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GEOPHYSICS, VOL. 72, NO. 1 �JANUARY-FEBRUARY 2007�; P. P9–P21, 19 FIGS.10.1190/1.2392789

nsupervised seismic facies analysis usingavelet transform and self-organizing maps

arcílio Castro de Matos1, Paulo Léo Manassi Osorio2,nd Paulo Roberto Schroeder Johann3

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ABSTRACT

Unsupervised seismic facies analysis provides an effectiveway to estimate reservoir properties by combining differentseismic attributes through pattern recognition algorithms.However, without consistent geological information, param-eters such as the number of facies and even the input seismicattributes are usually chosen in an empirical way. In this con-text, we propose two new semiautomatic alternative meth-ods. In the first one, we use the clustering of the Kohonenself-organizing maps �SOMs� as a new way to build seismicfacies maps and to estimate the number of seismic facies. Inthe second method, we use wavelet transforms to identifyseismic trace singularities in each geologically oriented seg-ment, and then we build the seismic facies map using theclustering of the SOM. We tested both methods using syn-thetic and real seismic data from the Namorado deepwater gi-ant oilfield in Campos Basin, offshore Brazil. The resultsconfirm that we can estimate the appropriate number of seis-mic facies through the clustering of the SOM. We alsoshowed that we can improve the seismic facies analysis byusing trace singularities detected by the wavelet transformtechnique. This workflow presents the advantage of beingless sensitive to horizon interpretation errors, thus resultingin an improved seismic facies analysis.

INTRODUCTION

Reservoir models are initially generated from estimates of specif-c rock properties and maps of reservoir heterogeneity. Estimates ofock properties, including porosity, permeability, fluid type, and li-

Manuscript received by the Editor February 28, 2005; revised manuscript r1BrazilianArmy, Military Institute of Engineering and Pontifícia Universi

razil 24.412-155. E-mail: [email protected] S. A.-Exploration and Production Department, Av. República

etrobras.com.br.2007 Society of Exploration Geophysicists.All rights reserved.

P9

hology, as well as reservoir compartmentalization and thickness,re fundamental for the exploration, development, and production ofetroleum fields. Small quantitative errors in the properties, as wells in the reservoir dimensions, can have great economic conse-uences.

Many types of information are used in reservoir model construc-ion. One of the most important sources of information comes fromells, including well logs, core samples, and production data. How-

ver, well log and core data are local measurements that may not re-ect the reservoir behavior as a whole. In addition, well data are notvailable at the initial phases of exploration. Models built using wellata require interpolation and extrapolation of local properties mea-ured in the wells to the entire prospect.

In contrast to sparse well data, 3D seismic data cover large areas.hanges in the lithology and fluids result in changes in amplitude,

hape, lateral coherence, and other seismic attributes. Seismic at-ributes can provide information for the construction of reservoir

odels. The description and interpretation of extracted seismic re-ection parameters, which include geometry, continuity, amplitude,requency, and interval velocity, is known as seismic facies analysisMitchum, 1977�.

Seismic facies analysis can be accomplished through the use ofattern recognition techniques. Given the appropriate combinationf seismic attributes, one can identify lateral changes in the reser-oir, which can then be calibrated with well information. The searchor an appropriate representation of petroleum reservoirs, using seis-ic data and pattern recognition techniques, has been the subject of

everal scientific publications �Dumay and Fournier, 1988; Schultzt al., 1994; Fournier and Derain, 1995; Walls et al., 1999; Johann etl., 2001; Saggaf et al., 2003�.

When the geological information is incomplete or nonexistent,eismic facies analysis is called nonsupervised and is performedhrough unsupervised learning or clustering algorithms �Duda et al.,

June 9, 2006; published online December 13, 2006.tólica do Rio de Janeiro, Rua Rangel, 140-Pita, São Gonçalo, Rio de Janeiro,

le, 65/1704B-Centro, Rio de Janeiro, Brazil 20035-900. E-mail: johann@

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001�. One of the most promising mathematical techniques appliedo nonsupervised pattern classification is the Kohonen self-organiz-ng map �SOM� �Kohonen, 2001� used by Coléou et al. �2003�, Tanert al. �2001�, Zhang et al. �2001�, and Matos et al. �2003a, b, 2004a,�.

Commercial implementations of the SOM are routinely used toerform seismic facies analysis. However, tasks such as the identifi-ation of the number of seismic facies in the analyzed data are stillerformed in an empirical way.

Regardless of the methodology for seismic facies analysis, theemporal and spatial segmentation of the reservoir area needs to beone carefully. The confidence in the interpretation depends on theeological system complexity, the seismic data quality, and the inter-reter’s experience �Rankey and Mitchell, 2003�, especially because

Selection of appropriate seismic attributes

Choice of the number of classes, facies, or patterns that will be used bythe algorithm

Training and classification of the selected attributes using anappropriate statistical or neural network tool

Construction and interpretation of the seismic facies maps andValidation using nonseismic data when they are available

Windowing of 3D seismic traces along the interpreted horizon

igure 1. Workflow for seismic facies analysis �Johann et al., 2001�.

he seismic facies analysis is sensitive to picking errors when usingeismic-trace waveforms in the reservoir area.

