We N Optimised Spectral Merge of the Background Model in Seismic Inversion

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1 We N Opimised Specral Merge of he Background Model in Seismic Inversion R.E. Whie* (Universiy of London - Birkbeck) & E. Zabihi Naeini (Ikon Science) SUMMARY Seismic inversion generaes low-frequency modulaions if he frequency conen of he background model does no merge smoohly ino he specrum of he invered seismic daa. An incorrec low-frequency phase is anoher source of inversion arefacs. We presen a sysemaic mehod of selecing a background model and low-frequency phase response ha minimises he misfi a he specral merge. The mehod compares well-log and seismic relaive impedances compued a bes-mach well-ie locaions over a range of background models and low-frequency phase correcions. The background models are specified by a lowcu corner frequency and he phase correcions by he phase inercep a zero frequency. The mehod is illusraed by applicaion o broad-band inversion. 77 h EAGE onference & Exhibiion 2015

2 Inroducion When invering seismic daa i is imporan ha he frequency conen of he low-frequency background model merges smoohly ino he specrum of he invered seismic daa. In a classic paper on seismic inversion Lindseh (1970) poins ou ha a gap in he specrum of he invered daa inroduces a modulaion ha makes he impedance oo high in some inervals and oo low in ohers. An overlap of frequency conen would have a similar effec. Modulaions are a serious problem because hey could, for insance, indicae high porosiy where i is poor, or vice versa. Wagner e al. (2006) show a sriking example of how a lack of low frequencies in invered sreamer daa causes arefacs ha are no seen when invering daa recorded wih ocean-boom hydrophones which is richer in lower frequencies. They also show ha inversion of he ocean-boom hydrophone daa inroduces less "curaining" (laerally variable amplificaion of low-frequency noise) han he sreamer daa. While broad-band recording miigaes he problem of designing a suiable background model o some exen, i does no eliminae i. Any mismach in ampliude simply moves he modulaion o a lower frequency and a second cause of low-frequency modulaions, an incorrec low-frequency phase, remains. In band-limied inversion a smooh merge can be arranged by ensuring ha he low-pass filer applied o he well-log impedances is he complemen of he effecive high-pass filer seen by he relaive impedance from he inversion; ha is, he wo filers add o one from zero frequency ou o a highcu frequency where noise exceeds signal. In model-based inversion he background model is ypically an iniial model ino which he inversion rouine compues he impedance conrass needed o consruc synheic seismograms ha closely fi he seismic daa. Whaever he inversion rouine, i is never cerain o wha exen he rouine compensaes he low-frequency roll-off seen in seismic specra or o wha exen i amplifies low-frequency noise. onsequenly choosing a corner frequency based on a seismic daa specrum does no guaranee a saisfacory merge. In addiion he phase in he viciniy of he roll-off is difficul o measure accuraely, especially in broad-band daa (Whie and Zabihi Naeini (2014)). The low-frequency phase from a well-ie wavele always ends o 0 o or 180 o a zero frequency, depending on wheher he sum of wavele coefficiens happens o be posiive or negaive. The acual phase is conrolled ab iniio by he phase of he seismic recording sysem, which ends o an ineger muliple of 90 o a zero frequency, and is subsequenly modified by phase shifs inroduced during processing. The merging of a background impedance ino impedance inversion herefore requires a proper choice of corner frequency and low-frequency phase response. In principle he merge requires a hird parameer, he low-frequency decay rae, bu i appears o be a less imporan one. In his paper we presen a sysemaic approach o selecing a corner frequency and low-frequency phase response ha minimises he misfi a he merge in a leas squares sense. Mehod In he following we assume ha he background impedance model is obained by low-pass filering he log impedances a wells which are hen exrapolaed, in manner consisen wih he seismic image and/or inerpreed horizons, o cover he whole survey area (e.g. Douma and Zabihi Naeini (2014), Zabihi Naeini and Hale (2015)). A specral mismach in he inversion oupu is more difficul o avoid when oher mehods of consrucing a background model, such as use of a smoohed inerval velociy field obained from processing velociies, are used. The esing for opimum corner frequency and low-frequency phase is carried ou a he bes-mach wellie locaion(s). Alernaively one could find he locaions ha bes mach he seismic daa o he well-log relaive impedances. For daa shown here he bes-mach locaions were he same. We Illusrae he mehod using a model-based inversion and compare is resuls wih hose from a band-limied inversion designed o opimise he merge of background and relaive impedances. The mehod consiss of hree seps: (1) invering he bes-mach race using a range of background models, (2) applying a se of low-frequency phase correcions o he relaive impedance from each inversion, and (3) comparing hese 77 h EAGE onference & Exhibiion 2015

