Hyperbolic Velocity Model

Transcription

1 Interntionl Journl of Geosienes doi:046/ijg Pulished Online June 0 ( Hyperoli eloity odel Igor Rvve Zvi Koren Prdig Geophysil Herzliy Isrel Eil: igorrvve@pdgo zvioren@pgdo Reeived rh 7 0; revised April 9 0; epted y 6 0 Copyright 0 Igor Rvve Zvi Koren This is n open ess rtile distriuted under the Cretive Coons Attriution Liense whih perits unrestrited use distriution nd reprodution in ny ediu provided the originl wor is properly ited ABSTRACT Asyptotilly ounded veloity profiles desrie the vertil veloity vritions in opted sedients in ore relisti wy thn unounded veloity odels nd llow presenting the susurfe y sller nuer of thier lyers The first nd the siplest syptotilly ounded odel is the Hyperoli veloity profile proposed y ust in 97 nd our pper is n extension of this erly study The Hyperoli odel hs n dvntge over other ounded odels: The veloity inreses with depth nd pprohes the liiting vlue with ore sooth nd grdul rte We derive the tie-depth reltionships forwrd nd wrd trnsfors etween the instntneous veloity profile nd the effetive odels (verge RS nd fourth order verge veloities) study the trjetories for pre-ritil nd post-ritil urved rys nd derive the equtions for trveltie lterl propgtion nd r length We opre the ry pths otined with the Hyperoli odel nd with the other ounded veloity profiles Keywords: eloity odels; eloity Trnsfors; Sedients Introdution The Hyperoli veloity odel ws first proposed y ust [] nd pulished in 97 However sine then the odel ws not extensively studied nd is unjustifily ignored in the literture The ojetive of this reserh is to extend the originl study nd to orret the inuries We show the ple of the Hyperoli odel ong the other syptotilly ounded odels nlyze its si reltionships nd ttept to develop oplete theory Asyptotilly ounded veloity odels desrie the veloity profile in opted sedients where the veloity grdully inreses with depth nd eventully pprohes liiting vlue These odels e it possile to desrie vertil veloity profile with sller nuer of intervls s opred to the lssil unounded odels suh s liner veloity vs depth [] unounded exponent [4] liner slowness [5] sloth (liner vrition of slowness squred) eg [6] proli odel [78] Fust veloity odel [90] with referene depth nd different root indies The unounded odels re desried y two preters: the instntneous veloity t the top interfe nd the vertil veloity grdient t the se level The Fust odel inludes lso the root index n norlly n 6 Asyptotilly ounded odels require n dditionl preter: the liiting vlue of veloity t infinite depth Two odels of this fily were studied y Rvve nd Koren: the Exponentil syptotilly ounded odel [] nd the Coni odel [] The syptotilly ounded profiles n e used in prtiulr s veloity trend funtions for the onstrined veloity inversion with the est (eg lest-squres) fit of the input dt [4] Exples of syptotilly ounded odels re presented elow For eh odel we first give the originl forultion of the veloity profile s it ppers in the originl wors y the uthors nd then we onvert it to nonil for in ters of the stndrd preters nd Preter ens the instntneous veloity rnge The Hyperoli veloity odel y ust [] z z A A z onst () In our nottion the Hyperoli profile reds z z The Exponentil veloity odel y ust [] () z z ln ln B onst B z We onvert it to our nottion () Copyright 0 SiRes

2 I RAE Z KOREN 75 where the preters re z tnh Az z o (4) Arosh zo The Exponentil slowness odel [55] z exp z zo It n e onverted to noni for o (5) (6) exp z zo (7) z z The Exponentil syptotilly ounded (EAB) veloity odel [] z z The Coni veloity odel [] z Q z h Q zh where exp Q h (8) (9) (0) A detiled review on unounded nd ounded veloity odels is given y Kufn [6] Figure shows grphs of the instntneous veloity vs depth for the five syptotilly ounded veloity odels entioned ove For ll odels we ssue the se veloity profile preters: /s s 6/s The vertil grdients of the veloity vs depth re plotted in Figure It is interesting to note tht ong the five odels presented the ust Hyperoli odel (Eqution () nd grey line on the plot) pprohes the liiting vlue in the slowest nd the ost grdul nner The seond slow is the Coni veloity odel (red line) nd the third slow is the EAB odel (lue line) An syptotilly ounded odel n e hrterized y its grdient-veloity reltionship whih is tully the governing differentil eqution of the veloity odel This pper is strutured s follows We define the Hyperoli odel using ) the originl ust [] forultion depth vs veloity ) the physil preters: xiu grdient R length sle Q nd vertil shift h nd ) the tehnil or geophysil preters: top interfe veloity top grdient nd syptoti veloity We introdue the diensionless syptoti ftor tht siplifies the trnsfor equtions First Depth () 0 Asyptotilly Bounded eloity odels 6 Coni eloity odel EAB eloity odel 9 Exponentil Slowness Exponentil ust Hyperoli ust eloity (/s) Figure Asyptotilly ounded veloity odels: ust hyperoli odel ust exponentil odel the Exponentil slowness odel the Exponentil syptotilly ounded odel nd the Coni odel: eloity vs depth For ll odels the profile preters re: the top interfe veloity = /s the top grdient = s nd the syptoti veloity = 6 /s Depth () Grdient vs Depth Coni odel 6 EAB odel Exp Slowness 7 Exp ust 8 Hyp ust ertil Grdient (/s) Figure ertil grdients vs depth for syptotilly ounded veloity odels we derive the tie-depth nd the depth-tie reltionships Next we proeed to forwrd trnsfors fro the instntneous veloities to the effetive odels suh s the verge the RS nd the fourth order verge veloity Then we study the inversion proles onsidering the inversion with the instntneous veloities nd grdients nd the inversion with the effetive odels ie the verge or the RS veloities given vs tie or depth Next we oent on the two types of urved rys existing in ll syptotilly ounded odels depending on the initil te-off ngle nd derive the trjetories of the ry pths for the ust veloity profile For oth types of the urved rys we derive the lterl propgtion the trveltie nd the r length The Hyperoli eloity Profile ust [] defined the Hyperoli odel y z z A z 6 () where is the instntneous veloity z is depth esured Copyright 0 SiRes

