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1 A HIDDEN ARKOV ODEL APPROACH FOR LIHOLOGY IDENIFICAION FRO LOGS ara Padron, Sona Garca-Salce, Danel Barraez, Bernadee Dorzz, Sylve hra Insu Naonal des élécouncaons (IN, Evry, France; Unversdad Cenral de Venezuela (UCV, Caracas, Venezuela; LODYC, Unversé Perre e are Cure, Pars, France ara.padron@n-evry.fr, Sona.Salce@n-evry.fr, dbarraez@euler.cens.ucv.ve, Bernadee.Dorzz@nevry.fr, Sylve.hra@lodyc.jusseu.fr. INRODUCION We presen a new sascal ehod of denfyng lhologes relyng on wrelne log easureens ade on wo holes fro he french se of arcoule. Snce several years, sascal echnques have appeared as a powerful ool o classfy coplex and heerogeneous reservor lhology: ulvarae Sascs [Doveon (994], Dscrnan Analyss [Busch (987] and, ore recenly, Neural Neworks have been appled o hs proble. Concernng Neural echnques, ullayer Perceprons (LPs are used o classfy lhologes, eher relyng on well-logs drecly [Sauel (992], or soees afer usng Kohonen aps o deerne he lhologes of he reservor [Saggaf (2000]. Also, Self-Organzng aps have been used o reconsruc he lhologc faces of a drllng hole [Frayssne (2000, Anouar (997]. he goal of hs sudy s o denfy lhologes fro logs, relyng on nforaon abou rocks porosy and pereably. o hs end, we propose an orgnal approach based on Hdden arkov odels (Hs. Indeed, we consder a log sere of a drllng hole as a sequence of easures, and propose o odel n he sascal fraework gven by Hs. he reason s ha, n hs way, we can ake no accoun conexual dependences beween easures ade a dfferen levels of he drllng hole, whle perforng lhology denfcaon. In parcular, n coplex reservors, several lhologes are xured, and s exreely dffcul, even for a huan exper, o deerne whch s he lhology relyng only on he log easures aken a a gven level. In hs fraework, conexual nforaon ay be of porance o prove classfcaon a a gven level. Hs are ndeed wellknown sascal odels n oher applcave areas (lke speech recognon, on-lne handwrng recognon, ec... hey appear o be a powerful ool o explo conexual nforaon when perforng classfcaon locally n a sequence of observaons. In such applcaons, he sgnal s eporal and non saonary; he conex of a sngle observaon brngs nforaon abou he evoluon of he sgnal n e. Our purpose n he presen work s o envsage hs approach for sedenal seres deposed durng e. hs work s srucured as follows: Hs are brefly presened n Secon 2, as well as her applcaon o lhology denfcaon. For ha, we sar descrbng he applcave conex n deal. hen, he odel s descrbed n Secon 3, and resuls are hen presened and dscussed n Secon H FOR LIHOLOGY IDENIFICAION 2. he applcave conex: he arcoule se he arcoule se s n he souh of France, n he Gard area, near Bagnols-sur-Cèze. A hundred llon years ago, hs se was covered by an ocean and asde he ounans of he assf Cenral. I s why he subsol s coposed of boh faces of connenal orgn (resulng fro eroson of he crsalln foraons of he assf Cenral and of arne orgn. he subsol s ade of clayey and sandy sedenal seres, whch have been deposed a Creaceous. Daa coe fro wo drllng holes, naed AR402 and AR203. he profle of he arcoule se shows a l of he sol beween hese wo holes; because of hs l, he faces encounered n AR203 are encounered n he nferor half of he well AR402. For ha reason, AR402 s n fac ore coplee han AR203 fro a geologcal pon of vew: soe faces presen n AR402 are absen of AR203. hs s an poran fac n our sudy. Also, core daa fro holes s only avalable a ceran levels of he holes. For hs reason, we use labels