S-transform analyss Varable factor S-transform sesmc ata analyss Toor I. Toorov an Gary F. Margrave ABSTRACT Most of toay s geophyscal ata processng an analyss methos are base on the assumpton that the sesmc sgnal s statonary an employ an etensve use of Fourer analyss. However ue to varous attenuaton mechansms of the Earth, the sesmc sgnal s not statonary. The short-tme Fourer transform Gabor transform has been evelope to eal wth nonstatonary sgnals. The varable factor S-transform s an etenson of the Gabor transform an proves better tme-frequency ecomposton of a nonstatonary sgnal for all frequences. An S-transform econvoluton metho s evelope as an etenson of the nonstatonary Gabor econvoluton reporte n the lterature. The new metho s teste on a constant Q synthetc ata an shows superor results over the tratonal statonary Wener econvoluton an an mprovement over the nonstatonary Gabor econvoluton. In a separate applcaton an f-t- CDP nose attenuaton metho s evelope. A synthetc eample proves the effectveness of the f-t- nose attenuaton for both, hgh-ampltue lnear nose an ranom nose. INTRODUCTION Most of toay s geophyscal ata processng an analyss are base on the assumpton that the sesmc sgnal s statonary an employ an etensve use of Fourer analyss. However ue to varous attenuaton mechansms of the Earth, the sesmc sgnal s not statonary. Margrave 998 has presente the theory of the nonstatonary lnear flterng, whch s a better appromaton to the physcal phenomena of sesmc wave propagaton n the earth over the tratonal statonary convolutonal moel. Margrave an Lamoureu 00 presente a nonstatonary econvoluton base on the Gabor transform, known as short-tme Fourer Transform as well Mertns, 999. Snce the ntroucton of the Gabor transform Gabor, 946 varous tme-frequency ecomposton methos has been propose to mprove on the tme-frequency resoluton an other propertes of the transform Mertns, 999. One such transform s the S-transform Stockwell, 996. Ths work s concerne wth the nvestgaton of the S-transform applcaton for nonstatonary sesmc econvoluton an nose attenuaton. The Fourer transform THE VARIABLE FACTOR S-TRANSFORM The Fourer transform s a funamental tool n sesmc ata processng an analyss. It apples to almost all stages of processng. A number of sesmc attrbutes use n nterpretaton are also base on the Fourer transform. It completely escrbes a sesmc trace as a sum of comple snusos each wth unque ampltue, frequency, an phase. The analyss of a sgnal ht s acheve by the forwar Fourer transform CREWES Research Report Volume 009
Toorov an Margrave πft = h t e t an ts synthess by the nverse Fourer transform H f, πft h t = H f e f, where t s the tme varable, f s frequency, an Hf s calle the Fourer spectrum. It s a comple functon an can be epresse n terms of ts real H R f an magnary H I f components H f = H f H f. R The Fourer spectrum can be wrtten as a functon of the ampltue spectrum Af an the phase spectrum Φf H f whch are compute from the followng equatons I Φ f = A f e 4 A f = H f H f 5 R I H I f Φ f = tan 6 H f One rawback of the Fourer transform s that t only prouces the tme-average spectrum. Fgure s a splay of a partcular sgnal an ts ampltue an phase spectra. By eamnng them one can conclue the frequency content of the sgnal,.e. the sgnal contans 0, 0, 50, 00, an 50 Hz frequency components. Fgure shows how ths sgnal was constructe. The Fourer transform correctly tells us whch frequences est n the sgnal, however they are not present all the tme. So the Fourer transform gves the frequency nformaton of the sgnal, but oes not tell us when n tme these frequency components est. In other wors, Fourer transform s approprate tool for frequency analyss of statonary sgnals,.e. sgnals whose frequency content oes not change n tme. For nonstatonary sgnals, whose frequency