3 76 I RAE Z KOREN fro the top interfe is the top interfe veloity A is the hrteristi distne (sle) tht ffets the top grdient nd is the syptoti veloity Inverting Eqution () we otin z A z () za The veloity grdient eoes d z dz A za () where At z 0 the top grdient is Therefore A nd A (4) Introdue Eqution (4) into Eqution () In our nottion the Hyperoli profile reds z z (5) z z We ll vlues nd the tehnil preters of the profile At definite height ove the erth surfe (ove the upper interfe) where z h the instntneous veloity vnishes Aording to Eqution (5) h (6) Introdue the solute fre z zh where the instntneous veloity vnishes t the origin z 0 Preter h is the shift etween the two fres of referene In the solute fre the veloity profile siplifies to Rz Rhz z (7) Qz Q h z where Q nd R Q (8) We ll vlues RQ nd h the physil preters of the profile Note tht the liner veloity profile where the ry trjetories re irulr rs is prtiulr se of the Hyperoli odel with Q 0 nd in suh wy tht their produt R Q reins finite vlue nd preter R eoes the onstnt veloity grdient of the liner odel The veloity grdient of the Hyperoli odel reds R z (9) Qz At the solute origin z 0 the veloity grdient rehes its xiu vlue x R Copring Equtions (7) nd (9) we onlude tht z z (0) Eqution (0) is the governing differentil eqution of the Hyperoli veloity profile It n e used to plot the grdient-veloity digr Suh digrs for severl syptotilly ounded veloity odels re studied in Appendix A Introdue the norlized (diensionless) veloity v the norlized grdient ˆ nd the norlized solute depth ẑ v ˆ R zˆ Qz () Note tht preter Q is the reiprol hrteristi length With these nottions the Hyperoli veloity profile siplifies to z v ˆ ˆ () zˆ zˆ x The tehnil preters of the veloity profile re relted to the physil preters Rh R R () Qh Qh Q The inverse reltionship is R R h Q (4) Asyptoti Ftor To siplify the equtions for veloity trnsfors it is suitle to introdue speil preter This preter n e defined t ny point of the profile nd in prtiulr t the top nd the otto interfes of n intervl z Qhz Qh (5) Q hz where z is the intervl thiness (the vertil distne etween the two interfes) susript is relted to the top interfe z 0 nd susript is relted to the otto interfe z z It follows fro Eqution (5) tht (6) where nd re the top nd otto instntneous veloities respetively Next it follows fro Eqution (6) tht preter is the inverse norlized esure of the differene etween the veloity t the given depth level nd the syptoti veloity Eqution (6) n e inverted Copyright 0 SiRes

4 I RAE Z KOREN 77 (7) The veloity grdient is lso relted to the syptoti ftor nd R R (8) It follows fro Eqution (5) tht Q z z Qh Qhz (9) (0) We use Equtions (8) nd (9) to get the interfe grdients nd through the inreent of the syptoti ftor z z It follows fro Eqution () z z () () Equtions (7) nd (9) result in the verge grdient on the intervl ve expressed either through the interfe syptoti ftors nd or through the interfe grdients nd ve z z z Q R () Introdution of Eqution (8) into Eqution () leds to ve (4) The verge grdient on the intervl with the Hyperoli veloity profile is the geoetri verge of the top nd otto interfe grdients Given the veloity nd its grdient t one interfe one n lulte these preters t the other interfe The lultions n e done either in depth or in tie Four proles of this ind re onsidered in Appendix C 4 Depth-Trveltie Reltionship Integrte the slowness to get the vertil trveltie vs the intervl thiness z hz hz dz dz Qz t dz 0 z h z h Qz z hz ln Q h (5) The trveltie eqution n e written in ters of syptoti ftors t the top nd otto interfes nd With the use of Equtions (5) nd (9) we otin t z ln (6) where the top syptoti ftor is lulted with Eqution (6) nd the otto syptoti ftor z- with the first eqution of Eqution Set () The intervl veloity (lol verge veloity) through the lyer etween the interfes eoes Int z (7) t ln To get the vertil distne vs trveltie we should invert Eqution (6) ie find t Introdution of Eqution (9) into (6) results in Qt ln (8) Eqution (8) should e solved for the unnown otto syptoti ftor ln ln Rt (9) Ting exponent fro oth sides of Eqution (9) we get exp Rt exp exp (40) Eqution (40) n e solved with the Lert funtion L0 exp exp Rt (4) where nottion L0 ens the zero rnh of the Lert funtion The Lert funtion y Lx delivers the solution of the trnsendent eqution x yexp y see Appendix B for detils In ters of the interfe veloities Eqution (4) redues to L0 exp exp Rt (4) After the otto syptoti ftor or the otto interfe veloity is found the intervl thiness n e estlished with Eqution (9) Copyright 0 SiRes

5 78 I RAE Z KOREN z Q Q (4) 5 Hyperoli nd Non-Hyperoli oveout In the sene of the intrinsi nelliptiity the hyperoli preter W nd the non-hyperoli preter H on the intervl re defined s t z W dt dz t z t z 4 H dt dz t z (44) Introdue the veloity profile fro Eqution (7) The hyperoli preter W eoes z z h z Rz dz W dz z z h Qz Qhz z ln Q Qh The non-hyperoli preter H eoes z z hz Rz dz d z z h Qz z H z Q Qh Q Qhz Q Qh Q Qhz Qhz ln Q Qh (45) (46) With the use of the top nd otto syptoti ftors the hyperoli preter eoes W ln Q z ln (47) Introduing Eqution (7) for the trveltie into Eqution (47) we otin the lol RS veloity U over the intervl By definition U W t so U ln ln The non-hyperoli preter eoes (48) H Q ln Q (49) With the use of Eqution (9) the non-hyperoli preter siplifies to H z (50) ln When the preters of the veloity profile re speified the top syptoti ftor is nown vlue The otto syptoti ftor n e presented either vs depth (intervl thiness) or vs trveltie Thus the hyperoli nd non-hyperoli preters eoe funtions of depth or trveltie ordingly The nelliptiity indued y the vertilly vrying veloity is defined s the frtionl differene etween the fourth-order verge veloity nd the RS veloity (5) Preter n e lso onsidered s funtion of depth or vertil tie For prtiulr se of single infinite lyer (hlf-spe) with ny vertil veloity profile W H H t W t t 8W 4 4 (5) The grph for the indued nelliptiity is plotted vs depth in Figure for three syptotilly ounded veloity odels: Exponentil Coni nd Hyperoli For ll the three odels the preters of the veloity pro- file re: /s s nd 6/s At the surfe the nelliptiity is zero s there re yet no uulted vritions of the instntneous veloity The indued nelliptiity is lwys positive It rehes xiu vlue definite depth nd then vnishes t the Depth () 0 6 Indued Anelliptiity of eloity odels 9 5 Liner EAB Coni Hyperoli Anelliptiity Figure Indued nelliptiity vs depth for syptotilly ounded veloity odels Copyright 0 SiRes

