05 Malvic.indd 37 � 1. INTRODUCTION Geos ta tis ti cs is a we ll-es tab lis hed tool in dif fe re nt geos cien- ces, pri ma ri ly de fi ned as ad van ced in ter po la tion al go rit hms. In mo st ca ses, geos ta tis ti cal met ho ds of fer se ve ral ad van ta ges com pa red to other mat he ma ti cal ly sim pler in ter po la tion tec hniques. The mo st im por ta nt ad van ta ge of geos ta tis ti cs is the va riog ram fun ction it se lf. The re are se ve ral spa tial in ter po- la tion met ho ds that ha ve the abi li ty of gi vi ng weig hts to da ta poin ts ac cor di ng to their re la ti ve po si tion to the lo ca tion to be es ti ma ted. Howe ver kri gi ng is the on ly met hod that can ta ke in to con si de ra tion the spa tial ani sot ro py of the pa ra me te rs bei ng es ti ma ted throu gh the ani sot ro py of di rec tion va riog- ram, and, what is mo re, wit hout the va riog ram it wou ld be near ly im pos sib le to es ti ma te a geo lo gi cal at tri bu te whi ch does not ha ve fi ni te va rian ce li ke frac tu re po ro si ty (János Gei ger, per so nal com mu ni ca tion, 2007). Mo reo ver, the va riog ram as a ba sic geos ta tis ti cal te rm for- ces ac cu ra te de ter mi na tion of spa tial re la tion shi ps amo ng ha- r d-da ta. Of cour se, it is not a ma gic tool, but an expe rien ced geo lo gi st can ve ry pre ci se ly de fi ne the weig hti ng coef fi cien ts be tween ha r d-da ta and their in fl uen ce at the new es ti ma tion poi nt. The geos ta tis ti cal idea was mos tly de ve lo ped in de pen den- tly of ma jor ac hie ve men ts in the fi e ld of spa tial sta tis ti cs. The wor ld-re nowned pio neer wo rk was do ne by KRIGE (1951). The fi r st wi de ly known re sul ts are tho se in the wor ks of Mat- héron (e.g. MATHÉRON 1962, 1963, 1965). The se in clu ded de tai led expla na tio ns of the kri gi ng, as a ba sic geos ta tis ti cal To mis lav Malvić1 AB STRA CT The data pre sen ted he re for the fi r st ti me in the Croa tian geo mat he ma ti cal com mu ni ty, ai m to es tab li sh se ve ral cri te- ria whi ch wi ll aid in the se lec tion of de ter mi nis tic or stoc has tic geos ta tis ti cal met ho ds of es ti ma tion. This re view as so cia tes the theo re ti cal bac kgrou nd of kri gi ng, cok ri gi ng or stoc has tic si mu la tio ns wi th so me re sul ts ob tai ned by map pi ng Ba de nian clas tic re ser voi rs in the Dra va Dep res sion. The se lec ted re ser voi rs are lo ca ted in the Sta ri Gra da c-Bar cs Nyu gat fi e ld, in the Wes te rn pa rt of the Dra va Dep res sion, and at the Be ni čan ci fi e ld in the Eas- te rn pa rt of this dep res sion. Bo th da ta se ts (ea ch wi th 14 poin ts) in clu de mean po ro si ty va lues ta ken from we ll log ana lysis in the re ser voir in ter val at the we ll si tes. This re sul ted in sig ni fi ca nt un cer tain ties in the va riog ram mo de ls, es pe cial ly wi th the de ter- mi na tion of ran ge (at the se con da ry va riog ram axis). The cri ti cal ad van ta ge was the avai la bi li ty of a seis mic at tri bu te that cou ld be used as a se con da ry and co-re- gio na li zed va riab le (po ro si ty is the pri ma ry va riab le). In the fi r st exam ple (the Be ni čan ci fi e ld) the seis mic at tri bu te was cor re la ted wi th the ave ra ge log ged po ro si ties. This ma de it pos sib le to ap ply the cok ri gi ng met hod as the be st in ter po la tion op tion. The cro ss-va li da tion re su lt was 2.19 for 14 wel ls. In con tra st, the exis ten ce of on ly a pri ma ry va riab le at the Sta ri Gra da c – Bar cs Nyu gat fi e ld, for ced the ap pli- ca tion of stoc has tic si mu la tio ns as the bet ter es ti ma tion tool, whi ch can bet ter des cri be the po ro si ty chan ges in in te r- we ll areas. Keywor ds: krigi ng, cok ri gi ng, Gaus sian si mu la tio ns, po ro si ty, Be ni čan ci fi e ld, Sta ri Gra da c – Bar cs Nyu gat fi e ld, Badenian, Dra va Dep res sion, Croatia 1 Re ser voir En gi nee ri ng & Fie ld De ve lop me nt, INA-Oil In dus try Plc., Zag reb, Croa tia (to mis lav.malvic@ina.hr) Kri gi ng, cok ri gi ng or stoc has ti cal si mu la tio ns, and the choi ce be tween de ter mi nis tic or sequen tial ap proac hes Geologia Croatica 61/1 37–47 7 Figs. 0 Tabs. Zagreb 2008 Geologia CroaticaGeologia Croatica � 05 Malvic.indd 3705 Malvic.indd 37 24.6.2008 11:10:1724.6.2008 11:10:17 Geologia Croatica Geologia Croatica 61/1 38 tec hnique equi va le nt to the met hod of mean squa res ap plied in a li near Gaus sian mo del. The se pa pe rs we re fol lowed by ma ny ot he rs from geos ta tis ti cal exper ts that im pro ved theory and prac ti ce of dif fe re nt geos ta tis ti cal met ho ds. A ve ry fa mous book was writ ten by JOURNEL & HUJIBREGTS (1978). An- dre Jour nel was Mat he ron’s stu de nt and he es tab lis hed geos ta- tis ti cal cour ses at Stan fo rd Uni ver si ty. Mo reo ver, RIPLEY (1981) showed the li nk be tween the mean squa res met hod and the li near Gaus sian mo del. Ten yea rs la ter CRESSIE (1991) des cri bed geos ta tis ti cs as one of the three main fi el ds in spa tial sta tis ti cs. ISAAKS & SRIVASTAVA (1989) pub lis- hed their we ll ac cep ted ea sy-lear ni ng geos ta tis ti cs book for re ser voir cha rac te ri za tion. DEUTSCH & JOURNEL (1997) pro du ced their geos ta tis ti cal lib ra ry for the mo st fa mous geos- ta tis ti cal prog ram GSLIB. The re are a lot of ot her excel le nt boo ks and pa pe rs. Geos ta tis ti cs re cei ved ve ry stro ng in put from mo de rn re- ser voir cha rac te ri za tion in the ni ne tee n-eig hties. The se re sul ts im pro ved our knowled ge and vi sua li za tion about the sub sur- fa ce, es pe cial ly in the in te r-we ll area. We ll and seis mic da ta can be ap plied in geos ta tis ti cal mo- de ls of any ki nd (map pi ng, stoc has tic or in di ca tor si mu la tio- ns). In ge ne ral, geos ta tis ti cal re sul ts can be di vi ded in to two grou ps – de ter mi nis tic and stoc has tic. The geo lo gi cal mo del can be defi ned as de ter mi nis tic if the sa me in pu ts always gi ve the sa me out put, e.g. for the sa me da ta set, usi ng iden ti cal in- ter po la tion met ho ds, the in te r-we ll area wi ll be map ped con- sis ten tly ea ch ti me. The mo del is stoc has tic if the in te r-we ll area, es pe cial ly far away from ha r d-da ta, is des cri bed by si mi- lar but dif fe re nt so lu tio ns ea ch ti me. Hypot he ti cal ly, the ran- dom mo de ls in ea ch ca se wi ll yie ld dif fe re nt re sul ts, but this is to tal ly use le ss and can be con si de red on ly as a pos si bi li ty in un con di tio nal si mu la tio ns. Ac cor di ng to the clas si fi ca tion of JENSEN et al. (2000) re ser voi rs can be con si de red as: 1. Com ple te ly de ter mi nis tic re ser voi rs whe re the in te r- we ll area is we ll known, cor re la ted and des cri bed by pa laeoen vi ron men ts. Howe ver, su ch lo ca li ties are ra re, and mos tly in clu de ma tu re fi el ds, wi th a lot of wel ls. The se are of in te re st on ly in the de ve lop me nt sta ge or when sear chi ng for »by-pas se d« oil; 2. Stoc has tic re ser voi rs are tho se that in clu de so me un cer- tain ties, but whe re the ar chi tec tu re is mos tly known at the mac ro-sca le. Howe ver, the in te r-we ll area can not be des cri bed for cer tain, and im pro ved in ter po la tion met ho ds, li ke geos ta tis ti cs, are ve ry de si rab le. 3. Ran dom re ser voi rs are ve ry ra re and are mos tly not a goal of geo lo gi cal mo del li ng. The re are on ly se ve ral wel ls and/or gra vi met ric and mag ne to met ric da ta that cou ld be des cri bed by Mon te Car lo sam pli ng or by ana- lo gy wi th si mi lar pros pec ts. The geos ta tis ti cal de ter mi nis tic ap proa ch in clu des dif fe re- nt kri gi ng and cok ri gi ng met ho ds. A stoc has tic ap proa ch al so en com pas ses dif fe re nt met ho ds that can rep re se nt the re ser- voir throu gh a set of rea li za tio ns. For the mo st pa rt, the stoc- has tic ap proa ch can be mo re des crip ti ve than de ter mi nis tic met ho ds. Howe ver, the re is al so a se co nd ad di tio nal sour ce of in for ma tion, whi ch can im pro ve any re ser voir pic tu re whet her the de ter mi nis tic or stoc has ti cal ap proa ch has been used. The- se are seis mic at tri bu tes. Cok ri gi ng is a de ter mi nis tic ap proa- ch that uses seis mi cs as a secon da ry sour ce of in for ma tion, des cri bi ng the