Recovering low frequencies for impedance inversion by frequency domain deconvolution

Transcription

1 Recoverig low frequecies for impedace iversio by frequecy domai decovolutio Sia. Esmaeili*, CREWES, Uiversity of Calgary, Gary. Frak. Margrave, CREWES, Uiversity of Calgary, Summary Acoustic impedace is a rock property that ca be derived from seismic data ad cotais importat iformatio about subsurface properties. Direct measuremets of acoustic impedace are available from acoustic ad desity well logs, but these well data ca provide the acoustic impedace oly at the well s locatio. Mathematically it is true that acoustic impedace ca be calculated from earth s reflectivity fuctio, ad this fuctio ca be estimated from seismic data. Additioally, estimatio of reflectivity from seismic data is always badlimited ad affects acoustic impedace sigificatly. Acoustic impedace iversio ca easily be computed by a stadard impedace iversio algorithm which uses well logs to fill i the low-frequecy iformatio that is missig i badlimited seismic data. I this study we ivestigate the performace of stadard decovolutio ad its ability to recover low frequecy cotet directly from seismic data. We fid that stadard decovolutio does ot perform well at low frequecies ad this is a limitig factor i impedace iversio. Usig frequecy domai decovolutio, we show that improvig the spectral smoothig process ad applyig a miimum phase spectral color operator to the decovolved seismic trace ca improve the performace of impedace iversio ad reduce the badwidth ecessary from well cotrol. Itroductio The ultimate goal of geophysics is to determie the earth s reflectivity as a fuctio of positio beeath a seismic survey. Oce the raw data is processed, it is possible to estimate the earth s reflectivity from them. The low frequecy seismic data is gettig cotamiated with low frequecy oises, ad it will result i missig low frequecy data i the recorded data. The questio is, ca we otherwise suppress low-frequecy oise without wastig good iformatio? Waters (1978) described a impedace iversio scheme which is a simple approach to derive impedace values from seismic data. A impedace estimate, from a well log or stackig velocities, is first combied with itegrated seismic data i the frequecy domai. Detailed impedace values are thus provided by the itegrated seismic data, ad the low-frequecy tred is provided by the well-log. Lidseth (1979) also added low frequecies derived from velocity aalysis, ad Oldeburg et al. (1983) itroduced two differet approaches for recoverig low frequecy iformatio. Acoustic impedace iversio ca also be computed easily by a BLIMP (BadLimited IMPedace) algorithm (Ferguso & Margrave, 1996) which uses well logs to fill i GeoCovetio 2014: FOCUS 1

2 the low frequecy iformatio that is missig i badlimited seismic data. Recoverig the low frequecies before passig through the impedace estimatio process ca be challegig. The key poit of this idea is estimatig the wavelet as accurately as possible durig decovolutio hece the low frequecy part ca be recovered from estimated reflectivity. We start by reitroducig the covolutioal model for ormal icidet seismograms ad the show how reflectivity ca be estimated by decovolutio. Two approaches will be discussed for recoverig low frequecies i a decovolutio algorithm, ad the result of impedace iversio derived from the ew decovolutio will be preseted. Theory ad Method A wave source ca put eergy ito the groud, ad the wave ca propagate through the earth ad be reflected by reflectors (Figure 1). Regardless of the effect of geometrical spreadig, trasmissio losses, a-elastic absorptio ad multiple reflectios, a simple geophysical model ca be itroduced by a covolutioal model. Seismic data recorded by a receiver are equal to a covolutio of earth s reflectivity fuctio with a kow wavelet (Sheriff & Geldart, 1995). s( t) r( t) w( t), (1) where st () is the recorded seismic data, rt () is the reflectivity fuctio, wt () is the wavelet ad is a covolutioal operator. FIG. 1. A homogeeous medium with a sigle reflector. It is also possible to add statioary ad white oise to equatio 1 to represet a recorded oisy seismogram. Equatio 1 the becomes s( t) r( t) w( t) ( t), (2) where t () is a statioary ad white oise which meas that it has a costat power at all frequecies. As was metioed before the mai goal of geophysics is determiig a reflectivity fuctio from recorded seismic data. Oce the seismic data is recorded by receivers, it is used to estimate reflectivity. However, the oly kow parameter i equatio 2 is st (), which is a fuctio of time, while all other parameters are ukow. Mathematically, decovolutio is a algorithm-based process which is used to reverse the effects of covolutio o the recorded data. The goal of a decovolutio scheme is to remove the effect of the wavelet from seismic traces ad the retrievig the earth s reflectivity GeoCovetio 2014: FOCUS 2

