Application of the radial basis function neural network to the prediction of log properties from seismic attributes

Transcription

1 Applicaion of he RBF Applicaion of he radial basis funcion neural newor o he predicion of log properies from seismic aribues Brian H. Russell, Laurence R. Lines, and Daniel P. Hampson ABSTRACT In his paper, we use he radial basis funcion neural newor, or RBF, o predic reservoir log properies from seismic aribues. We also compare he resuls of his approach wih he use of he generalized regression neural newor, GR, for he same problem, as proposed by Hampson e al (00). We discuss boh he heory behind hese mehods and he mehodology involved in applying neural newors o seismic aribues. We hen illusrae he mehod using he Blacfoo 3D seismic volume, a channel sand example from Albera. ITRODUCTIO The predicion of reservoir parameers from seismic daa has radiionally been done using deerminisic mehods, in which we assume a mahemaical model ha relaes our observed seismic daa o he reservoir parameer of ineres, and apply an inverse ransform o our seismic daa based on his model. Such mehods include deconvoluion, pos-sac race inversion, and linearized pre-sac race inversion, or AVO. These mehods are limied by a number of facors, including noise in he seismic daa, incorrec scaling of he seismic daa, and limiing assumpions in he model (for example, we ofen ignore he fac he seismic wavele is ime-varying). There is also a fundamenal ambiguiy in he derivaion of he long period componen of he earh properies if we use he seismic daa alone. To ry o avoid he las problem, we may use inversion mehods which are model-based, in ha hey use an iniial model of he earh compued from more deailed well log measuremens, and hen perurb his model o fi he seismic observaions. In more recen approaches, a saisical, raher han deerminisic, mehod is used o loo for a relaionship beween he seismic daa and he reservoir parameers. These mehods are based on radiional muli-regression, or on neural newors. For example, Ronen e al. (994) proposed he use of he radial basis funcion neural newor (RBF) for he seismic-guided esimaion of log properies. They applied he RBF o averaged inervals from log daa and he corresponding averaged inervals from he 3D seismic daa volumes which were ied by hese logs. In more recen wor (Hampson e al, 00), neural newors were used o predic reservoir log properies from he complee 3D seismic volume, using each log sample over he zone of ineres in he raining phase. In his laer approach, he generalized regression neural newor, or GR, was used for he predicion of log properies, alhough his mehod was referred o in his paper as he probabilisic neural newor, or P. (As we will see in he following heory secion he P approach is a classificaion echnique which is he basis upon which he GR mehod is buil). In his paper, we exend he RBF mehod o he compuaion of log properies from he full 3D seismic volume, and show he relaionship beween he RBF and GR mehods. Alhough boh mehods are based on he applicaion of Gaussian Hampson-Russell Sofware Services Ld, 50, 75 5 h Avenue SW, Calgary, Albera, TP X6 CREWES Research Repor Volume 4 (00)

