Milton J. Porsani

Seismic trace interpolation using half‐step prediction filters

A method to perform seismic trace interpolation known as the Spitz method handles spatially aliased events. The Spitz method uses the unit‐step prediction filter to estimate data spaced at Δx/2. The missing data are obtained by solving a complex linear system of equations whose unknowns are the coefficients at the interpolated location. We attack this problem by introducing a half‐step prediction filter that makes trace interpolation significantly more efficient and easier for implementation. A complex half‐step prediction filter at frequency f/2 is computed in the least‐squares sense to predict odd data components from even ones. At the frequency f, the prediction operator is shrunk and convolved with the input data spaced at Δx to predict data at Δx/2 directly. Instead of solving two systems of linear equations, as proposed by Spitz, only a system for the half‐step prediction filter has to be solved. Numerical examples using a marine seismic common‐midpoint (CMP) gather and a poststack seismic section w...

Imaging industrial contaminant plumes with resistivity techniques

A systematic geophysical procedure has been developed and applied to image groundwater contamination caused by industrial activities in Bahia, Brazil. The procedure combines the use of conventional Schlumberger sounding with a regular scheme of sampling the resistivity stratification in depth. This is achieved by traversing an area with multiple profiles measured at selected electrode spacing. By multiple profiles we mean closely spaced partial soundings made using only six electrode spacings. Partial and complete soundings are correlated and inverted assuming horizontally stratified models within the limits covered by each electrode array. Pseudo-resistivity sections constructed from these data are inverted and adjusted using a two-dimensional finite difference algorithm. Electrical and lithological well logs are used to constrain this interpretation. The procedure was successfully applied to investigate the groundwater conditions and to outline contaminant plumes within industrial areas of the Camacari Petrochemical Center, Reconcavo basin, Bahia. The study includes cases of electrically conductive plumes generated by infiltration of inorganic aqueous effluents and a resistive plume containing hydrocarbon contaminants.

Mixed‐phase deconvolution

We describe a new algorithm for mixed‐phase deconvolution. It is valid only for pulses whose Z-transform has no zeros on the unit circle. That is, the amplitude spectrum cannot be zero for any frequency. Using the Z-transform of a discrete‐time signal, and assuming that the signal has α zeros inside the unit circle, the inverse of its minimum‐delay component may be estimated by solving the extended Yule‐Walker (EYW) system of equations with the lag α of the autocorrelation function (ACF) on diagonal of the coefficient matrix. This property of the solution of EYW equations is exploited to derive mixed‐phase inverse filters and their corresponding mixed‐phase pulses. For different values of α, a suite of inverse filters is generated using the same ACF. To choose the best decomposition and its corresponding mixed‐phase inverse filter, we have used the value of α which gives the maximum value of the Lp norm of the filtered signal. The optimal value of α does not seem to be very sensitive to the choice of norm...

Direct multichannel predictive deconvolution

The Levinson principle generally can be used to compute recursively the solution of linear equations. It can also be used to update the error terms directly. This is used to do single-channel deconvolution directly on seismic data without computing or applying a digital filter. Multichannel predictive deconvolution is used for seismic multiple attenuation. In a standard procedure, the prediction-error filter matrices are computed with a Levinson recursive algorithm, using a covariance matrix of the input data. The filtered output is the prediction errors or the nonpredictable part of the data. Starting with the classical Levinson recursion,wehave derived new algorithms for direct recursive calculationof the prediction errors without computing the data covariance-matrix or computing the prediction-error filters. One algorithm generates recursively the one-step forward and backward predic-tion errors and the L-step forward prediction error, computing only the filter matrices with the highest index. A numeri...

A fast, modified parabolic Radon transform

We propose a fast and efficient frequency-domain implementation of a modified parabolic Radon transform (modified PRT) based on a singular value decomposition (SVD) with applications to multiple removal. The problem is transformed into a complex linear system involving a single operator after merging the curvature-frequency parameters into a new variable. A complex SVD is applied to this operator and the forward transform is computed by means of a complex back-substitution that is frequency independent. The new transform offers a wider curvature range at signal frequencies than the other PRT implementations, allowing the mapping in the transform domain of low-frequency events with important residual moveouts (long period multiples). The method is capable of resolving multiple energy from primaries when they interfere in a small time interval, a situation where most frequency-domain methods fail to discriminate the different wave types. Additionally, the method resists better to amplitude variations with o...

