5.1 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. 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.
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. 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.
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. 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.
5.2 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. 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.
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.
Putting it all together, seismic inversion can play the central role in an improved understanding of the reservoir. In Figure 1, upper panel, 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 5.1), 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 incised valley sand.
The value of the inversion process is illustrated again in (Figure 5.2). The facies of interest are 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.
(Figure 5.1) The upper panel is a seismic section from southeast Asia and represents a sequence of sands and shales. The Yellow horizon interpretation has not been completed. It is made difficult by the close proximity of events and the structure. The lower panel shows a post-stack, mixed-norm inversion. It is a much simpler section due to the attenuation of wavelet side-lobes. The completion of the Yellow horizon is now easy. The low impedance region above the Yellow, in the middle of the figure has been interpreted to be a valley fill.
Figure (5.2)-Above is the post-stack seismic inversion of the overlain seismic reflection. Smoothed well logs are also shown. Good sandstones are represented by the cyan colour. Note the shale in the region of interest at well 121-16. It is not obvious from the seismic (partially hidden by the logs in this figure). To the west, the inversion indicates different episodes of sandstone deposition.
These are revealed upon converting to depth and preparing impedance slices (Figure 5.3). The probability of sandstone deposition can also be formalized for any inversion (post-stack or AVO), as
demonstrated in (Figure 5.4). In 3D probability space, is the likelihood of occurrence of single sandstone? (Figure 5.5) 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.
5.3 AVO Inversion
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 – 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.
The example in (Figure 5.6) illustrates the classic problem of separating sandstones from shales when discrimination is not possible from P Impedance alone. In (Figure 5.6) are slices of P Impedance and Vp/Vs from a Simultaneous AV O 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 5.7) is a 3D perspective of the Vp/Vs volume where it can be seen that the valley development is essentially defined by Vp/Vs. The LambdaRho-MuRho (LMR) technology can offer advantages to interpretation by optimally separating fluid and rock effects. LMR volumes are easily computed from any AVO Inversion.
Figure (5. 3)-The inversion in Figure (5.2) has been converted to depth and mapped using time slices. These results reinforce the ideas of different episodes of deposition.
Figure (5.4) Using the observed distribution of sandstone impedances from logs and probability analysis, a sandstone probability volume was computed. Here the
probability of sand -stone exceeding 88% is shown for a particular portion of the survey.
Figure (5.5) The two panels compare interpretations done from the seismic using traditional methods and from the inversion. The authors judge that the accuracy of net pay estimations was improved by a factor of two with the inversion.
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 5.8). 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.
Figure 5.6. The two panels show slices of P Impedance and Vp/Vs from a
Simultaneous AVO Inversion of the Blackfoot data. Note the predominance of off-s t r u c t u r e and off-valley shales in the P Impedance map. The upper valley sandstone is clearly imaged on the Vp/Vs map.
Figure (5.7) Vp/Vs from AVO Inversion is shown in perspective view. The two episodes of sandstone deposition are clearly visible.
Figure (5.8) The logs in the left panel from a heavy oil play were used to compute the synthetic PP and PS gathers in the 2nd and 3rd panels. The background colours in the next three panels are the Joint Simultaneous AVO Inversions for P Impedance, Vp/Vs and Density. Smoothed logs are overlain in each panel. The wavelet band was 10-60 Hz for the PP data and 10-35 Hz for the PS. Despite the fact that the maximum angle was only 35 deg., density information was recovered.
5.4 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
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 5.9) shows an example of this for simulations of porosity. 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 5.9). 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 5.10). 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.
Figure (5.9) Geostatistical simulation of porosity was done over a Western Canadian Devonian 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 pro b a bilities themselves was viewed 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 map of (probability body thickness) is shown here. The colours are the thickness (ms) of bodies which should have greater than 10% porosity, 80% of the time.
Figure (5.10) - This is a perspective view showing the results of the simultaneous geosta -tistical simulation of density and lithotype (shale, tight sand, porous sand). The green bodies are low density sands. A set of simulations were computed and integrated at each CMP to establish the probability of occurrence of sand. Wells drilled on these analyses have shown significantly higher production rates.
5.3 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 5.11) 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 5.11) 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.
Figure (5.11)-The upper panel shows the results of an AVO Inversion at Blackfoot. The lower panel shows a high resolution AVO inversion using a technologies which brings together aspects of pattern recognition and inversion within a geosta -tistical type of framework.
5.4 Stochastic Seismic Inversion
Stochastic seismic inversion is one method that can provide the vertical resolution sufficient to generate detailed 3D reservoir property models.The log data (sonic and density) are used in the simulation of pseudo-logs at each trace within the seismic survey (figure 5.12). A synthetic seismogram is generated from the pseudo-impedance log and is compared with the actual seismic trace at that location. The simulation that produces the best match between the synthetic seismogram and the actual seismic trace, as defined by some quantitative measure of goodness of fit, is retained as the inversion solution at that location. The vertical resolution of the simulated log data is determined by the selection of the vertical cell size (determined by the user), not by the frequency content of the seismic data (as determined by Mother Nature).
The enhanced resolution of the stochastic inversion process is evident from a visual examination of the seismic data cubes displayed in (figures 5.13-5.15). (Figure 5.13) displays the input seismic volume. (Figure 14) is the result of inversion. This inversion result displays a resolution similar to that of the input data. That is to be expected, as the vertical resolution is derived from the seismic data, and is therefore subject to the inherent limitations of seismic resolution. (Figure 15)
depicts the stochastic inversion cube, which displays a much finer vertical resolution than is observed in the input data or the recursive inversion result (figures 13 and 15).
Fig (5 13) Input seismic data volume
Fig.(5.15). Stochastic inversion result. Note that the vertical resolution is significantly improved.
Fig.(5.14) Recursive inversion result. Note that vertical resolution is similar to that of the input data.
Figure (5.12) Schematic depiction of the stochastic inversion process.