In this paper, we first review some modern seismic facies analysisechniques. Then, we introduce the concept of the SOM, and we pro-ose the first method that uses the clustering of the SOM, based onhe trace waveform, as a new way not only to estimate the number ofeismic facies but also as a nonsupervised seismic facies analysisechnique �Matos et al., 2003b�. Next, we use joint time-frequencynalysis to characterize reservoirs because variations in frequencyontent are sensitive to subtle changes in reflective informationSteeghs and Drijkoningen, 2001�. In this context, we show that theontinuous wavelet transforms �CWTs� and the discrete waveletransform �DWT� without time decimation can be applied to detectingularities �Shensa, 1992�. We use the concepts of wavelet trans-orm modulus maxima �WTMM�; the lines that link the WTMMsre called wavelet transform modulus maxima lines �WTMMLs�.hen, we perform the analysis along the WTMML amplitudes,hich is called wavelet transform modulus maxima line amplitude

WTMMLA�, to characterize the seismic trace singularities. We alsoropose a second method that uses joint time-frequency properties,btained through the detection of singularities using wavelet trans-orms �Matos et al., 2003a, b�, as a tool for the detection and charac-erization of seismic events. These singularities are subject to analy-is and SOM clustering. Finally, we tested these two algorithms onynthetic and real data with and without Gaussian noise and seismicnterpretation errors.

SOM CLUSTERING ANALYSIS

One of the most important goals of seismic stratigraphy is to rec-gnize and analyze seismic facies with regard to the geologic envi-onment �Dumay and Fournier, 1988�. For Sheriff �2002�, seismicacies analysis is done by examining seismic traces to identify theharacteristics of a group of reflections involving amplitudes, abun-ance, continuity, and configuration of reflections and also to predicthe stratigraphy and depositional environment.

Independent of whether seismic facies analysis is supervised orot, it can be implemented using the workflow shown in Figure 1Johann et al., 2001�. In spite of its proven effectiveness, the processescribed above should be performed very carefully.

The SOM �Kohonen, 2001� is currently one of the most importantools for the nonsupervised seismic facies analysis �Coléou et al.,003�.

ohonen SOMs

An SOM clusters similar data in a manner that provides effectiveisualization of multidimensional data. An SOM converts statisticalelationships among multidimensional data into simple, typicallyD, geometric relationships of the corresponding points. Mathemat-cally, the SOM preserves the metric relationships and the topologiesf the multidimensional input in the output 2D net. This net can besed as a visualization tool to show the different data characteristics,ith the possibility of group structuring �Taner et al., 2001�.The SOM is closely related to vector quantization methods

Haykin, 1999�. We begin by assuming that the input variables, i.e.,he seismic attributes, can be represented by vectors in the space Rn,= �x1,x2, . . . , xn�. The objective of the algorithm is to organize the

ata set of input seismic attributes delineated by a geometric struc-ure called the SOM. Each SOM unit, defined as a vector prototype,s connected to its neighbors, which in 2D usually forms hexagonalr rectangular structural maps.

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We assume that the map has P elements; therefore, there will existprototype vectors mi, mi = �mi1, . . . , min�, i = 1,2, . . . ,P, where n

s the dimension of the input vector, i.e., the number of input seismicttributes. After SOM training, the prototype vectors represent thenput data set of seismic attributes.

The number of prototype vectors in the map determines its effec-iveness and generalization capacity. During the training, the SOMorms an elastic net that adapts to the cloud formed by the input seis-ic-attributes data. Data that are close to each other in the input

pace will also be close to each other in the output map. Because theOM can be interpreted as a mapping of the input n-dimensionalpace into a 2D grid that preserves the original topological structure,nd because seismic data measures the changes in geology, SOMreserves the topological relation of the underlying geology.

Usually, the SOM prototype vectors are updated iteratively. First,he prototype vectors can either be randomly initialized or initializedy using the projections in the two largest eigenvectors of the inputata. Second, for each learning step, an input vector x is randomlyhosen from the input data set. The distances be-ween x and all the prototype vectors are comput-d. The map unit with the smallest distance mb tohe input vector x is called the best matching unitBMU� and is computed by �Kohonen, 2001�

�x − mb� = mini

��x − mi�� . �1�

Next, the prototype vector corresponding tohe BMU and their neighbors are updated, which

eans that they are moved towards the input win-er vector in the input space. The updating ruleor the ith unit is given by

mi�t + 1� = mi�t� + ��t�hbi�t��x − mi�t�� ,

�2�here t denotes the iteration, ��t� is the learning

ate, and hbi�t� is the neighborhood size centeredt the winner unit. The value of hbi�t� decreasesith each iteration in the learning process and isiven by

hbi�t� = e−��rb − ri�2/2�2�t��, �3�

here rb and ri are the positions of the prototypeectors b and i in the SOM grid and �2�t� defineshe neighborhood width.

Figure 2 shows a very simple example with di-ensionless data using the SOM. The input data are the 3D coordi-

ates for three Gaussian-distribution functions, with their centerslightly shifted from each other. There are 3000 3D input vectorsorresponding to 3000 measurements of the x-, y-, and z-compo-ents. We follow Vesanto and Alhoniemi �2000� to define the 2Dap dimensions using the ratio between the two largest eigenvectors

f the input data covariance matrix and obtain a 13�7 hexagonaltructure giving rise to 91 prototype vectors �Figure 2�.