content changes n tme, we nee both the tme an the frequency epenence smultaneously. The short-tme Fourer transform To overcome the lmtatons of the Fourer transform n analysng nonstatonary sgnals, Gabor 946 ntrouce the short-tme Fourer transform, known as Gabor transform as well. The concept s smple: multply the sgnal ht wth an analyss wnow an then compute the Fourer transform of the wnowe sgnal. Ths gves us the local Fourer spectrum of the sgnal. By shftng the analyss wnow along the sgnal, we get a tme-frequency spectrum of the sgnal. R CREWES Research Report Volume 009
Mathematcally, the Gabor transform s efne as e.g. Mertns, 999 S-transform analyss πft = h t w t τ e t G τ, f 7 where wt s the analyss wnow an Gτ,f s the comple Gabor spectrum. In hs work Gabor has chosen a Gaussan wnow t / σ w t = e 8 πσ where σ s the stanar evaton. By applyng the Gabor transform to a sgnal, we ecompose one mensonal functon ht nto a two mensonal one Gτ,f. Ths process s known as tme-frequency analyss. The goal s to tell not only whch frequences est, but when they appear as well. We try to acheve the best possble tme an frequency resoluton as well. However, choosng a short tme wnow leas to goo tme resoluton, but low frequency resoluton. On the other han, a long tme wnow yels low tme resoluton, but goo frequency resoluton. The mamum possble resoluton s governe by the uncertanty prncple e.g. Mertns, 999 t ω 9 where t ω s the area n the tme frequency plane covere by a partcular wnow. The equty sgn,.e. the best tme-frequency resoluton, s acheve only f wt s a Gaussan functon e.g. Mertns, 999. The Gabor transform synthess s acheve by ntegraton over frequency an over wnow poston πft h t = G τ, f γ t τ e τf 0 where the synthess wnow γt must satsfy the conton w t γ t = Fgure s a splay of the Gabor tme-frequency ecomposton of the compose sgnal wth a varable wth wnows. The CREWES Project MATLAB toolbo functon fgabor was use. The wnow step was chosen to be equal to the samplng rate of the sgnal, 0.00 sec. The mamum reunancy s chosen for a far comparson wth the later scusse S-transform. Fgure a s the Gabor spectrum wth 0.0 secons wth, whch obvously s a poor choce an leas to smearng along the frequency as. Note the very hgh tme resoluton. Fgure b s the Gabor spectrum wth 0.05 secons wth. The frequency resoluton s much better, ecept for the 0 an 0 Hz components. By CREWES Research Report Volume 009
Toorov an Margrave ncreasng the wnow wth to 0. secons Fgure c we acheve a very goo frequency resoluton, however we have lost the goo tme resoluton for the 50 Hz component. By ncreasng the wnow wth to 0. we loose completely the tme resoluton of the hgh-frequency component. By eamnng the plots, one can conclue that a wnow wth of 0.05 sec gves a goo tme-frequency representaton of the 50 Hz component, however the wnow wth 0. wth gve us the goo tme-frequency representaton of the lower frequency components of 0 an 0 Hz. So wth the Gabor transform, for a partcular wnow choce, we can acheve goo tme-frequency representaton for a partcular frequency range, but poor tme or frequency resoluton outse of the ban. The S-transform an the varable factor S-transform To overcome the lmtatons of the short-tme Fourer transform a varety of tmefrequency ecompostons have been propose, notably the contnuous wavelet transform e.g. Mertns, 999. Ths paper s partcularly concerne wth the S-Transform Stockwell, 996. It unquely combnes a frequency epenent resoluton wth smultaneously localzng the real an magnary spectra. It s partcularly appealng to geophyscal sgnal analyss snce t generates tme-frequency ecomposton versus tmescale ecomposton n the contnuous wavelet transform case. Frequency s preferre snce t has a specfc physcal meanng. The contnuous S-transform of a sgnal ht s efne by Stockwell, 996 f S τ, f