6 I RAE Z KOREN 79 infinity where the ediu veloity is syptotilly onstnt 6 Forwrd Dix Trnsfor Consider pge of n lyers (vertil intervls) where the nodes (interfes) re enuerted fro zero nd lyers re enuerted fro Intervl n onnets nodes n (top interfe) nd n (otto interfe) The nodl verge veloity n RS veloity n nd fourthorder verge veloity re n n 4 4 n 4n t z t t t W t t t H t t n n n n n n n n n n 4 4 n n n n n (5) where t is the one-wy intervl trveltie n zn is the lyer thiness W n nd H n re the intervl hyperoli nd non-hyperoli preters respetively For n we set t 0 0 in Eqution (5) The effetive veloities (verge RS nd fourth-order verge) n e lso defined for ny internl point of the intervl 7 Inverse Dix Trnsfor Rell tht the Hyperoli veloity profile on the intervl is defined y the three preters: the top interfe instntneous veloity the top interfe grdient nd the syptoti veloity We onsider tht the syptoti veloity is lwys given priori When the two other preters nd re lso nown then veloity trnsfors re onsidered forwrd When one or oth preters re unnown (with nother dt speified insted) we del with the veloity inversion There re three groups of inverse trnsfors studied in Appendies D E nd F Appendix D onsiders the inversion tht does not involve the RS veloity These forultions del with the instntneous veloity nd its grdient only We solve prole where the two veloities re given t the interfes nd or lterntively the two grdients nd Another ind of prole is when the veloity nd its grdient re given t the different interfes of the intervl ie the veloity is given t the top interfe nd the grdient t the otto interfe nd or vie vers nd We solve lso prole where the instntneous veloity is given t the otto interfe nd t the interedite point of the intervl nd These proles re studied oth vs depth nd vs tie Appendix E onsiders the inversion with the RS ve- loity speified t the interfes vs depth or tie with single preter unnown either or We onsider lso prole with the trveltie speified vs the intervl thiness lso with single preter unnown Finlly we onsider the RS veloity speified vs oth depth nd tie with the two preters unnown nd In Appendix F we study the two-intervl inversion The RS veloity is given vs depth or tie t the two interfes nd t n internl point of the intervl Alterntively depth n e speified vs trveltie t the three points Both preters of the veloity profile re unnown This is so-lled three-point or two-intervl inversion 8 Ry Trjetories In this setion we estlish the trjetories of non-vertil rys Due to Snell s lw in D ediu the horizontl slowness p is onstnt nd the ry ngle (esured fro the vertil xis) eoes sin p z (54) Introdue the ry preter Q (55) pr p P where Q nd R re the physil preters of the Hyperoli veloity profile P p is the norlized ry slowness nd is its inverse vlue We ll preter eentriity of the ry trjetory s it is very siilr to the eentriity of the hyperoli nd ellipti rys of the Coni veloity odel [] With Eqution (7) the sine of the ry ngle eoes prz Qz sin (56) Qz Qz so tht the tngent of this ngle is tn sin sin Qz Qz Q z (57) where Preter (the onjugte eentriity squred) y e positive or negtive Introdue the diensionless oordintes xˆ Qx zˆ Qz (58) The tngent of the ry ngle eoes ˆ d ˆ tn z x ˆ ˆ d dx dˆ z z z z Integrting Eqution (59) we otin (59) Copyright 0 SiRes

7 70 I RAE Z KOREN zz ˆdˆ xˆ xˆ ˆ z zˆ (60) where x is the onstnt of integrtion This integrl n e redued to zz ˆdˆ ˆ z zˆ ˆ z zˆ dˆ z ˆ z zˆ (6) To otin the integrl on the right side of Eqution (6) we onsider two ses or two rnges of the eentriity: > (pre-ritil rys) nd < (post-ritil rys) z dˆ z ˆ z zˆ ˆ rosh for dˆ z ˆ z zˆ zˆ ros for For liiting se (ritil rys) (6) zz ˆdˆ zˆ zˆ xˆ xˆ (6) ˆ z We ephsize tht two inds of rys exist for ny onotonously inresing nd syptotilly ounded veloity odel nd in prtiulr for the Hyperoli odel The pre-ritil rys tht y strt on the erth surfe propgte to the infinite depth nd their urvture syptotilly vnishes The post-ritil rys hve liited propgtion depth Their r-lie trjetories hve finite iniu urvture t the turning point nd these rys return to the erth surfe Note tht t ny point of the trjetory the ry pth urvture depends on the veloity grdient only z p z (64) In prtiulr the liner veloity odel with onstnt veloity grdient leds to ry trjetories of onstnt urvtures ie to the irulr rs The ritil rys with the unit eentriity re the liit se etween the two types of rys Their teoff ngle (the ry ngle t the upper interfe) is lled the ritil ngle C It follows fro Equtions (54) nd (55) tht the ritil te-off ngle is C rsin (65) It follows fro Equtions (6) nd (6) tht the trjetories of the pre-ritil nd post-ritil rys re for xˆxˆ zˆ zˆ zˆ rosh for xˆ xˆ zˆ zˆ zˆ ros (66) At infinite depth pre-ritil rys eoe syptotilly stright Eqution (57) leds to tn sin p (67) However lthough the slope of these rys onverges to onstnt vlue nd their urvture eoes infinitesil the pre-ritil rys of the Hyperoli odel hve no syptoti stright line unlie the pre-ritil rys of the EAB nd the Coni odels The pre-ritil ritil nd post-ritil rys re plotted in Figure 4 for the Hyperoli the Coni nd the Exponentil (EAB) odels Preters of the veloity profile re the se Depth () Ry Trjetories Hyp Con EAB /s Distne () Figure 4 Pre-ritil (red lines) ritil (green lines) nd post-ritil (lue lines) ry trjetories for Hyperoli (lines 4 7) Coni (lines 5 8) nd EAB (lines 6 9) veloity odels Pre-ritil rys propgte to n infinite depth nd eoe syptotilly stright Critil ry propgte to n infinite depth nd the ry ngle pprohes π/ t lrge depth ut this ry hs no syptote Postritil rys pss the turning point nd return to the erth surfe their trjetories re syetri rs The Coni rys pss ove the Hyperoli odel rys euse their urvture is lrger For the se reson the EAB rys pss ove the Coni rys For ll odels preters of the ediu re the se s in Figure The te-off ngle of the pre-ritil rys is π/4 tht of the ritil rys π/6 nd tht of the post-ritil rys 5π/4 Copyright 0 SiRes