3 fuctio. Wieer spikig decovolutio (Leibach, 1995), maximum etropy (Burg) decovolutio (AuYeug, 1986), frequecy domai decovolutio (Margrave, 2002), ad Gabor decovolutio (Margrave & Lamoureux, 2002) are differet decovolutio methods that ca be applied to seismic data to estimate reflectivity. This report outlies a attempt to estimate the reflectivity by applyig frequecy domai decovolutio to zero-offset seismic data ad utilizig the result to calculate acoustic impedace iversio. Impedace Iversio Oce the reflectivity fuctio has bee estimated it is possible to calculate the impedace iversio. The product of desity ad acoustic velocity, which varies amog differet rock layers, is kow as acoustic impedace, commo symbols for it are I ad Z. Acoustic impedace idicates how much soud pressure is geerated by the vibratio of molecules of a particular acoustic medium. Therefore, i a oe dimesioal medium ad the acoustic case for the ormal icidet wavelet, the reflectio coefficiet ca be writte as (Margrave, 2002) where I 1 ad I r I 1 1 I I, (3) I represet the acoustic impedace of the (+1) th ad th layer respectively. To calculate the acoustic impedace istead of usig the impedace to compute reflectio coefficiets i equatio 3, it is possible to use reflectio coefficiets which are derived from seismic data, i order to determie acoustic iversio (Lidseth, 1979). The reflectio coefficiets ca be derived from recorded seismic data ad well logs. Mathematically, the impedace ca be writte i terms of reflectio coefficiets like By assumig r 1 equatio (4) ca be approximately writte as I 1 1 r I. (4) 1 r 2 I 1 I 1 r 1 r I 1 r I 1 2r. (5) Replacig by -1, equatio 5 ca be writte for I as 1 1 Usig the same procedure for upper layers, I 1 ca be rewritte as I I 1 2r. (6) 1 1 j1 ad by a simple calculatio, 1 2r j ca be estimated as I I 12rj, (7) GeoCovetio 2014: FOCUS 3

4 Therefore, equatio 7 becomes 2R j 1 2r e. (8) j j1 rj 2 2rj j 1 I I e I e (9) Equatio 9 is a type of iversio process which computes acoustic impedace from seismic reflectio iformatio ad is kow as impedace iversio (II). Therefore, give the impedace of the first layer ad the estimated reflectivity fuctio, acoustic impedace ca be calculated. The seismic sources do ot geerate useful power at all frequecies, therefore it is accepted that ay reflectivity estimate must be badlimited. I this situatio the badlimited reflectivity icludes fewer details tha the actual earth reflectivity. The broadbad ad badlimited reflectivity i the frequecy ad time domai is illustrated i figure 2. FIG. 2. Comparig broadbad ad badlimited reflectivity i both frequecy ad time domai The first diagram shows that the broadbad reflectivity cotais all frequecies from zero to 500 Hz, but the badlimited oe cotais oly the frequecies from 10 Hz to 120 Hz. I the secod diagram, the differeces betwee two reflectivity fuctios are oticeable. It ca be realized that the data which lack low ad high frequecies have less resolutio tha the broadbad data set. Frequecy Domai Decovolutio The method described here is based o a frequecy domai framework, which might be the easiest way to estimate reflectivity. Regardless of the phase spectrum of a seismic trace, the amplitude spectrum of seismic data is similar i shape to the amplitude spectrum of wavelet as show i Figure 3. GeoCovetio 2014: FOCUS 4