τ t f πft = h t e e t π where t an τ are tme varables an f s frequency varable. The S-transform can be erve from the Gabor transform by efnng the stanar evaton σ of the Gaussan wnow to be a functon of the frequency f σ f = f Snce Sτ,f s comple, we can efne ts ampltue Aτ,f an phase Φτ,f spectrum n a smlar way to the Fourer transform. The tme average of the Sτ,f gves the Fourer spectrum of ht Stackwell, 996. The nverse S-transform s gven by Stockwell, 996 πft h t = S τ, f τ e f 4 The lossless nvertblty of the S-transform allows for flterng n the tme-frequency oman. 4 CREWES Research Report Volume 009
S-transform analyss The orgnal S-transform propose by Stockwell set the stanar evaton of the Gaussan wnow proportonal to the nverse of the frequency eq.. Manshnha et al 997 ntrouce a constant factor k n the stanar evaton of the analyss wnow k σ f = 5 f By ncreasng the factor k they have acheve better frequency resoluton, wth a corresponng loss of resoluton n tme. In ther work they have suggeste k=. Fgure 4 a, b, c s the S-transform ampltue spectrum of the sgnal from Fgure wth factors,, an respectvely. We can notce goo frequency-tme resoluton of the lower frequency components wth factor, however poor frequency resoluton for the m an hgher frequency components. Factor mproves the frequency resoluton, however we start to loose the tme resoluton for the 0 Hz component. the mfrequency range has goo tme-frequency resoluton. Wth factor we acheve a very goo frequency resoluton, however the tme resoluton of the low-frequency components s completely lost. By eamnng the plots one can conclue that factor s a goo choce to acheve both tme an frequency for low-frequency components, factor for the m-range components, an for the hgher frequences. However n the Stockwell S-transform the factor s a constant number. We propose a Varable Factor S-transform, n whch the factor s a functon of the frequency as well k f σ f = 6 f τ t f f k f πft = h t e e t π k f S τ, f. 7 The smplest way to efne a varable k s by usng a lnear moel. I efne mnmum factor at zero frequency, mamum factor at Nyqust frequency an lnearly nterpolate for any frequency n between. Fgure 4 s the ampltue spectrum of the same sgnal usng lnear factor from at zero frequency an 6 at Nyqust. We can see a goo tmefrequency resoluton for all the present frequency components. The nverse transform can stll be efne by equaton 4. The convolutonal moel S-TRANSFORM DECONVOLUTION The goal of the sesmc econvoluton process s to compress the basc wavelet, remove multples, an yel a representaton of the subsurface reflectvty rt Ylmaz, 00. The theoretcal base of the majorty of toay s econvolutonal methos s the convolutonal moel, efne as CREWES Research Report Volume 009 5
Toorov an Margrave where s t = w t e t n t 8 st s the recore sesmc trace, wt s the sesmc wavelet, et s the earth mpulse response, nt s atve nose. Ths smple equaton may be nterprete n fferent ways Margrave, 005. In forwar moelng, for eample, one may choose to convolve a zero-phase wavelet wth reflectvty erve from a sonc an ensty logs. The result s zero-offset, prmares only sesmc trace,.e. the mpulse response n the above equaton s the reflectvty. However, for a recore sesmc trace the mpulse response s a superposton of varous phenomena n aton to reflectons only, lke multples, transmsson losses, moe conversons, an so on. In general, the source wavelet s unknown an we en up wth two unknowns to solve for: wt an rt. In aton, the sesmc wavelet can be regare as the source wavelet or as a more complcate functon combnng the source wavelet wth the near surface effects. To summarze, sesmc wave propagaton s a very comple phenomenon, an any econvoluton metho s base on some smplfcatons an assumptons. Wener Deconvoluton Statonary Wener econvoluton s one of the man