8 I RAE Z KOREN 7 s ove The three oluns of nuers to the right of the plot re re veloities for the three odels t the speified depth levels 9 xiu Penetrtion Depth Pre-ritil rys penetrte to infinite depth The xiu penetrtion depth of post-ritil rys follows fro Eqution (59) At the turning point the ry ngle π nd thus its tngent eoes infinite This leds to qudrti eqution with single positive root zˆ x (68) Rell tht ẑ is the diensionless depth esured fro the solute origin (ove the upper interfe) The xiu penetrtion depth in units of length esured fro the upper interfe reds z x p p sin sin sin sin C 0 Lterl Propgtion Trveltie nd Ar Length C (69) In D ediu it is onvenient to express the lterl propgtion distne x trveltie t S nd r length s through the ry ngle nd ngle-dependent grdient These reltionships re [6] (Kufn 95; Rvve nd Koren 006) sind x p t S d sin d s p (70) where nd re ry ngles t the deprture nd the destintion points respetively Equtions (7) nd (9) e it possile to eliinte depth nd to express the grdient through the veloity R (7) Next we pply Snell s lw nd otin the vertil grdient vs the ry ngle Note tht sin R (7) dx Q dx sin pr d d sin dts R d sin sin sin sin sin d sin sin sin (7) d ln tn sin (74) (75) d os sin sin The indefinite integrl on the right side of this eqution essentilly depends on the rnge of the eentriity resulting in pre d I sin tn roth for pre-ritil pst d I sin tn rot for post-ritil (76) For the ritil ry d sin sin sin d os sin sin sin (77) Let nd e the ry ngles t the strt point nd the destintion point of the ry pth respetively Eqution Set (76) n e re-rrnged s follows For the pre-ritil rys I pre tn roth (78) os sin roth sin For the post-ritil rys Copyright 0 SiRes

9 7 I RAE Z KOREN pst tn I rot (79) os sin rot sin pre pst I nd I re funtions of the ry ngles t the endpoints of the pth The following identities were used AB rot Brot Arot A B AB roth Broth Aroth A B (80) To siplify the nottions we introdue one ore funtion of the ry ngles t the endpoints J os sin os sin sin sin sin The norlized lterl propgtion eoes Qx I J The norlized trveltie is Rt S tn ln tn (8) (8) I J The norlized r length is Qs I C J (8) (84) For the ritil rys nd the deprture ngle is ritil The ry pth preters re os Qx sin sin sin Rt S ln tn sin C sin os sin os sin Qs C C C (85) The urrent depth n e lso expressed through the ry ngle It follows fro Eqution (7) tht Rell tht nd therefore z z z R Q (86) p sin (87) p sin sin Qz z sin sin sin sin sin sin sin os In Figure 5 we plot the grphs for the trveltie vs ry pth r length for the pre-ritil the ritil nd the post-ritil rys of the Hyperoli veloity profile The trigonoetri solution for the lterl propgtion nd trveltie of the post-ritil rys ws otined (in different for) y ust (97) However it ws not pointed out in this erly study tht the solution ws relted to the post-ritil rys only nd tht the other hyperoli solution exists for the pre-ritil rys (nd trnsient solution for the ritil rys whih re the liit se etween the two si types of rys) Note tht for the vnishing or infinitesil preter Q 0 the shpe of the trjetory the lterl propgtion the trveltie nd the r length of the Hyperoli odel ry pth onverge to the orresponding hrter- Tie (s) 4 0 Ry Trveltie vs Ar Length Ry r length () Pre-ritil ry Critil ry Post-ritil ry 0 (88) Figure 5 Trveltie vs r length of ry pth for the three inds of rys of the Hyperoli veloity odel: the preritil ry α = 5 the ritil ry α = α C = 0 nd the post-ritil ry α = 75 Copyright 0 SiRes

10 I RAE Z KOREN 7 istis of the liner veloity profile In this se the syptoti veloity eoes unounded so tht the produt R Q reins finite vlue nd onverges to onstnt grdient of the liner veloity odel The eentriity eoes infinitesil nd Q pr li li pr (89) 0 Funtions I nd J fro Equtions (79) nd (8) siplify to I J os os (90) Eqution (66) oes to xx p R z p R (9) C This is n eqution for the irulr r of rdius pr whose enter is loted t x x z 0 where p is the ry slowness The lterl propgtion Eqution (8) eoes prx os os (9) Eqution (8) yields the trveltie for this liiting se tn RtS ln (9) tn nd finlly Eqution (84) for the r length onverges to prs (94) Full Ar of Post-Critil Ry Consider two points on the erth surfe the trnsitter nd the reeiver loted x distne prt The gol is to tre the full r of the post ritil turning ry tht onnets the two points Note tht due to the syetry of the r the ry ngle t the destintion point is relted to the te-off ngle π (95) Applying Equtions (79) (8) nd (8) we otin sin os Qx rot os sin (96) where is the onjugte eentriity of the post-ritil ry pth Rell tht sin sin C (97) Eqution (96) siplifies to sin C Qx rot sinc sin sin C C (98) Eqution (98) should e solved nuerilly for the unnown eentriity To otin the initil guess we ssue tht the distne x is sll Then the te-off ngle pprohes π nd ording to Eqution (97) the eentriity exeeds the sine of the ritil ngle only slightly We ssue sin (99) where is sll positive vlue Next we expnd Eqution (98) into the Tylor series nd neglet the high order ters sinc sinc sinc 6 sinc sin C (00) 5 Qx O The ui Eqution (00) hs single positive root For exple for the veloity profile /s s nd 6/s nd the offset x 0 the ritil ngle eoes C π 6 Eqution (00) leds to 0 nd Eqution (99) yields the initil guess 07 Solving Eqution (98) with the Newton ethod we otin the eentriity The te-off ngle eoes The rs re plotted in Figure 6 for the three syptotilly ounded veloity odels In the shllow region the Hyperoli odel hs sller vertil grdient (nd thus sller urvture) thn the Coni nd the EAB odels nd thus the Hyperoli odel yields sller te-off ngle The ry pth r of the Hyperoli odel psses ove the Coni nd the EAB rs Boundry lue Ry Tring Given dt re the deprture point x z nd the rrivl point x z nd the gol is to tre the ry pth The ry pth is n expliit funtion of the eentriity nd this preter is so fr unnown Without ny loss of generlity we ssue here tht x x ie tht the lterl distne x x x nd the horizontl ry slowness Depth () Distne () C Full Ars of Post-Critil Rys EAB Coni Hyperoli Figure 6 Full rs of post-ritil rys for the EAB the Coni nd the Hyperoli veloity profiles 0 Copyright 0 SiRes