in te r-we ll area in mo re de tail than has been pos- sib le usi ng kri gi ng. Stoc has tic si mu la tio ns ba sed on cok ri gi ng as ze ro-rea li za tion, al so use seis mic at tri bu te(s) as se con da ry sour ces of in for ma tion. The con sta nt »prob le m« is the re lia bi- li ty of seis mic in pu ts, i.e. whet her they cor re la te wi th we ll da- ta or ju st show a tre nd. Se ve ral pa pe rs re cen tly pub lis hed in Croa tian jour na ls, in- tro du ced geos ta tis ti cal theo ry to the Croa tian geo lo gi cal com- mu ni ty, and show the dif fe re nt met ho ds ap plied at se ve ral hydro- car bon fi el ds. In his dis ser ta tion, MALVIĆ (2003), used da ta from the Bje lo var sub dep res sion and ana lyzed it by expe ri men- tal va riog ra ms. MALVIĆ & ÐUREKOVIĆ (2003) com pa red the re sul ts of po ro si ty map pi ng in li mi ted in put da ta se ts ob tai- ned by three dif fe re nt met ho ds – In ver se Dis tan ce Weig hti ng, Or di na ry Kri gi ng and Col lo ca ted Cok ri gi ng. The sum ma ry re- sul ts of po ro si ty map pi ng of the enti re wes te rn pa rt of the Dra- va dep res sion (Mol ve, Ka li no vac and Sta ri Gra dac) we re pub- lis hed by MALVIĆ (2005). Fi nal ly, the geos ta tis ti cal ap proa ch had been con nec ted to de po si tio nal mo de ls and po ro si ty dis tri- bu tion, (MALVIĆ, 2006). The fi r st stoc has tic ap proac hes we re ap plied by SMOLJANOVIĆ & MALVIĆ (2005). The ever pre se nt and le gi ti ma te en gi nee ri ng and scien ti fi c ques tion is, whi ch ap proa ch is mo re ap prop ria te – the de ter mi- nis tic li ke kri gi ng or cok ri gi ng or the stoc has tic that va ries ha- r d-da ta in sta tis ti cal ly al lowed li mi ts. The aim he re is to eit her an swer this ques tion, or pro vi de a mea ns of as ses si ng whi ch wou ld be the be st choi ce. 2. KRIGING AND COKRIGING THEORY The theo ry of va riog ram ana lysis has been explai ned in ma ny pa pe rs and boo ks, and rep re sen ts the ne ces sa ry in put for kri gi ng and cok ri gi ng inter po la tion met ho ds. De ter mi na tion of spa tial de pen den ce is mos tly do ne by a va riog ram fun ction cal cu la tion (Equa tion 1.0): h n z x z x h2 1 i i i n 1 $c = - + = _ ` `i j j9 C! (1.0) Whe re are: h2c _ i – va riog ram n – num ber of da ta pai rs com pa red on h dis tan ce z xi ` j– va riab le va lue on cho sen lo ca tion (xi) z x hi +` j – va riab le va lue on the lo ca tion wi th »h« dis- tan ce from the ini tial lo ca tion (xi+h) Va riog ram va lue de pen ds on ly on the spa tial dis tri bu tion of lo ca tio ns, e.g. on the num ber of known va lues over a cho sen dis tan ce. The re sul ts are expe ri men tal va riog ram cur ves, des- cri bed wi th theo re tic mo de ls, whi ch are the in put for geos ta tis- ti cal es ti ma tion met ho ds. The ba sic geos ta tis ti cal es ti ma tion theo ry is cal led kri gi ng, whi ch can be exten ded to cok ri gi ng and stoc has tic met ho ds. In ge ne ral, kri gi ng tec hniques are cal- led the be st li near un bia sed es ti ma to rs (ab br. BLUE), exce pt 05 Malvic.indd 3805 Malvic.indd 38 24.6.2008 11:10:1824.6.2008 11:10:18 Geologia Croatica 39 Tomislav Malvić: Kri gi ng, cok ri gi ng or stoc has ti cal si mu la tio ns, and the choi ce be tween de ter mi nis tic or sequen tial ap proac hes for Sim ple Kri gi ng whi ch is on ly the be st un bia sed es ti ma tor. The be st shows that weig hti ng coef fi cien ts are cal cu la ted from mi ni mum kri gi ng va rian ce, li near mea ns that es ti ma tion is do- ne by li near com bi na tion of ha r d-da ta, un bia sed es ti ma tion as- su mes that expec ta tion va lue is true for the en ti re ima gi na ry po pu la tion and es ti ma tion is met ho do lo gy. The ge ne ral kri gi ng for mu la is shown in Equa tion 1.1: z z xk i i n i 1 #m= = ` j! (1.1) The re gio na li zed va riab le is es ti ma ted at lo ca tion (Zk) from exis ti ng ha r d-da ta Z xi ` j at lo ca tion xi accom pa nied by a true weig hti ng coef fi cie nt (li ). It al so mea ns that in put da ta are cha rac te ri sed by a Gaus sian dis tri bu tion (whi ch is of ten not the ca se, and da ta are tran sfor med in a Gaus sia n-li ke or stan da rd nor mal pro ba bi li ty dis tri bu tion). Fi nal kri gi ng va lues are al so shown by a nor mal dis tri bu tion. This sim ple for mu la gi ven in Eq. 1.1 can be exten ded in kri gi ng mat rix equa tio ns that can be symbo li cal ly shown as Equa tion 1.2: W B# m =6 6 6@ @ @ (1.2) This is sim ple mat rix for mu la that des cri bes all kri gi ng tec hniques li ke Sim ple Kri gi ng, Or di na ry Kri gi ng, In di ca tor Kri gi ng, Uni ver sal Kri gi ng, Dis jun cti ve Kri gi ng and ot he rs. The cal cu la tion of the weig hti ng coef fi cie nt is the main ta sk of the kri gi ng equa tio ns. The hig her va lues wi ll be cal cu la ted for the ha r d-da ta lo ca ted at the prin ci pal axis of the ani sot ro- py. The se coef fi cien ts al so can be ne ga ti ve, but the con di tion 1m =! is sa tis fi ed for Or di na ry Kri gi ng that is used in this ana lysis. Mo reo ver, va riog ram ana lysis can poin ted at geo lo gi cal ele men ts li ke struc tu ral axes, de po si tio nal chan ne ls etc. The weig hti ng coef fi cie nt va lues fi nal ly de pe nd on ly on dis tan ces amo ng ha r d-da ta in clu ded in an el lip soid of spa tial de pen den- ce. Howe ver, ad van ta ges of geos ta tis ti cs can not be con fi r med in ca ses of ve ry li mi ted in put da ta se ts. Su ch a li mit can be set at 10 or 15 ha r d-da ta poin ts, be cau se smal ler da ta se ts can not be des cri bed by a va riog ram mo del. Cok ri gi ng, li ke kri gi ng, al so in clu des se ve ral es ti ma tion tec hniques, mos tly iden ti cal kri gi ng tec hniques (Sim ple Cok- ri gi ng, Or di na ry Cok ri gi ng etc.). The on ly con di tion is the in- tro duc tion of a se con da ry va riab le (or co-va riab le) that is: (a) in mea nin gful con nec tion wi th the pri ma ry va riab le, (b) sam- pled at ma ny lo ca tio ns and (c) sig ni fi can tly cor re la ted wi th the pri ma ry va riab le. The mat he ma ti cal exten sion is ve ry sim- ple, de ri ved from Eq. 1.1 and is gi ven in Equa tion 1.3: Z z x w yi i n i j j n j 1 1 z w $ $m |= + = = ` aj k! ! (1.3) Whe re: Z z xi i n i 1 z $m= = ` j! = the pa rt ap plied to the pri ma ry va riab le, w yj j n j 1 w $| = a k! = the pa rt ap plied to the se con da ry va- riab le. Cok ri gi ng met ho ds and co-re gio na li zed va riab les are re la ted in the sa me way as the theo ry of re gio na li zed va riab les and kri- gi ng. Co re gio na li za tion des cri bes the spa tial va ria tion of se ve- ral va riab les. This is exten ded from the theo ry of kri gi ng and a re gio na li sed va riab le to se ve ral cok ri gi ng va riab les. The se va riab les ha ve a mul ti va ria te spa tial cro ss-cor re la tion as we ll as the uni va ria te spa tial au to-cor re la tion. They are then cal led co-re gio na li zed va riab les and can be used to un der sta nd spa- tial pro ces ses that ma ni fe st them sel ves as se ve ral mea su red va riab les (ROSSITER, 2005). Two or mo re de pen da nt va riab- les can al so be in ter po la ted from col lo ca ted ob ser va tio ns usi- ng Reg res sion Kri gi ng or Kri gi ng wi th Exter nal Dri ft, but this is not dis cus sed he re. Mo re of ten the re are mo re ob ser va tio ns of the co-va riab le, i.e. at so me poin ts whe re the tar get va riab- le was not mea su red and requi res co re gio na li za tion. A typi cal exam ple is seis mic sig na ls re cei ved from lit ho lo gi cal boun da- ries and sec tio ns of a hydro car bon re ser voir. Co re gio na li za tion is a ge ne ral te rm for the theo re ti cal mo del of how se ve ral va riab les spa tial ly co-va ry. The idea is that the pro ce ss that gi ves ri se to one va riab le is the sa me, or at lea st re la ted, to the pro ce ss that is ma ni fe st in the ot her va- riab les. All the va riab les in vol ved are re gio na li zed va riab les and in ad di tion they are re la ted bo th in fea tu res and geog rap- hic spa ce, hen ce they are co-re gio na li zed. The pro ce du re of co re gio na li za tion can be refl ec ted in the va riog ram cal cu la- tion, whi ch mea ns that the re cou ld be two types of va riog ra ms mo del led: (a) di rec tio nal or om ni di rec tio nal va riog ra ms cal cu- la ted for a sin gle re gio na li zed va riab le, i.e. one va riog ram per va