Porosity and permeability prediction from wireline logs using artificial neural networks: a North Sea case study

Estimations of porosity and permeability from well logs are important yet difficult tasks encountered in geophysical formation evaluation and reservoir engineering. Motivated by recent results of artificial neural network (ANN) modelling offshore eastern Canada, we have developed neural nets for converting well logs in the North Sea to porosity and permeability. We use two separate back-propagation ANNs (BP-ANNs) to model porosity and permeability. The porosity ANN is a simple three-layer network using sonic, density and resistivity logs for input. The permeability ANN is slightly more complex with four inputs (density, gamma ray, neutron porosity and sonic) and more neurons in the hidden layer to account for the increased complexity in the relationships. The networks, initially developed for basin-scale problems, perform sufficiently accurately to meet normal requirements in reservoir engineering when applied to Jurassic reservoirs in the Viking Graben area. The mean difference between the predicted porosity and helium porosity from core plugs is less than 0.01 fractional units. For the permeability network a mean difference of approximately 400 mD is mainly due to minor core-log depth mismatch in the heterogeneous parts of the reservoir and lack of adequate overburden corrections to the core permeability. A major advantage is that no a priori knowledge of the rock material and pore fluids is required. Real-time conversion based on measurements while drilling (MWD) is thus an obvious application.     

Using Artificial Neural Networks to Predict the Presence of Overpressured Zones in the Anadarko Basin,Oklahoma

Many sedimentary basins throughout the world exhibit areas with abnormal pore-fluid pressures (higher or lower than normal or hydrostatic pressure). Predicting pore pressure and other parameters (depth, extension, magnitude, etc.) in such areas are challenging tasks. The compressional acoustic (sonic) log (DT) is often used as a predictor because it responds to changes in porosity or compaction produced by abnormal pore-fluid pressures. Unfortunately, the sonic log is not commonly recorded in most oil and/or gas wells. We propose using an artificial neural network to synthesize sonic logs by identifying the mathematical dependency between DT and the commonly available logs, such as normalized gamma ray (GR) and deep resistivity logs (REID). The artificial neural network process can be divided into three steps: (1) Supervised training of the neural network; (2) confirmation and validation of the model by blind-testing the results in wells that contain both the predictor (GR, REID) and the target values (DT) used in the supervised training; and 3) applying the predictive model to all wells containing the required predictor data and verifying the accuracy of the synthetic DT data by comparing the back-predicted synthetic predictor curves (GRNN, REIDNN) to the recorded predictor curves used in training (GR, REID). Artificial neural networks offer significant advantages over traditional deterministic methods. They do not require a precise mathematical model equation that describes the dependency between the predictor values and the target values and, unlike linear regression techniques, neural network methods do not overpredict mean values and thereby preserve original data variability. One of their most important advantages is that their predictions can be validated and confirmed through back-prediction of the input data. This procedure was applied to predict the presence of overpressured zones in the Anadarko Basin, Oklahoma. The results are promising and encouraging.     

Rainfall‐runoff modelling using artificial neural networks: comparison of network types

Growing interest in the use of artificial neural networks (ANNs) in rainfall‐runoff modelling has suggested certain issues that are still not addressed properly. One such concern is the use of network type, as theoretical studies on a multi‐layer perceptron (MLP) with a sigmoid transfer function enlightens certain limitations for its use. Alternatively, there is a strong belief in the general ANN user community that a radial basis function (RBF) network performs better than an MLP, as the former bases its nonlinearities on the training data set. This argument is not yet substantiated by applications in hydrology. This paper presents a comprehensive evaluation of the performance of MLP‐ and RBF‐type neural network models developed for rainfall‐runoff modelling of two Indian river basins. The performance of both the MLP and RBF network models were comprehensively evaluated in terms of their generalization properties, predicted hydrograph characteristics, and predictive uncertainty. The results of the study indicate that the choice of the network type certainly has an impact on the model prediction accuracy. The study suggests that both the networks have merits and limitations. For instance, the MLP requires a long trial‐and‐error procedure to fix the optimal number of hidden nodes, whereas for an RBF the structure of the network can be fixed using an appropriate training algorithm. However, a judgment on which is superior is not clearly possible from this study. Copyright © 2004 John Wiley & Sons, Ltd.     

Test of rock physics models for prediction of seismic velocities in shallow unconsolidated sands: a well log data case‡

This paper tests the ability of various rock physics models to predict seismic velocities in shallow unconsolidated sands by comparing the estimates to P and S sonic logs collected in a shallow sand layer and ultrasonic laboratory data of an unconsolidated sand sample. The model fits are also evaluated with respect to the conventional model for unconsolidated sand. Our main approach is to use Hertz‐Mindlin and Walton contact theories, assuming different weight fractions of smooth and rough contact behaviours, to predict the elastic properties of the high porosity point. Using either the Hertz‐Mindlin or Walton theories with rough contact behaviour to define the high porosity endpoint gives an over‐prediction of the velocities. The P‐velocity is overpredicted by a factor of ～1.5 and the S‐velocity by a factor of ～1.8 for highly porous gas‐sand. The degree of misprediction decreases with increasing water saturation and porosity.Using the Hertz‐Mindlin theory with smooth contact behaviour or weighted Walton models gives a better fit to the data, although the data are best described using the Walton smooth model. To predict the properties at the lower porosities, the choice of bounding model attached to the Walton Smooth model controls the degree of fit to the data, where the Reuss bound best captures the porosity variations of dry and wet sands in this case since they are caused by depositional differences. The empirical models based on lab experiments on unconsolidated sand also fit the velocity data measured by sonic logs in situ, which gives improved confidence in using lab‐derived results.     