11 74 I RAE Z KOREN p re positive Assue lso z z whih is lso not liittion (one n reverse the endpoints otherwise) Sine the ry tring equtions depend on the type of ry we need to deterine whether the ry is pre-ritil or post-ritil For this one n plot ritil pth tht strts t the deprture point t the ritil te-off ngle C rsin If the destintion point lys to the left fro the ritil trjetory then the ry pth is pre-ritil The ry pth is post-ritil if the destintion point lys to the right The ritil lterl propgtion x C is delivered y Eqution (6) whih n e rerrnged s QxC Q z h Q z h (0) Q z h Q z h Given the vertil oordintes of the soure nd the reeiver z nd z we lulte the ritil lterl propgtion nd then pply the riterion xxc pre-ritil ry xxc ritil ry (0) xx post-ritil ry C The veloities t the end points of the trjetory z nd z re nown vlues It follows fro Snell s lw tht the ry ngles t the end points of the trjetory re the funtions of the eentriity lone sin sin (0) Note tht for the pre-ritil rys nd for the post ritil rys efore the turning point the ry ngle is ute while for the turning rys fter the turning point the ry ngle is otuse z z rsin efore turning point (04) z zπ rsin fter turning point Eqution (8) reltes the lterl propgtion x to the ry ngles t the endpoints whih in turn depend on the eentriity ording to Equtions (0) nd (04) where J Funtion J I Q x sin os sin (05) (06) I I (07) is delivered y Equtions (78) nd (79) It ws initilly defined s funtion of the endpoints ry ngles ut due to Equtions (0) nd (04) it n e onsidered s funtion of the eentriity lone Next we solve nonliner Eqution (05) nuerilly for the unnown eentriity Then the ry ngles t the endpoints n e estlished nd the ry pth n e plotted with Eqution (66) Nueril exples for the oundry vlue ry tring with the Hyperoli veloity profile re presented in Appendix G Conlusion The Hyperoli syptotilly ounded exponentil veloity odel hs een studied nd opred to other syptotilly ounded odels in prtiulr the Exponentil nd the Coni The forwrd nd the inverse veloity trnsfors re derived The Hyperoli odel llows etter representtion of the vertil veloity vritions in opted sedients espeilly in the se of thi lyers An dvntge of the Hyperoli odel is tht the instntneous veloity rehes the syptoti vlue in ore slow nd grdul fshion s opred to other syptotilly ounded odels Ry tring equtions hve een derived The ry trjetories trvelties nd r lengths hve een studied nlytilly nd the oundry vlue ry tring prole hve een solved We hve tried to present oplete theory for oth vertil nd non-vertil rys propgting through the Hyperoli odel Applition of the Hyperoli veloity distriution enles us to present relisti geologil odels using fewer preters s opred to the lssil liner veloity funtion We showed tht the liner veloity funtion is liiting prtiulr se of the Hyperoli odel 4 Anowledgeents We re grteful to Prdig Geophysil for the finnil nd tehnil support of this study nd for the perission to pulish its results REFERENCES [] ust A Note on Propgtion of Seisi Wves Geophysis ol No 4 97 pp 9-8 doi:090/48098 [] Slotni On Seisi Coputtions with Applitions Prt I Geophysis ol No 96 pp 9- doi:090/47084 [] Slotni Lessons in Seisi Coputing Soiety of Explortion Geophysiists Olho 959 [4] Slotni On Seisi Coputtions with Appli- Copyright 0 SiRes

12 I RAE Z KOREN 75 tions Prt II Geophysis ol No 96 pp doi:090/47 [5] Al-Chli Instntneous Slowness versus Depth Funtions Geophysis ol 6 No 997 pp 70-7 doi:090/4447 [6] C H Chpn nd H Keers Applition of the slov Seisogr ethod in Three Diensions Studi Geophysi et Geodeti ol 46 No 4 00 pp doi:00/a: [7] C E Houston Seisi Pths Assuing Proli Inrese of eloity with Depth Geophysis ol 4 No 4 99 pp -6 doi:090/ [8] Al-Chli Preter Non-Uniqueness in eloity versus Depth Funtions Geophysis ol 6 No 997 pp doi:090/4440 [9] L Y Fust Seisi eloity s Funtion of Depth nd Geologi Tie Geophysis ol 6 No 95 pp 9-06 doi:090/47658 [0] L Y Fust A eloity Funtion Inluding Lithologi rition Geophysis ol 8 No 95 pp 7-88 doi:090/47869 [] I Rvve nd Z Koren Exponentil Asyptotilly Bounded eloity odel Prt I: Effetive odels nd eloity Trnsfortions Geophysis ol 7 No 006 pp T5-T65 doi:090/960 [] I Rvve nd Z Koren Exponentil Asyptotilly Bounded eloity odel Prt II: Ry Tring Geophysis ol 7 No 006 pp T67-T85 doi:090/94897 [] I Rvve nd Z Koren Coni eloity odel Geophysis ol 7 No 007 pp U-U46 doi:090/7005 [4] Z Koren nd I Rvve Constrined Dix Inversion Geophysis ol 7 No pp R-R0 doi:090/4876 [5] E Roein eloities Tie-Iging nd Depth-Iging in Refletion Seisis: Priniples nd ethods EAGE Pulitions Houten the Netherlnds 00 [6] H Kufn eloity Funtions in Seisi Prospeting Geophysis ol 8 No 95 pp doi:090/4787 [7] R Corless G H Gonnet D G Hre D J Jeffrey nd D D Knuth On the Lert W Funtion Advnes in Coputtionl thetis ol 5 No 996 pp 9-59 doi:0007/bf04750 Copyright 0 SiRes

13 76 I RAE Z KOREN Appendix A Grdient-eloity Digrs In this ppendix we derive the grdient-veloity digrs for the five syptotilly ounded veloity odels In ll ses we pss to shifted fre z in whih the governing equtions re essentilly siplified The vlue of the vertil shift h is different for ll odels For ll odels the solute origin orresponds to point of xiu grdient For ll odels exept the Exponentil slowness this is lso the point of vnishing instntneous veloity As we show elow for the Exponentil slowness odel the solute origin orresponds to the hlf-liiting veloity z A The Hyperoli ust odel The veloity profile is given y z z (A-) Estlish the depth level where the veloity vnishes This point is loted ove the erth surfe z h h (A-) Introdue the shifted fre z zh The veloity profile eoes z z (A-) z The vertil grdient is z z At the solute origin (A-4) z 0 the grdient is xil (A-5) x Introdution of Eqution (A-5) into (A-) nd (A-4) leds to z x z z (A-6) z z x x x Finlly eliintion of depth z fro the two equtions of Eqution Set (A-6) results in x (A-7) A The Exponentil ust odel The veloity odel is desried y The vertil shift is z tnh A z z o (A-8) h Az o nd in the shifted fre z z h Eqution (A-8) siplifies to z tnh z z o (A-9) The veloity grdient is z z osh z z with x zo so tht z osh z z x o o o (A-0) (A-) Eliinte the solute depth fro Equtions (A-9) nd (A-) nd otin the grdient-veloity reltionship x A The Exponentil Slowness odel The profile eqution reds (A-) exp z z o (A-) z zo Unlie the other syptotilly ounded odels entioned in the introdution the veloity in the Exponentil slowness odel does not vnish t finite negtive depth The veloity vnishes t z nd pprohes to the syptoti vlue t z The grdient of the veloity is exp zz o z (A-4) z exp o z zo The grdient z vnishes t oth reote ends z nd z hs single ritil point: the xiu of the grdient ours t z z z zo ln (A-5) We ephsize tht the logrith in Eqution (A-5) y prove to e oth positive nd negtive At the point z z the xiu grdient nd the veloity re x (A-6) 4z o Note tht in se when the depth of the xiu grdient is positive z 0 this point relly exists underground nd the grdient first inreses then epts the xiu vlue x 4 (A-7) Below this point the grdient egins to dey nd eventully vnishes t the infinite depth In se when Copyright 0 SiRes