resulng fro a prevous research work obaned wh a Kohonen ap on he arcoule se [Frayssne (2000]. AR203 s drlled unl 89 s and AR402 unl 530 s. We use hree logs ha are PEF (phooelecrc effec, RHOB (relave densy n gr/c3, and GR (Gaa-Ray n API nubers. In he drllng, he easures are aken every half-foo (5.24 c. Neverheless, he sudy of sgnals shows ha her vercal resoluon s raher of around 50 cs. In he drllng hole AR203, we have 5590 easures' levels, and n AR402, 9962 easures' levels. Accordng o [Frayssne (2000], welve lhologes were deerned n he arcoule se: Lesones (C, arls (, glauconc Sandsone (Gga, Shales (A, Oher Shales (A, Sls (S, Oher Sls (S, coarse Sandsone (Ggs, Sandsone (G, sandy Lesones (Cg, sandy Brecca (B and Lgne (L. 2.2 Hdden arkov odels In he las years, Hdden arkov odels have becoe a useful ool n non saonary sgnal recognon [Rabner (989, Rabner (993]. Hs are sascal odels based on he classcal arkov chans. Consder a sochasc process (q whch s descrbed a e as beng n one of a se of N saes, S, S 2,,S N. he process (q s a arkov chan f, n order o ake a predcon a e on wha s gong o happen n he

2 fuure, s useless o know anyhng ore abou he whole pas up o e -,.e. ( q s j q s, q 2 sk, q s P( q s j q s P..., ( We only consder he hoogeneous arkov Chan, ha s hose processes n whch he rgh-hand sde of ( (naely he ranson probably fro sae q - o sae q s ndependen of e. he arx of sae ransons probables A{a j } and he nal dsrbuon π are he relevan nforaon n order o descrbe he e evoluon of he process: j ( q s q s a P,,j N (2 P ( q s j π, N (3 he arkov Chan defned n hs way s called an observable arkov odel snce he oupu of he process are he saes. In any neresng probles n whch he sgnal s non saonary, he saes of he arkov Chan are hdden, no drecly observable, and he observaons are he rando sgnals eed by he saes. A Hdden arkov odel (H s herefore a double sochasc process characerzed by: - he nuber N of saes {S, S 2,,S N } n he odel; - he sae ranson probably dsrbuon; a N j j P a j ( q s q s - he nal dsrbuon; ( q s,j N (4 π P, N. (5 - he se of observaon sgnal denses, B{b j }, where b j s he observaon sgnal densy when he process s n sae j. A H provdes he echans for a rando syse whch ay be descrbed as follows. A e, he nal sae q S wll be chosen a rando, accordng o he nal dsrbuon probably π. In hs sae S, a sgnal O wll be observed accordng o he observaon sgnal densy b. A e 2, he process changes o anoher sae S j accordng o he ranson arx a j, and so on. Noe ha a coplee specfcaon of a H s gven by he specfcaon of probably easures A, B and π. In he followng, λ(a,b,π denoes he coplee se of paraeers specfyng he H λ. For a coplee descrpon of ranng procedures n a H, see [Rabner (989, Rabner (993]. o denfy lhologes, we frs ran a H per class (per lhology. Durng hs sep, called ranng, for each lhology, we opze he odels' paraeers (A, B, π ha bes explan a gven se of observaon sequences, called ranng daabase. Aferwards, n a second sep, called recognon, a sequence of logs s, a he sae e, segened and recognzed by he lhologes' Hs. he Verb algorh [Rabner (989] gves ndeed he sequence of saes wh hghes lkelhood for hs observaon sequence. hs allows o segen he observaon sequence correspondng o a log sere of a hole, n dfferen lhologes, whle such lhologes are recognzed. hese wo seps wll be dealed n secon Srucure of he odel 3. RAINING here are dfferen ypes of Hs: dscree or connuous Hs, regardng he naure of he sae esson probably laws, lef-rgh Hs, or parallel ones, or ergodc Hs, regardng o he opology of