riab le (see Eq. 1.1) or (b) cro ss va riog ra ms cal cu la ted for the pair of re gio na li zed va riab les e.g. as des cri bed in ROS- SITER, (2005). From Equa tion 1.3, de ter mi na tion of weig htin gs for λ (the tar get va riab le) and χ (for the co-va riab le), mi ni mi zes the pre- dic tion er ror. It is why cok ri gi ng can be des cri bed, li ke kri gi ng, by the ac ro nym BLUE (be st li near un bia sed es ti ma tor). This sho rt re view of de ter mi nis tic geos ta tis ti cal too ls is on ly a mi ni mal sum ma ry ne ces sa ry for un der stan di ng the re- sear ch des cri bed in this pa per. Mo re de tai ls can be fou nd in ma ny pub li ca tio ns in clu di ng GOOVAERTS (1997), HOHN (1988), ISAAKS & SRIVASTAVA (1989), JOURNEL & HU- IJBREGTS (1978), LIEBHOLD et al. (1993), XU et al. (1992) and ot he rs. 3. THE CONCEPT OF STOHASTIC SIMULATIONS Si mu la tio ns, wri ti ng li te ral ly, pre ser ve the fl a vour of real sub sur fa ce va ria bi li ty, map pi ng all pos si bi li ties be tween the extre me va lues of the re gio na li zed va riab le. If si mu la tion is »con di tio na l«, it ho nou rs the ob ser ved da ta exac tly, wit hout the at ten da nt smoot hi ng of the in ter po la ted es ti ma tes, as in kri gi ng. Si mu la tion pro vi des vi sual and quan ti ta ti ve mea su res of the un cer tain ty of the es ti ma ted va riab le (»co si mu la tio n«, ana lo gous to cok ri gi ng). Da ta are ho nou red whe re the un der- lyi ng tre nd or pat te rn is rep ro du ced (e.g. as in kri gi ng), whe reas da ta are rep ro du ced whe re pre dic ted va lues are for ced to be equal to ob ser ved va lues. The re is, of cour se, a dif fe ren ce be tween si mu la tio ns and kri gi ng. Kri gi ng pro du ces ju st one map of es ti ma tes whi ch is 05 Malvic.indd 3905 Malvic.indd 39 24.6.2008 11:10:1924.6.2008 11:10:19 Geologia Croatica Geologia Croatica 61/1 40 »bes t« (a nd hen ce unique) in so me sta tis ti cal sen se, but whi ch does not rep ro du ce glo bal sta tis ti cs (his tog ram, va rian ce, co- va rian ce). Kri gi ng is a glo bal es ti ma tor, and its es ti ma te rep- re sen ts all the da ta wit hin a de fi ned va riog ram el lip soid. The kri ged grid ho nou rs da ta poi nt-va lues on ly if the par- ti cu lar poi nt coin ci des wi th a grid no de. The kri gi ng map is use ful to show smoo th va ria tio ns and un der lyi ng tren ds, but not the in he re nt lo cal va ria bi li ty of the da ta. Kri gi ng pro vi des an in com ple te mea su re of lo cal ac cu ra cy (i.e. kri gi ng »va rian- ce« may not rep re se nt a Gaus sian dis tri bu tion of er ro rs), and no mea su re of joi nt ac cu ra cy when se ve ral lo ca tio ns are con- si de red to get her. All the se fac ts are al so va lid for cok ri gi ng tec hniques. The si mu la tion is al so a lo cal es ti ma tor. The goal is the rep ro duc tion of glo bal pat ter ns of spa tial con ti nui ty and glo- bal sta tis ti cs (his tog ram, co va rian ce), rat her than lo cal ac cu ra- cy. Si mu la tion is used to de mon stra te the na tu re of lo cal va ria- bi li ty and glo bal pat ter ns of lo cal va ria bi li ty, su ch as the con nec tio ns and tren ds amo ng dif fe ren tial ly va lued re gio ns. The si mu la tion pro du ces any num ber of sta tis ti cal ly equi va le- nt ma ps whi ch, when ta ken to get her, de fi ne the lo cal es ti ma- tion un cer tain ty as we ll as the glo bal pat te rn of un cer tain ty. Un der ta ki ng si mu la tion, se ve ral ap proac hes can be se lec ted (WELHAN, 2001): • Si mu la ted va riab les can be con ti nuous or ca te go ri cal; • Con di tio nal si mu la tion rep ro du ces the ob ser ved da ta; • Un con di tio nal si mu la tion on ly rep ro du ces glo bal sta tis ti cs (his tog ram, va riog ram, etc.); • Ca te go ric va riab les can be si mu la ted to honour spe ci- fi c geo met ric pat ter ns (ob je ct-ba se d); • Con ti nuous va riab les can be si mu la ted to ho nour a spe- ci fi ed co va rian ce mo del; • Va riab les can be si mu la ted throu gh op ti mi za tion pro- ces ses star ti ng from scrat ch or from pre vious ly-si mu la- ted ima ges, to ho nour ad di tio nal, exter nal, con strai ni ng da ta. Gaus sian Si mu la tion, usi ng con ve nien ce (bo th theo re ti- cal and prac ti cal) wi th whi ch Gaus sian dis tri bu tio ns can be des cri bed and han dled sta tis ti cal ly ma kes a Gaus sian ran dom fun ction mo del ve ry ap pea li ng. This cla ss of si mu la tion al go- rit hms is res tric ted to con ti nuous va riab les. Gaus sian si mu la- tion can be ap plied if the pro ba bi li ty dis tri bu tion fun ctio ns (PDF’s) of all va riab les that par ti ci pa te in the si mu la tion, (e.g. a re gio na li zed va riab le plus its lo cal un cer tain ty or va rian ce) are nor mal ly dis tri bu ted. The mo st im por ta nt con di tion is that the va riab les are uni va ria te and bi va ria te nor mal and in de pen- de nt. The as sum ption of nor ma li ty is not res tric ti ve for rea l- wor ld da ta, sin ce a nor ma l-sco re tran sfo rm can be ap plied to ma ke any da ta set nor mal ly-dis tri bu ted. The mo st se rious li mi- ta tion for real da ta is that a re gio na li zed va riab le at a gi ven lo- ca tion is of ten not in de pen de nt. It may be cor re la ted wi th its lo cal va rian ce, (via a pro por tio nal ef fe ct) and/or it may ha ve stro ng lo cal spa tial cor re la tion, (e.g. in ter con nec ted hi gh or low va lues). WELHAN (2001) in di ca ted stro ng lo cal spa tial cor re la tion as the ma jor pos sib le prob lem, and re com men ded a met hod des cri bed in DEUTSCH & JOURNEL (1997) for chec ki ng and ne ces sa ry ap pli ca tion if the Gaus sian si mu la tion ap proa ch is to be cor rec tly de fen ded. The pro por tio nal ef fe ct can be neg lec ted if it is weak or is re mo ved by mo vi ng the va- riog ram el lip soid (sear ch neig hbor hood). Des pi te su ch ca vea- ts, the Gaus sian si mu la tion ap proa ch is wi de ly used; its in tui- ti ve and al go rit hmic sim pli ci ty de fi ne its wi des pread ap peal. Sequen tial Si mu la tion Al go rit hm (e.g. DEUSTSCH & JOURNEL, 1997) con tai ns a po pu lar cla ss of al go rit hms that is in wi des pread use. Su ch an approa ch is ve ry of ten in the Gaus sian ver sion as Sequen tial Gaus sian Si mu la tion, but can al so be ap plied in si mu la tio ns of ot her types. The sequen tial si mu la tion ap proa ch can be bro ken down con cep tual ly in to three par ts (WELHAN, 2001): • The ini tia li za tion pro ce ss; • A glo bal ran dom wa lk pro ce ss and • A lo cal sear ch and con di tio nal ran dom es ti ma tion pro- ce ss. Ini tia li za tion de ter mi nes the uni va ria te cu mu la ti ve dis tri- bu tion fun ction (CDF) from the en ti re stu dy area, usi ng dec- lus te ri ng if ne ces sa ry. A nor ma l-sco re tran sfo rm of the da ta usi ng a stan da r d-nor mal CDF mu st be per for med, and a re gu- lar grid ne two rk of no des de fi ned, at whi ch es ti ma tes wi ll be ma de. Da ta poin ts may be re-lo ca ted to the se grid no des (for fas ter com pu ta tion) or not (for maxi mum ac cu ra cy). Typi cal- ly, this grid ne two rk wi ll be mu ch fi ner than was used in kri gi- ng, be cau se si mu la tion at tem pts to rep ro du ce the na tu re of lo- cal va ria bi li ty and un cer tain ty. The Glo bal ran dom wa lk pro ce ss uses a ran dom num- ber seed. The es ti ma tion star ts at a ran dom lo ca tion on a grid ne two rk wi th the ran dom num ber ge ne ra tor, mo ves to sub- seque nt ran dom lo ca tio ns af ter a local sear ch has been com- ple ted, but ne ver vi si ts the sa me grid no de twi ce. The es ti ma- ted da ta are con si de red as the new ha r d-da ta va lues. Lo cal sear ch and con di tio nal es ti ma tion pro ce ss loo ks wit hin a de sig na ted sear ch neig hbor hood for a pres cri bed num- ber of nea re st neig hbo rs, com pri si ng bo th ori gi nal da ta and/or pre vious ly si mu la ted va lues (WELHAN, 2001). It uses the va- riog ram mo del of nor ma l-sco re da ta wi th Sim ple Kri gi ng to es ti ma te the mean and va rian ce at the grid no de lo ca tion. This de fi nes the cu mu la ti ve dis tri bu tion fun ction (CDF) of the re- gio na li zed va riab le at that sear ch neig hbor hood lo ca tion. In the ca se of neit her ha r d-da ta or pre vious ly si mu la ted no des exis ti ng wit hin the sear ch neig hbour hood, the CDF is con di- tio nal on ly to the glo