processes apple n toay s sesmc ata processng. It s base on the assumpton that the sesmc trace can be escrbe as a convolutonal process of the earth reflectvty rt an a wavelet wt s t = w t r t n t 9 The wavelet s escrbe as the embee wavelet,.e. a wavelet that fts to the convolutonal moel. Snce the prme goal of the econvoluton process s the recovery of the reflectvty, the actual components of the wavelet are not escrbe an treate as a package. Our goal s to erve an nverse flter t of the wavelet to erve the reflectvty. A number of further assumptons are mae to accomplsh ths, the most mportant lste as follows Ylmaz, 00; Margrave, 005: the wavelet oes not change as t travels,.e. we eal wth a statonary process the wavelet s causal, mnmum-phase the reflectvty s ranom, whte sequence,.e. the autocorrelaton of the sesmc trace s a scale verson of the autocorrelaton of the wavelet the nose s whte an statonary 6 CREWES Research Report Volume 009
S-transform analyss The process of fnng an m-length causal, nverse flter, base on the above assumptons, reuces to solvng a lnear system of equatons φ0 φ φ φ m φ φ φ φ 0 m φ φ φ φ 0 m φm 0 φ m 0 φ m = 0 φ 0 0 m where φ s the -th lag of the sesmc trace autocorrelaton. The system s solve usng the least-squares approach. In practce, a small value s ae to the zero-lag agonal to ensure stablty. Gabor Deconvoluton Due to the subsurface attenuaton, the recore sesmc trace s nonstatonary by nature. The Wnner econvoluton however s base on the statonary convolutonal moel, so the current sesmc processng requres atonal ampltue correctons. One can apply a geometrcal spreang correcton, but an absorpton correcton usually s not apple. A common way to go aroun the problem s to apply AGC. However, AGC smears the true relatve ampltues an the nterpreters en up wth ncorrect ampltue maps an AVO analyss. So a nonstatonary moel s requre. The statonary convolutonal moel can be generalze to a nonstatonary one Margrave, 998 0 s t = w t τ, τ r τ τ Base on the theory of the nonstatonary flterng Margrave, 988 Margarave an Lamoureu 00 efne a nonstatonary convolutonal moel base on the constant Q theory πft = W f α f, t r t e t S f where captal letters enote the forwar Fourer transform an πt α t, f = ep f H f Q t The man avantage of the nonstatonary formulaton s the separaton of the source wavelet an the attenuaton process. The above nonstatonary moel may be can be epresse n Gabor oman CREWES Research Report Volume 009 7
Toorov an Margrave S G τ, f = W f α τ, f R τ, f 4 where S G an R G are the Gabor transform of the sesmc trace an the reflectvty. By nvokng the some assumptons of the statonary econvoluton, lke mnmum-phase wavelet an whte reflectvty, one can recover R G, an by the nverse Gabor transform rt. S-transform econvoluton The Gabor econvoluton methoology s easly mofe to perform a Varable Factor S-transform econvoluton. The only fference s that we substtute the Gabor transform wth the S-transform. The followng summarses the actual mplementaton compute the Varable Factor S-transform S S τ,f of the nput sesmc trace apply hyperbolc smoothng to SSτ,f S S τ,f Margrave an Lamoureu, 00 estmate nverse operator OSτ,f usng mnmum-phase assumpton an S S τ,f multply S S τ,f wth O S τ,f nverse S-transform To compare the performance of the scusse econvoluton methos a synthetc trace eample wth a constant Q moel s evelope. Fgure 5 s a splay of a ranomly generate reflectvty sequence an the corresponng synthetc trace wth a mnmumphase wavelet an Q=00. Fgures 6 an 7 are splays of the Gabor an the Varable Factor S-transform of the synthetc trace respectvely. The clear ecay of the spectrum n frequency an tme shows the nonstatonary nature of the sgnal. Fgures 8, 9, an 0 splay the results from performng the three types of econvoluton: Wener, Gabor, an Varable Factor S-transform. The Wener econvoluton fals to recover any of the hgh magntue reflectvty. The Gabor econvoluton has recovere the reflectvty much better compare wth the Wener metho, especally the hgh magntue reflectvty. The Varable Factor shows an mprovement over the Gabor metho. Fgures -6 are splays of the same eperment wth Q=60. Smlar conclusons can be mae. Table summarzes the results of the eperments. G Metho Total Abs. Error, Q=00 Total Abs. Error, Q=60 Wener 7.67 8.997 Gabor.056.0757 S-transform.680.749 8 CREWES Research Report Volume 009
