**Authors**

**Abstract**

Reservoir models are initially generated from estimates of specific rock properties and maps of reservoir heterogeneity. Many types of information are used in reservoir model construction. One of the most important sources of information comes from wells, including well logs and core samples. Unfortunately well log and core data are local measurements that may not reflect the reservoir behavior as a whole. In addition, well data are not available at the initial phases of exploration. In contrast to sparse well data, 3D seismic data cover large areas. Seismic attributes extracted from 3D seismicdata can provide information for the construction of reservoir models. Seismic facies analysis can be accomplished through the use of pattern recognition techniques. When the geological information is incomplete or nonexistent, seismic facies analysis can be done using unsupervised learning techniques. One of the most promising mathematical techniques of unsupervised learning is the Kohonen's Self Organizing Map (SOM) (Kohonen, 2001).

In this paper we use the SOM and time-frequency analysis to characterize reservoirs. Since variations in frequency content are sensitive to subtle changes in reflective information. In this context, we show that the wavelet transform modulus maxima line amplitudes (WTMMLA) that extracted from continuous wavelet transforms (CWT) can be applied to detect singularities. These singularities are analyzed and clustered by SOM.

The SOM networks map points of input space to points in an output space while preserving the topology. Topology preservation means that points which are close in the input space should also be close in the output space (map). Normally, the input space is of high dimension while the output is two-dimensional. The seismic attributes, can be represented by vectors in the space Rn, x = [x1,x2...xn].We assume that the map has Pelements; therefore, there will exist P prototype vectors mi, mi= [mi1. . . min], i = 1, 2, . . . ,P, where n is the dimension of the input vector. After the SOM training, prototype vectors represent the input data set of seismic attributes, the distances between x and all the prototype vectors are computed. The mapunit with the smallest distance mbto the input vector x is called the best matching unit (BMU) and is computed by, . The prototype vector corresponding to the BMU and their neighbors are moved towards the input winner vector in the input space. Since one of the main objectives of this work was the identification of data clusters, we displayed the distances between the neighbor prototype vectors to identify similarities among the vector prototypes. We used the U-matrix (Ultsch, 1993), to represent these distances. After the SOM learning, the U-matrix was generated by computing, for each SOM prototype vector, the distance between the neighbor prototype vectors and their average.

For estimation of the number of existing seismic facies in the data, we used a K-means partitive clustering algorithm. We clustered the prototype vectors instead of the original data. In this manner, large data sets formed by the SOM prototype vectors can be indirectly grouped. Results showed that the proposed method not only provides a better understanding about the group formations, but it is also computationally efficient. Another benefit of this methodology is noise reduction because the prototype vectors represent local averages of the original data without any loss of resolution. To automate the classification process, we used the Davies and Bouldin (1979) index (DBI) as means of evaluating the results of the K-means partitioning.

Transitions, or irregular structures, present in any kind of signals carry information related to its physical phenomena (Mallat, 1999). Besides the horizon locations, the identified transition characterization in the interpretation is associated with geological processes. In this way, a possible transition classification could be linked to the seismic facies. Detection of transitions or singularities in signals is based on simple mathematical concepts. The signal inflection points are associated with the first-derivative extremes which correspond to the second-derivative zero crossings. For the signal inflection-point positions, using the CWT local peak locations, a wavelet should be chosen as the first-derivative of the smoothing function . One of the wavelet functions that fulfill this requirement is the first-derivative of the Gaussian function, called the Gauss wavelet. We can extract scalogram's local peaks coincide from the signal inflection points. It can be proven that these lines, which are called WTMMLA, can be used to characterize the signal irregularity. The signal irregularities can be characterized mathematically through the WTMMLA and H?lder exponent (Mallat, 1999). The exponent can be obtained from the slope estimation of the curve created by the log2 of the WTMMLA coefficients divided by the log2 of the scales. In This study we used WTMMLA as a direct seismic attribute. We calculated CWT coefficients and WTMMLA for sixteen seismic data samples around the picked reservoir horizon. The extracted WTMMLA can show the possible heterogeneity and singularity within the reservoir. We used these attribute as input vector for the SOM step and obtained the U_matrix. The K-mean and DBI estimate the number of seismic facies.Utilizing of CWT to locate events in time through the identification of signal singularities also proved to be useful as an appropriate tool for detection of seismic events. Therefor this method proved to be less sensitive to interpretation errors. The performance of the method was tested on Kangan formation at one of the Iranian oil fields.