2 Russell, Lines, and Hirsche weighing funcions o disances measured in muliple seismic aribue space, hey are quie disinc in heir calculaion of he final resul. Following a discussion of he heory behind he neural newor echniques used here, we will discuss he mehodology for applying he mehods o seismic daa and hen illusrae he mehods wih a channel sand case sudy from Albera. ITRODUCTIO TO EURAL ETWORKS Firs of all, wha is a neural newor? The simples answer is ha a neural newor is a mahemaical algorihm ha can be rained o solve a problem ha would normally require human inervenion. Alhough here are many differen ypes of neural newors, here are wo ways in which hey are caegorized: by he ype of problem ha hey can solve and by heir ype of learning. eural newor applicaions in seismic daa analysis generally fall ino one of wo caegories: he classificaion problem, or he predicion problem. In he classificaion problem, we assign an inpu sample o one of several oupu classes, such as sand, shale, limesone ec, whereas in he predicion problem we assign a specific value o he oupu sample, such as a porosiy value. A companion paper in his volume (Russell e al, 00) discusses he applicaion of wo differen neural newors, he muli-layer percepron (MLP) and he radial basis funcion neural newor (RBF) o a sraighforward AVO classificaion problem. In his paper, we will be concerned wih he predicion problem, specifically he predicion of P-wave velociy from seismic aribues. eural newors can also be classified by he way hey are rained, using eiher supervised or unsupervised learning. In supervised learning he neural newor sars wih a raining daase for which we now boh he inpu and oupu values. The neural newor algorihm hen learns he relaionship beween he inpu and oupu from his raining daase, and finally applies he learned relaionship o a larger daase for which we do no now he oupu values. Examples of he supervised learning approach are he MLP, he P, he GR, and he RBF, of which he laer hree mehods will be used oday. In unsupervised learning, we presen he neural newor wih a series of inpus and le he neural newor loo for paerns iself. Tha is, he specific oupus are no required. The advanage of his approach is ha we do no need o now he answer in advance. This disadvanage is ha i is ofen difficul o inerpre he oupu. An example of his ype of unsupervised echnique is he Kohonen Self Organizing Map (KSOM) (Kohonen, 00). THEORY As menioned in he las secion, in his paper we will be using neural newors o perform he supervised predicion of reservoir parameers. Our raining daase will consis of a se of nown raining samples i, which in our case will be some well log derived reservoir parameer such as V P, SP, S W, ec. Each raining sample, which is a scalar quaniy, is in urn dependan on a vecor of L seismic aribue values, correlaed in ime wih he raining samples. (The issues of which seismic aribues o use and how o opimize he correlaion beween he raining samples and he seismic daa are CREWES Research Repor Volume 4 (00)

3 Applicaion of he RBF hemselves difficul problems, and will be deal wih in he nex secion.) These seismic aribue vecors can be wrien s i (s i, s i,, s il ) T, i,,,. The obecive of our neural newor is o find some funcion y such ha: y(s i ) i, i,,, () Once his funcion has been found, i can be applied o an arbirary se of M seismic aribue vecors x,,,, M, where he aribues in he x vecors are idenical o hose in he s i vecors. This is illusraed in Figure for wo arbirary raining samples and a single applicaion sample. The ey poin o noe is ha we now he oupu sample for he raining vecors, bu are predicing an oupu sample for he applicaion vecors. (We have chosen o use a differen leer o represen he raining and applicaion daa vecors o emphasize he fundamenal difference beween hem, and also o laer clarify he difference beween RBF and GR mehods. In mos exboos his is handled simply by varying he subscrips on he daa vecors.) FIG. : A schemaic illusraion of he differences beween he raining vecors, s i and s, in which he oupu samples i and are nown and are used in he raining process, and he applicaion vecor x, in which he oupu sample y is no nown. Before discussing he RBF and GR algorihms, we will firs discuss he simpler P algorihm. The P algorihm is based on he concep of disance in aribue space. To beer undersand his concep, consider Figure, in which we have drawn, in graphical form, he hree arbirary wo-dimensional seismic aribue vecors shown in Figure. oe ha disance on hese graphs is aribue ampliude raher han Caresian disance. Recall ha wo of hese vecors are from he raining daase (s i and s ) and one is from he applicaion daase (x ). CREWES Research Repor Volume 4 (00) 3

4 Russell, Lines, and Hirsche FIG. : A schemaic graph of he vecors, s i, s, and x, from Figure, where he coordinae axes represen aribue ampliude raher han Caresian disance. We can define he hree possible disances beween hese vecors, as displayed in Figure. These are given by d i i s ( si s ) + ( si s ) s, and d d i i x si x ) + ( si x ) s, x ( s x ) + ( s x ) s. I is imporan o disinguish wo fundamenally differen ypes of aribue disance in he above equaions. The d i disance can be called he iner-raining disance and he d i and d disances can be called applicaion disances. (oe ha his erminology is unique o his paper, since we feel ha his concep is crucial o he undersanding of he hree mehods of P, GR, and RBF.) A second imporan concep is ha i is no he disances hemselves ha will be used in our neural newor applicaions, bu some funcion of he disances, ϕ(d), called a basis funcion. Alhough here have been a number of forms proposed for he basis funcion (Bishop, 995), he mos common form, and he one we will be using in his paper, is he Gaussian funcion, which can be wrien as d ( d ) exp, () σ where σ is a smoohness parameer. oice ha σ can also be inerpreed as he variance of a Gaussian disribuion cenered on d. Thus, as we decrease σ, he widh of he 4 CREWES Research Repor Volume 4 (00)

