Impedance inversion of post-stack seismic data is routinely done to obtain reliable information about the reservoir lithological properties. The usual practice is to estimate the compressional (P) impedance from the seismic traces via model-based inversion, which requires an initial model and an estimated wavelet. The model is recursively updated until the data misfit reaches the user-defined value. The updated model is the accepted P impedance volume.

A comparison is made between P impedance estimated from model-based inversion and neural network-based estimation. Arcis Corp.’s neural network-based estimation uses probabilistic neural networks (PNN).A conjugate gradient algorithm is used to train and validate the PNN for the optimum set of parameters used for estimation. The company trains and validates the PNN to estimate density and P-wave velocity separately. The P impedance is computed from the estimated density and P-wave velocity.Two case studies from Alberta, Canada, show that the neural network-based estimation provides more subtle P-impedance information with respect to post-stack inversion estimation.

Model-based inversion
P impedance is an important attribute for interpreters as it provides more accurate and reliable information about the

The P-impedance is obtained from model-based post-stack inversion (a). The P-impedance also is obtained from PNN analysis (b). P-impedance logs at two locations are inserted. The black ellipse shows the zone of shale play. Notice the detailed and accurate correlation of the impedance values with the impedance log curves as seen on the neural network-estimated impedance. (Images courtesy of Arcis Corp.)

lithological properties of the reservoir. Conventionally, P impedance is obtained from the seismic data via model-based inversion, which requires an initial model and estimation of a wavelet from the data. The initial model generally is obtained from the available well log by interpolation and application of a low-pass filter (approximately 10 Hz). The wavelet is estimated from the data. The shortcoming of this method is that the solution is affected by the non-uniqueness of the problem. Thus, the solution depends on the chosen initial model.

Neural network-based estimation
Neural networks have been in use for geophysical applications since the early 1990s. M.D. McCormack describes some of the early geophysical applications of neural networks by predicting lithology logs for an entire well using back-propagation multilayer feed-forward networks. Subsequent to this work, P.S. Schultz et al. proposed the application of neural networks in estimating the log properties from the seismic data in a data-driven interpretation framework. J. Liu and Z. Liu applied the neural networks for the inversion of sonic and shale content logs using well log and seismic data. K.P. Dorrington and C.A. Link describe an approach based on a combination of genetic algorithms and neural networks to predict the porosity log for 3-D data. A hybrid strategy is used to determine the optimal number and type of attributes that can accurately predict the porosity in the reservoir zone.

The neural network analysis estimates the target log by making use of several attributes chosen from a suite of attributes. The selection of the optimum number of attributes usually is done by the linear multi-attribute regression analysis.Also, so that the seismic data and the target well logs are scaled to the same resolution level, a convolutional approach is used to estimate the target logs, according to D.P. Hampson et al. The optimum number of attributes and the operator length for each of the P-wave velocity and density estimations are obtained from the linear multi-attribute regression analysis. The optimum attributes and the operator length thus obtained are used to train the probabilistic neural network. The trained network is further validated by computing the prediction error between the target log and the predicted log by sequentially hiding the target logs. The trained and validated network is subsequently applied over the entire 3-D data volume to individually estimate the P-wave velocity and density parameters for each of the data volumes under study. The estimated P-wave velocity and density are used to compute the P impedance over the two 3-D volumes.

Alberta analyses
A case study in Alberta, Canada, shows the conventional P impedance obtained from the model-based post-stack

The P-impedance is obtained from model-based post-stack inversion (a). The P-impedance also is obtained from PNN analysis (b). The P-impedance log is inserted. The black ellipse shows the zone of shale-sand play.

inversion and the PNN analysis-based P impedance estimation. Arcis inserted two black curves in the image to represent the P impedance logs. The general low-frequency trends in the two figures compare well. However, the analysis with PNN provides more detailed impedance profile information compared with conventional inversion that, in turn, correlates well with the impedance logs. The elliptical zone lies within the shale play. The conventional post-stack inversion shows that most of this zone has low impedance and is hence porous. This result is not consistent with the well log. The PNN analysis shows a tight shale zone sandwiched between thin porous layers, which is consistent with the well log information.

A second case study also shows the P impedance obtained from the model-based post-stack inversion and the P impedance estimated by the PNN analysis. The inserted black curve is the P impedance log.As in the previous case, the general P impedance trends are comparable. However, the result obtained with the PNN analysis shows more detailed information about the subsurface impedance structure. Furthermore, the well logs correlate well with the result obtained from the PNN analysis. The zone marked by the ellipse falls within a shale-sand sequence of relatively lower porosity. The conventional P impedance inversion shows that the zone is characterized by a continuous high-porosity sand, which is misleading. However, the P impedance obtained by the PNN analysis indicates a moderately tight sand consistent with the well log signature.

Results from the case studies show the estimated P impedance obtained from PNN analysis and post-stack inversion can be compared within a reasonable accuracy as far as the general low-frequency trends are concerned. The results differ in quantitative comparison. The P impedance estimated by the PNN analysis yields better correlation with the well logs as compared to the P impedance obtained from the model-based post-stack inversion. This is an expected result because the neural network-based estimation has a bias toward a solution that is consistent with the available well logs whereas the model-based post-stack inversion is subjected to a large extent to the non-uniqueness in the inversion problem.

Acknowledgement
We acknowledge Arcis Corp. for the data examples included in this article as well as for the permission to publish it.

References available.