO is many dB down from the Shear contribution in normal field acquisition. It only begins to become important at angles greater than 50 deg. In addition, Anisotropy is also a significant contributor at these large angles and must be accounted for in any attempt to invert for density. When large angles are not recorded, density needs to be softly constrained to something like the Gardner relation or perhaps, the relationship observed in logs between it and P Impedance.

The so called Joint inversions are variants of this technology. The theory readily accommodates PP-PS or any other possible combination. We have already noted the importance of accurate alignment and the correct alignment of PS modes to PP takes these challenges to a new level. Nevertheless, this technique contains the potential for density estimation at low angles, as illustrated by the heavy oil synthetic example in Figure 8. The figure shows PP and PS gathers and their simultaneous inversion to P Impedance, Vp/Vs and Density. The maximum angle used to make the synthetic gathers shown was only 35 deg. The band of the PP gather was 10-60 Hz while that of the PS was restricted to 10-35 Hz. Comparing the overlain logs to the inversion results shows that density information can be extracted.

Curiously, Joint Inversions find application to 4D projects. When the low frequencies below the seismic band are believed to be constant, then a Joint PP inversion of all vintages will provide the most stable baseline, against which to measure differences. The method also works for 4D AVO.

#### Geostatistical 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 distributions. As in the other model-based methods, the logs are assumed to represent the correct solution at the well locations. It is useful to run a mixed-norm inversion first, to establish this. Historically, away from wells, geostatistics has had problems. It is the inversion aspect of geostatistics which has finally guaranteed its use as a modern inversion tool. The geostatistical inversion algorithm simply accepts or discards simulations at individual grid points depending upon whether they imply synthetics which agree with the input seismic. The decision 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 can be 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 hold the possibility of resolution beyond that of traditional inversion methods.

Important end results of 3D Geostatistical modelling are property probability volumes. A set of volume simulations of 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 9 shows an example 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%. 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 9. 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 directly into risk management analyses.

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 10 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 set 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.

#### Merging Technologies – The New Inversions

Concatenating seismic inversion with other technologies, such as neural nets, seismic attributes or pattern recognition has been a strategy employed by some explorationists over the years. The idea has been to try and extract every last bit of information from the input data sets. We are now seeing disparate technologies beginning to be combined within the same algorithm. Figure 11 is such an example from Blackfoot. It brings together aspects of pattern recognition and post-stack and AVO Inversion. The top is a traditional Simultaneous AVO Inversion for Vp/Vs while the bottom is the new high resolution technology. It was run in a "blind-to-the-wells" mode, so the agreement to the logs is not perfect. As resolution is pushed to its limits, we must understand that there can be no single answer, only a collection of probable answers. The new technologies recognize this and in fact, the bottom panel in Figure 11 is an average of six such realizations. The variability between the realizations could have been used to compute a probability of occurrence for the low Vp/Vs sandstones. All of this sounds very geostatistical, although upon closer examination, there are differences.

#### Summary

I hope that I have been able to convey the wide range of possibilities in modern seismic inversions. Careful consideration should be given in selecting the best tool. Interpreters need to consider seismic inversion whenever interpretation is complicated by interference from nearby reflectors or when the end result is to be a quantitative reservoir property such as porosity. Outputs in the format of geologic cross-sections of rock properties (as opposed to seismic reflection amplitudes) are putting geologists, geophysicists, petrophysicists and engineers "on the same page".

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 productivity. The process requires and integrates input from all members of the asset team. Horizons should be re-assessed, models re-built, log processing reviewed and inversion steps iterated toward the best result. After drilling, new information should be used to create a living volume, always up-to-date with all available information. It is this partnership directed to the solution of real reservoir characterization problems which leads to success.

#### References

*Caulfield, C., Feroci, M., Yakiwchuk, K., Seismic Inversion for Horizontal Well Planning in Western Saskatchewan, CSEG Ann. Mtg., 2005*

*Goodway, B., Chen, J., Downton, J., 1997, AVO and Prestack Inversion, CSEG Ann. Mtg. Abs. p.148*

*Latimer, R.B., Davison, R., Van Riel, P., 2000, An Interpreter's Guide to Understanding and Working with Seismic-Derived Acoustic Impedance Data, The Leading Edge, 19 #3, p.242*

*Pendrel, J., Debeye, 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.1070*

## Biography

**John Pendrel's** academic career included a B.Sc. at The University of Saskatchewan in Regina (1968), and an M.Sc. from The University of Saskatchewan, Saskatoon, (1972). While at Saskatoon, he was part of a team investigating auroral magnetic fields from rocket-borne magnetometers. John also holds a Saskatchewan Class "A" teacher's certificate and served a brief stint as a high school physics teacher in Saskatoon. Leaving teaching, he obtained a Ph.D. in Geophysics from York University, Toronto in 1977 where his interests were in two-dimensional time series and spectral analysis.

He began his career in the oil industry in 1977 with Gulf Science and Technology Company in Pittsburgh, PA, the research arm of the former Gulf Oil Corp. His first direct report was Ralph Shuey, whose classic paper led to the wider use of AVO. Another colleague was Gerry Gardner, most-famous for his relation between density and velocity. While at Pittsburgh, John was a member of Gulf's tomography team and conducted research in the areas of pattern recognition and principal components analysis. From 1981 to 1995, he was Sr. Geophysicist and then Manager, Geophysical Technology with Gulf Canada Resources in Calgary.

John joined Jason Geosystems as Chief Geophysicist in 1995. There, he did applied research and published papers in Seismic Inversion, Geostatistical Analysis and AVO. In 2002 he became Business Manager of the new Fugro-Jason Canada.

Away from work, beside playing on a hockey team, John volunteers with the Calgary Exhibition and Stampede where he serves on the Executive Committee of the Calgary Stampede Showband. He is also their photographer and has traveled extensively with the Band on overseas trips.