3 relaive impedances wih a se of well-log relaive impedances. The background models are formed using he complemens of he high-pass filers, specified by corner frequencies f (m), ha are applied o he well-log impedance Z o form he well-log relaive impedances y (m). hoice of he f (m) range is guided by he muli-aper seismic daa specrum. The phase correcions φ (k,m) ( f ) are defined by he phase a zero frequency φ (k) (0) and he corner frequency f (m). The correcion decays o 0 wihin he seismic bandwidh where he phase can be esimaed fairly accuraely from a well ie. A lowcu phase response exends above he corner frequency. A simple exrapolaion of phase φ (k,m) ( f )= φ (k) (0)exp( f /(2 f (m) )), wih 90o incremens for φ (k) (0), was used. Phase responses of Buerworh low-cu filers were ried and gave similar resuls. Each phase-shifed relaive impedance x ( j,k) from he jh inversion of he bes-mach race is compared wih he se of well-log relaive impedances y (m), resuling in a 2D scan of goodness-of-fi over f (m) and φ (k) (0) for each x ( j). The goodness-of-fi is defined as he RMS error of fi Σ N 1 (y(m) bx ( j,k) ) 2 /N where b is a scaler ha minimises he RMS error. I is imporan o use he RMS error and no he correlaion coefficien which could mask a difference in he means of y (m) and x ( j,k). A difference in means would show up as a separaion of he seismic and well-log impedances. The cross-correlaion is no appropriae here because i measures he goodnessof-fi of a regression, viz. y (m) = a + bx ( j,k) where a is he difference in means of y (m) and bx ( j,k). When merging he background and relaive impedances we do no wan o end up wih a difference in means. The RMS error corresponds o fiing y (m) = bx ( j,k). Why is a scaler needed when he wavele has already been scaled o mach he well-log synheic seismogram? The leas-squares scaler ha fis y (m) = bx ( j,k) is no he reciprocal of he leas-squares scaler b ha fis he inverse relaion x ( j,k) correlaion coefficien beween x ( j,k) and y (m) is ±1. 77 h EAGE onference & Exhibiion 2015 = b y (m) unless he The procedure for opimising he corner frequency for band-limied inversion is as follows. A high-cu frequency, say 80 Hz, is se o aenuae high-frequency noise. For each low-cu corner frequency f ( j) : 1. filer he well-log impedance Z wih a f ( j) 80 Hz filer o produce a well-log relaive impedance; 2. design and apply he maching filer ha convers he seismic race a he bes mach locaion o his f ( j) 80 Hz relaive impedance a he well; 3. compue he RMS misfi beween he seismic well-log relaive impedances. The procedure can be elaboraed a sep 3 in order o esimae he low-frequency phase by measuring he RMS misfi afer applying phase correcions o he relaive impedance. However i is more informaive o scan he seismic relaive impedances as described nex for a model-based inversion since i indicaes when f ( j) is oo low o capure signal. The procedure for esing model-based inversion uses he same se of he low-frequency background models. Each is used in urn o compue a model-based relaive impedance x ( j) which is compared wih he se of well-log relaive impedances y (m). A each comparison (designaed by f (m) ) a se of phase correcions (designaed by φ (k) (0)) is applied o x ( j) and he goodness-of fi is compued. The scan over f (m) and φ (k) (0) is repeaed for each x ( j). The phase correcion applied o he relaive impedance implies he opposie phase correcion applied o he wavele. Example We illusrae he mehod using a broad-band daa se. There is a close well ie wih he daa (Figure 1). Over a s ime window he ie predics a proporion 0.75 of he race energy, corresponding o a correlaion coefficien of Table 1 liss he resuls from rials of corner frequency for he bandlimied inversion. The RMS error decreases a a slowing rae. A coninuing decrease is ineviable as he inversion moves wihin he seismic bandwidh and has o predic less of he well log.