After each iteration of the algorithm, the prototype winner vectornd its neighbors are updated using equations 2 and 3. After theearning, the prototype vectors represent not only a means of map-ing the input data in the SOM but also a good approximation to thenput vectors �Figure 2�. Owing to the simplicity of this example, it isery easy to visualize the prototype vectors. Because the number ofttributes is often larger than three, the visualization in the inputpace is usually difficult for general cases. To avoid this limitation,

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e could project the vector prototypes onto the two or three eigen-ectors which correspond to the two or three eigenvalues in the setKohonen, 2001�.

Because one of the main objectives of this work is the identifica-ion of data clusters, we will display the distances between the neigh-or prototype vectors to identify similarities among the SOM vectorrototypes. Specifically, we will use the unified matrix of distance,r U-matrix �Ultsch, 1993�, to represent these distances.

After the SOM learning, the U-matrix is generated by computing,or each SOM prototype vector, the distance between the neighborrototype vectors and their average. For the prototype vector 33 onhe SOM map shown in Figure 3a, we compute the distances to eacheighbor �prototype vectors 25, 26, 32, 34, 39, and 40� as well as theverage of these distances. In the U-matrix image, the intensity ofach pixel corresponds to the respective estimated distance. There-ore, the U-matrix not only shows the average distance between eachlement, it also shows the gradient between them. Figure 3b showshe U-matrix for the example in Figure 2 in which red or yellow rep-

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cation example showing a map with 91 �13�7� prototype vectorsof three Gaussian distributions in 3D space as input attributes.

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igure 3. �a� U-matrix generation; �b� U-matrix for the example inigure 2. Color bar is the distance between the prototype vectors.

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esent the prototype vectors that are far apart, whereas blue repre-ents prototype vectors that are close to each other. This 2D topolog-c image can be visualized as three dense blue valleys which are sep-rated by red, green, and yellow ridges; in other words, they formhree outstanding clusters. The U-matrix allows us to visualize theeometry and the number of existing groups in the data. One signifi-ant advantage of the SOM is the use of similar colors to map similarlasses.

lustering of the SOMIn nonsupervised seismic facies analysis, the estimation of the

umber of existing seismic facies in the data is typically determinedn an empirical way �Johann et al., 2001�.

We propose to estimate the number of seismic facies throughOM visualization. We begin by choosing a number for the SOMrototype vectors that is larger than the number of expected groupsn the data. Even though only qualitative information is generated,y using concepts of geomorphology, this procedure can be a quiteowerful interpretation tool.


Selection of appropriate seismic attributes

Generation of the SOM with a larger number of prototype vectors than theexpected number of seismic facies

Estimation of the number of seismic facies based on the DBI and SOM U-matrix visualization

Clustering and labeling of the SOM prototype vectors using the K-means partitive algorithm

Classification of each seismic attribute vectors to the closest prototypevector and, thus, to each seismic facies

Construction and interpretation of the seismic facies maps

igure 4. Workflow for nonsupervised seismic facies analysis basedn SOM clustering using waveform attributes �first method�.

To obtain a more quantitative clustering of data properties, SOMroups could be visualized using the U-matrix and chosen manually.owever, the manual selection of the clusters could be tedious and

mprecise. Agglomerative, or partitive, SOM clustering or U-matrixegmentation using image processing algorithms �Costa and Netto,999� provides an automated means of clustering. We will use a-means partitive clustering algorithm. In contrast to conventional-means, we will cluster the prototype vectors instead of the origi-al data �Vesanto and Alhoniemi, 2000�. In this manner, large dataets formed by the SOM prototype vectors can be indirectlyrouped. The proposed method not only provides a better under-tanding about the group formations, but it is also computationallyfficient �Vesanto et al., 1999�. Another benefit of this methodologys noise reduction because the prototype vectors represent local aver-ges of the original data without any loss of resolution.

he K-means clustering

An optimal clustering algorithm should minimize the distance be-ween the elements of each group and, at the same time, maximizehe distance between the different clusters. There are several ways of

easuring distance �Theodoridis and Koutroumbras, 1999�; forimplicity, we will use the Euclidian norm. To compute the distanceetween the elements of each group, we use the average distance Sk

etween each element xi and its group centroid ck:

Sk =

�i

�xi − ck�

Nk, �4�

here Nk is the number of elements in the group. The distance be-ween the k and l groups is computed as the distance between theirentroids:

dkl = �ck − cl� . �5�

The partitive clustering algorithm divides the data set into a pre-efined number of clusters, trying to minimize some error functionPandya and Macy, 1995, chapter 8�, with the number of groups cho-en and verified through SOM visualization. To automate the classi-cation process, we use the Davies and Bouldin �1979� index �DBI�s a means of evaluating the results of the K-means partitioning. Theest clustering corresponds to the minimum DBI given by

DBI =1

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K

maxl�k

Sk + Sl

dkl , �6�

here K is the number of groups, Sk and Sl are defined by equation 4,nd dkl is defined by equation 5. DBI values smaller than unity repre-ent separate groups, whereas values larger than unity representroups that may overlap.

SOM CLUSTERING APPLIED TOSEISMIC FACIES ANALYSIS

rinciples of the methodology — Seismic faciesith the SOM

The flowchart in Figure 4 shows the proposed methodology foronsupervised seismic facies analysis based on the SOM clustering.he main contribution of this methodology is to assist in the choicef the number of seismic facies in a semiautomatic way.

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eismic facies analysis of synthetic data —urbidite system modelTo verify the effectiveness of the proposed procedure, we used theodel shown in Figure 5a �Santos, 1997� to generate the convolu-

ional synthetic cube shown in Figure 5b. The reservoir is represent-d by three different seismic facies characterized by the their P-waveropagation velocities of 3100 m/s, 3200 m/s, and 3300 m/s.

