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Environmental and Exploration Geophysics II Amplitude Frequency and Bandwidth and their relationship to Seismic Resolution tomhwilson wilsongeowvuedu Department of Geology and Geography West Virginia University Morgantown WV transmissivitmetorsx en Partitioning fEnergy 0 er o i159 5FHERIGAL DIVERGENCE A ATS95 I fs 3 u 5 quot m 8 g E 2 m 39 012m rm M HEWr 1951 52 Am 39 3 PARTITION OF ENERGY 2 RR z I A 219m A1El FILM n A i s Z a a 439 M HEWrt 195 Aseozmou or 5215mm ENERGY Ax Aged A Aunpllkudz a distance 1 A9 Ampliludz d Manna cl AH nua idn Factor M HE WM 1951 AEEDQP39TION CoMstANT Q I Q39dh i 1 Represents the Energg Dissipaf d in am Wave Langih 1 M HE mm 1951 m 13 Jillma a Mir 5 Ax moon v 10000 nsu M HEWff 1951 ltmore absorptive less absorptive emu Granite and Dbrllu leulam so no mo 100 QValuz M HEWff 1951 When surface irregularities have wavelengths similar to the wavelen ths comprising an 9 37 J e signific When surface irregularities are small compared to t are small The surface appears smoother to the incident wavelet We developed a simple relationship between source amplitude and wavelet amplitude at a distance r from the source A T e 39 r Ideally we would like to have A OCR To do this we must r to move or correct for the various loss mechanisms Design The operator ApplicaTion of This operaTor m re A m T A e7 mR r yields Am ASR Hence The raTio of ampIiTudes Al A2 The raTio of reflecTion coefficienTs RIRZ The co recTed ampIiTude Ac r is proporTionaI To reflecTion coefficienT mama ramM xm nmwrlvmmtln The reIaTive ampIiTude differences shown in The high fideiTy secTion IefT Top provide more accuraTe informaTion abouT acTuaI differences in reflecTion coefficienT and Thus iThoogy Than does The secTion below L iquot CM 1 law Lv P From the start you have been aware that seismic data is recorded as a series of measurements made at constant time intervals the sample interval How do we decide what the sample rate should be How often should ground motion be sampled to accurately represent it The example of a rotating wheel with spokes serves as the best example of the possible effect of sample rate on the conclusions of our observations concerning the motion of the wheel Start with a wheel that rotates at constant frequency and then sample it at varying rates Assume the period of rotation 17 is 1 second and that we sample the motion of the wheel every 14 second In this case I observed Tactual LeT AT 12 second We estimate 5 1 but we are unable To deTermine The direcTion of wheel roTaTion LeT AT 34 second Our estimate of E 3 seconds is in error by 300 When AT 1 second The wheel appears To remain aT The sTarTing poinT buT acTually makes a single roTaTion beTween samples In The second case we deal wiTh siTuaTions where The sample inTerVal AT is consTanT and The period of roTaTion E varies Take The case where AT 12 second and E 4 seconds We have no problem in This case accuraTely delineaTing The period of roTaTion NoTe ThaT f 1 E 14 second and ThaT 1 AT 18Th 1 AT is The fracTion of a cycle The wheel has moved in Time AT When AT equals 12 we also correc Hy idenTify roTaTion periods of 15 seconds As before when E 1 and AT 12 we are still able To idenTify The period as 1 second However when E 34 sec The frequency f 1 1 133 cycles per second BeTween consecuTive samples The wheel will Turn Through fA l39 cycles check uniTs fAT 23rds cycle beTween samples The oquuT period is 15 seconds and The oquuT frequency is 23rds cycle per second 4 r 19 Va Our esTimaTe of E in This case is Twice iTs acTual value When I 23 seconds The frequency of roTaTion f15 cyclessec Thus The wheel Turns Throu9h fA l39 or 34 cycles beTween samples The wheel appears To move counTerclockwise wiTh frequency equal To 12 cycle per second When I 12 second The frequency of roTaTion f2 cyclessec f AT 1 Thus The