5 FIG. 3. Amplitude spectrum of white spectrum reflectivity (blue), seismic data (gree) ad a miimum phase wavelet (red). If the amplitude spectrum of the wavelet ca be computed by smoothig the amplitude spectrum of seismic data, the amplitude spectrum of the wavelet could be extracted ad thus the reflectivity ca be estimated. The perfect decovolutio operator ca be defied as: w( t) d( t) ( t), (10) so dt () is iverse of wt (). By substitutig the iverse of wt () ito equatio 1, rt () becomes r( t) s( t) d( t), (11) where rt () is the exact reflectivity fuctio. But i practice, because of the badlimited ature of wavelets ad the uavoidable presece of oise, eve if we could fid dt () as a fuctio to make equatio 10 equal to () t, such a operator would simply produce oise at frequecies where oise domiates sigal. This importat fact leads us to the cocept that the estimated reflectivity fuctio is ever exactly the same as the true reflectivity fuctio. Mathematically, it ca be writte as: s ( t) r( t) w ( t), (12) where sd () t is the estimated reflectivity, ad wd () t ca be represeted as d d w( t) d( t) w ( t), (13) where wd () t is the estimated wavelet ad is a approximatio of true wavelet. Figure 4 is illustratig a example of a actual wavelet ad its estimatio i the time ad frequecy domai. d FIG. 4. True ad estimated wavelet i the time domai (left) ad frequecy domai (right) from a oise-free seismogram. GeoCovetio 2014: FOCUS 5

6 To costruct a frequecy domai decovolutio operator that ca be applied to seismic data, some assumptios are required: 1. The wavelet should be miimum phase. 2. The wavelet spectrum should be smooth. 3. The wavelet should be statioary. 4. The reflectivity is assumed to be radom, therefore its amplitude spectrum is assumed to be white. O the other had, by writig equatio 2 i frequecy domai, S( f ) R( f ) W( f ) N( f ), (14) It is possible to defie a specific regio of frequecy ( fmi f fmax ), i which the R( f ) W( f ) term domiates over N( f) ad the oisy ad oise-free seismograms are almost the same (figure 5). The white reflectivity assumptio meas R( f) 1, (15) where the overbar idicates smoothig. Therefore, the amplitude spectrum of a estimated wavelet ca be expressed as S( f ) W( f ). (16) The amplitude spectrum of a decovolutio operator ca be calculated from equatio 16 ad equatio 13 as followig 1 1 D( f ) W( f ) S( f ), (17) estimated which idicates that the amplitude spectrum of the decovolutio operator is the iverse of the estimated wavelet or iverse of the smoothig of the seismic amplitude spectrum. Therefore, the better smoothig of the seismic data we have the better reflectivity estimatio. FIG. 5. Amplitude spectrum of oisy ad oise-free seismograms. GeoCovetio 2014: FOCUS 6

7 Accordig to the miimum phase assumptio of a wavelet ad all above results, the complete form of a decovolutio operator becomes (Margrave, 2002) D f 1 A id ( f ) ( ) e, W ( f ) where is called the stability factor or white oise factor, a small positive umber usually betwee 0.01 ad , ad A max is the maximum value of the spectrum of W( f ) est. Also, D( f ) phase of the decovolutio operator ad ca be defied as est max (18) is the f H D l D ( f ), (19) where H is a liear trasform ad is called Hilbert trasform. By applyig equatio 18 to a seismic trace, the reflectivity fuctio ca be estimated. Example REAL WELL DATA DECONVOLUTION RESULTS Geophysics attempts to make a model of the subsurface that is as accurate as possible. To reach this goal sythetic data, which is the modelled data, are created for the purpose of study, ad ca be compared to the actual seismic data results. Oe optio to produce very realistic sythetic seismic data is to use soic ad desity logs. Soic ad desity logs provide detailed ad accurate iformatio about the subsurface. However, this iformatio is valid oly at the well s locatio. By havig velocity ad desity at the well s locatio, it is possible to calculate a exact reflectivity fuctio by equatio 3. By covolvig the reflectivity with a specific wavelet it is possible to sythesize seismic data. I September 2011, CREWES iitiated a seismic experimet with the goal of pushig the low-frequecy cotet of seismic dow as low as possible. This project was located ear Hussar, Alberta, which is about 100km east of Calgary, Alberta. The lie was 4.5km log ad itersected three wells, 12-27, ad 14-35, show i Figure 6 (Margrave, et al., 2012). I this study, the log data from well have bee used. FIG. 6. Locatio of the seismic lie area ear Hussar, Alberta, Caada, idicated by the red marker. (Lloyd, 2013) GeoCovetio 2014: FOCUS 7