We used, as seismic attributes �Taner et al., 1994�, the seismic am-litudes within a 20-sample window around the reservoir base, i.e.,he marked area between blue dotted lines in Figure 6a. The use ofhe contiguous seismic trace amplitudes as input attributes is equiva-ent to a waveform classification in the area of interest. The analysisesults with the proposed algorithm are shown in Figure 6. Figuresb–d show the U-matrix, the DBI, and the resulting facies map, re-pectively. In this example, three groups or facies are easily identi-ed from the U-matrix, and the classification result was excellent.owever, the minimum DBI of 4 did not correspond to the numberf existing facies. Therefore, the choice of the number of facieshould, whenever possible, be done in a semiautomatic way; in otherords, the estimate of the facies number should be confirmed by the-matrix visualization.This analysis was repeated by adding Gaussian noise to the syn-

hetic seismic model. The result in Figure 7 confirms the good per-ormance obtained with the proposed methodology and reaffirms themportance of the U-matrix visualization for the estimation of theumber of seismic facies.

However, when simulating a noisy horizon interpretation in therea of the reservoir, the proposed methodology gave a bad result, ashown in Figure 8. Such a result is related to the chosen seismic at-ribute, which is known to be sensitive to time displacements �Ran-ey and Mitchel, 2003�. Therefore, the choice of the seismic at-ributes for the classification of seismic patterns is fundamental tobtain coherent results �Poupon et al., 2004�.

Seismic facies analysis of real data —Namorado field, Campos Basin

We now apply the proposed methodology foreismic facies analysis to a 3D seismic real dataet from the Campos Basin that was made avail-ble by the Brazilian National Petroleum AgencyANP� and Petrobras. Figure 9 illustrates a dipine and the interpreted reservoir top and base.ine samples of the reservoir’s upper stratigraph-

c unit �Johann, 1997� were used as an input at-ribute. Because the variation in reservoir thick-ess is quite large, nine samples were interpolatednside this stratigraphic unit to avoid the use ofamples outside of the interest area. The seismicacies analysis is illustrated in Figure 10. The re-ults show that it is not possible to distinguish anyell-defined groups through the U-matrix visual-

zation. It seems that there is a single group withmall possible divisions, which means that a clearivision doesn’t exist among the preponderantrace waveforms in the data. The minimum DBIn Figure 10b indicates the existence of six seis-

ic facies, which is acceptable but not totally jus-ifiable. It is known from the petrophysics analy-is of these data that the most probable number ofacies is four �Johann, 1997�. The maps with four

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nd six facies are shown in Figures 10c and d, respectively.Thus, the number of seismic facies can not always be identified

learly through SOM clustering. In this case, the choice of seismic-race amplitudes was inappropriate for seismic facies identification.eologically, we expect a wide range of waveform variations in the

rea of interest because the seismic data were extracted from a com-lex sandstone turbidite system. To address this difficulty, we devel-ped a second method based on time-frequency techniques to locatend emphasize reservoir characteristics.

DETECTION AND CHARACTERIZATION OFSINGULARITIES USING TIME-

FREQUENCY TECHNIQUES

Seismic signals are a low-frequency representation of the subsur-ace reflectivity. Specifically, seismic signals measure changes in

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lastic impedance, and they can be approximated, depending on theavefront incident angle, by the convolution of the reflectivity func-

ion with a seismic wavelet. Therefore, the waveform at the strongeflection peaks is also influenced by the geological facies variationetween the stratigraphic units.

Subsurface analysis of turbidite sandstone systems shows that theeflectivity is not necessarily regular and uniform. Interpretation ofaveforms in the time domain is overly sensitive to interpretation

rrors, which results in facies misclassifications.Frequency-domain analysis using power-spectrum techniques

as been used in seismic interpretation mainly for thin-bed detection

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igure 7. �a� Seismic signal temporal segmentation with Gaussian aar is amplitude; �b� U-matrix; �c� DBI; �d� seismic facies map �50�

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igure 8. �a� Seismic signal temporal segmentation with interpretatimplitude; �b� U-matrix; color bar is distance; �c� DBI; �d� Seismic fepresents the classes.

Partyka et al., 1999; Mafurt and Kirlin, 2001; Johann et al., 2003�.owever, the power spectrum does not reveal how the frequency

ontent varies in time. On the other hand, joint time-frequency algo-ithms can be used to locate and to emphasize reservoir characteris-ics �Matos et al., 2003a, b; Castagna et al., 2003; Matos et al., 2004a,�.

ontinuous wavelet transform (CWT)

As formally introduced by Grossmann and Morlet �1984�, a func-ion ��x��L2�R� with zero mean is called a wavelet if it satisfiesome well-established conditions.Afamily of wavelet functions can

be obtained from the mother wavelet ��t� by scal-ing it by s and shifting by u:

�u,s�t� =1�s

�� t − u

s . �7�

If we compress a wavelet, its spectrum willspread and move to higher frequencies. On thecontrary, if we dilate a wavelet, its spectrum willcompress and move to lower frequencies �Mallat,1999�. The wavelet spectrum has no DC, or zero-frequency, component.

In addition, the spectrum of each wavelet in thefamily maintains a constant ratio between its cen-tral frequency and the corresponding bandwidth.Once a wavelet family is chosen, then the conti-nous wavelet transform �CWT� W of a functionf�t��L2�R� at time u and scale s is defined as

Wf�u,s� =1�s�−�

+�

f�t��*� t − u

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where the superscript * indicates the complexconjugate.