In this paper we use the SOM and time-frequency analysis to characterize reservoirs. Since variations in frequency content are sensitive to subtle changes in reflective information. In this context, we show that the wavelet transform modulus maxima line amplitudes (WTMMLA) that extracted from continuous wavelet transforms (CWT) can be applied to detect singularities. These singularities are analyzed and clustered by SOM.

The SOM networks map points of input space to points in an output space while preserving the topology. Topology preservation means that points which are close in the input space should also be close in the output space (map). Normally, the input space is of high dimension while the output is two-dimensional. The seismic attributes, can be represented by vectors in the space Rn, x = [x1,x2...xn].We assume that the map has Pelements; therefore, there will exist P prototype vectors mi, mi= [mi1. . . min], i = 1, 2, . . . ,P, where n is the dimension of the input vector. After the SOM training, prototype vectors represent the input data set of seismic attributes, the distances between x and all the prototype vectors are computed. The mapunit with the smallest distance mbto the input vector x is called the best matching unit (BMU) and is computed by, . The prototype vector corresponding to the BMU and their neighbors are moved towards the input winner vector in the input space. Since one of the main objectives of this work was the identification of data clusters, we displayed the distances between the neighbor prototype vectors to identify similarities among the vector prototypes. We used the U-matrix (Ultsch, 1993), to represent these distances. After the SOM learning, the U-matrix was generated by computing, for each SOM prototype vector, the distance between the neighbor prototype vectors and their average.

For estimation of the number of existing seismic facies in the data, we used a K-means partitive clustering algorithm. We clustered the prototype vectors instead of the original data. In this manner, large data sets formed by the SOM prototype vectors can be indirectly grouped. Results showed that the proposed method not only provides a better understanding about the group formations, but it is also computationally efficient. Another benefit of this methodology is noise reduction because the prototype vectors represent local averages of the original data without any loss of resolution. To automate the classification process, we used the Davies and Bouldin (1979) index (DBI) as means of evaluating the results of the K-means partitioning.

Transitions, or irregular structures, present in any kind of signals carry information related to its physical phenomena (Mallat, 1999). Besides the horizon locations, the identified transition characterization in the interpretation is associated with geological processes. In this way, a possible transition classification could be linked to the seismic facies. Detection of transitions or singularities in signals is based on simple mathematical concepts. The signal inflection points are associated with the first-derivative extremes which correspond to the second-derivative zero crossings. For the signal inflection-point positions, using the CWT local peak locations, a wavelet should be chosen as the first-derivative of the smoothing function . One of the wavelet functions that fulfill this requirement is the first-derivative of the Gaussian function, called the Gauss wavelet. We can extract scalogram's local peaks coincide from the signal inflection points. It can be proven that these lines, which are called WTMMLA, can be used to characterize the signal irregularity. The signal irregularities can be characterized mathematically through the WTMMLA and H?lder exponent (Mallat, 1999). The exponent can be obtained from the slope estimation of the curve created by the log2 of the WTMMLA coefficients divided by the log2 of the scales. In This study we used WTMMLA as a direct seismic attribute. We calculated CWT coefficients and WTMMLA for sixteen seismic data samples around the picked reservoir horizon. The extracted WTMMLA can show the possible heterogeneity and singularity within the reservoir. We used these attribute as input vector for the SOM step and obtained the U_matrix. The K-mean and DBI estimate the number of seismic facies.Utilizing of CWT to locate events in time through the identification of signal singularities also proved to be useful as an appropriate tool for detection of seismic events. Therefor this method proved to be less sensitive to interpretation errors. The performance of the method was tested on Kangan formation at one of the Iranian oil fields.

**Keywords**