bal his tog ram. The va lue es ti ma ted at su- ch a grid no de wi ll be a va lue drawn at ran dom from the CDF es ti ma ted from the glo bal his tog ram (WELHAN, 2001). At the end of this pro ce ss, the si mu la ted nor ma l-sco re va lues are tran sfor med ba ck in to si mu la ted va lues for the ori gi nal va riab- le. The des cri bed al go rit hm is re pea ted for mul tip le rea li za tio- ns, star ti ng ea ch ti me at a dif fe re nt ini tial grid lo ca tion and vi- si ti ng grid no des in a dif fe re nt or der. The Gaus sian Si mu la tion ten ds to ove res ti ma te lo cal spa tial en tro py, crea ti ng mo re ap pa re nt va ria bi li ty than is ac- tual ly pre se nt, this bei ng the mo st ob vious li mi ta tion of si mu- la tion. It is al so not suf fi cien tly fl exib le to han dle a mixtu re of 05 Malvic.indd 4005 Malvic.indd 40 24.6.2008 11:10:1924.6.2008 11:10:19 Geologia Croatica 41 Tomislav Malvić: Kri gi ng, cok ri gi ng or stoc has ti cal si mu la tio ns, and the choi ce be tween de ter mi nis tic or sequen tial ap proac hes dif fe re nt nor mal po pu la tio ns. Fur ther mo re, sequen tial si mu la- tion does not al low the in cor po ra tion of ca te go ri cal da ta (e.g. 1 and 0), but this is over co me by in tro du ci ng an indi ca tor si- mu la tion al go rit hm. The Gaus sian sequen tial si mu la tion ap- proa ch is wi de ly used des pi te the se li mi ta tio ns. Anyo ne in te res ted in geos ta tis ti cal theo ry, bo th de ter mi- nis tic or stoc has tic, as ap plied in the Croa tian pa rt of the Pan- no nian Ba sin is ad vi sed to fol low the re sear ch pre sen ted in MALVIĆ (2003, 2005, 2006) or SMOLJANOVIĆ & MALVIĆ (2005). 4. EXAMPLE 1: COKRIGING OR STOCHASTIC CO-SIMULATION The Beni čan ci fi e ld is the lar ge st hydro car bon re ser voir in the Be ni čan ci oil zo ne, and the lar ge st oil pro du ci ng fi e ld in the Croa tian pa rt of the Dra va Dep res sion (Fi gu re 1). The oil re ser voir is lo ca ted in brec cia of Ba de nian age, and is over lain by struc tu ral ly shal lower and smal ler gas re ser voi rs. This fi e ld and the enti re zo ne lo ca ted in the eas te rn pa rt of the Dra va Dep res sion, has been stu died for de ca des. The dif fe re nt geo lo- gi cal and pet ro leum set tin gs ha ve been pub lis hed in ma ny pa pe rs. ŠIMON & BATUŠIĆ (1974) des cri bed the lit hos tra- tig rap hic sec tion of the Be ni čan ci fi e ld. TURAJLIĆ et al. (1979) com pa red the de ve lop me nt of the Be ni čan ci, Obod and La dis lav ci-Ku čan ci fi el ds. HERNITZ et al. (1993, 1995) des cri bed shal low on sho re clas ti cs of Mio ce ne age and sour ce roc ks in the wi der area of the Be ni čan ci fi e ld, as we ll as in ot her par ts of the Dra va Dep res sion. The de po si tio nal mo del of Neo ge ne clas ti cs was in de tai l described by TIŠLJAR (1993). Struc tu ral ly, the fi e ld is de fi - ned by brac hian tic li ne axes of 8.0 by 1.3 km, faul ted mos tly by dis pla ce men ts cha rac te ri sed by a NW-SE stri ke. Howe ver, the main re ver se fau lt clo ses the struc tu re in the sou th (Fi gu- re 2). The en ti re fi e ld is an antic li no rium that in clu des four smal ler an tic li nes, sin ki ng towa rd the ea st. The stra tig rap hy in clu des ba se men ts roc ks of Per mian and Trias sic age and Neo ge ne clas tic se di men ts. Dif fe re nt ba se me- nt schis ts are de ter mi ned on ly in wel ls dee per than 2500 met- res. The Mio ce ne stra tig rap hy is ve ry he te ro ge neous. The ol de- st roc ks are spo ra di cal ly rep re sen ted by ef fu si ves be lon gi ng to the Mio ce ne mag ma tic cycle, whi ch ha ve been par tial ly re-de- po si ted. Their la te ral equi va len ts are Trias sic li my brec cia and quar tzi tes that ha ve al so been re de po si ted. The main se di men- ta ry sequen ce is of Ba de nian age, rep re sen ted by do lo mi tic and li mes to ne brec cia. Det ri tus is mos tly do lo mi tic and can rea ch met re di men sio ns. The mat rix is al so of mic ro to crypto do lo- mi te, tec to ni cal ly crus hed, pet ri fi ed and crystal li zed. Re ser voir car bo na te brec cias are cha rac te ri sed by re la ti ve ly low pri ma ry po ro si ty, but se con da ry pro ces ses (in clu di ng dis so lu tion, faul- ti ng and frac tu ri ng) re sul ted in in crea sed se con da ry po ro si ty. The shal low pa rt of the re ser voir is cha rac te ri sed by a mixtu re of brec cia and con glo me ra te, wi th mixed car bo na te and si li cic- las tic det ri tus. In ge ne ral, Ba de nian brec cias we re de po si ted in a shal low ma ri ne en vi ron me nt, and con sequen tly cou ld be re- de po si ted and weat he red by cur ren ts and wa ve fron ts. TIŠLJAR (1993) des cri bed this Ba de nian se di men ta ry sequen ce throu gh four types of pos sib le re ser voi rs as fol lows: Fi gu re 1: Lo ca tion map of the Be ni čan ci and Sta ri Gra da c-Bar cs Nyu gat fi el ds in Dra va Dep res sion 05 Malvic.indd 4105 Malvic.indd 41 24.6.2008 11:10:1924.6.2008 11:10:19 Geologia Croatica Geologia Croatica 61/1 42 1. Cli no fo rm bo dies of car bo na te roc kfa ll and deb ri te brec- cias; 2. Near sho re con glo me ra tes; 3. Chan nel and fan bo dies ma de of brec cia-con glo me ra- tes and san dsto nes. 4. Tec to nic brec cias, ge ne ra ted by crus hi ng and tec to ni za- tion of pre vious ly lit hi fi ed Mio ce ne tur bi di te se di me ta- ry roc ks. The ob ser ved re ser voir mos tly be lon gs to type 4 and par- tial ly to type 1. Ba de nian se di men ts we re over lain by a typi cal, youn ger se di men ta ry sequen ce in the Dra va dep res sion. This be gan wi th Lower Pan no nian cal ci tic mar ls (»Croa ti ca bed s«), whi ch we re ero ded, on the sout he rn pa rt of the fi e ld. Then, La te Pan- no nian se di men ts (»Ba na ti ca bed s«) of si mi lar lit ho lo gy we re de po si ted, whi ch are in te res tin gly cha rac te ri sed by grea ter thic- knes ses in the sout he rn pa rt of the struc tu re. Ear ly Pon tian se- di men ts (»A bic hi bed s«) are mos tly san dsto nes (mixtu re det- ri tus) in ter ca la ted wi th ha rd mar ls. La te Pon tian (»R hom boi dea bed s«) se di men ts are mos tly weak san dsto nes and me dium to weak mar ls, so me ti mes clayey, in clu di ng coal. The youn ge st se di men ts of Plio ce ne age are mos tly un con so li da ted clay and sa nd, as we ll as Qua ter na ry gra vel, sa nd and clay wi th li mes- to ne con cre tio ns. The tar get, or pri ma ry, va riab le that had been geos ta tis ti- cal ly ana lyzed in the brec cia re ser voir of Ba de nian age was the ave ra ge re ser voir po ro si ty. Four teeen lo ca tio ns (or wel ls) we re cho sen as the mo st rep re sen ta ti ve ha r d-da ta poin ts, de ri- ved from the newe st lo g-cur ve ana lyses. The se va lues are mos- tly re gu lar ly dis tri bu ted ac ro ss the re ser voir area, and can be ob ser ved on the po ro si ty ma ps. Mo reo ver, the ana lysed re ser voir is al so co ve red by 3D seis mic cu be and in ter pre ted at tri bu tes. At tri bu te ana lysis was tar ge ted for the inter val that rep re sen ts the re ser voir de fi ned by per meab le lit ho lo gy. It be gi ns 20 m from the seal ro ck in the youn ge st Ba de nian se di men ts and con ti nues to eit her the re ser voir ba se de fi ned by Pa laeo zoic ba se me nt roc ks, or the ba se of the we ll whe re the ba se me nt roc ks we re not dril led. Seis mic at tri bu te ana lysis en com pas ses am pli tu de at tri bu tes, frequen cy and pha se. The exis ten ce of seis mic at tri bu tes ma de ap pli ca tion of geos ta tis ti cal in ter po la tion of the cok ri gi ng pos sib le. The co- k ri gi ng requi res the de ter mi na tion of an ad di tio nal, se con da ry va riab le that des cri bes the be ha viour of the pri ma ry one. Six at tri bu tes we re cor re la ted wi th po ro si ty, sear chi ng for a con- nec tion. Un for tu na te ly, the va lues of the Pear son cor re la tion coef fi cie nt did not show any sig ni fi ca nt cor re la tion, due to the absen ce of a Gaus sian dis tri bu tion in the ana lyzed va riab les. It was mo re ap prop ria te to use the no n-pa ra met ric Spear man ra nk cor re la tion coef fi cie nt, ba sed on a median va lue (in stead of nor mal dis tri bu tion pa ra me te rs li ke stan da rd de via tion and mean). The hig he st cor re la tion was cal cu la ted for the pair po- ro si ty-refl ec tion stren gth (as a de ri va ti