Because the wavelet ��t� has zero average, theCWT can be interpreted as the crosscorrelationbetween the signal and the wavelet, shifted by uwith scale s. Rewriting equation 8, we can also in-terpret the CWT as a convolution operation:

Wf�u,s� = �−�

+�

f�t�1�s

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= f � �̄s�u� , �9�

where

�̄s�t� =1�s

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s . �10�

Because the spectrum of ��t� resembles aband-pass filter, then equation 9 can be interpret-ed as the convolution of the signal f�t� with theband-pass filter response scaled by s.

It can be proven that the CWT preserves thesignal energy and is invertible, such that, the sig-nal can be reconstructed from the wavelet-trans-form coefficients �Mallat, 1999�. The energy rep-resentation of the CWT coefficients is called ascalogram.

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etection and measurement ofingularities through the CWT

The transitions, or irregular structures, present in any kind of sig-al carry information related to its physical phenomena. For in-tance, the image boundaries of pictures are often enough to identifynd to characterize objects.

Seismic interpretation can be described as the identification of theeflectivity transitions smoothed by the seismic wavelet. Besides theorizon locations, the identified transition characterization in the in-erpretation is associated with geological processes. In this way, aossible transition classification could be linked to the seismic fa-ies.

The detection of transitions or singularities in signals is based onimple mathematical concepts. The signal inflection points are asso-iated with the first-derivative extremes, which correspond to theecond-derivative zero crossings. On the basis of this concept, mostf the edge detectors search to identify the transitions in multiplecales after the previous smoothing of the signal �Canny, 1986�. Theetection of singularities through multiple scales is related to theWT, as described below.Let’s define a doubly differentiable smoothing function ��x� �Jaf-

ard et al., 2001, p. 123; Mallat and Hwang, 1992, Figure 1�, with in-egral equal to unity, that converges to zero when x tends to ±�. Aaussian curve is an example of such a smoothing function. Let usefine

� a�x� =d��x�

dx, and � b�x� =

d� 2�x�dx2 . �11�

Because the integrals of �a�x� and �b�x� are zero in the interval��x��, they can be considered to be wavelets. In this way, theWT of a signal f�x� in the scale s can be obtained by convolving the

ignal with a scaled wavelet. The CWTs of f�x�, using the two wave-ets defined by equation 11, are

W saf�x� = f � � s

a�x�, and �12�

W sbf�x� = f � � s

b�x� . �13�

Inserting the smoothing-function derivativesnto equations 12 and 13 yields

saf�x� = f � �s

d�s

dx �x�

= sd

dx�f � �s��x�, and �14�

sbf�x� = f � �s2d2�s

dx2 �x�

= s2 d2

dx2 �f � �s��x� . �15�

Thus, the wavelet transform Wsaf�x� and the

ransform Wsbf�x� are, in the scale s, the

moothed-signal first- and second-derivative, re-pectively. Therefore, the local extremes of

saf�x� relate to the zero crossings of Ws

bf�x�,hich correspond to the inflection points of

f ��s�x�. In the case that ��x� is a Gaussian curve,he image-boundary detection process is equiva-ent to the method proposed by Canny �1986�.

map size [

SClu

Number of c

a)

b) 1.5

1.1

0.8

DB

I

0 4

Figure 10. ReDBI; �c� seismlocation showunit with fourure 9.

avelet transform modulus maxima (WTMM)For the signal inflection-point positions, using the CWT local

eak locations, a wavelet should be chosen as the first-derivative ofhe smoothing function ��x�. One wavelet that fulfills this require-

ent is the first-derivative of the Gaussian function, called the Gaussavelet. Figure 11b illustrates the scalogram obtained from theWT using the Gauss wavelet for the test signal shown in Figure1a. We can see that the scalogram local peaks coincide with the sig-al inflection points. Figure 11c shows the lines formed by linkinghese inflection points at each scale. It can be proven that these lines,hich are called Wavelet Transform Modulus Maxima Lines �WT-MLs�, can be used to characterize the signal irregularities through

0 1km 2kmDistance (m)

TW

T (

ms)

2320

2360

2400

2440

2480

2520

2560

2600

2640

2680–8590.0 –4290.0 0.0 4290.0 8590.0

igure 9. Seismic line extracted from the 3D real data. The blue linesepresent the reservoir base and top interpretations, whereas the yel-ow and the green lines correspond to intermediate stratigraphic ho-izons. Color bar is amplitude.

Classification with four facies

Classification with six faciesSeismicfacies

Seismicfacies

Distance (m)

Distance (m)

Dis

tanc

e (m

)D

ista

nce

(m)

4730

2720

715

d)

c) n

4

3

2

1

n

6

5

4

3

2

1

Distance (m)0 1km 2km

0 1km 2km

seismic facies analysis of the top of the reservoir; �a� U-matrix; �b�es map with six facies; the black line corresponds to the seismic linegure 9; �d� seismic facies map of the reservoir’s upper stratigraphicthe black line corresponds to the seismic line location shown in Fig-

29,24]

ix sters

lusters 8

al-dataic facin in Fifacies;

tTl

t�sc

ittwttc

t

11dee

D

lvtc

fi�tlibdafsatedi

mw

cZa

kl

FlcM�W

Ft

P16 Matos et al.