4 TWT (ms) Figure 1 Well ie showing he bes-mach well-log synheic seismogram spliced ino he seismic daa. low-cu corner frequency f ( j) (Hz) RMS error in relaive impedance (km/s)(g/cc) Table 1 Band-limied inversion: variaion of goodness-of-fi wih corner frequency. As he aim of he inversion is o maximise he informaion from he seismic daa, he choice is beween a low-cu f ( j) of 1 or 2 Hz. omparison of he absolue (background+relaive) impedance esimaes using hese wo low-cu frequencies wih he well log impedance (Figure 2) indicaes ha he 2 Hz low-cu is marginally beer. A scan over f (m) and φ (k) (0) reveals wheher he 2Hz low-cu inversion could be improved by a phase correcion (Table 2). Evidenly no phase correcion is needed. The 2 Hz low-cu Analysis ime window: s low-cu corner frequency f (m) (Hz) phase inercep φ (k) (0) RMS error Table 2 Band-limied inversion: variaion of goodness-of-fi wih corner frequency and phase. inversion is consisen in fiing he 2 Hz well-log relaive impedance bes. The same is rue of he 1 Hz low-cu impedance bu he able for he 0.5 Hz low-cu relaive impedance shows ha he 1 Hz well-log impedance fis bes, indicaing ha he inversion bandwidh canno be exended below 1 Hz. The model-based relaive impedances for corner frequencies ranging from 0.25 o 2 Hz are all very similar and all correlae bes wih he 2 Hz well-log relaive impedance and even yield he same crosscorrelaion coefficien (0.83). The 1 Hz seismic correlaes well (0.80) wih 1 Hz well-log impedance bu he lower like-wih-like correlaions are poor, confirming ha he inversion bandwidh canno be pushed below 1 Hz. The RMS errors from he f (k) φ ( j) (0) scan for he 2 Hz low-cu inversion are abulaed in Table 3. The bes fi comes from use of a 2 Hz low-cu and a zero phase inercep, implying no phase correcion in he wavele. The fi o he well-log impedance is shown in he lower panel of Figure 2. onclusion Even if he correc low-frequency response were buil ino he seismic wavele, i would imply a long and noise-prone inverse. I is herefore debaable how well seismic inversion can compensae he lowfrequency roll-off and conrol low-frequency noise amplificaion. The approach proposed here is o use he seismic wavele in inversion, hereby recovering frequencies in he seismic bandwidh and possibly beyond, and deal wih he low-frequency response in he relaive impedance domain. Because inversion booss low frequency componens, relaive impedance is more sensiive o he low-fequency response. 77 h EAGE onference & Exhibiion 2015

5 Analysis ime window: s low-cu corner frequency f (m) (Hz) phase inercep φ (k) (0) RMS error June 2015 IFEMA Madrid Table 3 Model-based inversion: variaion of goodness-of-fi wih corner frequency and phase. AI (km/s)(g/cc) AI (km/s)(g/cc) omparison of band limied seismic and well log AI esimaes: well log AI band limied AI from s mach using 2 Hz low cu band limied AI from s mach using 1 Hz low cu ime (ms) omparison of model based seismic and well log AI esimaes: well log AI bes fi model based AI using 2 Hz low cu ime (ms) Figure 2 omparison of seismic impedance wih he well-log impedance. In addiion a sysemaic analysis of he low-frequency cu-off frequency and low-frequency phase correcion makes possible an opimally smooh and consisen merge of he background impedance ino he inversion. Acknowledgemens We hank Ikon Science for heir suppor of his research and Dolphin Geophysical for providing he broad-band daa and processing suppor. References Douma, J. and Zabihi Naeini, E. [2014] Applicaion of image-guided inerpolaion o build low frequency background model prior o inversion. 76h EAGE onference and Exhibiion, Exended Absrac, We G Lindseh, R.O. [1979] Synheic sonic logs - a process for sraigraphic inerpreaion. Geophysics, 44(1), Wagner, S.R., Penningon, W.D., and MacBeh,., [2006] Gas sauraion predicion and effec of low frequencies on acousic impedance images a Foinaven Field. Geophysical Prospecing, 54(1), Whie, R.E. and Zabihi Naeini, E., [2014] Broad-band well ie. 76h EAGE onference and Exhibiion, Exended Absrac, Tu EL Zabihi Naeini, E. and Hale, D., [2015] Image- and horizon-guided inerpolaion. Geophysics, Acceped for publicaion. 77 h EAGE onference & Exhibiion 2015