**Authors**

**Abstract**

In seismic exploration studies different types of techniques are used to recognize seismic features in terms of their temporal and spatial spectra. Variations in frequency content are sensitive to subtle changes in reflection information (Castro de matos et al., 2003). In this study the joint time-frequency analysis is used for seismic texture recognition.

Discrete wavelet transform (DWT) without decimation is implemented to identify the seismic trace singularities in each geologically oriented segment along a seismic line within a time window. The DWT involves a decimation operation and a down sampling by a factor of two. This process causes DWT not to be invariant to displacement in time and applicable for detecting and characterizing the singularities. To obtain an invariant DWT, the down sampling operation is removed from the process. DWT without decimation, known as wavelet a trous, is a signal convolution with filter bank coefficients with zeros inserted between the samples (Matos, 2007).

The mathematical concepts are the basis of detection of singularities in signals. The signal inflection points are associated with the first derivative extremes. A differentiable smoothing function , with integral equal to one, is defined, which converges to zero when tends to . Since the integral of and are zero in the interval of they can be considered as a wavelet. In this way, the WT of a signal , in the scale , can be obtained by convolving the signal with a scaled wavelet; . If the wavelets are substituted by the derivatives of smoothing function into the last equations, they change to; . Hence, the wavelet transforms and are the first and second derivatives of the signal. The local extremes of and zeros of correspond to the inflection points of . The local changes in the seismic event manifest themselves as changes in the extremes, which are the minimum and maximum amplitudes of each decomposition level of seismic traces derived by DWT without decimation. Hence lateral changes to seismic reflections are characterized by the extremes.

The extremes obtained from decomposition are used as seismic attributes and classified to define the seismic texture variation. A support vector machine (SVM), a learning machine based on statistical learning theory (Vapnik, 1995), is used to classify the attributes. The concept of SVM is based on finding an optimal separating hyperplane, which could be derived either in the input space or in a more generalized feature space.

SVM uses the structural risk minimization principle to construct decision rules that generalize well (Burges, 1998). The SRM method is based on the fact that the test error rate is bounded by the sum of the training error rate and a term which depends on the VC dimension of the learning machine, and generalized by minimizing the summation. For a linear hyperplane the decision function is; . The VC dimension can be controlled by controlling the norm of the weight vector . Giving training data , a separating hyperplane which generalized well can be found by subject to , , .

The method can be generalized to a nonlinear decision surface by mapping the input nonlinearly into some high dimension space, and finding the separating hyperplane in the space. This is achieved by using different types of kernel functions, , instead of ordinary scalar product . Consequently, the generalized decision function for a nonlinear input will be; .

Classification can be performed in several dimensions using two or more attributes derived from decompositions. In this instance two attributes are used as inputs for the SVM classifier and the hyperplane separates the regions where the seismic reflections are changing. The hyperplane was constructed by half of the selected training data, and then its accuracy was cross checked by the other half. Finally, all test data is used as input to the classifier. For instance, the seismic textures are classified into two types within a time window of a phantom horizon on a 2D seismic section. The color code of green and white indicate different seismic textures on the color band.

Discrete wavelet transform (DWT) without decimation is implemented to identify the seismic trace singularities in each geologically oriented segment along a seismic line within a time window. The DWT involves a decimation operation and a down sampling by a factor of two. This process causes DWT not to be invariant to displacement in time and applicable for detecting and characterizing the singularities. To obtain an invariant DWT, the down sampling operation is removed from the process. DWT without decimation, known as wavelet a trous, is a signal convolution with filter bank coefficients with zeros inserted between the samples (Matos, 2007).

The mathematical concepts are the basis of detection of singularities in signals. The signal inflection points are associated with the first derivative extremes. A differentiable smoothing function , with integral equal to one, is defined, which converges to zero when tends to . Since the integral of and are zero in the interval of they can be considered as a wavelet. In this way, the WT of a signal , in the scale , can be obtained by convolving the signal with a scaled wavelet; . If the wavelets are substituted by the derivatives of smoothing function into the last equations, they change to; . Hence, the wavelet transforms and are the first and second derivatives of the signal. The local extremes of and zeros of correspond to the inflection points of . The local changes in the seismic event manifest themselves as changes in the extremes, which are the minimum and maximum amplitudes of each decomposition level of seismic traces derived by DWT without decimation. Hence lateral changes to seismic reflections are characterized by the extremes.

The extremes obtained from decomposition are used as seismic attributes and classified to define the seismic texture variation. A support vector machine (SVM), a learning machine based on statistical learning theory (Vapnik, 1995), is used to classify the attributes. The concept of SVM is based on finding an optimal separating hyperplane, which could be derived either in the input space or in a more generalized feature space.

SVM uses the structural risk minimization principle to construct decision rules that generalize well (Burges, 1998). The SRM method is based on the fact that the test error rate is bounded by the sum of the training error rate and a term which depends on the VC dimension of the learning machine, and generalized by minimizing the summation. For a linear hyperplane the decision function is; . The VC dimension can be controlled by controlling the norm of the weight vector . Giving training data , a separating hyperplane which generalized well can be found by subject to , , .

The method can be generalized to a nonlinear decision surface by mapping the input nonlinearly into some high dimension space, and finding the separating hyperplane in the space. This is achieved by using different types of kernel functions, , instead of ordinary scalar product . Consequently, the generalized decision function for a nonlinear input will be; .

Classification can be performed in several dimensions using two or more attributes derived from decompositions. In this instance two attributes are used as inputs for the SVM classifier and the hyperplane separates the regions where the seismic reflections are changing. The hyperplane was constructed by half of the selected training data, and then its accuracy was cross checked by the other half. Finally, all test data is used as input to the classifier. For instance, the seismic textures are classified into two types within a time window of a phantom horizon on a 2D seismic section. The color code of green and white indicate different seismic textures on the color band.

**Keywords**