5 Applicaion of he RBF disribuion becomes narrower. The P is hen defined for each of he x poins as he sum over all he possible ( d ) funcions, or x s p ( x ) exp,,,, M (3) σ If we use all he poins in he raining daase, P will resul in a single number, which will be used laer as a normalizing facor in he GR mehod, bu does no on is own give us a very useful discriminaion echnique. However, if we brea he raining poins ino a number of classes, P becomes a very good classificaion mehod. In fac, i can be hough of as an implemenaion of Bayes Theorem. Le s consider he simples case, ha of wo classes. If we have class C wih poins, and class C wih poins, where +, hen we can define and p x (4) ( ), p x. (5) ( ) The p values can be inerpreed as he probabiliy of membership in a class. Tha is, if p (x ) > p (x ), hen x is a member of class C, or, if p (x ) < p (x ), hen x is a member of class C. This can be generalized o any number of classes. ex, we will see how he GR can be hough of as an exension of P. A well nown resul from saisical heory is ha he condiional expecaion of a coninuous scalar variable y given a coninuous vecor x can be wrien as - EY X ( x ) (6) f ( x, y )dy - yf X Y X Y ( x, y )dy Transforming he coninuous form seen in equaion (6) o he discree form of he probabiliy funcion ha we are using here (Masers, 995), gives x s x s exp exp σ σ y( x ),,,,M, (7) x p( ) s x exp σ Equaion (7) is nown in saisical esimaion heory as he adaraya-wason esimaor (adaraya, 964; Wason, 964), and was re-discovered in he conex of neural newors by Spech (990) and named he generalized regression neural newor, CREWES Research Repor Volume 4 (00) 5

6 Russell, Lines, and Hirsche or GR. This name is due o he fac ha he raining values hemselves are weigh values and hus his is indeed a ype of generalized regression. oe ha he normalizaion facor in he denominaor of equaion (7) is he P esimae of he complee raining daase. We can also observe how well his equaion wors on he raining daa by re-wriing equaion (7) using he raining daa, or si s exp σ y( s i ), i,,,. (8) p( s ) i s s exp σ i i oice ha he fi will be quie good since, when i, exp[ 0] Anoher way o chec he effeciveness of his mehod is by cross-validaion, where we leave each well in urn ou of he predicion, predic is values blindly, and hen compue he error wih he nown values. This will be discussed more fully in he nex secion. Equaion (8) also gives us a saring poin for undersanding he RBF approach, which was originally developed as a mehod for performing exac inerpolaion of a se of daa poins in muli-dimensional space (Powell, 987). In his mehod, we also use Gaussian basis funcions. However, insead of calculaing he weighing values on he fly using he raining samples, we compue he weighs iniially from he raining daa using he equaion s i s ( s i ) w exp w i, i,,,. (9) σ Equaion (9) can be derived from basic principles using he heory of regularizaion, which also involves he inroducion of pseudo-differenial operaors (Poggio and Girosi, 990), bu will no be derived here. To solve equaion (9), noice ha i can be wrien as a se of equaions in unnowns, or. w w w + w + w + w + + w + + w + + w (0) Equaion (0) can be wrien more compacly as he marix equaion: Φ w, () 6 CREWES Research Repor Volume 4 (00)

7 Applicaion of he RBF CREWES Research Repor Volume 4 (00) 7 where and, w w, # $ # Φ w. The soluion o equaion () is simply he marix inverse [ ], λ w + Ι Φ () where λ is a pre-whiening facor and I is he ideniy marix. This equaion can be solved efficienly by noing ha he marix is symmeric. Once he weighs have been compued, hey are applied o he applicaion daase using,m,,, σ exp w ) y( s x x. (3) oe ha equaion (3) can be hough of as he general form of boh RBF and GR if we rewrie he weighs in he GR case as ) p( σ exp w x s x. (4) This observaion will allow us o examine he relaionship beween RBF and GR. If we assume ha he scaling facors σ are he same in boh mehods and ha p(x ), hen, o equae he wo mehods, we simply need o loo a he relaionship beween and w.. By expanding equaion (), we ge λ λ λ w w w # $ # # # $ # # (5) where i i σ exp s s, and i is an elemen of he invered marix. Thus, we find ha: i w (6)