he analysis along the WTMML amplitudes, which is called Waveletransform Modulus Maxima Line Amplitudes �WTMMLAs� �Mal-

at and Hwang, 1992�.The signal irregularities can be characterized mathematically

hrough the Lipschitz exponent ��, also called the Hölder exponentMallat, 1999, chap. 6�. The � exponent can be obtained from thelope estimation of the curve created by the log2 of the WTMMLAoefficients divided by the log2 of the scales �Mallat, 1999, chap. 6�:

log2�Wf�u,s�� log2 A + �� +1

2 log2 s . �16�

It should be observed that WTMML must be generated by obey-ng the wavelet cone of influence. The cone of influence is the area ofhe signal around a point at time t0 that is taken into consideration forhe CWT computation when the scale s is varied. Considering that aavelet has a compact support in the interval �−C,C�, it is said

hat the CWT cone of influence along the s scales, for a certain loca-ion in the time t0, is equal to �t0 − Cs,t0 + Cs�. Figure 12 illustrates aone-of-influence example with respect to a location t0.

Figure 11d illustrates the WTMMLAbehavior along the scales forhe discontinuity around sample 64 of the test signal shown in Figure

20 40 60 80 100 120

20 40 60 80 100 120

20 40 60 80 100 120

2 4 6 8 10 12 14 16

0 1 2 3 4

Time (s)

Time (s)

Time (s)

Scale

log2 (scale)

log2

(W

TM

MLA

)W

TM

MLA

Sca

les

Sca

les

Am

plitu

de

1 0.5 0–0.5

1.5

1

0.5

0.5 0

–0.5

–1

1612 8 4 1

1612 8 4 1

a)

b)

c)

d)

e)

+ 1/2 = 1/2

igure 11. �a� Test signal; �b� CWT scalogram using the Gauss wave-et, in which colors close to white represent positive coefficients andolors close to black correspond to negative coefficients; �c� WTM-L; �d� WTMMLAalong the scales for the discontinuity detected in

a� around sample 64; �e� Lipschitz exponent obtained throughTMMLAaccording to equation 16.

1a and detected by WTMML, as illustrated by Figure 11c. Figure1e shows the Lipschitz � exponent curve corresponding to the sameiscontinuity. We can see in Figure 11e that the curve slope comput-d by the log2 of the WTMMLAis equal to 0.5. Therefore, as expect-d for a discontinuity, the Lipschitz exponent � is equal to zero.

iscrete wavelet transform without time decimation

The CWT uses families of functions created by varying continual-y the scale s and the displacement u parameters. This continuousariation of two parameters simultaneously results in a computa-ionally intensive operation. In practice, this is accomplished byomputing the CWT for a great number of scales.

The fastest and most common method to compute the CWT useslter banks. In this case, it is called the Discrete Wavelet TransformDWT� �Burrus et al., 1997�. The DWT coefficients are computed inhe analysis filter bank, which is a set of two filters, one half-bandow pass and one half-band high pass. In the forward transform, thenput signal is filtered by the two filters followed by a downsamplingy a factor of two. In this context, these two last operations are calledecimation. The output of the low-frequency branch is then filteredgain using the same filter bank of analysis. This process continuesor the number of desired levels of decomposition. This process isimilar to the CWT, except for the fact that the scales change in a dy-dic way, i.e., in powers of two, and owing to the decimation opera-ions, the DWT is not invariant to displacements in time. Althoughfficient for data compression and some filtering operations, this lastetail renders the DWT unfeasible for the detection and character-zation of singularities.

An alternative way to obtain a DWT that is invariant to displace-ents is to remove the downsampling operation from the filter bank,hich results in having only the scale as a dyadic sequence.Mallat �1999� showed that the dyadic form makes a stable and

omplete representation that preserves the signal energy. Mallat andhong �1992� proposed a quadratic-spline wavelet family appropri-te for the detection and characterization of signal singularities.

The implementation of the DWT algorithm without decimation isnown as wavelet à trous �WAT�, and it consists, in the signal convo-ution with the bank filter, of coefficients with �2 j − 1� zeros inserted

Cone of influence

t0 = 0 Time (s)

Sca

le

12

8

4

1

–4 –3 –2 –1 1 2 3 4

igure 12. Wave transform cone of influence for a point located atime t .
0

bwr

ttoot

sTts

tcets

Pw

cstcflt

FiWt

Fbm


etween the samples �Shensa, 1992�. For that reason, it is calledavelet à trous, which in French means a wavelet with holes or ze-

os.We can also use the WTMM, computed through the WAT, for de-

ection and characterization of signal singularities. Figure 13b illus-rates singularity detection through WTMMLs for a test signal usingnly four dyadic scales, whereas Figure 13c illustrates the evolutionf the WTMMLA for the singularities located around sample 100 ofhe test signal and detected in Figure 13a.

Hoekstra �1996� proposed to use the Lipschitz � exponent as aeismic attribute by applying the WAT to compute the WTMMLA.he Lipschitz � exponents can be computed for each WTMMLA. In

he Figure 13c example, the � exponents can be estimated from fouramples through a simple linear regression.

Liner et al. �2004� proposed to calculate the Hölder exponent fromhe CWT of a seismic trace as an instantaneous seismic attribute andalled this SPICE, which became a commercial product �Li and Lin-r, 2004�. Instead of using the WTMMLA, they proposed to estimatehe slope of the curve along the scales for each localized time of theeismic trace.