S-transform analyss Table : Absolute an relatve errors for teste econvoluton methos. F-T-X S-TRANSFORM NOISE ATTENUATION The comple one-step-ahea precton flter for ranom nose attenuaton n f- oman on stacke ata was ntrouce be Canales 984. Gulunay 986 however states that the f- metho wll not work on ata wth conflctng ps. Sptz 99 evelope the f- trace nterpolaton. The above technques transform the ata from t- to f- oman usng the Fourer transform. We have alreay argue that the sesmc trace s nonstatonary, so a tme-frequency transforms are better sute to escrbe t. The nose, especally source nose an low / hgh frequency bursts n ata, s not statonary. Ths type of nose s present for a lmte pero of tme. The groun roll s hghly spersve n nature, so fferent frequency component wll rese n fferent locatons on the f-t- space. Ths makes the S-transform a very attractve tool to esgn a nose attenuaton algorthm. We propose a new metho for nose attenuaton on NMO-correcte CDP gather base on the S-transform. The metho nvolves transform the CDP ata from t- oman to f-τ- oman usng the S-transform for each f, τ efne the array f=const, τ=const, for a ata pont esgn a comple precton flter n -recton from the surrounng samples..., -,,... keep the error between the actual an the precte value for each substtute the actual value wth the precte one for the sample wth the largest error net τ, f nverse S-transform Note that only one sample for a partcular τ, f s change. Ths assures that the nonnosy samples are not altere an the nose oes not creep out to the nose-free samples. The same process can be repeate as a secon teraton on the fltere secton f necessary. The followng eample eplans the flter esgn. Problem: for a partcular f, τ prect usng L surrounng samples by esgnng a precton flter g: form the followng system of equatons n matr form, L=6 CREWES Research Report Volume 009 9
Toorov an Margrave 0 CREWES Research Report Volume 009 = 0 0 0 0 0 0 g g g g g g 5 solve for g usng the truncate sngular value ecomposton Aster et al, 005 prect A synthetc CDP moel was generate to test the concept Fgure 7. Fgure 8 shows the prmares only an Fgure 9 the ae nose, whch contans a low-frequency, hgh ampltue lnear event an a nosy trace. Fgure 0 s the result of the f-t- nose attenuaton wth a sngle teraton an Fgure s the remove nose. Most of the lowfrequency hgh ampltue nose has been remove an the ranom nose has been remove as well even wth a sngle teraton snce they has fferent frequency content. Fgure s the fltere CDP gather after a secon teraton an Fgure s the remove nose after both teratons. We can conclue that the nose has been remove successfully whle the non-nosy ampltue has not been altere. The escrbe metho can be use as a trace nterpolator n NMO-correcte CDP-gathers an shot-gathers as well. CONCLUSIONS The Varable Factor S-transform shows a better smultaneous tme-frequency resoluton than the Gabor transform an the tratonal S-transform. The econvoluton metho base on the Varable Factor S-transform s superor over the tratonal Wener econvoluton an mprovement over the recently evelope Gabor econvoluton. The f-t- nose attenuaton algorthm has shown a goo potental an further testng s neee. However the S-transform s computatonally epensve an further work nees to be one to nvestgate possble performance mprovements. FUTURE WORK The Varable Factor S-transform have shown some encouragng result so far. We conser t a valuable tool for sesmc ata processng an analyss an we are plannng some further work wth t. Some eamples are: mplement a frequency oman S-transform computaton for better computatonal performance nvestgate the reunancy n the S-transform evelop a surface-consstent S-transform econvoluton