he odel [Rabner (993], ha s he ransons ha are auhorzed beween he saes of he H. We odel each lhology by an ergodc and gaussan connous H. Ergodcy pers o envsage ransons fro every sae o any oher sae of he H (see Fgure. Also, we used a xure of gaussan denses o approxae he dsrbuon of he observaons, represened by he logs. hs eans ha each observaon O (each log a e, a vecor of denson 3 (he PEF, RHOB and GR logs, s eed by sae j wh probably: b j ( O c [ η O, µ, U ], j N (6 where c s he xure coeffcen for he h xure coponen n sae j and η he gaussan densy funcon, wh ean and covarance arx U for he h µ ( xure coponen n sae j, ha s η O, µ,u : η ( O, µ, U ( 2π n 2 (de( U - 2 exp ( o µ U ( o µ he xure coeffcens c sasfy he sochasc consrans: c c 0 N, (8 s a N a N a 2 a 2 a 2N s N s 2 a N2 Fgure. Ergodc H of a gven lhology (N saes 3.2 Isolaed ranng 2 (7

3 We frs consder a ranng paradg n whch solaed log sequences of each class (lhology are used o ran he correspondng H. We call hs parcular ranng paradg "Isolaed ranng". For hs, we cu he coplee sequence of logs of hole 402 (he hole used for ranng purposes no segens, where each segen corresponds o a dfferen lhology. Each resulng sequence of observaons n a gven lhology has a sze 6. We used he Bau-Welch algorh o esae he paraeers of each lhology H λ(a,b,π. o suarze, hs algorh axzes eravely P(O λ, he lkelhood of he observaon gven he odel. A local axu s aaned afer a gven nuber of eraons of he ranng daabase. hs algorh works n he followng erave for:.- Inalzaon of he odel: he ranson and nal probables are defned equprobable. Also, he nuber of observaons n each log sequence are dsrbued equably n he saes of he H. he sae s done n each sae regardng he nuber of gaussan denses of he sae esson probably law. 2.- Afer each eraon of he ranng daabase, we reesae ˆ λ Aˆ, B ˆ, ˆ as follows: ( Π - for he nal probably, he expeced frequency n sae s a e s copued as: ˆ π ( N (9 where ( s he a poseror probably of beng n sae a e : ( P ( q O λ, (0 - for ranson probables, he expeced nuber of ransons fro sae s o sae s j, dvded by he expeced nuber of ransons fro s s copued: aˆ j where: ε (, j, (,j N ( (, j P ( q, q j O λ ε +, (2 ha s, he a poseror probably o be n sae a e and n sae j a e +; - for he sae esson probables, he paraeers of each gaussan densy funcon and he xure coeffcens are reesaed. he xure coeffcen reesaon for he gaussan densy k n sae j s he followng [Rabner (989]: cˆ k ( j, k ( j, k (3 where ( j, k s he probably of beng n sae j a e wh he kh xure coponen accounng for O ( j, k ( j η [ O, c η µ ] [ O, µ, U ], U (4 Fnally, he ean and he covarance arx of he gaussan densy k n sae j are reesaed as follows: ˆ µ Uˆ ( ( j, k O ( j, k j, k( o µ ( o µ ' ( j, k (5 (6 3.- ranng s sopped when he average log-lkelhood on he ranng daabase s sablzed, ha s when: P r + r ( O P ( O λ r P ( O λ λ < c (7 where c s 0-3 or 0-4 accordng o he class (lhology ha we consder. Bau [Bau (970] showed ha a he rh eraon of hs algorh, we have: P ( O λ r + P r ( O λ for each observaon sequence, unl a local axu s reached. 