Mud-rock line estimation via robust locally weighted scattering smoothing method

Transcription

1 Mud-roc lne va LOESS Mud-roc lne estmaton va robust locally weghted scatterng smoothng method A. Nassr Saeed, Laurence R. Lnes, and Gary F. Margrave ABSTRACT The robust locally weghted scatterng smoothng (LESS) s another regresson method that not only smoothes the scatter plot but also guard aganst outlers that dstort the smoothed ponts. The mud-roc lne produced by the robust locally weghted smoothng scatterng method, LOESS, has shown superorty over the Castagna lnear regresson method n mappng the Glaucontc sand reservor of Blacfoot area. The produced graph has demonstrated to be a good vsualzng tool for an effectve drect hydrocarbon ndcator (DHI) as well as n dscrmnatng sand and carbonate lnes. INTRODUCTION The lnear regresson method s substantally nfluenced by the presence of nose or outlers that mght gve naccurate estmaton of the slope and ntercept ponts; thus erroneous predcted data. The man objectve of ths study s to research for another regresson method that s less affected by the presence of nose n order to estmate the mud-roc lne more precsely. MUDROCK LINE ESTIMATION Castagna et al., (1985) used the Vp-Vs lnear relaton to predct V p from V s va lnear regresson method. The Castagna s mud-roc lne equaton s wrtten as V p 1.16V s (1) where sonc veloctes are n Km/sec. A classcal procedure for nosy data usually requre some edtng before performng lnear regresson so as to reduce nfluences of outlers on estmatng slope and ntercept ponts used n best ft-lne equaton. Another approach s by fttng polynomal of hgher degrees, such as quadratc equaton, where the curvature factor s ncluded n the best ft-lne equaton. The latter procedure wll volate the smple Vp-Vs lnear relaton sought n predctng mssng log data. Hence, n order to map mud-roc lne more precsely, we should loo for another regresson method that tae nto consderaton the effect of neghbor ponts n the scatterng graph as well as to prove t s robustness aganst the outlers. CREES Research Report Volume 22 (2010) 1

2 Saeed, Lnes, and Margrave ROBUST LOCALLY EIGHTED SMOOTHING SCATTERING (LOESS) Cleveland, (1979) has ntroduced the robust locally weghted scatterng smoothng (LOESS) method. Ths s another regresson method that not only smooths the scatter plot but also guard aganst outlers that normally dstort the smoothed ponts n the scatterng plot. Gven (x, y ) ponts, let d are the dstance between x and t s b th nearest neghbors along the x-axs. b s the nearest nteger to ( f n / 2), where n s the number of data pont and f s the smoothng factor ( 0 < f < 1 ). A greater value of f leads to smoothng the curve lne, whle small values of f produce rougher curve. A practcal value of f s usually between 0.2 and 0.8. The weghted slope ( b estmate ) and the ntercept ( aestmate ) of lne derved from robust locally weghted regresson method can be wrtten as, b estmate ( x x ) ( y y) ( x x ) (2) aestmate y bestmate. x (3) here, y and x are weghted mean values, and x ϖ x d, { ϖ( x) p p (1 x ) forx < 1 0 forx 1 The best ft lne equaton s then wrtten as yw aestmate+ bestmate. x (4) 2 CREES Research Report Volume 22 (2010)

3 Mud-roc lne va LOESS The weght functon can be ether b-weght functon (p2) or, tr-cube weght functon (p3). In ths study, tr-cube weght functon s appled. Note that the sze of weght depends on the magntude of resdual y yˆ. Large resduals results n small weghts and small resduals result n large weghts (Cleveland, 1979). The robust locally weghted smooth scatterng, LOESS, s used n ths study n order to map the mud-roc lne. Fgure (1) shows the mud-roc lne produced by LOESS method. As expected, ths method s less nfluenced by nose that substantally affects the estmaton of slope and ntercept of fttng lne. Fgure (2) shows the mud-roc lnes produced by LOESS (n red color), locally lnear-regresson (n magenta color) and by usng Castagna s mud-roc equaton (n green color). The data ponts are colored by gamma ray values, where sand usually has low gamma ray values. The slope and ntercept values estmated va LOESS method are also used n equaton (1), and produced mud-roc lne (n purple color) n fgure (2). The resultng weghted mud-roc lne shows better estmate for the ntercept value compared to the locally derved lnear regresson method, and demonstrated to be less nfluenced by the outlers. The mud-roc lne estmated by LOESS method has successfully delneated channel sand, whereas lnear regresson and ARCO mud-roc equaton dd not. Furthermore, the carbonate lne can be easly dstngushed by the LOESS method, where t starts from the ntersecton pont of Vs (around 2.8 Km) and Vp of 4.8Km/sec and above. In fgure (3), velocty values of the sonc shear (Vs) log of the productve reservor sand channel (blac dots) are supermposed n the secton. Indeed, the mudroc lne estmated by LOESS regresson method has proven to be effectve DHI tool. In fgure (4), the sonc P-wave of the log and P-wave velocty produced va LOESS method show good resemblances. The curve n green color represents the P-wave velocty produced n smoothng scatterng doman. CONCLUSIONS The robust locally weghted smoothng scatterng method successfully maps the productve sand reservor as well as the carbonate lne. It proves to be a good nterpretve tool to be used n hydrocarbon and lthology dscrmnaton. The predcted log of P-wave velocty shows excellent resemblance to the orgnal sonc P-wave log REFERENCES Castagna, J. P. Batzle, M.L. and Eastwood, R.L., 1985, Relatonshps between compressonalwave and shear-wave veloctes n elastc slcate rocs: Geophyscs, 50, Cleveland,.S., 1979, Robust locally weghted regresson and smoothng scatterplots: Journal of the Amercan Statstcal Assoc., 74, CREES Research Report Volume 22 (2010) 3

4 Saeed, Lnes, and Margrave FIG.1.Robust locally weghted regresson method, LOESS, for estmatng mud-roc lne. FIG.2. Robust locally weghted regresson method for estmatng mud-roc lne. Data ponts are colored by Gamma ray 4 CREES Research Report Volume 22 (2010)

5 Mud-roc lne va LOESS FIG.3. Robust locally weghted regresson method for estmatng mud-roc lne. V s values from the productve sand channel reservor (blac dots) are supermposed. FIG.4. P-wave sonc log, and derved P-wave velocty va the LOESS method of well CREES Research Report Volume 22 (2010) 5