14 I RAE Z KOREN 77 the depth z of the xiu grdient is negtive this point is ove the erth surfe nd throughout the whole depth rnge 0 z the veloity grdient z is tully onotonously deresing funtion Note tht z 0 or (A-8) Next we ssue the shift hz nd pss to the shifted fre z zh The grdient epts now xiu vlue t the origin Rerrnge Eqution (A- ) exp z zo 4exp zz o x exp zz z z o (A-9) Note tht the veloity profile in Eqution (A-9) n e set in n lterntive wy z expz expzexpz (A-0) osh zsinh z tnh z osh z where so tht z (A-) o x z z z x z osh x tnh x (A-) Finlly we eliinte depth fro Eqution (A-) nd otin 4x (A-) A4 The Exponentil Asyptotilly Bounded eloity odel (EAB) In this se the veloity profile is exp z z (A-) The veloity vnishes ove the erth surfe t z h In the shifted fre h ln (A-4) z zh the veloity profile siplifies to z z z exp z exp At the shifted origin the grdient is xil x (A-5) (A-6) Introdue Eqution (A-6) into (A-5) z z x exp z z x x exp (A-7) Finlly we eliinte depth fro Eqution (A-7) nd otin the governing eqution (A-8) x Note tht only for the EAB veloity odel the digr Eqution (A-8) is liner A5 Coni eloity odel For the Coni profile the veloity nd its grdient in the solute fre re given y Rz z Qz (A-9) R z Qz This eqution n e rerrnged s z x z x z z x z x (A-0) Next we eliinte the solute depth z fro Eqution (A-9) nd get the governing differentil eqution of the Coni veloity odel x (A-) The Coni veloity profile Eqution (A-0) n e lso set in n equivlent for through hyperoli nd n inverse hyperoli funtion z x tnh rsinh z (A-) Copyright 0 SiRes

15 78 I RAE Z KOREN A6 Coents on Digrs Surize the grdient-veloity digrs for the five syptotilly ounded veloity odels The governing differentil equtions re x x x ust Hyperoli odel ust Exponentil odel x Exponentil Slowness odel 4x EABeloity odel Coni eloity odel (A-) The grdient-veloity digrs for the five syptotilly ounded odels re plotted in Figure 7 Note tht in the originl fre of referene the syptotilly ounded odels re desried y the three preters As we entioned in the shifted fre where the veloity vnishes t the origin (or the vertil grdient epts xiu vlue t the origin) only two preters re needed These two preters y e the xiu grdient x nd the syptoti veloity s in Eqution Set (A-) The onstnt vlue tht ppers upon the integrtion of eh eqution is not new preter s it should e djusted to th the xiu vlues x nd We ephsize tht only for the EAB odel the grdient-veloity reltionship is liner: Derivtive of n exponent is proportionl to the se exponent Norlized eloity Grdient - eloity Digrs for Asyptotilly Bounded eloity odels Norlized Grdient Coni EAB Exp Slowness Exp ust Hyp ust Figure 7 Digrs Grdient-eloity for syptotilly ounded veloity odels Only for the EAB odel the digr is liner For ll odels exept the Exponentil slowness odel the grdient dereses with the inrese of veloity For the Exponentil slowness odel the grdient rehes xiu when the veloity eoes one hlf of the syptoti vlue Note the entrl syetry etween the two ust odels Hyperoli nd Exponentil For ll odels exept the Exponentil slowness [8] the grdient dereses with the inrese of veloity (nd depth) In se of the Exponentil slowness the vertil grdient inreses long with the veloity until the veloity rehes one hlf of the syptoti vlue At this point the grdient rehes its xiu vlue x nd then egins to derese with depth The point of xiu grdient in the Exponentil slowness odel y relly exist in the susurfe or it y e n iginry point loted ove the erth surfe (or ove the upper interfe of lyer) This point is rel in se when the initil veloity does not exeed the hlfliiting vlue Furtherore we oent on the speil entrl syetry etween the two ust (97) odels: Hyperoli (H) nd Exponentil (E) see Eqution (A-) nd the two orresponding plots in Figure 7 F F H E x x (A-4) where F H nd F E re the orresponding grdientveloity funtions in Eqution (A-) for these two odels Although these two odels re desried y essentilly different vertil veloity profiles there is n pprent siilrity in the grdient-veloity digrs Appendix B Lert Funtion The Lert funtion [7] y Lx delivers the solution of the trnsendent eqution x y exp y (B-) Its grph is plotted in Figure 8 nd onsists of two rnhes: rnh zero L0 x nd rnh inus one L x The rguent rnge is exp x for rnh zero nd exp x 0 for rnh inus one The vlue rnge is y for rnh zero nd y for rnh inus one In prtiulr this y = L (x) Lert Funtion Brnh Zero Brnh inus One x = y exp (y) Figure 8 Lert funtion y = L(x) is the solution of the trnsendent eqution y exp(y) = x for given vlue x nd n unnown vlue y The funtion hs two rnhes Brnh zero is plotted in red nd rnh inus one is plotted in lue Copyright 0 SiRes