SVD filtering applied to ground-roll attenuation

We present a singular value decomposition (SVD) filtering method for attenuation of the ground roll. Before the SVD computation, normal move-out (NMO) correction is applied to the seismograms, with the purpose of flattening the reflections. SVD is performed on a small number of traces in a sliding window. The output trace is the central trace of the first few eigenimages. These contain mostly horizontally aligned signals, and other noise in the data will be suppressed. By performing this action with the sliding window moving in steps of one trace, the number of output traces is equal to the number of input traces. The new method preserves the character and frequency content of the horizontal reflections and attenuates all other type of events. We illustrate the method using land seismic data of the Tacutu basin, located in the northeast part of Brazil. The results show that the proposed method is effective and is able to reveal reflections masked by the ground roll. The new SVD filtering approach provides the results of better quality, when compared with the results obtained from the conventional f-k filtering method.

A combined genetic and linear inversion algorithm for seismic waveform inversion

We combine a genetic algorithm (GA) with a linearized inversion (LI) scheme to develop a new approach to seismic waveform inversion. By incorporating the LI method into GA we intend (i) to overcome the limitations of the knowledge of a good starting model in LI and (ii) to reduce the computational cost of GA. The new method takes advantage of the convergence properties and local search approach of the linear method while the global search is carried out using GA. The two methods working together improve the directivity of the model ensemble increasing the fitness and accelerating the convergence to near the global optimum. To illustrate the procedure, we derive estimates of vp, and density for a 1-D elastic earth structure by modeling plane wave decomposed seismic data.

Mixed-phase deconvolution and wavelet estimation

The seismic convolutional model describes seismic wave propagation as a linear system. A seismic trace then consists of a wavelet or pulse convolved with the reflection response or reflectivity series plus additive noise. An important problem in seismic data processing is the removal of the effect of the wavelet (spiking deconvolution or inverse filtering) in order to estimate the reflectivity series. When the wavelet is known, designing an optimal spiking filter by a least-squares approach is straightforward. For a mixed-delay wavelet, we must delay the position of the desired output spike. This delay can later be removed so that the effective filter is no longer causal but operates on both past and future input data values with respect to the output position. When the wavelet is unknown, we must determine both the wavelet and the reflectivity series from the input data. Solving this ambiguous problem requires additional assumptions about the wavelet, the reflectivity series, and the noise. Common assumptions are that the reflectivity series is a stationary random process and that it is uncorrelated to the stationary random noise. Furthermore, if it is assumed that the signal-to-noise ratio is large, the autocorrelation function (ACF) of the seismic trace can be used as an estimate of the ACF of the wavelet. If, in addition, it is assumed (as it commonly is) that the wavelet is minimum delay, meaning that all roots of its associated polynomial have amplitude greater than 1, an optimal least-squares inverse filter can be computed by solving the normal equations using a Levinson algorithm. The normal equations are also called the Yule-Walker (YW) equations. However, problems can arise with minimum-phase inverse filtering when the wavelet is not minimum delay. In a tutorial on spiking deconvolution ( TLE , 1995), Leinbach showed that applying a minimum-phase filter to an air-gun …

Ground-roll Attenuation Based On SVD Filtering

We present a singular value decomposition (SVD) filtering method for attenuation of the ground roll. Before the SVD computation, the normal move-out (NMO) correction is applied to the seismograms, with the purpose of flattening the reflections. SVD is performed on a small number of traces in a sliding window. The output trace is the central trace of the first few eigenimages. These contains mostly horizontally aligned signals, and other noise in the data will be suppressed. The new method preserves the character and frequency content of the horizontal reflections and attenuates all other type of events. We illustrate the method using land seismic data of the Tacutu basin, located in the north-east part of Brazil. The results show that the proposed method is effective and is able to reveal reflections masked by the ground-roll.

An Adaptive Local-slope SVD Filtering Approach to Enhance Events On Seismic Sections

We present an adaptive singular value decomposition (SVD) filtering method for enhancement of the spacial coherence of the reflections and for the attenuation of the uncorrelated noise. The SVD filtering is performed on a small number of traces and a small number of samples collected around each data component. The method uses the local slope of the reflections to re-sample the data set surrounding each data component and the SVD filtering is locally applied to compute the filtered data. The filtered data component is obtained by stacking the components of the first K eigenimages along the slope. The method is applied in two steps: (i) before the SVD computation, the normal move-out (NMO) correction is applied to the seismograms, with the purpose of flattening the reflections. We use the local slopes equal to 90◦ to preserve the horizontal coherence of the primary reflections and (ii) for the second step the SVD filtering uses as input the filtered data of step-1 and the method is applied in the common-offset domain. Now the local slopes of the reflections are used in order to drive the SVD filtering. We illustrate the method using land seismic data of the Tacutu basin, located in the Northeast of Brazil. The results show that the proposed method is effective and is able to reveal reflections masked by the ground-roll.