wheel Turns Through one cycle beTween samples The wheel appears fixed The oquuT frequency is O A plot of input versus oufpuf sampled frequency reveals saw Too fh like variations of output frequency as input frequency increases The output frequency never increases beyond 1 cycle per second in The case where motion is sampled every 12 second The Nyquis139 frequency is The highest observable frequency at a given sample rate i lN 94 AN 1 ri NL Lmriquot Cc cul lie M91ch Higher inpu r frequencies appear To have lower frequency in The output recording This phenomena is referred To as frequency or signal aliasing v a xZE iEEE i lt The range of frequencies present in the wavelet controls its ability to resolve the top and bottom of a layer of given thickness Recall our general introduction to the concept of the wavelet earlier in the semester The wavelet or transient mechanical disturbance generated by the source can be thought of as a superposition or summation of sinusoids with varying frequency and amplitude sink SYNTHESIS a xZEEii i i lt Hiferman 19295 The examples below illusTraTe The effecT of increasing The frequency range or bandwidTh of The waveleT W 39 mmquot l Mom3 r m V mu Frequency my 0 Imaz 1957 The following simple example helps illusTraTe The concepT of an ampliTude specTrum Below is a signal consisTing of Two sinusoids mrur DATA 2 AND a SAMPLEIGVCLE camuan s a H4 mmrun o m an an 4a an an m w an um Hi 11 mo swnwasn 5 sum coumnsm u mmmue 1 Au 0 m an 3a a an no 7n IO n ma nu m an smmuuuaaa a sum canEm Ln Aw AMPLIYUDE I o w 2 a n so a 10 m an m m 120 10 sAIIDLEmIMm Each sinusoid is associated with a specific frequency There are two frequency components The 32 sample per cycle component has a frequency of 4 and the 8 samples per cycle component has a frequency of 16 The amplitude of the 32 samplecycle component is twice that of the 8 samplecycle component mmuos a 0 s m 395 smns MUMIER rnzuuzucv on wAvENuMBER AMPLITUDE swzcmuu or mm mm a s a 20 25 The frequency spectrum above of the signal at the top of the previous slide is an equivalent representation of the signal Hill IHRVELEY Time domain nPLlluDE Frequency domain Ilillllll H39IHHIH llll raauuzucy Im a Imaz 1957 Amplitude spectrum Phase spectrum Individual frequency components Ti medomain wavelets AE H1 1 55 Zero Phase Minimum Phase I 5mm 9 Extracting information about FULSEWIDTH AND BANDWIDTH NAVELEY wavelet frequency content from an isolated reflection event Flquot w s The dominant perIod TC of the response corresponds to quotl the time from one peak to the next or from one trough to the next The reciprocal quotmm mm of this dominant period is a measure of the dominant frequency f of the signal or wavelet spectrum IK 711 an The reciprocal of the halfwidth of the responseenvelop Tb provides an estimate of the bandwidth fb of the signal Hi Ireman I 9175 spectrum AVELET SPECTRUM The dominanT r39epr39eserITaTion of The signal waveIeT can be used To provide a skeTch of The waveIeT specTr39um AMPLITUDE a rnsbusum JusT as impor39TanTIy These measures can nommmr be r39eIaTed direcTIy To quotquotquotw39quot The rest uTion comm proper Ties of The FREGUEMCV seismic waveIeT Hfe ma 1985 LeT39s come back To This issue in a minuTe buT firsT eT39s pull some ideas TogeTher To develop a basic undersTanding of how The seismic signal arises in Terms of reflecTion coefficienTs and waveIeTs MW IDN moms our INTERFACE mun 10 LOW vuoan uawiu39luvum 2mm Rum run mummy VI an 39 L IIEI39I INTERFACE ulgtLC v2lnaon Ec v24 7 IMoo xsnmi Exxan inhause caurse 770125 on mnnnm Luv1390 nmn vtwcrrv OWNTRAVELING cum numeral Mann 1 vlmooosc mm wrwm v vnIsmwsu XM mnwse a me In 125 arJas uu in grraranse numericquot mucus TWO luTEnFAuu INTERFERENCE v mun SEC composrn sum 11m mmmca n ummu 1 vqsnrmssc snn39r urn nee 1 nanxmow z Vquml 15m 50m mnwse a me In 125 menuquot rnockss TWquot INTERFACE