8 Russell, Lines, and Hirsche In oher words, w is a weighed sum of all he raining values. If he invered marix consised only of values along he main diagonal, we could rewrie he above equaion as w (7) If his is he case,φ would consis only of he values along he main diagonal. Bu noe ha s s exp σ Thus, Φ I, he ideniy marix, and Φ I also. Therefore, for his rivial case, he RBF and GR mehods are idenical o a scale facor. This would occur in wo siuaions: () si s 0, for all i and, and () σ 0 In he more general case, he off-diagonal elemens in he covariance marix are all non-zero. Indeed, hey will only be equal o zero for he case in which he seismic aribues are saisically independen. Thus, we can hin of he GR mehod as a subse of he RBF mehod for he case of saisical independence of he aribues. We would herefore expec he RBF mehod o give a more high resoluion resul han he GR mehod, since he off-diagonal covariance elemens are being used. As we shall see in our case sudy, his is indeed he case. METHODOLOGY The mehodology for reservoir predicion using muliple seismic aribues has been exensively covered by Hampson e al (00) and will only be briefly reviewed here. The wo ey problems in he analysis can be summarized as follows: which aribues should we use, and which of hese aribues are saisically significan? To sar he process, we compue as many aribues as possible. These aribues can be he classical insananeous aribues of Taner e al. (974), frequency and absorpion aribues based on windowed esimaes along he seismic race, inegraed seismic ampliudes, model-based race inversion, or AVO inercep and gradien. Once hese aribues have been compued and sored, we measure a goodness of fi beween he aribues and he raining samples from he logs. Using he noaion of he previous secion, we can compue his goodness of fi by minimizing he leas-squares error given by E ( i w0 wsi wlsli ). (8) i oe ha he weighs can also be convoluional, as discussed by Hampson e al. (00), which is equivalen o inroducing a new se of ime-shifed aribues. The aribues o use in he neural newor compuaions, and heir order is hen found by a echnique called sepwise regression, which consiss of he following seps: 8 CREWES Research Repor Volume 4 (00)

9 Applicaion of he RBF () Find he bes aribue by an exhausive search of all he aribues, using equaion () o compue he predicion error for each aribue (i.e. L ) and choosing he aribue wih he lowes error. () Find he bes pair of aribues from all combinaions of he firs aribue and one oher. Again, he bes pair is he pair ha has he lowes predicion error from equaion (8), wih L. (3) Find he bes riple, using he pair from sep () and combining hem wih each oher aribue. (4) Coninue he process as long as desired. This will give us a se of aribues ha is guaraneed o reduce he oal error as he number of aribues goes up. So, when do we sop? This is done using a echnique called cross-validaion, in which we leave ou a raining sample and hen predic i from he oher samples. We hen re-compue he error using equaion (8), bu his ime from he raining sample ha was lef ou. We repea his procedure for all he raining samples, and average he error, giving us a oal validaion error. This compuaion is done as a funcion of he number of aribues, and he resuling graph usually shows an increase in validaion error pas some small number of aribues such as five or six. In acual fac, we do no perform his procedure for all samples, bu raher on a well by well basis. The nex secion will give an illusraion of his procedure. One final poin o discuss is he opimizaion of he sigma values. The firs poin o observe is ha, in genera, he opimum sigma value for he GR mehod will be differen han he opimum value for he RBF mehod. A second imporan poin is ha, for he GR mehod, we also deermine a differen sigma value for each aribue. This could be inroduced in equaion (3) or (7) by changing he erm o ( x s ) ( x s ) ( x s ) L L exp + + +, σ σ σ L where L is equal o he number of aribues. The opimizaion echnique used for GR is described by Masers (995). A CHAEL SAD CASE STUDY We will now illusrae our mehodology using a channel sand case sudy from Albera, as discussed earlier by Russell e al. (00). This sudy involved he predicion of porosiy in he Blacfoo field of cenral Albera. A 3C-3D seismic survey was recorded in Ocober 995, wih he primary arge being he Glauconiic member of he Mannville group. The reservoir occurs a a deph of around 550 m, where Glauconiic sand and shale fill valleys incised ino he regional Mannville sraigraphy. The obecives of he survey were o delineae he channel and disinguish beween sand-fill and shale-fill. The well log inpu consised of welve wells, each wih sonic, densiy, and calculaed porosiy logs, shown in Figure 3. CREWES Research Repor Volume 4 (00) 9