ve of am pli tu de) whe re the va lue of r’ = – 0.64. The re le va nt va riog ram mo del had been de fi ned alo ng fi e- ld struc tu ral axes. The di rec tion of the pri ma ry va riog ram ax- is was 90-270o (E-W), whi le the se con da ry sprea ds oc cur red alo ng the di rec tion 0-180o. Due to the low num ber of da ta, the an gle to le ran ce was set at a maxi mum va lue of 45o. The ran- ges are al mo st isot ro pic – for the main axis the ran ge is 1750 met res de ter mi ned from 7 or mo re da ta pai rs. The ran ge of the se con da ry axis is 1500 met res, de ter mi ned sub jec ti ve ly con si- de ri ng on ly 5 or le ss da ta pai rs, and con sequen tly on ly 4 exis- ti ng poin ts. Expe ri men tal va riog ra ms had been ap proxi ma ted by sphe ri cal theo re ti cal mo de ls. The cok ri gi ng re sul ts are pre sen ted in Fi gu re 3b. The se- lec ted tec hnique was col lo ca ted cok ri gi ng. The wo rd »col lo ca- te d« mea ns that the pri ma ry and se con da ry va riab les we re ob- ser ved at the sa me lo ca tio ns, i.e. the se con da ry va riab le had been extra po la ted at the sa me lo ca tio ns whe re the ha r d-da ta exi- st. The cok ri gi ng re sul ts we re sig ni fi can tly bet ter and mo re des- crip ti ve in the in te r-we ll areas than the kri gi ng re sul ts (MALVIĆ & ÐUREKOVIĆ, 2003). This cou ld ea si ly be con clu ded from re sul ts of the cro ss-va li da tion met hod (DAVIS, 1987), whe re the cok ri gi ng map was cha rac te ri sed by a mean squa re er ror of 2.19 and the kri gi ng map by the va lue of 2.97. Fi gu re 2: Palaeo re lief map of the Be ni čan ci fi e ld (after FUTIVIĆ & PLEIĆ, 2002) 05 Malvic.indd 4205 Malvic.indd 42 24.6.2008 11:10:2024.6.2008 11:10:20 Geologia Croatica 43 Tomislav Malvić: Kri gi ng, cok ri gi ng or stoc has ti cal si mu la tio ns, and the choi ce be tween de ter mi nis tic or sequen tial ap proac hes Al thou gh the to tal dif fe ren ce be tween the kri gi ng and cok- ri gi ng ma ps (Fi gu re 3a and 3b), in a nu me ri cal sen se, is about 25%, this dif fe ren ce is mos tly re gu lar ly dis tri bu ted ac ro ss the en ti re re ser voir area. The se con da ry ap proa ch for po ro si ty dis tri bu tion at the sa me re ser voir in clu ded usi ng Sequen tial Gaus sian Si mu la- tio ns (e.g. DUBRULE, 1998 and KELKAR & PEREZ, 2002), as an ap proa ch that cou ld in clu de the ma jo ri ty of un cer tain ties con nec ted to the in put da ta and in te r-we ll area, es pe cial ly in zo nes dis ta nt from ha r d-da ta. A set of on ly four stoc has ti cal equal ly pro bab le rea li za tio ns we re cal cu la ted, whi ch we re fou- nd to expre ss the noi sy na tu re of the ha r d-da ta. Ge ne ral ly, the- re mu st be at lea st 100-200 rea li za tio ns for ac cep tab le mea su- re me nt of the deg ree of un cer tain ty. The se rea li za tio ns we re ba sed on a cok ri gi ng map as »ze ro« rea li za tion. Due to stoc has tic theo ry, da ta we re nor ma li zed to Gaus- sian dis tri bu tio ns and their PDF cur ve was used to sam ple new po ro si ty va lues. The new, da ta set ke pt the sa me sta tis ti cs cha- rac te ris tic as the in put da ta set. Ea ch ce ll has a po ro si ty va lue sam pled from its own nor mal ly sha ped PDF de fi ned wi th mean and va rian ce va lues cha rac te ris tic for the cok ri gi ng »ze- ro« rea li za tion (e.g. DEUTSCH, 2002). The or der of the si mu- la ted cel ls was ran dom and sequen tial. The »extre me« rea li za tion P1 and P99 are shown on Fi- gu re 4. The symbol P1 mea ns that 99 % of all ot her pos sib le rea li za tio ns wou ld des cri be lar ger to tal re ser voir po ro si ty. The mea ni ng of P99 rea li za tion is the op po si te. The com pa ri son by cok ri gi ng map, or ze ro »so lu tio n«, re vea ls that the P99 rea- li za tion loo ks ve ry si mi lar to the cok ri gi ng map. Fur ther mo re, it can be con clu ded that the cok ri gi ng map is a ve ry good rep- re sen ta tion of po ro si ty dis tri bu tion, and the stoc has tic so lu tion sug ges ts that map ped un cer tain ties in the in te r-we ll area can be con si de red pret ty ho mo ge neous for the en ti re sta tis ti cal ran- ge. Due to the fa ct that Gaus sian si mu la tion ove res ti ma tes lo- cal va ria bi li ty, to get her wi th the ob ser ved si mi la ri ty be tween the de ter mi nis tic and stoc has tic rea li za tio ns in the ca se of the exis ti ng seis mic at tri bu te and li mi ted da ta set (about 15 ha r d-da- ta), the cok ri gi ng ap proa ch is to be re com men ded rat her than the use of Gaus sian si mu la tio ns. 5. EXAMPLE 2: KRIGING OR STOCHASTIC SIMULATION The Sta ri Gra da c-Bar cs Nyu gat ga s-con den sa te fi e ld is lo ca- ted on the Croa tia n-Hun ga rian bor der, alo ng the Dra va Ri ver, ap proxi ma te ly 150 km ea st of Zag reb. The fi e ld is si tua ted in Fi gu re 3: Kri gi ng (Fi gu re 3a – abo ve) and cok ri gi ng (Fi gu re 3b – be low) ma ps of po ro si ty dis tri bu tion 05 Malvic.indd 4305 Malvic.indd 43 24.6.2008 11:10:2124.6.2008 11:10:21 Geologia Croatica Geologia Croatica 61/1 44 the Nor thwes te rn pa rt of the Dra va dep res sion (Fi gu re 1). A mas si ve (an tic li ne) re ser voir is trap ped by a com bi ned struc tu- ra l-stra tig rap hic clo su re (Fi gu re 5). The lit ho lo gy of the re ser- voir is ve ry com plex, di vi ded in to four lit ho fa cies: ‘Clas ti tes’ of Ba de nian age, ‘Do lo mi tes’ of Lower Trias sic age, ‘Quar tzi- tes’ of Lower Trias sic age and ‘Me ta vol ca ni tes’ of Per mian, De vo nian and (pos sib ly) Car bo ni fe rous age. The re sul ts of geos ta tis ti cal po ro si ty map pi ng in lit ho fa- cies ‘Clas ti tes’ of Ba de nian age that in clu ded mos tly car bo na- te brec cia are ana lysed he re. The se re ser voir lit ho fa cies are lit ho lo gi cal ly com pa rab le to coar se-grai ned Bade nian lit ho fa- cies ana lysed from the Be ni čan ci fi e ld. The in put da ta set was re la ti ve ly sma ll in num ber, whi ch con sequen tly exclu ded the exis ti ng arit hme tic mean as bei ng va lid for a glo bal mean (i.e. exclu ded con di tion of fi r st-or der sta tio na ri ty ne ces sa ry for Sim ple Kri gi ng). Or di na ry Kri gi ng was se lec ted as the mo st ap prop ria te in ter po la tion tec hnique. The re was no es tab lis hed cor re la tion be tween seis mic at tri bu- tes and po ro si ty. The va riog ram ana lysis was do ne again alo ng the main struc tu ral axes of the fi e ld (Fi gu re 5). The prin ci pal va riog ram axis has a stri ke of 120-300o and the su bor di na te stri ke is 30- 210o. The expe ri men tal va riog ra ms we re cal cu la ted for ea ch lit ho fa cies, and par ti cu lar ly for the ‘Clas ti tes’. The ran ges are 3500 met res (prin ci pal) and 2000 met res (su bor di na te axis). The prin ci pal axis mo del li ng is fair ly re liab le ba sed on 8 or mo re da ta pai rs. The mo del li ng of the se con da ry axis is hig hly sub jec ti ve. All va riog ra ms are again ap proxi ma te by sphe ri cal theo re ti cal mo de ls. The kri gi ng map shows a sig ni fi ca nt in ter po la tion im pro- ve me nt com pa red to tho se pre vious ly pro du ced by In ver se Dis- tan ce Weig hti ng (Fi gu re 6). If the se two so lu tio ns are com pa red nu me ri cal ly throu gh cro ss-va li da tion re sul ts, the im pro ve me nt is mo re than 30% – for kri gi ng map 3.91 and for In ver se Dis tan- ce Weig hti ng map 5.28. It is not dif fi cu lt to explain why the kri gi ng map is mo re ac cu ra te and ac cep tab le for the geo lo gi cal mo del. It is clear that the In ver se Dis tan ce Weig hti ng map (Fi gu re 6, le ft) is un- rea lis ti cal ly too de tai led. Ba sed on on ly 14 ha r d-da ta poin ts, the smoot hed kri gi ng map (Fi gu re 6, rig ht) is mo re rea lis tic and in ter pre tab le. Howe ver, the chal len ge is to try to des cri be in te r- we ll area un cer tain ties by stoc has tic si mu la tion as we ll. Usi ng res pec ted ha r d-da ta poin ts, i.e. con di tio nal si mu la tio ns, 100 si- mu la tio ns we re cal cu la ted des cri bi ng re ser voir ran dom ne ss (Fi- gu re 7). The re ser voir spa ce is always cha rac te ri sed by un cer tain- ties. The per ma ne nt prob lem is how to expre ss the se »hid de n« cha rac te ris ti cs in the map ped va riab le. This is es pe cial ly em- pha si sed in the ca se of a ve ry li mi ted in put da ta set. In this ca- se, for the ‘Clas ti tes’ lit ho fa cies at the Sta ri