S-transform analyss Q estmaton test the f-t- nose attenuaton on a more complcate moel wth AVO an NMO-stretch, an on real ata REFERENCES Aster, R., Borchers, B., an Thurber, C., 005, Parameter Estmaton an nverse problems: Elsever Acaemc Press. Canales, L., 984, Ranom nose reucton: 54 th Annual SEG Meetng, Epene Abstracts, 55-57. Gabor, D., 946, Theory of communcaton: J. IEEE Lonon, 9, 49-457. Gulunay, N., 986, F-X econ an comple Wener precton flter: 56 th Annual SEG Meetng, Epene Abstracts, 79-8. Margrave, G., 998, Theory of nonstatonary lnear flterng n the Fourer oman wth applcaton to tmevarant flterng: Geophyscs, 6, 44-59. Margrave, G., an Lamoureu, M., 00, Gabor econvoluton: The CREWES Project Research Report,, 4-76. Margrave, G., 005, Sesmc Processng Funamentals: CSEG Course Notes. Mansnha, L., Stockwell, R., an Lowe, R., 997, Local S-spectrum analyss of -D an -D ata: Physcs of the Earth an Planetary Interors, 0, 9-6. Mertns, A., 999, Sgnal Analyss: John Wley an Sons. Sptz, S., 99, Sesmc trace nterpolaton n the f- oman: Geophyscs, 56, 785-794. Stockwell, R., Mansnha, L., an Lowe, R., 996, Localzaton of the comple spectrum: The S-transform: IEEE Trans. Sgnal Processng, 44, 998-00. Ylmaz, O., 00, Sesmc ata analyss: SEG. CREWES Research Report Volume 009
Toorov an Margrave Fgure : Fourer transform of a partcular sgnal. Fgure : Comple snusos use to construct the sgnal. CREWES Research Report Volume 009
S-transform analyss Fgure : Gabor transform of the sgnal n Fgure wth fferent wnows: a 0.0 sec., b 0.05 sec., c 0. sec., 0. sec. Fgure 4: S-transform wth factor: a, b, c, 0.7-6 CREWES Research Report Volume 009
Toorov an Margrave Fgure 5: A ranom reflectvty moel an a synthetc trace wth a mnmum-phase wavelet an Q=00. Fgure 6: Gabor transform of the synthetc trace wth Q=00. 4 CREWES Research Report Volume 009
S-transform analyss Fgure 7: Varable factor S0transform of the synthetc trace wth Q=00. Fgure 8: Wener econvoluton of the synthetc trace wth Q=00. CREWES Research Report Volume 009 5
Toorov an Margrave Fgure 9: Gabor econvoluton of the synthetc trace wth Q=00. Fgure 0: Varable factor S-transform econvoluton of the synthetc trace wth Q=00. 6 CREWES Research Report Volume 009
S-transform analyss Fgure : A ranom reflectvty moel an a synthetc trace wth a mnmum-phase wavelet an Q=60. Fgure : Gabor transform of the synthetc trace wth Q=60. CREWES Research Report Volume 009 7
Toorov an Margrave Fgure : Varable factor S-transform of the synthetc trace wth Q=60. Fgure 4: Wener econvoluton of the synthetc trace wth Q=60. 8 CREWES Research Report Volume 009
S-transform analyss Fgure 5: Gabor econvoluton of the synthetc trace wth Q=60. Fgure 6: Varable factor S-transform econvoluton of the synthetc trace wth Q=60. CREWES Research Report Volume 009 9
Toorov an Margrave Fgure 7: A synthetc CDP moel wth nose. Fgure 8: Prmares only of the CDP synthetc moel. 0 CREWES Research Report Volume 009
S-transform analyss Fgure 9: Nose only of the synthetc CDP moel. Fgure 0: CDP moel after a sngle terartn of the f-t- nose attenuaton. CREWES Research Report Volume 009
Toorov an Margrave Fgure : Nose remove after a sngle teraton. Fgure : CDP moel after a secon teraton of the f-t- nose attenuaton. CREWES Research Report Volume 009
S-transform analyss Fgure : Nose remove after a secon teraton. CREWES Research Report Volume 009