3.3 Conexual ranng afer Isolaed ranng Afer solaed ranng s perfored, we consder hs as an nalzaon for anoher ype of ranng, ha we call "Conexual ranng".he neres of such ranng s o nroduce n he paraeer esaon process, conexual nforaon presen n he hole. Indeed, solaed ranng only rans each lhology odel on solaed sequences of each lhology. In hs new ranng paradg, we consder nsead longer subsequences of logs of he hole, conanng several lologes. hen, for ranng purposes, we us concaenae he Hs correspondng o he lhologes presen n each of hese subsequences. hs way, paraeer esaon for each of hese lhology odels wll be nfluenced by he neghborng lhologes, as follows : he Verb algorh [Rabner (989] s used o segen he whole sequence no dfferen lhologes, and hs segenaon s exploed for ranng purposes, as we explan below. he Verb algorh copues he opal pah shown n Fgure 2 (he "hdden" sae sequence n ers of axal lkelhood of he whole logs-subsequence (conanng C,, and L, presened o he correspondng

4 Hs (hose of C, and L (see Fgure 2. he opaly creron used on he sequence of logs s herefore global. Conexual nforaon s nroduced n he ranng process precsely when usng he resulng opal pah o reesae he paraeers of he correspondng Hs. Fgure 2. Verb pah copuaon durng Conexual ranng on a logs-sequence alernang C, and L he Verb algorh [Rabner (989] s based on dynac prograng; hs algorh fnds he bes sae sequence n he sense of axal lkelhood, for a gven observaon (logs sequence. For he frs observaons, he probably: ax (... q, OO... O ( q q q Pqq λ (8 gves he bes score along he pah perng a e o reach sae s : A recurrence s hen saed as follows: [ ( a ] b ( O + ( j ax j j + (9 Also, a varable ψ (j conans he bes precedng sae for sae j a e. hs algorh has hree seps:. Inalzaon ( π b (20 ( O Ψ ( 0 2. Recurson: ( j ax [ ( aj ] b j ( o ( j ax [ ( a ] N 2 N (2 ψ arg j j N 3. ernaon P q ax [ ( ] N (22 arg ax [ ( ] he opal sae sequence s obaned by "backrackng", as follows : q Saes of he odels λ L λ λ C ψ ( q + + O..O 5 O 6.. O -3 O Opal pah Logs CCCCLLLLLLL Lhologes -,-2,., (23 As enoned before, afer hs segenaon sep, he reesaon of he Hs' paraeers s perfored. hs ranng paradg s well-known as he Segenal K- eans algorh [Rabner (993]. Accordng o he segenaon gven by he opal sae sequence, all he observaons (logs arbued o a gven sae are affeced o a gven gaussan densy n hs sae. hs s done by he copuaon of he dsance of each observaon o he eans of all he gaussans n hs sae. For each observaon, he gaussan realzng he nu dsance s affeced o such observaon. hen, paraeer reesaon can be perfored for he Hs, as follows: - for he ean of he xure coponen, for a gven sequence of he ranng daabase, ˆ µ ( q ( q j O ( O j ( O c c (24 here s he Kronecker funcon, c denoes cluser (he observaons affeced o he xure coponen, and O denoes he curren observaon a e. - for he covarance arx of xure coponen : Uˆ ( q j ( O c ( O ˆ µ ( ( q j ( O c O ˆ µ (25 he xure coeffcen s reesaed as he nuber of observaons (logs affeced o cluser of sae j dvded by he nuber of observaons affeced o sae j : cˆ ( q j ( O ( q j c 3.4 Conexual ranng afer rando nalzaon (26 Anoher ype of ranng was also esed: consss of Conexual ranng by he Segenal K-eans algorh descrbed n Secon 3.3, when he Hs are no nalzed by Isolaed ranng (prevously descrbed n Secon 3.2. In hs fraework, he ranng daabase (a se of sequences of lengh 25 of AR402 s used once (one epoch o nalze he Hs, usng he "correc pah" nsead of he opal pah copued by he Verb algorh. he "correc pah" s n fac he pah ha we oban when we assocae o each observaon (a log s correc label (he correspondng lhology. Wh hs "correc segenaon" of each sequence of he ranng daabase, we oban a frs esaon of he Hs' paraeers by he Segenal K-eans algorh. 3.5 Convergence crera wo convergence crera are consdered: he frs one s based on he sablzaon of he average log-lkelhood per class; he second one s based on he sablzaon of he perforance of each lhology H (he percenage of correc classfcaon.