10 Russell, Lines, and Hirsche FIG 3: The figure above shows he disribuion of wells wihin he 3D seismic survey, as well as seismic inline 95, shown in Figure 4. One of he lines from he 3D survey, inline 95, is indicaed on Figure 3, and shown in Figure 4. Figure 4(a) shows he saced seismic daa iself, and Figure 4(b) shows he invered impedance, which was invered using a model-based inversion approach. The P-wave sonic log from well has been insered a cross-line 5, and shows he exra resoluion presen in a well log. We are going o use he P-wave sonic as our arge on which o rain he seismic aribues. This daase was also used by Hampson e al. (00) o illusrae heir approach o muli-aribue predicion using he GR neural newor. They obained excellen resuls using all welve wells in heir analysis. However, in his sudy, we wish o also see how he use of a subse of he oal number of wells will affec he final resul. We will sar by using all he wells in he raining, as well as a convoluional operaor lengh of seven poins and seven possible aribues. The resuls of he raining are shown in Figure 5, where Figure 5(a) shows he aribues ha were seleced by he raining procedure. oe ha, as we would expec, he inversion aribue has he bes correlaion. Figure 5(b) hen shows he resuls of cross-validaion. This ells us ha only he firs 4 aribues are saisically significan. 0 CREWES Research Repor Volume 4 (00)

11 Applicaion of he RBF (a) FIG. 4: The figure above shows inline 95 from he 3D seismic survey, wih he P-wave sonic from well overlain a cross-line 5 (see Figure 3), where (a) shows he final CDP sac, and (b) shows he invered impedance inversion. (b) CREWES Research Repor Volume 4 (00)

12 Russell, Lines, and Hirsche (a) FIG 5: The figure above shows he resul of raining he muli-aribue algorihm using all welve wells, where (a) shows he bes seven aribues, as chosen by sep-wise regression, and (b) shows he resul of cross-validaion, which indicaes ha only he firs four aribues are saisically significan. (b) ex, we will use he GR and RBF algorihms o predic he P-wave velociy. Figure 6 shows he resul of applying he GR algorihm he predicion of P-wave velociy, using all welve wells in he raining. From he analysis shown in Figure 5; his involved using he firs four aribues from he able in Figure 5(a). In Figure 6, which shows only four of he welve wells, noice ha we see an excellen fi beween he original and modeled logs. As seen a he op of he figure, we ge an excellen correlaion coefficien of and an average error of 79 m/s. Figure 7 hen shows he resul of applying he RBF algorihm o he same se of four aribues used in he GR approach, again using all welve wells in he raining. The resul is close o ha seen in Figure 6, alhough he fi is no quie as good, eiher visually or in erms of he correlaion coefficien (0.836) or he average error (96 m/s). CREWES Research Repor Volume 4 (00)

13 Applicaion of he RBF FIG. 6: Applicaion of he GR algorihm o four of he P-wave sonic logs in he welve log suie, where all he raining samples are used in he predicion. FIG. 7: Applicaion of he RBF algorihm o four of he P-wave sonic logs in he welve log suie, where all he raining samples are used in he predicion. ex, we loo a he cross-validaion plos, in which we will leave he prediced well ou of he raining. This is shown in Figure 8 for he GR algorihm and in Figure 9 for he RBF algorihm. In boh plos, he fi is no as good as i was when we used all he wells in he raining, as we would expec. Again, we also see a slighly beer resul for he GR algorihm, boh visually and analyically. oe ha correlaion coefficien for CREWES Research Repor Volume 4 (00) 3