20 40 60 80 100 120

0 20 40 60 80 100 120

1 1.5 2 2.5 3 3.5 4

Time (s)

Time (s)

Scales

Sca

leS

igna

la4

d4d3

d2d1

0.5 0–0.5

0.40.20

1 0–1

2

1

0

–1

–2

1 0–1

1 0–1

1 0–1

1

2

3

4

a)

b)

c)

igure 13. �a� Test signal decomposition in four levels; vertical scales in amplitude; a4 is the approximated wavelet; �b� WTMML; �c�

TMMLA curve for the singularities located around sample 100 ofhe test signal detected in �a�.

RESERVOIR CHARACTERIZATION USING JOINTTIME-FREQUENCY ANALYSIS

PLUS SOM CLUSTERINGrinciples of the methodology — Seismic faciesith WT and the SOM

It can be observed that the WTMMLA curves used for the Lips-hitz exponent computation can be interpreted as patterns by them-elves; in other words, instead of using � as an attribute, we proposeo use the whole WTMMLA curve as an input attribute for a patternlassification system. The proposed algorithm is detailed in theowchart of Figure 14 and is shown schematically in Figure 15 �Ma-

os et al., 2003a, b�.


Decomposition of the seismic traces using the WAT with the desirednumber of levels

Computation of each trace line WTMMLA and initialization of the attributevector with the most significant WTMMLA (usually two)

Formation and training of the SOM with a larger number of prototypevectors than the expected number of seismic facies

Estimation of the number of seismic facies by comparing the SOM U-matrix visualization with the DBI for different numbers of groups

Clustering and labeling of the SOM prototype vectors using the K-meanspartitive algorithm

Classification of each seismic attribute vectors to the closest prototypevector and, thus, to each seismic facies

Construction and interpretation of the seismic facies maps

igure 14. Workflow for nonsupervised seismic facies analysisased on SOM clustering using WTMMLA attributes �secondethod�.

ST

scccca

SN

oatrsi

b

P18 Matos et al.

eismic facies analysis of synthetic data —urbidite sandstone system model

We used the same synthesized data with interpretation noise,hown in Figure 6a and repeated in Figure 16a, to test the seismic fa-ies analysis with the second method. Figure 16d shows the resultinglusters. Comparing Figure 16c and e, we can see that the result is ex-ellent and that it confirms the expectation about the capacity to lo-ate seismic events in time and the consequent characterizationlong the scales.

Seismic trace

WTMMLAseismic

attribute inpvector

WAT decomposition with three levels

Sam

ples

Singularities

WTMMLA

–16513

–32828

–25449

14104

30063

23735

a) b) c) 0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

16

–1 0 1 2 3 4 5 6

SOMmap size[10 x 5] Distance (m)

Distance (m)

Two-

way

tim

e (s

)

0

0.5

1

1.5

1.5 1 0.5 0–0.5–1–1.5–2

0 500 1000 1500 2000

a)

)

c)

d)

e)Seismicfacies

Dis

tanc

e (m

) n

3

2

1

Distance (m)

Dis

tanc

e (m

)

Distance (m

1.54

0.892

0.241

U-matrix U-matrix

eismic facies analysis of real data —amorado field, Campos Basin

Seismic interpretation of Namorado field reservoirs, on the basisf 45 well log data sets and 3D seismic calibration, allows the char-cterization of a reliable stratigraphic model with three high-resolu-ion stratigraphic units that defined the internal architecture of theseeservoirs �Johann et al., 1996; Johann, 1997�. Three depositionalequences were correlated across the field, each of them correspond-ng to cycles of relative sea level falls.

Clusteringof theSOM

Figure 15. �a� Seismic trace from Figure 9 with 16time samples extracted around the base horizon;the seismic trace is repeated four times; �b� WATwith three decomposition levels, d1, d2, and d3 andWTMMLA associated with the seismic trace sin-gularities; �c� the WTMMLAseismic attribute vec-tor example shows the two most significant WTM-MLAs that maintain the time sequence. This vectoris used as input to the SOM clustering algorithm.

SOMmap size

[7 x 7]

ce (m)

Seismicfacies

n

3

2

1

1.54

0.769

0

Figure 16. Seismic facies analysis of synthetic datawith interpretation noise; �a� seismic data; color baris amplitude; �b� U-matrix; �c� seismic facies map�50�2500 m� using the waveform as attribute; �d�U-matrix; �e� seismic facies map �50�2500 m�using WTMMLAattributes.

ut

Distan

)

pim

swhy1i

ta

rpstvgt

Ftmli


The basal unit is characterized by confined turbidite sediments de-osited in the salt-controlled topographic lows �paleocanyons�. Thentermediate unit and the top unit are more related to classical Bou-

a turbidites interbeded with marls and shales �Johann, 1997�.We now apply the proposed methodology to the Campos Basin

eismic data used previously. For this analysis, we used a windowith 16 samples, e.g., a 64-msec time-interval window, around theorizon that delimits the top of the reservoir. The seismic facies anal-sis result is illustrated in Figure 17c. It can be observed in Figure7b that the four groups formed for the analysis were well-enhancedn the U-matrix; they coincided with the facies number suggested in

U-matrix(SOM map size

a)

WTMMLAattributes

igure 17. Real-data seismic facies analysis of theop of the reservoir using the second method; �a� U-

atrix; �b� DBI; �c� seismic facies map; the blackine corresponds to the seismic line location shownn Figure 9.