5 4. ESING HE SYSE Classfcaon s perfored on he oher hole (AR203 usng he Verb algorh: he coplee sequence of logs easured n AR203 s presened o he 2 lhology Hs a he sae e, o copue he opal sae sequence n he Verb sense. o hs end, ransons are auhorzed fro any sae of any H o any sae of any oher H. 4. esng afer Isolaed ranng he followng able (able shows he nuber of log daa avalable per lhology n AR402 ("Daa" colun n able and he nuber of resulng solaed sequences of each lhology ("oal" colun n able, afer daa n AR402 s cu no segens (of axu lengh 6 of each lhology. Noce ha lhologes S, and Cgs have he fewes nuber of sequences for ranng her respecve Hs. On he oher hand, lhology L has he hghes nuber of sequences for ranng purposes. Class Daa oal Class Daa oal C S Cgs Cga G A Cg A B S L able. Daa descrpon for Isolaed ranng Resuls on AR203 afer Isolaed ranng n sequences of each class exraced fro AR402, are presened n able 2: colun "C" s he class (lhology now nubered fro o 2 (desgnng respecvely C,, Gga, A, A, S, S, Ggs, G, Cg, B, L. Colun "S" s he nuber of saes n he H, colun "" s he nuber of gaussan denses (xure coponens per sae of he H, colun "Daa" gves he nuber of logs per class n AR203, colun "LL" s he nuber of ranng eraons ade, colun "E" s he value of consan c n forula (7 o sop ranng by he Bau-Welch algorh, and colun "%" s he percenage of logs correcly classfed n AR203. Accordng o he nuber of sequences per class, he creron o sop ranng uses a dfferen value of consan c (0-3 or 0-4. Dfferen ess were ade changng he nuber of gaussans (, bu we presen only he resuls wh, as hey are he bes. Resuls are globally good, hey vary fro 44.44% o 96.39%. hs can be explaned by he fac ha here are no enough solaed sequences o ran he Hs n he confguraon n whch esson probables are xures of gaussan denses. Indeed, hs fraework ples uch ore paraeers o esae (several covarance arces, several eans, xure coeffcens. I s why n he followng (Secons 4.2 and 4.3, all he experens are perfored n he fraework of one gaussan densy per sae of he Hs. C S Daa LL E % e e e e e e e e e e e e able 2. es resuls for afer Isolaed ranng 4.2 esng afer Conexual ranng wh Hs nalzed by Isolaed ranng In hs fraework, ranson probables beween dfferen lhology Hs are nroduced n he copuaon of he opal pah by he Verb algorh. hese ranson probables are fxed durng ranng and esng; hey are esaed on he hole AR402 by relave frequences. her role s o favour soe ner-odel ransons, accordng o wha s observed n AR402. For Conexual ranng, we used sequences of lengh 25 of AR402. When ranng s sopped accordng o he perforance creron per H (sablzaon of he percenage of correc classfcaon per class, convergence s reached afer 39 epochs (eraons of he ranng daabase, and afer 5 epochs for he creron of lkelhood sablzaon. able 3 shows resuls on AR203 for boh convergence crera: colun "Q" gves n fac he values n whch he perforance becoes sable n he "ranng hole" (AR402, and colun "%Q" gves he correspondng resuls n he "es hole" (AR203. Analogously, colun "LL" gves he average value of he log-lkelhood per class a convergence (when hs value becoes sable, and colun "%LL" gves he correspondng resuls n he "es hole" (AR203. We frs noce ha he percenage of correc classfcaon s proved n half of he lhologes (classes 2,3,6,7,8, copared o he resuls obaned afer Isolaed ranng.

6 he oher lhologes (classes, 4, 5, 9, 0, 2 for whch resuls are degraded, are very xured n he drllng holes, ha s a sngle or very few observaons (logs of such lhologes are ofen found beween oher lhologes. For hs reason, only few sequences of such lhologes have a sgnfcan lengh durng Conexual ranng. hs s parcularly vsble for lhology 2 (L, for whch pleny of daa are avalable, bu such daa are spread n he drllng holes a os easures' levels. Indeed, hs lhology, of vegeal orgn, s very weak and ends o ge daaged durng he daa acquson process, spreadng self a os easures' levels. In oher words, Conexual ranng s effecve when subsequences of each lhology appearng n he conex of oher classes are of sgnfcan lengh. C S Daa Q %Q LL %LL able 3. es afer Conexual ranng when Hs are nalzed by Isolaed ranng 4.3 esng afer only Conexual ranng In hs fraework, as dealed n