wnn mmxrenmcm d amrosl rz vmua su mmms quotquot2 vso 039sxc inquot x 1 VnnnD SEC T nr r 2 STAIH As The Two layers move closer and closer TogeTher we geT To a poinT where The second cycle in The waveleT reflecTed from The Top of The layer overlaps wiTh The arrival of The ead cycle in The waveleT reflecTed from The base of The layer This occurs aT Twoway Time equal To 12 The dominanT period of The waveleT won who DUSE uff nmmmu Pnackss TWKI mummy wn n mmenmz Iowann S39I AK I 39 aw5x6 mmmn Tmcz mama I v5mu s c i m n39z vmmm Ec TEquot m 6 AT This poinT There is maxmmquot mns 39uc ve I39IIIeI39feI39eIIcebeTween The reflecTians from The T0 and baTTam of The layer The campasiTe reflecTian evenT aT righT above aTTains peak ampliTLIde Ewanmimosa CDUFSE DfES The peak period of The waveleT can be deTermined using peakToTrough Times which can be ThoughT of as corresponding To one half The dominanT period of The waveleT MulTiply Those Times by Two To geT The dominanT period COMP 0 l lf millill lwx 3mm glam Maximum consTrucTive inTerference illusTraTed for The zero phase waveleT The peakToTrough Time equals 152 21 l WAVELEI39S CDMPDSITE FIG 5A CONSTRUCHVE INTERFERENCE I R WAVELEIS BOMPIISITE I 1 2 1 FIG 68 BESTHUC HVE INTERFEHHICE Exxon in 41min aursz nail Once The separaTion in Time drops To less Than half The dominanT period of The waveleT desTrLIcTive inTerference in The reflecTions from The Top and boTTom of The layer will occur However as The layer conTinLIes To Thin The dominanT period of The composiTe reflecTion evenT does noT drop below 116 However The ampliTLIde of The composiTe conTinLIes To drop BLIT noT The period Maximum C t These ampliTude relationships are summarized below in The model seismic response of a Thinning layer similar To ThaT which you will generaTe in lab Today The ampliTude difference TroughTopeak remains consTanT for Twoway Travel Times much greaTer Than half The dominanT period Thickness vr2 As The Top and boTTom of The layers merge closer and closer TogeTher The lead cycle in The reflecTion from The base of The layer overlaps wiTh The followcycle in The I reflecTion from The Top r l and The amplITude of The gallmma mummi i i composiTe reflecTion y evenT begins To increase i lllll Illlll n W 23 inTerval TransiT To one half The velociTy of The mm urmvnl Layer Thickness is simply VTZ where T is The Two way Time Tuning occurs aT Two way Times equal dominanT period Tc2 If The inTerval layer in quesTion is known The dominanT period can be converTed inTo The Tuning Thickness PEFLEC MJM AMPUTU E Di mg was rm w mun umgw Difference of arrival Time beTween The reflecTions from The Top and boTTom of The layer decreases abruple aT abouT 8 milliseconds T e Two way Travel Time Through The layer iT is also The Time aT which Tuning occurs and is half The dominanT period of The seismic waveleT The layer Thus 154 is The layer 8 milliseconds represenTs 8 milliseconds is ECZ and The TWO way Time Through ismcmspnnsmmum mammnglaymngmsi APPARENT quotHicKNESS E 5 E 3 smu wounds 17 The one way Time Through APPARENT THicKNESS mum msw M urmslcms smu morass um 154 The oneway Time Through The layer equals 4 milliseconds The inTerval velociTy in The layer is 11300 fs Hence The Thickness of The layer aT This poinT is 45 feeT This is The Tuning Thickness or minimum resolvable Thickness of The layer obTainable wiTh The given seismic waveleT WhaT is The ampliTude specTrum of waveleT 5 P f n 5 T2 1 u an Famiva Inmz 1957 25 SpecTral bandwidTh waveleT duraTion in The Time domain and resoluTion EC is only one parameTer ThaT affecTs resoluTion 1 is also an imporTanT parameTer BANDWIDTN EFFEcrs Hiferman 1955 1 Minimum lJnASc Wavelelv 0me The minimum phase waveleT has iTs energy concenTraTed Toward The from end of The waveleT The ampliTude of The disTurbance decays exponenTially This waveleT is a causal waveleT and The ocaTion of