Gra da c-Bar cs Nyu- Fi gu re 4: P1 (Fi gu re 4a – abo ve) and P99 (Fi gu re 4b – be low) equip ro bab le stoc has tic rea li za tio ns 05 Malvic.indd 4405 Malvic.indd 44 24.6.2008 11:10:2224.6.2008 11:10:22 Geologia Croatica 45 Tomislav Malvić: Kri gi ng, cok ri gi ng or stoc has ti cal si mu la tio ns, and the choi ce be tween de ter mi nis tic or sequen tial ap proac hes gat fi e ld, the stan da rd de via tion of in put po ro si ty da ta set was hig her than 0.5 of the pos sib le maxi mum. The sta tis ti cal un cer- tain ty is mos tly in clu ded in the variog ram mo del, for ci ng the lar ge pri ma ry ran ge and es ti ma ted ran ge of the su bor di na te axis from the ra tio of the struc tu ral axes. The se fac ts ju st in di- ca te a stoc has tic ap proa ch as the mo st fa vou rab le for in te r-we- ll area des crip tion. The dif fe ren ces in the set of si mu la ted rea- li za tio ns (dif fe ren ce be tween P1 and P99) we re about 14% (or mi ni mum vo lu me of ‘Clas ti tes’ was 5,241,097 and maxi mum 5,973,280 m3). Mo reo ver, it is im por ta nt to em pha sis that bo- th kri gi ng and stoc has tic rea li za tio ns led to a dec rea se in re ser- voir vo lu me, over that pre vious ly cal cu la ted by In ver se Dis tan- ce Weig hti ng. In any ca se, whe re the in put da ta set is ve ry li mi ted, wi th the absen ce of an ad di tio nal, sig ni fi can tly cor re- la ted seis mic va riab le, the stoc has tic ap proa ch is pre fer red. The Sta ri Gra da c-Bar cs Nyu gat da ta set is ve ry li mi ted and it is ha rd to des cri be lo cal va ria bi li ty. A stoc has tic ap proa- ch is mo re des crip ti ve and al lows geo lo gi st to spe cu la te about fea tu res that in fl uen ce po ro si ty dis tri bu tion. Fi nal ly it led to bet ter cha rac te ri sa tion of po ro si ty in the in te r-we ll area. 6. CONCLUSIONS Po ro si ty in ter po la tion re sul ts we re com pa red at two si mi lar re ser voi rs from the Be ni čan ci and Sta ri Gra da c-Bar cs Nyu gat fi el ds. Bo th are lo ca ted in the Dra va Dep res sion, and oc cur in coar se-grai ned se di men ts (car bo na te brec cia) of Ba de nian age. The re sear ch goal was to es tab li sh qua li ta ti ve cri te ria to aid the Fi gu re 5: Struc tu re map of the top of the ‘Clas ti te s’ lit ho fa cies (after GAĆEŠA at al., 20013) Fi gu re 6: Sche ma tic po ro si ty ma ps in ter po la ted by In ver se Dis tan ce Weig hti ng (le ft) and kri gi ng (rig ht) 2 GAĆEŠA, S., FUTIVIĆ, I, FERENCZ, G. & HORVATH, Z. (2001): Barcs Nyugat-Stari Gradac fi eld; Geological evaluation-summary study. Unpublished report, company internal fi les, INA Naftaplin, Sektor za razradu, 27 p. 05 Malvic.indd 4505 Malvic.indd 45 24.6.2008 11:10:2324.6.2008 11:10:23 Geologia Croatica Geologia Croatica 61/1 46 se lec tion of eit her de ter mi nis tic or stoc has tic geos ta tis ti cal ap proac hes for po ro si ty in ter po la tion in ana lyzed re ser voi rs. The fi r st ca se in clu ded use of an ad di tio nal seis mic va riab- le and con sequen tly the cok ri gi ng met hod. The se co nd exam- ple was orien ted on ly on a pri ma ry va riab le in ter po la ted by kri gi ng and com pa ri son of stoc has ti cal rea li za tio ns usi ng the kri gi ng map. In the fi r st ca se, i.e. po ro si ty ana lysis at the Be ni čan ci fi e ld, seis mic da ta led to a lower es ti ma tion er ror (cok ri gi ng va rian ce er ror) and for tu na te ly this ad van ta ge was re gu lar ly dis tri bu ted ac ro ss the en ti re re ser voir. Mo reo ver, the seis mic at tri bu te at te nua ted the ‘bu ll-eye-ef fe ct’, so ob vious ly pre se nt on the kri gi ng map. The cok ri gi ng map can be con si de red as a ve ry re liab le rep re sen ta tion of the dis tri bu tion of po ro si ty in the brec cia re ser voir in the Be ni čan ci fi e ld. The set of stoc has- tic rea li za tion re sul ts on ly con fi r med that the cok ri gi ng map can be used as a ‘de fi ni te’ po ro si ty so lu tion, ve ry op ti mis ti cal- ly des cri bi ng the re ser voir, (the cok ri gi ng map is ve ry si mi lar to the stoc has tic map ob tai ned by P99 rea li za tion). Mo reo ver, the cok ri gi ng rea li za tion wi ll not te nd to ove r-em pha sis lo cal va ria bi li ty, whi ch cou ld be cha rac te ris tic of co-si mu la tion. The se co nd ca se, ana lysis at the Sta ri Gra da c-Bar cs Nyu gat fi e ld, was dif fe re nt be cau se it was not pos sib le to se- le ct a se con da ry seis mic sour ce of in for ma tion. Ma ny of the un cer tain ties we re again in cor po ra ted in a va riog ram mo del, in a si mi lar way to the Be ni čan ci re ser voir. Howe ver, the Sta- ri Gra da c-Bar cs Nyu gat da ta set was cha rac te ri sed by a very hi gh stan da rd de via tion, al thou gh the re we re on ly 14 ha r d-da- ta poin ts. It mea ns that the mean and va rian ce can not be con- si de red as rep re sen ta ti ve, and ran dom ne ss is one of the mo st ob vious re ser voir cha rac te ris ti cs. It stron gly in di ca tes a stoc- has tic ap proa ch for tes ti ng and cal cu la tion of a re la ti ve ly wi de set of rea li za tio ns (100 ma ps we re cal cu la ted). Bo th ap proac- hes, the kri gi ng and Sequen tial Gaus sian Si mu la tio ns, yiel ded bet ter so lu tio ns than tho se ob tai ned by the Inver se Dis tan ce Weig hti ng met hod. Mo reo ver, the stoc has tic ap proa ch led to bet ter re ser voir cha rac te ri za tion, be cau se it de pic ted mu ch mo- re va ria tion in the zo nes dis ta nt from ha r d-da ta than was vi sib- le on the kri gi ng map. Com pa ri ng P1, P50 and P99 rea li za tio- ns on ly the bor der of zo ne of low po ro si ty (red zo ne shown at Fi gu re 7) and con sequen tly of hi gh po ro si ty (lig ht blue area Fi gu re 7: Po ro si ty es ti ma ted in me dian stoc has tic rea li za tion (P50) at Fi gu re 7) can be pre dic ted. This zo ne al so in di ca tes the boun da ry be tween bet ter and poo rer par ts of the re ser voir, i.e. at the boun da ry be tween proxi mal and dis tal par ts of the al lu- vial fan, whe re coar se-grai ned clas ti cs of Ba de nian age were de po si ted (MALVIĆ, 2006). Fi nal ly, bo th exam ples rep re se nt ve ry li mi ted da ta se ts of 14 ha r d-da ta va lues. The re sul ts in di ca ted that if a se con da ry sour ce of in for ma tion (li ke seis mic) is avai lab le then the de ter- mi nis tic cok ri gi ng ap proa ch is re com men ded. Howe ver, if the se con da ry va riab le is not avai lab le the stoc has tic Gaus sian si- mu la tion is a mo re des crip ti ve tool. ACKNOWLEDGEMENT This wo rk is pa rt of mul ti dis cip li na ry geos ta tis ti cal re sear ch that was per for med in 2006 and 2007 wit hin the pro je ct en tit led »Stra tig rap- hi cal and geo mat he ma ti cal re sear ches of pet ro leum geo lo gi cal syste ms in Croa tia« (pro je ct no. 195-1951293-0237), fi nan ced by the Mi nis try of Scien ce, Edu ca tion and Spor ts of the Re pub lic of Croa- tia. The aut hor wou ld al so li ke to tha nk to Ma rio Weis ser, MSc, the di rec tor of Re ser voir En gi nee ri ng and Fie ld De ve lop me nt De par- tme nt (as a pa rt of INA-Oil In dus try Plc.) for per mis sion to pub li sh fi gu re 5. 7. REFERENCES CRESSIE, N. (1991): Sta tis ti cs for Spa tial Da ta. – Wi ley & So ns Ltd, 928 p. DAVIS, B. (1987): Uses and Abu ses of Cro ss Va li da tion in Geos ta tis ti- cs.– Mat he ma ti cal Geo lo gy, 19/3, 241–248. DEUTSCH, C.V. (2002): Geos ta tis ti cal Re ser voir Mo de li ng.– Oxfo rd Uni ver si ty Pre ss, 376 p. DEUT SCH, C.V. & JOUR NEL, A.G. (1997): GSLIB: Geos ta tis ti cal Sof- twa re Lib ra ry and Use r’s Gui de (Ap plied Geos ta tis ti cs Se ries), Se- co nd Edi tion.– Oxfo rd Uni ver si ty Pre ss, 369 p. DUBRULE, O. (1998): Geos ta tis ti cs in Pet ro leum Geo lo gy.– AAPG Edu ca tion Cour se No te, Se ries #38, AAPG and Geo lo gi cal So cie ty Pub lis hi ng Hou se, 210 p. FUTIVIĆ, I. & PLEIĆ, M. (2002): 3D Seismic Interpretation of the Be ničanci Oil Field. Naftaplin,1 (19), 17-25. GOOVAERTS, P. (1997): Geos ta tis ti cs for na tu ral re sour ce eva lua tion (ap plied Geos ta tis ti cs Se ries). Oxfo rd Uni ver si ty Pre ss Inc., 483 p. HERNITZ, Z., TIŠLJAR, J., KALAC, K., VELIĆ, J. & JELIĆ, K. (1993): Mio cen ski si par ni i prio bal ni klas ti ti, te ma tič ne sti je ne u drav skoj dep re si ji [Mio ce ne roc kfa ll and coas tal clas ti cs, and sour- ce roc ks in the Dra va dep res sion – in Croatian]. – Sav je to va nje o ko lek tor skim sti je na ma u Pa non skom ba ze nu, 20.