16 I RAE Z KOREN 79 ens tht for positive rguent x only rnh zero exists while for negtive rguent oth rnhes do exist Therefore in the ltter se the rnh index should e speified to void iguity The derivtive of the Lert funtion is dl L x for x 0 dx x L x nd for the infinitesil rguent (B-) dl x li (B-) x0 dx Coent A generl oent is relted to Appendies C to F The trnsfor equtions re forulted in the diensionless for with the unnown top nd otto syptoti ftors nd After the trnsfor eqution or eqution set is resolved we pply Eqution (7) to find the top nd otto instntneous veloities nd If the trnsfor is forulted in depth (ie the intervl thiness z is speified) we pply Eqution () to find the top nd otto grdients of veloity nd If the trnsfor is forulted in tie (ie the intervl trveltie t is speified) then we first pply Eqution (7) to estlish the intervl thiness z nd then Eqution () to find the top nd otto grdients Appendix C Swpping Interfes In this ppendix we find the instntneous veloity nd its grdient t the otto interfe given these preters t the top interfe nd vie vers nd onsider these proles oth vs depth nd vs tie Prole C Given the veloity nd its grdient t the top interfe nd the lyer thiness z one n estlish the orresponding preters t the otto interfe For this we lulte the top syptoti ftor with the first eqution of Eqution Set (6) The otto s- yptoti ftor n e otined with the first eqution of Eqution Set () Prole C The instntneous veloity nd its grdient re speified t the otto interfe nd the intervl thiness z is given eloity nd grdient should e found t the top interfe Thus preters re given nd preters re to e found In this se lulte the otto syptoti ftor with the seond eqution of Eqution Set (6) nd pply the seond eqution of Eqution Set () to get the top get Prole C The veloity nd its grdient t the top interfe nd re given the intervl trveltie t is nown nd the otto interfe preters nd should e found Coining the first eqution of Eqution Set () nd Eqution (6) we eliinte the intervl thiness z nd otin t (C-) ln where the top syptoti ftor is nown fro Eqution (6) Eqution (C-) should e solved for the unnown otto syptoti ftor We use ui pproxition for Eqution (C-) to get the initil guess ssuing the inreent of the syptoti ftor is sll t (C-) Prole C4 The veloity nd its grdient re speified t the otto interfe long with the intervl trveltie nd the profile preters should e found t the top interfe Thus preters nd t re given while preters nd re to e estlished For this we oine the seond eqution of Eqution Set () nd Eqution (6) ln t (C-) The otto syptoti ftor is nown fro Eqution (6) nd Eqution (C-) should e solved for the unnown top syptoti ftor The initil guess for n e found fro t (C-4) Appendix D Inversion with Instntneous eloity In this ppendix we onsider inversion proles tht do not involve the effetive odels (verge nd RS veloity) In this inversion group one or oth preters t the top interfe nd re unnown with the other dt given insted The group inludes four proles vs depth (intervl thiness) nd four siilr proles vs intervl trveltie Prole D Instntneous veloities vs depth Given dt re the top nd otto interfe instntneous veloities nd the syptoti veloity nd the intervl thiness z Find the top interfe grdient Solution Apply Eqution (6) to lulte the top nd otto syptoti ftors nd Prole D Instntneous veloities vs tie Given dt re the top nd otto interfe instntneous veloities nd the syptoti veloity nd the intervl trveltie t Find the top interfe grdient Solution Apply Eqution (6) to lulte the top nd otto syptoti ftors nd Prole D Grdients vs depth Given dt re the top nd otto interfe vertil grdients nd Copyright 0 SiRes

17 740 I RAE Z KOREN the syptoti veloity nd the intervl thiness z Find the top interfe instntneous veloity Solution Solve Eqution Set () for the unnown syptoti ftors nd A A whe re A z Prole D4 Grdients vs tie Given dt re the top nd otto interfe vertil grdients nd the syptoti veloity nd the intervl trveltie t Find the top interfe instntneous veloity Solution Introdue solution (D-) into the trveltie Eqution (6) This leds to nonliner eqution vs the unnown intervl thiness z (D-) z B ln z z t B B z (D-) where B Eqution (D-) should e solved nuerilly It is suitle to norlize the grdients nd the intervl thiness z Int t t z (D-) t The norlized eqution eoes z B z z ln B B z (D-4) where B To get n initil guess we expnd Eqution (D-4) into power series nd otin ui pproxition z ln z z B B (D-5) After the intervl thiness is found pply solution (D-) Prole D5 eloity nd grdient vs depth t different interfes Given dt re the top interfe veloity the otto interfe vertil grdient the syptoti veloity nd the intervl thiness z Find the top interfe grdient Solution Apply Eqution (6) to lulte the top syptoti ftor Introdue the norlized otto grdient ˆ z (D-6) With this nottion the seond eqution of Eqution Set () e- oes ˆ 0 (D-7) Ting into ount tht for n intervl of vnishing thiness z 0 the top nd otto syptoti ftors oinide one n estlish the single physil root of the qudrti eqution 4 ˆ (D-8) ˆ Prole D6 eloity nd grdient vs depth t different interfes Given dt re the top interfe grdient the otto interfe instntneous veloity the syptoti veloity nd the intervl thiness z Find the top interfe veloity Solution Clulte the otto syptoti ftor with eqution 6 Norlize the top grdient ˆ z (D-9) Get the top syptoti ftor 4ˆ (D-0) ˆ Prole D7 eloity nd grdient vs tie t different interfes Given dt re the top interfe veloity the otto interfe vertil grdient the syptoti veloity nd the intervl trveltie t Find the top interfe grdient Solution Apply Eqution (6) to lulte the top syptoti ftor Then solve the nonliner Eqution (C-) for the unnown otto syptoti ftor To get the initil guess we ssue Expnd Eqution (C-) for sll inreent of the syptoti ftor to get ui pproxition t t t (D-) Prole D8 Given dt re the top interfe grdient the otto interfe instntneous veloity the syptoti veloity nd the intervl trveltie t Find the top interfe veloity Solution Apply Eqution (6) to lulte the otto syptoti ftor To lulte the unnown top syptoti ftor we solve the nonliner Eqution (C-) To get the initil guess ssue This leds to t t t (D-) Prole D9 Given dt re the instntneous veloity t the otto interfe nd t n interedite Copyright 0 SiRes