14 Russell, Lines, and Hirsche GR is , compared wih a value of for RBF, and he average error is 73 m/s compared wih 90 m/s. FIG. 8: Validaion of he four P-wave sonic logs from Figure 6 using he GR algorihm, where he prediced well has been lef ou of he raining. FIG. 9: Validaion of he four P-wave sonic logs from Figure 7 using he RBF algorihm, where he prediced well has been lef ou of he raining. Alhough i would appear ha GR has done a slighly beer ob han RBF form our analysis of he predicion of he well logs, le s see how he algorihms compare when 4 CREWES Research Repor Volume 4 (00)

15 Applicaion of he RBF we loo a he applicaion o real seismic daa. This is shown in Figures 0 and, where Figure 0 shows he applicaion of he GR algorihm and Figure shows he applicaion of he RBF algorihm. I is clear from hese figures ha he RBF approach has inroduced a lo more frequency conen ino he final resul han he GR approach. And his high frequency conen appears o be realisic, since here is laeral coninuiy o he exra evens ha appear on he secion. FIG. 0: Applicaion of he GR algorihm o line 95 of he 3D volume, afer raining using all he wells. FIG. : Applicaion of he RBF algorihm o line 95 of he 3D volume, afer raining wih all he wells. ex, le us repea he same analysis, bu using fewer wells in he raining. Figure shows he resuls of he raining, where Figure (a) shows he bes five aribues chosen by he sep-wise regression algorihm discussed earlier, and Figure (b) shows ha only CREWES Research Repor Volume 4 (00) 5

16 Russell, Lines, and Hirsche he firs four aribues are saisically significan, based on cross-validaion. Wha is ineresing o noe in Figure (b) is ha he error on he las aribue shoos up almos verically. I is herefore imporan o only use he firs four aribues. (a) (b) FIG. : The figure above shows he resul of raining he muli-aribue algorihm using only hree wells, where (a) shows he bes five aribues, as chosen by sep-wise regression, and (b) shows he resul of cross-validaion, which indicaes ha only he firs four aribues are saisically significan. We will now loo a he resuls of applying he raining from all hree wells o he wells hemselves, which is shown in Figure 3 for he GR algorihm and in Figure 4 for he RBF algorihm. oe ha he visual mach is almos he same for boh mehods, as are he correlaion coefficiens and average error. I acual fac, RBF has done slighly beer han GR in he average error (30.7 m/s vs 34. m/s) and slighly worse for he correlaion coefficien (0.760 vs 0.763). However, when we loo a he validaion resuls in Figures 5 and 6, RBF has definiely done beer han GR, wih an average error of m/s agains 63.89, and a correlaion coefficien of 0.68 agains for GR.. 6 CREWES Research Repor Volume 4 (00)

17 Applicaion of he RBF FIG. 3: Applicaion of he GR algorihm o he hree P-wave sonic logs used in he raining, where all he raining samples are used in he predicion. FIG. 4: Applicaion of he RBF algorihm o he hree P-wave sonic logs used in he raining, where all he raining samples are used in he predicion. CREWES Research Repor Volume 4 (00) 7

18 Russell, Lines, and Hirsche FIG. 5: Validaion of he GR algorihm o he hree P-wave sonic logs used in he raining, where he each log has been lef ou of he raining. FIG. 6: Validaion of he RBF algorihm o he hree P-wave sonic logs used in he raining, where he each log has been lef ou of he raining. Finally, we will apply he resuls of he raining o he 3D seismic daase iself. Figure 7 shows he applicaion of he GR mehod o he seismic daa. oice ha he frequency conen of he resul is very low. However, when we apply he RBF mehod, as shown in Figure 8, much more high frequency deail has come hrough. We would 8 CREWES Research Repor Volume 4 (00)

19 Applicaion of he RBF herefore conclude ha, as he number of wells in our raining daase goes down, he RBF algorihm becomes preferable o he GR algorihm FIG. 7: Applicaion of he GR algorihm o line 95 of he 3D volume, afer raining using only hree of he wells. FIG. 8: Applicaion of he RBF algorihm o line 95 of he 3D volume, afer raining using only hree of he wells. COCLUSIOS In his paper, we have compared several neural newor approaches for he predicion of log properies using muliple seismic aribues. These mehods consised of he generalized regression neural newor, or GR, and he radial basis funcion neural newor, or RBF. In he heory secion, we showed how boh of hese mehods are based on Gaussian basis funcions of disance in aribue space, and boh can be developed from he probabilisic neural newor, or P. The ey difference beween CREWES Research Repor Volume 4 (00) 9