8 FIG. 7. P-wave ad desity log of well ear Hussar. FIG. 8. Illustratio of the reflectivity fuctio, the 15Hz miimum phase wavelet, ad the sythetic seismic trace, i the frequecy domai. Figure 8 shows that the reflectivity spectrum does ot have costat power which meas that it is ot a white reflectivity, ad is called colored. This type of spectrum displays the pricipal character of amplitude spectra of the real earth s reflectivity. However, a white spectrum is represetative of radom reflectivity. Therefore, applyig a stadard decovolutio operator, equatio 18, to real seismic data causes a icorrect estimate, which will be described i further detail i the proceedigs. I this study, we have tried to improve the reflectivity estimate at low frequecies. The followigs are two differet approaches to this issue: 1. Improvig the spectral smoothig process used to estimate the wavelet. 2. Applyig a spectral color operator to the decovolved data to correct for the white reflectivity assumptio. GeoCovetio 2014: FOCUS 8

9 Spectral smoothig Returig to the decovolutio process, the first step was to take the magitude of the complexvalued Fourier spectra, discardig all phase iformatio. If the reflectivity is white, the resultig amplitude spectra for a typical case with sythetic data are show i Figure 3. The geeral shape of the amplitude spectrum of the seismic sigal comes etirely from the spectral shape of the source waveform. Put aother way, if we smooth the amplitude spectrum of a seismic sigal, we will estimate the amplitude spectrum of the source waveform. Oce the amplitude spectrum of the source waveform has bee estimated, the waveform s phase spectrum eeds to be estimated. This is where the assumptio of miimum phase comes i. Therefore the most importat stage of a frequecy domai decovolutio process is the smoothig of seismic amplitude spectrum. Geerally, smoothig or filterig techiques have bee applied to suppress oise from oisy data. A commo smoothig techique is to replace a give data poit with the mea value of poits i its eighborhood. The size of the eighborhood defies the size of the smoothig operator. A equally whiteed local average is achieved by covolvig the spectrum with a boxcar fuctio. This process aturally results i a smoother sigal. I this study, we ivestigate the use of a Gaussia smoother istead of a boxcar smoother. O the other had the smoother should deped o the frequecy because the legth of smoother for the low frequecies ad the high frequecies are differet. The smoother legth for the low frequecies should be small ad for the high frequecies should be large. The reaso is for istace the low frequecy compoets whe chagig ito the time coordiate compoets the spaces betwee elemets beig shorter ad it should be covolvig with the smaller smoother i the legth. Therefor for creatig decovolutio operator we used the frequecy depedat Gaussia smoother. Applyig this decovolutio operator to our sythetic seismic data ca cause some issues, sice the sythetic seismic data used i this study are related to a colored spectrum reflectivity, ad they are cotamiated with white radom oise where its sigal to oise ratio is equal to two. For oisy seismograms we oly eed that part of a spectrum where sigal domiates oise, as show i Figure 5. This area is foud i the same figure i the regio betwee f mi ad f max. Therefore, the decovolved seismogram should be filtered by appropriate low-cut ad high-cut frequecy filters but here oly the high cut filter was applied sice the low frequecy wat to be recovered. Figures 9 ad 10 illustrate the decovolutio results for oise-free ad oisy seismogram with boxcar smoother ad frequecy depedat Gaussia smoother, respectively. At each of these figures the right plot shows zoomig o the low-frequecy part of left plot. It is obvious from these two figures that the results from a Gaussia smoother match the spectrum of reflectivity much better tha the results from a boxcar smoother. Here the legth of the frequecy smoother is 65Hz ad 40Hz for the oise-free ad oisy seismogram respectively. However the ew results have still some fudametal problems which come from assumptios o the decovolutio that have bee already metioed ad will be reitroduced i the ext part. GeoCovetio 2014: FOCUS 9