–24.4.1993., Zag- reb, Croa tia. Znan stve ni sav jet za naf tu HAZU, ab stra ct, 1–2. HERNITZ, Z., VELIĆ, J., KALAC, K. & SAFTIĆ, B. (1995): Mio cen- ske ta lo ži ne u ši rem pod ruč ju Be ni ča na ca i nji ho vo naf tno geo loš- ko zna če nje [Mio cene se di men ts in the wi der area of the Be ni čan- ci and their sig ni fi can ce for pet ro leum geo lo gy – in Croatian]. 1. hr vat ski geo loš ki kon gres, Opa ti ja 18.–21.10.1995., Zbor nik ra do- va 1, 195–200, Zag reb. HOHN, M.E. (1988): Geos ta tis ti cs and Pet ro leum Geo lo gy.– Van Nos- tra nd Rein ho ld, 264 p. ISAAKS, E. & SRIVASTAVA, R. (1989): An In tro duc tion to Ap plied Geos ta tis ti cs. – Oxfo rd Uni ver si ty Pre ss Inc., 580 p. JENSEN, J.L., LAKE, L.W., CORBETT, P.W.M. & GOGGIN, D.J. 05 Malvic.indd 4605 Malvic.indd 46 24.6.2008 11:10:3124.6.2008 11:10:31 Geologia Croatica 47 Tomislav Malvić: Kri gi ng, cok ri gi ng or stoc has ti cal si mu la tio ns, and the choi ce be tween de ter mi nis tic or sequen tial ap proac hes (2000): Sta tis ti cs for Pet ro leum En gi nee rs and Geos cien tis ts.– Pren- ti ce Ha ll PTR, 390 p. JOURNEL, A.G. & HUIJBREGTS, C.J. (1978): Mi ni ng Geos ta tis ti cs.– Aca de mic Pre ss, p. 600. KELKAR, M. & PEREZ, G. (2002): Ap plied Geos ta tis ti cs for Re ser voir Cha rac te ri za tion.– So cie ty of Pet ro leum En gi nee rs, 264 p. KRIGE, D.G. (1951): A Sta tis ti cal Ap proa ch to So me Ba sic Mi ne Va lua- tion Prob le ms on the Wi twa ter sra nd.– Jour nal of the Che mi cal, Me- tal lur gi cal and Mi ni ng So cie ty of Sou th Af ri ca, 52, 119–139. LIEBHOLD, A.M., ROSSI, R.E. & KEMP, W.P. (1993): Geos ta tis ti cs and Geog rap hic In for ma tion System in Ap plied In se ct Eco lo gy. – An nual Re view of En to mo lo gy, 38, 303–327. MALVIĆ, T. (2003): Vje ro jat no st pro na las ka no vih za li ha ug lji ko vo di ka u bje lo var skoj ulek ni ni [Oi l-Geo lo gi cal Re la tio ns and Pro ba bi li ty of Dis co ve ri ng New Hydro car bon Re ser ves in the Bje lo var Sag – in Croatian]. – Un pub lis hed PhD The sis, Fa cul ty of Mi ni ng, Pet ro- leum and Geo lo gy, Uni ver si ty of Zag reb, 123 p. MALVIĆ, T. (2005): Re sul ts of geos ta tis ti cal po ro si ty map pi ng in Wes- te rn Dra va Dep res sion fi el ds (Mol ve, Ka li no vac, Sta ri Gra dac).– Naf ta, 56/12, 465–476. MALVIĆ, T. (2006): Mid dle Mio ce ne De po si tio nal Mo del in the Drava Dep res sion Des cri bed by Geos ta tis ti cal Po ro si ty and Thic kne ss Ma- ps (Ca se stu dy Sta ri Gra da c-Bar cs Nyu gat Fie ld).– Ru dar sko-geo- loš ko-naf tni zbor nik, 18, 63–70. MALVIĆ, T. & ÐUREKOVIĆ, M. (2003): Ap pli ca tion of met ho ds: In- ver se dis tan ce weig hti ng, or di nary kri gi ng and col lo ca ted cok ri gi ng in po ro si ty eva lua tion, and com pa ri son of re sul ts on the Be ni can ci and Sta ri Gra dac fi el ds in Croa tia.– Naf ta, 54/9, 331–340. MATHÉRON, G. (1962): Trai té de géos ta tis tique ap pliquée.– To me 1, Edi tio ns Tec hnip, 334 p. MATHÉRON, G. (1963): Prin cip les of geos ta tis ti cs.– Econ. Geol., 58/8, 1246–1266. MATHÉRON, G. (1965): Les Va riab les Régio na li sées et leur Es ti ma- tion.– Mas son & Cie, 306 p. RIPLEY, B.D. (1981): Spa tial Sta tis ti cs.– Jo hn Wi ley & So ns Inc., 252 p. ROSSITER, D. G. (2005): An in tro duc tion to ap plied geos ta tis ti cs – Pa- rt 7 – Using se con da ry in for ma tion: Co-kri gi ng. Lec tu re no tes.– De- par tme nt of Ear th Syste ms Ana lysis, In ter na tio nal In sti tu te for Geo- in for ma tion Scien ce & Ear th Ob ser va tion (ITC) (http://www.itc. nl/personal/rossiter/teach/lecnotes.html#l6) (20 Feb 2008). SMOLJANOVIĆ, S. & MALVIĆ, T. (2005): Impro ve men ts in re ser voir cha rac te ri za tion ap plyi ng geos ta tis ti cal mo del li ng (es ti ma tion & stoc has tic si mu la tio ns vs. stan da rd in ter po la tion met ho ds), Ca se stu- dy from Croa tia.– Naf ta, 56/2, 57–63. ŠIMON, J. & BATUŠIĆ, V. (1974): O li tos tra tig raf skom stu pu naf tnog po lja Be ni čan ci [About lit hos tra tig rap hic sec tion of the oil fi e ld Be- ni čan ci – in Croatian]. – Zbor nik ra do va I. god. znan stve nog sku pa I. sek ci je ZSN HAZU (Opa ti ja 1973), 75–89. TIŠLJAR, J. (1993): Se di men ta ry bo dies and de po si tional mo de ls for the Mio ce ne pro du ci ng areas La dis lav ci, Be ni čan ci and Obod (Croa- tia).– Naf ta, 44/10, 531–542. TURAJLIĆ, V., MATELJAK, M. & DEVIĆ, T. (1979): Us po red ni pri- kaz i prob le ma ti ka raz ra de le žiš ta naf tnih po lja Be ni čan ci, Obod i La dis lav ci – Ku ča nci [Paral lel re view and prob le ms in de ve lop me- nt of the oil fi e ld Be ni čan ci, Obod and La dis lav ci-Ku čan ci]. – in Croatian – Zbor nik ra do va 3. god. nauč ni skup Sek ci je za prim je nu geo lo gi ju, geo fi zi ku i geo ke mi ju Znan stve nog sav je ta za naf tu, No- vi Sad (1977), 2, 643–658. WELHAN, J. (2001): Geo lo gy 606: Geos ta tis ti cs and Spa tial Mo de li ng. Sylla bus and Cour se Lec tu re No tes.– Spri ng se mes ter 2001, Ida ho Sta te Uni ver si ty, http://giscenter.isu.edu/training/ (20 Feb 2008). XU, W., TRAN, T.T., SRIVASTAVA, R.M. & JOURNEL, A.G. (1992): In teg ra ti ng Seis mic Da ta in Re ser voir Mo del li ng: The Col lo ca ted Cok ri gi ng Al ter na ti ve.– SPE pa per 24742. REVIEW PAPER Manuscript received June 6, 2007. Revised manuscript accepted January 18, 2008. 05 Malvic.indd 4705 Malvic.indd 47 24.6.2008 11:10:3124.6.2008 11:10:31 05 Malvic.indd 4805 Malvic.indd 48 24.6.2008 11:10:3224.6.2008 11:10:32 Gui de for con tri bu to rs Ori gi nal scien ti fi c pa pe rs dea li ng wi th eve ry as pe ct of Ear th scien ces, es pe cial ly con cer ni ng the Di na ri des, the Ad ria tic/Mediterra- nean re gion, Pan no nian Ba sin, and/or kar st is sues are con si de red for pub li ca tion in Geo lo gia Croa ti ca, as we ll as no tes and re views in te res ti ng to a wi der au dien ce (e.g. re view pa pe rs, book re views, no tes). Ma nus crip ts shou ld be writ ten in En gli sh. For Croa tian aut ho rs an ad di tio nal Ab stra ct in Croa tian may al so be in clu ded. Ma nus crip ts Elec tro nic ver sion of ma nus cri pt shou ld be se nt by e-mail to edi to rial.offi ce@geologia-croatica.hr. Ad di tio nal ly, pa per co py of the ma nus cri pt ac com pa nied by co pies of all il lus tra tio ns shou ld be se nt to the Geo lo gia Croa ti ca, Croa tian Geo lo gi cal Sur vey, Sac hso va 2, P.O. Box 268, HR-10000 Zag reb, Croa tia. Ma nus cri pt shou ld be in doub le spa ced types cri pt wi th am ple mar gi ns. Plea se, pro vi de ca re ful lan gua ge re vi sion be fo re sub mis- sion of the ma nus cri pt. The tit le shou ld be in for ma ti ve, but sho rt, and shou ld be fol lowed by the com ple te Fir st Na mes and Sur na mes of all aut ho rs. The fu ll pos tal and e-mail ad dres ses of the aut ho r(s) shou ld be gi ven at the bot tom of the fi r st pa ge. A brief in for ma ti ve abstra ct of 50–300 wor ds shou ld be in clu ded, as we ll as ca re ful ly cho sen Key wor ds pro po sed by the aut hor(s). Text The text shou ld be in wor dpro ces si ng fi le. We can not ac ce pt text and fi gu res in pdf for mat. We ac ce pt mo st wor dpro ces si ng for ma ts, but Wo rd is pre fer red. Always keep a bac kup co py of the elec tro nic fi le for re fe ren ce and sa fe ty. Sa ve your fi les usi ng the de fau lt exten sion of the prog ram used. It is im por ta nt that the fi le be sa ved in the na ti ve for mat of the wor dpro ces sor used. The text shou ld be in sin gle-co lu mn for mat. Keep the layout of the text as sim ple as pos sib le – most for mat ti ng co des wi ll be re mo ved and rep la ced on pro ces si ng the ar tic le. In par ti cu lar, do not use the wor dpro ces so r’s op tio ns to jus ti fy text or to hyphe na te wor ds. Howe ver, do use bo ld fa ce, ita li cs, sub scrip ts, su per scrip ts etc. Do not em bed ‘grap hi cal ly de sig ne d’ equa tio ns or tab les, but pre pa re the se usi ng the wor dpro ces so r’s fa ci li ty. The elec tro nic text shou ld be pre pa red in a way ve ry si mi lar to that of con ven tio nal ma nus crip ts. To avoid un ne ces sa ry er ro rs you are stron gly ad vi sed to use the ‘spel lchec ke r’ fun ction of your wor dpro ces sor. Fi gu re cap tio ns and tab les Pre se nt fi gu re cap tio ns and tab les at the end of the ar tic le. When pre pa ri ng tab