20 Russell, Lines, and Hirsche GR and RBF is ha he GR predicion is a weighed sum of he basis funcions and he raining values, whereas in he RBF he weighs are pre-compued using a generalized marix inversion of he basis funcions weighed by he raining values. Afer a discussion of he mehodology used in finding he bes se of aribues, and hose aribues ha are saisically significan, we applied he wo mehods o a channel sand case sudy from Albera. We firs did he raining using all welve wells in he sudy area. The resuls of he raining showed he GR was marginally beer han RBF in predicing he well log values, boh using all he raining wells and in he cross-validaion ess. This was quanified by looing a he raining error and correlaion coefficien values. However, when we applied he raining o he seismic daase, he RBF mehod produced high frequency resuls han he GR mehod, wih good laeral coninuiy. We hen applied he mehods using only hree of he wells in he raining phase, he resuls were differen. In his case, he RBF mehod was quaniaively beer han he GR mehod, for boh he full applicaion and cross-validaion resuls. Tha is, he raining error was smaller and correlaion coefficien larger. The effec on he seismic daa was even more dramaic. The applicaion of he RBF mehod displayed much higher frequency conen on he seismic daa han he applicaion of he GR mehod. Our conclusion is herefore ha, as he number of wells in he raining daase goes down, RBF provides a beer echnique for predicing log properies from seismic aribues, boh in he fi a he raining wells and in he applicaion o he seismic. As he number of wells increases, he wo mehods produce fairly consisen resuls, wih GR doing a slighly beer ob of predicing he raining wells, and RBF doing a beer ob of brining ou he high frequencies in he seismic daa. FUTURE WORK Fuure wor will involve research ino alernae forms of neural newors ha can be applied o his problem, and a search for a faser algorihm o solve for he opimum sigma value in he RBF mehod. ACKOWLEDGEMETS We wish o han our colleagues a he CREWES Proec and a Hampson-Russell Sofware for heir suppor and ideas, as well as he sponsors of he CREWES Proec. REFERECES Bishop, C.M., 995, eural ewors for Paern Recogniion: Oxford Universiy Press, Oxford, UK. Hampson, D., Schuele, J.S., and Quirein, J.A., 00, Use of muliaribue ransforms o predic log properies from seismic daa: Geophysics, 66, 0-3. Kohonen, T., 00, Self Organizing Maps: Springer Series in Informaion Sciences, 0, Springer Verlag, Berlin. Masers, T., 995, Advance Algorihms for eural ewors, John Wiley and Sons, Inc. adaraya, E.A., 964, On esimaing regression: Theory of Probabiliy and is Applicaions 9 (), CREWES Research Repor Volume 4 (00)

21 Applicaion of he RBF Poggio, T., and Girosi, F., 990, ewors for approximaion and learning: Proceedings of he IEEE 78 (9), Powell, M.J.D., 987, Radial basis funcions for mulivariable inerpolaion: a review. In J.C. Mason and M.G. Cox (Eds.), Algorihms for Approximaion, Oxford: Clarendon Press. Ronen, S., Schulz, P.S., Haori, M., and Corbe, C., 994, Seismic-guided esimaion of log properies, Pars,, and 3: The Leading Edge, 3, , , and Russell, B., Hampson D.P., Lines L.R., and Todorov, T., 00 Combining geosaisics and muliaribue ransforms A channel sand case sudy, CREWES Research Repor 3, Russell, B.H, Lines, L.R., and Ross, C.P., 00, AVO classificaion using neural newors: A comparison of wo mehods: CREWES Research Repor 4, his volume. Spech, D.F., 990, Probabilisic neural newors: eural ewors 3 (), Taner, M. T., Koehler, F. and Sheriff, R. E., 979, Complex seismic race analysis: Geophysics, 44, Wason, G.S., 964, Smooh regression analysis: Sanhya, he Indian Journal of Saisics. Series A 6, CREWES Research Repor Volume 4 (00)