AVO Course Notes, Part 4. Three-term AVO Inversion
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Shuey R. Williams M. Etris, Nick J. Crabtree and Jan Dewar Errors and Omissions A large volume of data is being converted to make this online archive.
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If you notice any problems with an article examples: incorrect or missing figures, issue with rendering of formulas etc. The CSEG does not endorse or warrant the information printed. Article References Print. Examples of Azimuthal AVO The literature abounds with numerous excellent examples and observations of AVAZ some of which are shown here in order to better understand the expectation and potential in pursuing this type of analysis Lynn et al.
Figure 1. Figure 2. AVAZ variation: parallel is strong flat gradient and perpendicular is weak -ve gradient. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7a. Figure 7b. Note that unlike figure 6, the relative stack response from these azimuthally anisotropic gradients would be stronger than the isotropic equivalent. Figure 8a. Figure 8b. Table 1. Figure 9a. Figure Figure 11a. Figure 11b. Figure 11c. Zoom of figure 11a for 3 term vs. Figure 11d. Figure 12a. Figure 12b. Figure 13a.
AVO Modeling in Seismic Processing and Interpretation Part 1. Fundamentals | CSEG RECORDER
HTI vertical fracture model. Figure 13b. VTI horizontal layering model. Pre-drill MuRho inversion from 3D at proposed Well A location showing highest rigidity Colorado B zone hot colours that is most likely to support natural or induced fractures. In our first example,the methodis tested on a simple synthetic. This example was usedinitially becauseit truly represents a "blocky" impedance and therefor. In this casewe haveuseda smoothedversion of the sonic velocities to provide the constraint.
A visual comparisonwoulU indicate that the extracteU velocity profile corresponds very well to the input. A moredetailed comparisonof the two figures showsthat the original and extracted logs do not matchperfectly. It is doubtful that a perfect match could ever be obtai neU. At the' top of the figure we see a sonic log with 'its reflectivity sequencebelow.
In this example,we have assumedthat the density is constant, but this is not a necessary restriction. The reflectivity wascbnvolvedwith a zero-phasewavelet,bandlimitedfrom10 to 60 Hz, andthe final syntheticis shownat the bottomof the figure. The results of the maximum-likelihood inversion method are sbown in Figure 6. In this calculation, the waveletwasassumed known. Notethe blocky nature of the estimatedvelocity profile compared with the actual sonic log profile. Again, the input and output logs do not matchperfectly. The fact that the two do not perfectly matchis due to slight errors in the reflectivity sizes whichare amplified by the integration process,and is partially the effect of the constaintused.
Theconstraintshownin Figure 6. In practice, this information could be derived from stacking velocities or from nearby well control. This blocky impedance canbe contrastedwith the more traditional narrow-band. Finally, Figure 6. In summary, maximum-likelihoodinversion is a procedurewhich extracts a broad-band estimate of the seismic reflectivity and, by the introduction of 1inear constraints, al lows us to invert to an acoustic impedancesection which retains the major geological features of boreholelog data.
Another method of- recursive, single trace inversion which uses a "sparse-spike" assumption is the L1 normmethod, developed primarilyby Dr. DougOldenburgof UBC. This method is also often referred to as the linear programming method,and this can lead to confusion.
Actually, the two namesrefer to separateaspectsof the method. Themathematical modelusedin the construction of the algorithm is the minimizationof the L1 norm. However,the methodusedto solve the problem is linear programming. The basic theory of this methodis found in a paper by Oldenburg, et el The authors point out that if a high-resolution aleconvolution is performedon the seismictrace, the resulting estimateof the reflectivity can be thought of as an averagedversion of the original reflectivity, as shownat the top of Figure6.
Now, the layered earth model equates to a "blocky" impedancefunction, which in turn equates to a "sparse-spiKe" reflectivity function. The above constraint will thus restrict our inverted result to a "sparse" structure so that extremely fine structure, such as very small reflection coefficients, will not be fully inverted. The other key difference in the linear programmingmethod is that the L1 norm is minimized rather than the L2 norm. The L1 norm is defined as the sum of the absolute values of the seismic trace.
The two norms are shownbelow, applied to the trace x: x1 : xi and x2: xi i i:1 The fact that the L1 norm favours a "sparse" structure is shown in the following simple example. Taken from the notes to Dr.
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Oldenburg's CSEG convention course' "Inverse theory with application to aleconvolution and seismograminversion". Hence, minimizing the L1 norm would reveal that g is a "preferred" seismic trace based on it's sparseness. Oldenburget al. That is, the reliable frequencyband is honored whileat the same timea sparsereflectivity is created.
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The results of their. The data consist of 49 traces with a sample rate of 4 msecand a Hz bandwidth. The figure showsthe linear programming reflectivity and impedanceestimates below the input seismic section. It should be pointed out that a three trace spatial smootherhas been applied to the final results in both cases. Finally, let us consider a dataset fromAlberta which has been processeU through the LP inversion method. The input seismic is shownin Figure 6. The constraints useU here were from well log data.
In the final inversion notice that the impedance has been superimposed on the final reflectivity estimate using a grey level scale. The sequence! BaseUon the these data handouts, do the following interpretation exerc i se: [ Tie the synthetic to the seismicline at SP Hint- use reverse polari ty syntheti c. Use a blocked off version of the sonic log. As the time separation between reflection coefficients becomessmaller, the interference between overlapping wavelets becomesmore severe.
Indeed, in Figure 6. In fact, the effect is more of a differentiation of the wavelet, which alters the amplitude spectrumas wel1 as the phase spectrum. In this section we will look closer at the effect of wavelets on thin beds and how. The first comprehensivel'ook at thin bed effects was done by Widess In this paper he used a model which has becomethe standard for discussing thin beds, the wedgemodel. That is, consider a high velocity laye6 encasedin a low velocity layer or vice versa and allow the thickness of the layer to pinch out to zero.
Next create the reflectivity responsefrom the impedance,and convolvewith a wavelet. The thickness of the layer is given in terms of two-waytime through the layer and is then related to the dominantperiod of the wavelet. The usual wavelet used is a Ricker becauseof the simpl i city of its shape. Figure 7.
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Note that what is refertea to as a wavelengthin his plot i s actually twice the dominantperiod. A few important points can be noted from Figure 7. First, the wavelets start interfering with eackotherat a thicknessjust below two dominant periods,but remain Clistinguishable down to about one period.