18 I RAE Z KOREN 74 level inside the intervl nd the syptoti veloity Two vertil distnes re speified: the full intervl thiness z (the distne etween the top nd otto interfes) nd the prtil intervl thiness z (the distne etween the top interfe nd the interedite level) Find the top interfe veloity nd grdient nd Solution Use Eqution (6) to lulte the otto syptoti ftor nd the interedite syptoti ftor Apply Eqution (9) for the full intervl nd for the prtil intervl Q (D-) z z Solve Eqution (D-) for the top syptoti ftor z z (D-4) z z Prole D0 Given dt re the instntneous veloity t the otto interfe nd t n interedite level inside the intervl nd the syptoti veloity Two vertil trvelties re speified: the full intervl trveltie t (the trveltie etween the top nd otto interfes) nd the prtil trveltie t (the trveltie etween the top interfe nd the interedite level) Find the top interfe veloity nd grdient nd Solution Use Eqution (6) to lulte the otto syptoti ftor nd the interedite syptoti ftor Apply Eqution (8) for the full intervl nd for the prtil intervl preter R eoes ln ln (D-5) t t Introdue preter oes i ln t t i Eqution (D-5) e- ln (D-6) Eqution (D-6) n e solved with the Lert funtion rnh zero f f L 0 exp f f t t f nd f t t t t Appendix E Two-Point Effetive odel Inversion (D-7) In this ppendix we onsider the two-point (single-intervl) inversion where the RS veloity is speified t the interfes of n intervl vs depth or trveltie or lterntively depth is speified vs trveltie insted of the RS veloity One of the two preters t the top interfe-either the top veloity or the top grdient -is nown vlue while the other one is unnown nd should e estlished In ll ses the syptoti veloity nd the top interfe solute trveltie t re ssued nown vlues We onsider lso speil se when the RS veloity is speified vs oth depth nd tie nd oth preters nd re unnown Prole E RS vs depth with unnown grdient Given dt re the RS veloities t the top nd otto interfes nd the intervl thiness z nd the top interfe veloity Find the top grdient Solution It follows fro the definition of the hyperoli preter Eqution (44) W t t t t t (E-) Rell tht t nd t re one-wy solute top nd otto interfe trvelties (esured fro the erth surfe) t is the one-wy intervl trveltie nd W is the hyperoli preter through the intervl Eqution (E-) n e rrnged s W t t (E-) Introdution of Eqution (7) for the trveltie t nd Eqution (47) for the hyperoli preter W into Eqution (E-) results in B ln ln A (E-) where A nd B re nown diensionless preters A B t B z (E-4) The top syptoti ftor is delivered y Eqution (6) nd the nonliner Eqution (E-4) should e solved for the unnown otto syptoti ftor To get the initil guess we ssue sll inreent of the syptoti ftor on the intervl nd expnd Eqution (E-) into power series The ui pproxition reds where C C C C A (E-5) 0 i B Ci i 0 (E-6) i i i Eqution (E-) should e solved for Prole E RS vs depth with unnown veloity Given dt re the RS veloities t the top nd otto interfes nd the intervl thiness z nd the top interfe grdient Find the top interfe Copyright 0 SiRes

19 74 I RAE Z KOREN veloity Solution Use Eqution (E-) nd the first eqution of Eqution Set () B ln ln A z (E-7) We solve Eqution Set (E-7) for the unnown syptoti ftors nd To otin the initil guess for ssue tht the inreent of the syptoti ftor is sll nd linerize Eqution Set (E-7) This leds to A B ˆ AB B A B ˆ ˆ 4 Prole E RS vs tie with unnown grdient Given dt re the RS veloities t the top nd otto interfes nd the intervl trveltie t nd the top interfe veloity Find the top grdient Solution First we pply Eqution (E-) nd lulte the hyperoli preter W At this tie W is nown vlue Next we pply Eqution (48) ln C (E-9) ln (E-8) where C is nown diensionless preter the norlized lol RS veloity U W C t (E-0) nd U is the non-norlized lol RS veloity on the intervl The top syptoti ftor is delivered y Eqution (6) nd the otto syptoti ftor is estlished fro Eqution (E-9) To solve this nonliner eqution n initil guess is needed Assue tht the inreent of the syptoti ftor on the intervl is sll nd expnd Eqution (E-9) into power series The ui pproxition reds (E-) C Prole E4 RS vs tie with unnown veloity Given dt re the RS veloities t the top nd otto interfes nd the intervl trveltie t nd the top interfe grdient Find the top interfe veloity Solution We solve set onsisting of two equtions: the first is Eqution (E-9) oined with Eqution (C-) nd the seond is (C-) itself ln C t ln t (E-) To get the initil guess we linerize Eqution Set (E- ) for sll inreent of the syptoti ftor nd otin U tu U (E-) Prole E5 Depth vs tie with unnown grdient Given dt re the intervl thiness z the trveltie t nd the top interfe veloity Find the top grdient Solution Use Eqution (6) to find the top syptoti ftor Next we pply Eqution (7) to estlish the otto syptoti ftor ln D (E-4) where D is nown diensionless preter the norlized intervl veloity z D t Int (E5) To get the initil guess for Eqution (E-4) we expnd it into power series for sll inreent of the syptoti ftor D (E-6) Eqution (E-4) should e solved for the otto syptoti ftor Prole E6 Depth vs tie with unnown veloity Given dt re the intervl depth z the intervl trveltie t nd the top interfe grdient Find the top interfe veloity Solution Apply Eqution (E-4) nd the first eqution of Eqution Set () D ln D (E-7) z Next we solve Eqution Set (E-7) for the unnown syptoti ftors nd To otin the initil Copyright 0 SiRes

20 I RAE Z KOREN 74 guess we linerize Eqution Set (E-7) for sll inreent of the syptoti ftor nd onsider the first eqution of the set li ln (E-8) Hene we otin the initil guess for the top syptoti ftor t (E-9) D tz Prole E7 RS vs depth nd tie with unnown veloity nd grdient Given dt re the RS veloities t the top nd otto interfes nd the intervl trveltie t nd the intervl thiness z Find the top interfe veloity nd grdient Solution The resolving set follows fro Equtions (E-9) nd (E-4) ln A (E-0) ln B where A nd B re nown diensionless preters Int U A Int B Int Int (E-) To otin the initil guess we ssue tht the top nd otto vlues of the syptoti ftor nd re lose nd solve the two equtions of Eqution Set (E-0) prt Eh eqution yields root; the sller root is the top syptoti ftor nd the lrger root is the otto syptoti ftor The initil guess eoes B Int (E-) Int A U Appendix F Three-Point Inversion Int In this ppendix we onsider three proles where the RS veloity is given t the interfes nd t n interedite (inner) point of the intervl vs depth or tie or depth is given vs tie t the interfes nd t n interedite point nd oth preters of the veloity profile nd re unnown Prole F RS vs depth with the unnown top interfe veloity nd top grdient Given dt re the RS veloities t the top nd otto interfes nd nd the RS veloity t the inner point of the intervl the full intervl thiness (etween z the top nd otto interfes) nd the prtil thiness z (etween the top interfe nd the interedite level) Find the top veloity nd the grdient Solution The resolving eqution set follows fro Eqution (E-) B ln ln A (F-) B ln ln A Coeffiients A A B nd B re nown vlues; they follow fro Eqution (E-4) A B t B z (F-) A B t B z where the syptoti ftors nd re to e found (F-) It follows fro Eqution () z (F-4) z Divide the seond eqution of Eqution Set (F-4) over its first eqution z (F-5) z Thus we solve Eqution Set (F-) together with Eqution (F-5) to otin the three unnown syptoti ftors To otin the initil guess we ssue tht the lyer is thin ie tht the differenes in the syptoti ftors nd re sll This ssuption llows lineriztion of Eqution Set (F-) whih in turn results in B A B (F-6) B A B Introdution of solution (F-6) into Eqution (F-5) leds to fourth order polynoil eqution for the top interfe syptoti ftor B A B A (F-7) z B z B Copyright 0 SiRes