U-matrix(SOM map size [

a)U

WTMMLAattributes

Figure 18. Real-data seismic facies analysis of thebase of the reservoir using the second method; �a�U-matrix; �b� DBI; �c� seismic facies map; theblack line corresponds to the seismic line locationshown in Figure 9; the dotted black lines corre-spond to the basal paleocanyon.

he petrophysics analysis and sedimentary facies analysis �Johann etl., 1996; Johann, 1997�.

The seismic facies analysis was also implemented around the ho-izon that delimits the base of the reservoir, also taking sixteen sam-les. The results, illustrated in Figure 18c, are also very good; theyhow an interpreted paleocanyon in the Namorado field sandstoneurbidite system �Figure 19�. Additionally, even using an unsuper-ised approach, it is possible to identify an interesting petrophysicsradation that helps define the internal geometry of this high-resolu-ion stratigraphic unit.

Seismic facies map withfour classes


K-m

eansclustering

Seismicfacies

Number of clusters

Dis

tanc

e (m

)

16800

8630

488

b)

c) n

4

3

2

1Distance (m)

Distance (m)

0 1km 2km

0 5 10

Dav

ies-

Bou

ldin

inde

x

0.74

0.72

0.7

0.68

0.66

0.64

0.62

0.6

0.58

rix

A

Seismic facies map withfive classes


K-m

eansclustering

Seismicfacies

Number of clusters

b)

n

5

4

3

2

1Distance (m)

Distance (m)

0 1km 2km

0 5 10

Dav

ies-

Bou

ldin

inde

x

0.85

0.8

0.75

0.7

0.65

[42, 16])

U-mat

50, 14])

Dis

tanc

e (m

)

c)

-matrix

A

Mttt

mcUtpi

tscamteoscem

otsfi

mvaap

bkgahg

Fe

P20 Matos et al.

The excellent result of the seismic facies analysis, using WTM-LAfor synthetic data, and the coherent results obtained for realda-

a suggest that the second method proposed in this work is an impor-ant tool for reservoir characterization as well as for seismic explora-ion, mainly, because it is less sensitive to interpretation errors.

CONCLUSIONS

In this paper we proposed two methods to estimate, in a semiauto-atic way, the number of seismic facies and also to create seismic fa-

ies maps. The first method estimates the number of facies from the-matrix visualization and the DBI, computed from the SOM clus-

ering. The main drawback of this method is its sensitivity to inter-retation errors, especially when seismic-trace amplitudes are the

7 km

0 km

3 km

TWT (ms)

igure 19. 3D view of structural time map of the Namorado field res-rvoir basal turbidite sandstone system.

nput data to the algorithm. s

LIST OF SYM

B

C

C

C

C

In the second method, we employed time-frequency techniqueshat use the WAT as a preprocessing tool to a pattern classificationystem, which also uses the SOM clustering. The use of WAT to lo-ate events in time through the identification of signal singularitieslso proved to be useful as an appropriate tool for detection of seis-ic events. Because the evolution of the WTMMLA characterizes

he signal singularities, we used it as a seismic attribute vector relat-d to the detected events. This approach has clear advantages overther techniques that use, for instance, the Lipschit coefficient as aeismic attribute. The SOM integration, as a classification tool asso-iated with the patterns generated using the WAT, has proven to be anffective tool in 3D seismic reservoir characterization. The secondethod also proved to be less sensitive to interpretation errors.It should be emphasized that the objective of this work was not to

btain an optimal way to cluster the data but, instead, to identify ando separate different characteristics in large data sets, as is the case ofeismic data. Obviously, this approach is only valid when the identi-ed groups, using the SOM, are similar to those of the original data.The results obtained with 3D seismic data suggest that the secondethod can also be used for 4D analyses, which is currently being in-

estigated. Such a statement is based on the fact that the anomalieslready explored in a certain area would probably have their associ-ted singularities altered because of the changes in the reservoirroperties related to the oil and/or gas production.

ACKNOWLEDGMENTS

The authors thank ANP �Brazilian Petroleum Agency� and Petro-ras for permission to use the seismic data contained herein and ac-nowledge Petrobras financial support through the Petrobras Strate-ic Research Program in Reservoir Geophysics, PRAVAP 19. Theuthors are grateful to GEOPHYSICS associate editor Kurt Marfurt foris patience in providing a thorough review of the manuscript, whichreatly improved its quality. Thanks to the reviewers for their con-
tructive comments.
BOLS

� Lipschit or Hölder exponentK number of SOM groupsck group k centroid in the K-means partitive

clusteringdkl distance between group centroids in the K-means

partitive clustering��x� smoothing function in the time-frequency method

D number of SOM map elementshbi�t� neighborhood size entered at the winner unit in

the SOM algorithm��t� learning rate in the SOM algorithm

mi prototype vectors i = 1,2, . . . ,P in the SOMalgorithm

mb smallest distance to the input vector x in theSOM algorithm

n number of input seismic attributesP number of prototype vectors in the SOM algorithmSk average distance between each element and its

group centroid in the K-means partitive clustering��t� mother wavelet

�a�x� first derivative of ��x��b�x� second derivative of ��x�

rb position of the prototype vector b in the SOMgrid

ri position of the prototype vector i in the SOM grids continuous wavelet scale coefficientu continuous wavelet translation coefficientx input vector of dimension n in the SOM algorithm

Wu,s�t� family of continuous wavelet functions

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