10 FIG. 9. Amplitude spectrum of decovolved oise-free ad oisy seismogram by usig boxcar smoother (left) ad zoomig i low-frequecy part (right). FIG. 10. Amplitude spectrum of decovolved oise-free ad oisy seismogram by usig Gaussia smoother (left) ad zoomig i low-frequecy part (right). Miimum phase spectral color operator Oe of the importat assumptios regardig the decovolutio operator was white reflectivity which meas that the power of the reflectivity s spectrum should be costat for all frequecies. The spectrum of estimated reflectivity matches very well with the spectrum of true reflectivity whe this operator is beig applied to the seismic data which is the covolutio of white reflectivity with a miimum phase wavelet (Figure 11). FIG. 11. Amplitude spectrum of decovolved oise-free ad oisy seismograms for a white spectrum (left) ad zoomig i the low-frequecy part (right). The reflectivity is well predicted by the decovolutio operator for the white spectrum case, especially i a oise-free seismogram. However, real seismic data are the cosequece of colored GeoCovetio 2014: FOCUS 10

11 spectrum reflectivity covolved with a miimum phase wavelet. O the other had, both oise-free ad oisy decovolved seismogram spectra for frequecies below 150Hz, have a white spectrum as show i Figure 9. This is oe of the most importat defects of applyig frequecy domai decovolutio to a real seismic trace. Lookig at the spectrum of reflectivity of frequecies below tha 150 Hz, it ca be realized that it is possible to derive the tred of reflectivity i this part ad apply it to the decovolved seismogram as a operator. This operator is called as a miimum phase spectral color operator which is show i Figure 12 i the frequecy domai ad ca be foud by fittig a curve ito the absolute value of reflectivity s spectrum. It should be also miimum phase as it was o of our assumptio which meas that its wavelet i time domai should be miimum phase. FIG. 12. Amplitude spectrum of colored spectral reflectivity (blue) ad the spectral color operator (red). Applyig a spectral color operator to a decovolved seismic trace ca be doe by covolvig a decovolved seismic trace ad color spectral operator i the time domai or equivaletly by multiplyig these two vectors i the frequecy domai. The results of the amplitude spectrum of a oise-free ad a oisy seismic trace after this process are show i Figure 13 ad 14, respectively. It ca be see from these two figures that the ew operator corrects the previous results effectively, ad its amplitude spectrum is matchig the spectrum of reflectivity much better tha before the correctio. I the time domai, the improvemet of results is obvious as well. Figure 16 shows the true reflectivity, oise-free ad oisy seismic trace i the time domai. It shows that our correctios i the time domai ad frequecy domai were quite effective. It should be oted that i Figure 15 the diagrams of the oise-free seismogram ad reflectivity are boosted to show the results the better. It should also be cosidered that, whe reflectivity is covolved with a miimum phase wavelet, other attributes of a wavelet such as polarity ad wavelet shifts ca be trasferred to the seismic data, ad these will appear i the estimated reflectivity later. I this study both oise-free ad oisy estimated reflectivity fuctios do ot have ay polarity chages, however, the oise-free oe had a 0.3 lag ad the oisy oe had 1.6 lag. After correctio their lags were reduced to 0.1 ad 0.5 respectively. The GeoCovetio 2014: FOCUS 11

12 complete compariso for maximum correlatio ad lags betwee the old versio of frequecy domai decovolutio, which was without ay smoothig ad color spectral correctio, ad ew versio which was after applyig the metioed correctio to both oise-free ad oisy estimated reflectivity, are represeted i table 1. FIG. 13. Spectrum of the oise-free estimated seismogram after applyig the spectral color operator. FIG. 14. Spectrum of the oisy estimated seismogram after applyig the spectral color operator. FIG. 15. Comparig oise-free ad oisy estimated reflectivity with true reflectivity i the time domai. GeoCovetio 2014: FOCUS 12