les, if you are usi ng a tab le grid, use on ly one grid for ea ch in di vi dual tab le and not a grid for ea ch row. If no grid is used, use ta bs, not spa ces, to ali gn co lum ns. Num ber tab les con se cu ti- ve ly in ac cor dan ce wi th their ap pea ran ce in the text. Pla ce foot no tes to tab les below the tab le bo dy and in di ca te them wi th su per scri pt lower ca se let te rs. Avoid ver ti cal ru les. Be spa ri ng in the use of tab les and en su re that the da ta pre sen ted in tab les do not dup li ca te re sul ts des cri bed el sewhe re in the ar tic le (in clu di ng fi gu res). Grap hi cs A hi gh stan da rd of il lus tra tion qua li ty is de si red. Ea ch il lus tra tion mu st ha ve a cap tion, whi ch is re le va nt and expla nato ry. Cap tio ns shou ld be typed on se pa ra te shee ts. Bar sca les are pre fer red. All il lus tra tio ns shou ld be num be red and the aut hor’s na me shou ld be writ ten on the ba ck of pa per co py in pen cil. The ap proxi ma te po si tion in the text may be in di ca ted in the mar gin of the ma nus cri pt. The maxi mum si ze of pla tes is 175 x 255 mm. The maxi mum wid th of il lus tra tio ns in the text (pho tog rap hs, tab les and drawin gs) is 80 mm (for one co lu mn) or 165 mm (for two co lum ns). For mo re than three pla tes, plea se con ta ct the edi to rs! Hi gh-re so lu tion grap hi cs fi les mu st always be pro vi ded se pa ra te from the main text fi le. Do not im po rt the fi gu res in to the text fi le, ju st in di ca te their ap proxi ma te lo ca tio ns di rec tly in the elec tro nic text and on the ma nus cri pt. Ge ne ral poin ts • Ma ke su re you use uni fo rm let te ri ng and si zi ng of all your ori gi nal ar two rk. • Sa ve text in il lus tra tio ns as »cur ves« or en clo se the fo nt. • On ly use the fol lowi ng fon ts in your il lus tra tio ns: Arial, Hel ve ti ca, Ti mes, Symbol. • Num ber the il lus tra tio ns ac cor di ng to their sequen ce in the text. • Use a lo gi cal na mi ng con ven tion for your ar two rk fi les, and sup ply a se pa ra te lis ti ng of the fi les and the sof twa re used. • Pro vi de cap tio ns to il lus tra tio ns se pa ra te ly. • Pro du ce ima ges near to the de si red si ze of the prin ted ver sion. For ma ts Re gar dle ss of the ap pli ca tion used, when your elec tro nic ar two rk is fi na li sed, plea se »sa ve as« or con ve rt the ima ges to one of the fol lowi ng for ma ts (No te the re so lu tion requi re men ts for li ne drawin gs, hal fto nes, and li ne/halftone com bi na tio ns gi ven be low.): EPS: Vec tor drawin gs. Em bed the fo nt or sa ve the text as »grap hi cs«. TIFF or hig he st-qua li ty JPEG – Co lour or greysca le pho tog rap hs (hal fto ne s): fi gu re in the fi nal si ze shou ld ha ve a mi ni mum re so- lu tion of 300 dpi. 05 Malvic.indd 4905 Malvic.indd 49 24.6.2008 11:10:3224.6.2008 11:10:32 TIFF or hig he st-qua li ty JPEG – Bit map ped li ne drawin gs: fi gu re in the fi nal si ze shou ld ha ve a mi ni mum re so lu tion of 1000 dpi. TIFF or hig he st-qua li ty JPEG – Com bi na tio ns bit map ped li ne/half-tone (co lour or greysca le): fi gu re in the fi nal si ze shou ld ha ve a mi ni mum re so lu tion of 500 dpi. DOC, XLS or PPT: If your elec tro nic ar two rk is crea ted in any of the se Mic ro so ft Of fi ce ap pli ca tio ns plea se sup ply »as is«. Plea se do not • Sup ply em bed ded grap hi cs in your wor dpro ces sor (spread sheet, pre sen ta tion) do cu me nt when sub mit ti ng the fi nal ver sion of the ac cep ted ar tic le; • Sup ply fi les that are op ti mi sed for screen use (li ke GIF, BMP, PICT, WPG); the re so lu tion is too low; • Sup ply fi les that are too low in re so lu tion; • Sub mit grap hi cs that are dis pro por tio na te ly lar ge for the con te nt. Ge ne ra and Spe cies na mes Ge ne ra and Spe cies na mes shou ld be writ ten in Ita li cs, e.g. Chrysa li di na gra da ta D’ORBIGNY. Ab bre via tio ns and uni ts SI uni ts are pre fer red. All mea su re men ts shou ld always be gi ven in fi gu res, e.g. 12 m, exce pt whe re the num ber be-gi ns the sen ten ce. The wo rd »fi gu re« shou ld be shor te ned to »Fig.« un le ss at the be gin ni ng of a sen ten ce. Re fe ren ces Re fe ren ces in text shou ld be lis ted by gi vi ng the aut hor’s sur na me (in ca pi tal let te rs) fol lowed by the year of pub li ca tion in pa ren the- ses, e.g. VELIĆ (2007) or KOROLIJA & JAMIČIĆ (1988). When re fe ren ce is ma de to wo rk by mo re than two aut ho rs, the fi r st na me shou ld be fol lowed by »et al.«, e.g. REIMANN et al. (2008). If se ve ral pa pe rs by the sa me aut hor and from the sa me year are ci ted, a, b, c, etc. shou ld be put af ter the year of pub li ca tion, e.g. VRSALJKO (2003b). Com ple te re fe ren ces shou ld be lis ted in al pha be ti- cal or der at the end of the pa per (aut ho rs mu st pro vi de En gli sh tran sla tio ns of ar tic le tit les writ ten in ot her lan gua ges), as in the fol lowi ng exam ples: FUČEK, L., JELASKA, V., GUŠIĆ, I., PRTOLJAN, B. & OŠTRIĆ, N. (1991): Pa din ski tu ron ski se di men ti uva le Br biš ni ca na Du gom oto ku (Tu ro nian slo pe de po si ts in the Br biš ni ca co ve, Du gi Otok is la nd, Croa tia – in Croa tian). – Geol. vjes nik, 44, 55–67. VELIĆ, I. (2007): Stra tig rap hy and Pa laeo bio geog rap hy of Me so zoic Ben thic Fo ra mi ni fe ra of the Kar st Di na ri des (SE Eu ro pe). – Geol. Croat., 60, 1–113. KOROLIJA, B. & JAMIČIĆ, D. (1988): Os nov na geo loška kar ta SFRJ 1:100000, li st Našice (Ba sic geo lo gi cal map of SFRJ 1:100000, Na ši ce Sheet). – In st. za geol. is traž., Zag reb, Sav. geol. za vod, Beog rad. PALMER, A.N. (2006): Di gi tal mo de li ng of kar st aqui fe rs – Suc ces ses, fai lu res, and pro mis ses.– In: HARMON, R.S. & WICKS, C.M. (eds.): Per spec ti ves on kar st geo mor pho lo gy, hydro lo gy, and geoc he mis try – A tri bu te vo lu me to De rek C. Fo rd and Wil liam B. Whi te. The Geo lo gi cal So cie ty of Ame ri ca, Spec. Pa per 404, 243–250. REIMANN, C., FILZMOSER, P., GARRETT, R. & DUTTER, R. (2008): Sta tis ti cal Da ta Ana lysis Explai ned: Ap plied En vi ron men- tal Sta tis ti cs wi th R. – Jo hn Wi ley & So ns, New Yo rk, Lon don, 362 p. VRSALJKO, D. (2003): Bios tra tig ra fi ja mio cen skih nas la ga Žum be rač kog i Sa mo bor skog gor ja na te me lju me ku ša ca (Bios tra tig- rap hy of Mio ce ne de po si ts from Žum be rač ko and Sa mo bor sko gor je Mts. on the ba sis of Mol lus cs – in Croa tian, wi th an En gli sh Ab stra ct). – Un pu bl. PhD The sis, Fa cul ty of Scien ce, Uni ver si ty of Zag reb, 147 p. Un pub lis hed re por ts from in sti tu tion ar chi ves shou ld be ci ted in text as e.g. (ANDRIĆ, 20012) and re fe red as foot no te. 2ANDRIĆ, M. (2001): Ro bin so n-Du bo ka Lju ta, seiz mič ka to mog raf ska mjerenja [Ro bin zo n-Du bo ka Lju ta to mog rap hic mea su re men ts – in Croatian]. – Un pub lis hed re po rt, Ar chi ve of Ci vil En gi nee ri ng In sti tu te of Croa tia, Zag reb, 30 p. In ter na tio nal Re view Edi to rs fi r stly che ck ea ch ma nus cri pt for com plian ce wi th the jour nal stan dar ds. All ori gi nal scien ti fi c pa pe rs whi ch mig ht be pub lis- hed in Geo lo gia Croa ti ca are then reviewed in ter na tio nal ly. The du ra tion of the re viewi ng pro ce ss is va riab le, de pen di ng on nu me rous fac to rs. We wi ll do our be st do re du ce it as mu ch as pos sib le. Howe ver, typi cal ly the en ti re pro ce ss from sub mis sion of the fi r st ver sion of the ma nus cri pt to pub li ca tion ta kes 5 to 10 mon ths, and ca re ful fol lowi ng of this gui de for con tri bu to rs wi ll usual ly sig ni fi can tly dec rea se du ra tion of this pro ce ss. Af ter re vi sion and cor rec tion of the ma nus cri pt has been com ple ted the aut hor shou ld pre pa re the fi nal ver sion of the ma nus cri pt, on ly in elec tro nic fo rm. This ver sion wi ll be se nt to the Uni ted Kin gdom to our lan gua ge edi tor, and on ly af ter this cor rec tion is do ne the ma nus cri pt is rea dy for pub li ca tion. Pa ge proo fs Pa ge proo fs wi ll be se nt to the aut ho rs. They shou ld be read ca re ful ly and re tur ned wit hin a week. Of fprin ts Twen ty fi ve of fprin ts of ea ch pa per and one vo lu me of the jour nal are sup plied free of char ge. Ad di tio nal of fprin ts may be or de red. 05 Malvic.indd 5005 Malvic.indd 50 26.6.2008 9:23:5826.6.2008 9:23:58