Secon 3.4, he ranng daabase s used once (one epoch o nalze he Hs, usng he "correc pah" nsead of he Verb pah. Wh hs "correc segenaon" of he sequences of he ranng daabase, we oban a frs esaon of he Hs' paraeers by he Segenal K-eans algorh. Our goal s o evaluae he nfluence of hs nalzaon, done n a conexual way, when followed by Conexual ranng. Resuls are gven n able eraons (epochs of he ranng daabase were necessary o sop ranng, for boh convergence crera (descrbed n Secon 3.5. In boh cases, resuls are he sae. able 4 shows ha for 2/3 of he classes, he resuls are proved copared o hose presened n able 3. Also, he degradaon of class 2 (L s confred afer hs conexual nalzaon. Copared o Isolaed ranng, wo classes are srongly proved: class 2 ( and class (B, and soe classes lke classes 6 and 7 (S and S, and class 0 (Cg are globally unchanged. hs ay reveal ha he laer are que dffcul o odel. C S Daa Q %Q LL %LL able 4. es afer only Conexual ranng 5. CONCLUSIONS We have proposed an orgnal approach based on Hs o denfy lhologes n a drllng hole. hs sascal approach consders a sequence of logs easured n a drllng hole as a e sere. hs pers o nroduce soe conexual nforaon presen n he sequence of logs when esang he paraeers of he sascal odels of each lhology. A lhology s odeled by a gaussan ergodc H and raned n hree dfferen ways: Isolaed ranng (n whch daa are separaed per lhology for ranng, Conexual ranng afer nalzng he Hs by eans of Isolaed ranng, and only Conexual ranng n whch he Hs are even nalzed n a conexual way. he las paradg proves resuls for 2/3 of he classes relavely o he second one. Soe classes are dffcul o odel n any of such paradgs: class 6 and 7 (S and S, and class 0 (Cg. Also, we noced ha Conexual ranng s uneffecve for hose classes whose sequences are no of sgnfcan lengh when aken n he conex of oher lhologes. For ha reason, only wo classes show he real neres of Conexual ranng relavely o Isolaed ranng: class 2 ( and class (B. On he oher hand, class 2 (L shows anoher l of our approach: hs class appears n he conex of all he ohers because s spread a all he levels of he drllng holes. Resuls for hs class ge degraded wh conexual ranng, and when he nalzaon s also done conexually, hey are even ore degraded. hese prelnary ess show ha, whle n general he nroducon of conex proves he classfcaon

7 accuracy, one has o be careful wh soe classes wh a very changng conex. Our furher work wll explore hs aspec n ore deals and propose soe effcen sraegy o cope wh hs phenoenon. 6. ACKNOWLEDGEENS hs work was parally suppored by he French- Venezuelan Acon ECOS-Nord N V990 (N , CDCH-UCV gran N and Agenda Peroleo Accon odelaje Esocasco Aplcado. 7. REFERENCES Anouar F., Badran F., hra S., 997: Self-Organzed ap, A Probablsc Approach, Proceedngs of he Workshop on Self-Organzed aps, Helsnk Unversy of echnology, Espoo, Fnland, June 4-6. Bau, L H., Pere., Soules G., Wess N., 970: A axzaon echnque ocurrng n he sascal analyss of probablsc funcons of arkov chans, Annals of aheacal Sascs, 4, nº, Busch, J.., Forney, W. G., Berry L. N., 987: Deernaon of lhology fro well logs by sascal analyss, SPE Foraon Evaluaon, vol. 2, Davs, J. C., 986: Sasc and daa analyss n geology, 2 nd ed..new York, John Wley, 273p. Delfner, P., Peyre O., Serra O., 987: Auoac deernaon of lhology fro well logs. SPE Foraon Evaluaon, vol. 2, Doveon J. H., 994: Geologc log analyss usng copuer ehods. A Assoc. Peroleu Geologss, Copuer ehods n Geology, No 2, 69p. Frayssne D., hra S., Badran F., Brqueu L., 2000: Use of Neural Neworks n Log's Daa Processng-Predcon and Rebuldng of lhologc faces. Perophyscs ees Geophyscs, Pars, France, Noveber 6-8. Juang. B. H., L. R. Rabner, 990: he Segenal K- eans algorh for esang paraeers of Hdden arkov odels. IEEE ransacons on Acouscs, Speech and Sgnal Processng, 38, nº 9, Rabner L.R., 989: A uoral on Hdden arkov odels and Seleced Applcaons n Speech Recognon. Proceedngs of he IEEE, 77, No 2, Rabner, L.R., 993: Fundaenals of Speech Recognon. Prence Hall Sgnal Processng Seres, Prence Hall. Rogers. S. J., J. H. Fang, C. L. Karra, and D. A. Sanley, 992: Deernaon of Lhology fro Well Logs usng a Neural Nework. A Assoc. Peroleu Geologss Bullen, 76, No 5,