The reflecTion coefficienT is placed aT The waveleT onseT which can be difficulT for The inTerpreTer To pick 26 The zero phase waveleT is symmeTrical This waveleT is cenTered over The reflecTion coefficienT The zero phase waveleT is produced Through daTa processing and is noT generaTed naTurally IT is non causal half of The waveleT arrives before The reflecTor appears in Time IT is easy for an inTerpreTer To pick reflecTion Times using The zero phase waveleT since highesT ampliTude occurs aT The reflecTion boundary THlCKNESS Mooas MODEL wwwwu we mm The zerophase waveleT is also considered To have higher resolving nicks power IT is generally more compacT Than The equivalenT minimum waveleT and is overall easier To inTerpreT mmwm PHASE The exploraTion dam is in a zero phase formaT 27 The default wavelet in Struct is the Ricker wavelet The Ricker wavelet is zero phase AMPLITUDE TIME SEPARATION ww39vuus m m ms nu Hi Ireman I955 COMPUTATIONAL PHASE GEOLOGIG ACOUSTIC IMPEDANCE LOG f Vz rm I z gt g T l REFLECTION SEISMIC COEFFICIENTS TRA E FWD 1139 50 7 T gt gt g 1 WAVELET 1 E w 4 I Hibernian 1985 Physical nature of the seismic response 28 5 V1 1 g m 739 a I mm a a a son as m Ion no unm mmwwu umvmul quot39 mmnmu mmm mu Exxonin case The output IS a superposmon coursenofes reflections from all acoustic interfaces 5Ezr Nm Exxon in Jim2 course nofes 29 One addi rional Topic To consider in general is wavele r deconvolu rion and how wavele r shape can affec r geologic in rerpre ra rions Consider The following s rruc rur al model Subsurface s rruc rure Nor rh Sea 520 5000 IBIOD lD0 0 METERS nu um um alumna mL mm39n If 1 MA v we mum 75 IcILW M szl39 quotnumm m ncau u m a m um39ll gainIE mun mum ume quotIn Any Wu mo mum 5444 Wow 39M mm 39 quot 139 Lu cw39m mal r l m mum 39 w mama mum no w 39my mu 7 Neidel 1991 Below is the synthetic seismic response computed for the North Sea model Potential hydrocarbon trap PPPPPP UT A hWMWVWlWMle39 if i ll lllllllllllllllllllllllllllllllllll 391 1 2 will ll IllSlimmm m f w l g mmMmml 111 siiimllmlil i lililllllliiiw llllllllliljl illl illllllllll llllllllllllllllglggililllllii minim m llllllllllll illlllllllllllillgl M3 ll lmmmwl Neidel 1991 l ill ill i Consider the effect of wavelet shape on the geologic interpretation of seismic response In the case shown below the primary reflection from the base of the Jurassic shale crosses a sidelobe in the wavelet reflected from the overlying basal Cretaceous interval PPPPPP UT llliliil ili quotlquot Illll iquot lllllmllilllllillllllllllilll l lllll v ill Jillilli gaagiggam 339 tillllllilllilii ill iiiiiiililllllillllligliill liilillmim illll illljllilll l militiailllllllllllll lllllsllzzls Hillillllll 1mmimmillllllilllllgill mill Il lyl yy i l sams quotl l y W illllllll in WHWHWMMMWJK Neidel 1991 Deconvolution is a filter operation which compresses and simplifies the shape of the seismic wavelet Deconvolution improves seismic resolution and simplifies interpretation 39 III M iiiquot ll Ivl lli i 39 lilao i i Neidel 1991 Will l39 l39 u 5 53 5 5331me mm willilllllllllliili m it ill kill l llllllllmlllllllllllll illll illlill mlii ml 39 lquot i iiiil l llllllll quot sq lll39lg llllllll quoti l quot39 in 1 North Sea Seismic display after deconvolution The geometrical interrelationships between I I lllllll K 39 l 1 iii all m l l 1 l lfdrllili quot I Ill H A BASIC WAVELET ll reflectors are clearly portrayed Consider the following problem You are given the seismic wavelet shown below Time ms Using the estimation procedure discussed in class today measure the appropriate feature on the above seismic wavelet and answer the following questions What is the minimum resolvable thickness of a layer having an interval velocity of 10000fps Show work on your handout What is the phase of the wavelet Why do you say that If you haven39t already finish reading chapter 4