13 Estimated reflectivity (Noise-free) Estimated reflectivity (Noisy) Frequecy Domai Decovolutio (Boxcar smoother before applyig color spectral operator) Maximum Correlatio = Lag = Maximum Correlatio = Lag = Frequecy Domai Decovolutio (Gaussia smoother after applyig color spectral operator ) Maximum Correlatio = Lag = Maximum Correlatio = Lag = Table 1. Table of maximum correlatio betwee estimated reflectivity ad true reflectivity i two differet cases. Impedace iversio results Fially, after approximatig reflectivity it is possible to estimate acoustic impedace from the estimated reflectivity. As was metioed before the acoustic impedace of each layer ca be calculated from equatio 9. All we eed is the first layer acoustic impedace ad reflectivity fuctio which was estimated i the last sectio. The acoustic impedace iversio for a oise-free seismic trace is illustrated i Figures 16 ad 17. The first oe is the results for the approximated reflectivity by the old decovolutio method ad the secod oe is the results for the approximated reflectivity by the ew approach. It is obvious that the differeces are sigificat. The results for the oisy seismogram are also show i Figures 18 ad 19. Agai it is clear that the result of acoustic impedace after applyig the ew smoother ad color spectral operator is matchig the well acoustic impedace much better tha the results before applyig them. FIG. 16. Acoustic impedace estimatio from the oise-free seismic trace, before applyig ew smoother ad the color spectral operator. GeoCovetio 2014: FOCUS 13

14 FIG. 17. Acoustic impedace estimatio from the oise-free seismic trace, after applyig ew smoother ad the color spectral operator. Fig. 18. Acoustic impedace estimatio from oisy seismic trace, before applyig ew smoother ad the color spectral operator. FIG. 19. Acoustic impedace estimatio from oisy seismic traces, after applyig ew smoother ad the color spectral operator. GeoCovetio 2014: FOCUS 14

15 Coclusios The most challegig part i reflectivity estimatio ad acoustic impedace iversio is determiig the low frequecy compoets of seismic data. Although there are a umber of approaches to extract low frequecy iformatio idirectly from other sources such as well log data, i this study we tried to ivestigate a ew approach to reach this goal directly from seismic data. Usig a proper smoother i the decovolutio process is a effective part of that procedure. It is realized that better seismic data smoothig ca result i more realistic reflectivity estimates. However, our ew smoother still eeds some correctios to achieve much better results. O the other had, as discussed, the decovolutio operator was desiged for white spectrum reflectivity, ad it had some issues i frequecies below 150Hz for colored reflectivity. Cosequetly, the spectral operator was applied to the decovolved seismogram, ad the result was well matchig the amplitude spectrum of the colored reflectivity. A optimal smoother determiatio, as well as a reasoable color spectral operator, has sigificat effects o reflectivity estimatio results. Returig to the acoustic impedace results shows us that suggested approaches ca affect the impedace estimatio cosiderably. However, our model still has a problem i determiatio of true impedace tred which meas that the low frequecy iformatio is still missig. Ackowledgemets I would like to thak my supervisor, Gary F. Margrave, for his exemplary guidace ad techical assistace durig this study ad also I would also like to thak the sposors of CREWES project for their fiacial support ad all the studets ad staffs i CREWES. Refereces AuYeug, C. (1986). Maximum etropy decovolutio. IEEE Iteratioal Coferece o ICASSP '86. (Volume:11 ), (pp ). Ferguso, R. J., & Margrave, G. F. (1996). A simple algorithm for badlimited impedace iversio. CREWES research report, Vol. 8, No. 21. Leibach, J. (1995). Wieer spikig decovolutio ad miimum-phase wavelets: A tutorial. Geophysics. Lidseth, R. O. (1979). Sythetic soic logs-a process for stratigraphic iterpretatio. Geophysics, Lloyd, H. E. (2013). A Ivestigatio of the Role of Low Frequecies i Seismic Impedace Iversio. Calgary: Uiversity of Calgary. Margrave, G. F. (2002). Methods of seismic data processig. Calgary: Departmet of Geosciece, Uiversity of Calgary. Margrave, G. F., & Lamoureux, M. P. (2002). Gabor decovolutio. CSEG Geophysics. Margrave, G. F., Mewhort, L., Phillips, T., Hall, M., Bertram, M. B., Lawto, D. C., et al. (2012). The Hussar low-frequecy experimet. CSEG Recorder, Sept., Oldeburg, D. W., Scheuer, T., & Levy, S. (1983). Recovery of the acoustic impedace from reflectio seismograms. Geophysics, Sheriff, R. E., & Geldart, L. P. (1995). Exploratio Seismology. Cambridge Uiversity Press. Waters, K. H. (1978). Reflectio seismology, a tool for eergy resource exploratio. New York: Joh Wiley ad Sos. GeoCovetio 2014: FOCUS 15