Fracture-frequency prediction from borehole wireline logs using artificial neural networks

Borehole-wall imaging is currently the most reliable means of mapping discontinuities within boreholes. As these imaging techniques are expensive and thus not always included in a logging run, a method of predicting fracture frequency directly from traditional logging tool responses would be very useful and cost effective. Artificial neural networks (ANNs) show great potential in this area. ANNs are computational systems that attempt to mimic natural biological neural networks. They have the ability to recognize patterns and develop their own generalizations about a given data set. Neural networks are trained on data sets for which the solution is known and tested on data not previously seen in order to validate the network result. We show that artificial neural networks, due to their pattern recognition capabilities, are able to assess the signal strength of fracture-related heterogeneity in a borehole log and thus fracture frequency within a borehole. A combination of wireline logs (neutron porosity, bulk density, P-sonic, S-sonic, deep resistivity and shallow resistivity) were used as input parameters to the ANN. Fracture frequency calculated from borehole televiewer data was used as the single output parameter. The ANN was trained using a back-propagation algorithm with a momentum learning function. In addition to fracture frequency within a single borehole, an ANN trained on a subset of boreholes in an area could be used for prediction over the entire set of boreholes, thus allowing the lateral correlation of fracture zones.     

A general approach for porosity estimation using artificial neural network method: a case study from Kansas gas field

This study aims to design a back-propagation artificial neural network (BP-ANN) to estimate the reliable porosity values from the well log data taken from Kansas gas field in the USA. In order to estimate the porosity, a neural network approach is applied, which uses as input sonic, density and resistivity log data, which are known to affect the porosity. This network easily sets up a relationship between the input data and the output parameters without having prior knowledge of petrophysical properties, such as porefluid type or matrix material type. The results obtained from the empirical relationship are compared with those from the neural network and a good correlation is observed. Thus, the ANN technique could be used to predict the porosity from other well log data.     

Modeling and interpretation of Q logs in carbonate rock using a double porosity model and well logs

Attenuation data extracted from full waveform sonic logs is sensitive to vuggy and matrix porosities in a carbonate aquifer. This is consistent with the synthetic attenuation (1 / Q) as a function of depth at the borehole-sonic source-peak frequency of 10 kHz. We use velocity and densities versus porosity relationships based on core and well log data to determine the matrix, secondary, and effective bulk moduli. The attenuation model requires the bulk modulus of the primary and secondary porosities. We use a double porosity model that allows us to investigate attenuation at the mesoscopic scale. Thus, the secondary and primary porosities in the aquifer should respond with different changes in fluid pressure. The results show a high permeability region with a Q that varies from 25 to 50 and correlates with the stiffer part of the carbonate formation. This pore structure permits water to flow between the interconnected vugs and the matrix. In this region the double porosity model predicts a decrease in the attenuation at lower frequencies that is associated with fluid flowing from the more compliant high-pressure regions (interconnected vug space) to the relatively stiff, low-pressure regions (matrix). The chalky limestone with a low Q of 17 is formed by a muddy porous matrix with soft pores. This low permeability region correlates with the low matrix bulk modulus. A low Q of 18 characterizes the soft sandy carbonate rock above the vuggy carbonate.This paper demonstrates the use of attenuation logs for discriminating between lithology and provides information on the pore structure when integrated with cores and other well logs. In addition, the paper demonstrates the practical application of a new double porosity model to interpret the attenuation at sonic frequencies by achieving a good match between measured and modeled attenuation.     

NMR Parameters Determination through ACE Committee Machine with Genetic Implanted Fuzzy Logic and Genetic Implanted Neural Network

Free fluid porosity and rock permeability, undoubtedly the most critical parameters of hydrocarbon reservoir, could be obtained by processing of nuclear magnetic resonance (NMR) log. Despite conventional well logs (CWLs), NMR logging is very expensive and time-consuming. Therefore, idea of synthesizing NMR log from CWLs would be of a great appeal among reservoir engineers. For this purpose, three optimization strategies are followed. Firstly, artificial neural network (ANN) is optimized by virtue of hybrid genetic algorithm-pattern search (GA-PS) technique, then fuzzy logic (FL) is optimized by means of GA-PS, and eventually an alternative condition expectation (ACE) model is constructed using the concept of committee machine to combine outputs of optimized and non-optimized FL and ANN models. Results indicated that optimization of traditional ANN and FL model using GA-PS technique significantly enhances their performances. Furthermore, the ACE committee of aforementioned models produces more accurate and reliable results compared with a singular model performing alone.     

Prediction of effective porosity from seismic attributes using locally linear model tree algorithm

In this study, a locally linear model tree algorithm was used to optimize a neuro‐fuzzy model for prediction of effective porosity from seismic attributes in one of Iranian oil fields located southwest of Iran. Valid identification of effective porosity distribution in fractured carbonate reservoirs is extremely essential for reservoir characterization. These high‐accuracy predictions facilitate efficient exploration and management of oil and gas resources. The multi‐attribute stepwise linear regression method was used to select five out of 26 seismic attributes one by one. These attributes introduced into the neuro‐fuzzy model to predict effective porosity. The neuro‐fuzzy model with seven locally linear models resulted in the lowest validation error. Moreover, a blind test was carried out at the location of two wells that were used neither in training nor validation. The results obtained from the validation and blind test of the model confirmed the ability of the proposed algorithm in predicting the effective porosity. In the end, the performance of this neuro‐fuzzy model was compared with two regular neural networks of a multi‐layer perceptron and a radial basis function, and the results show that a locally linear neuro‐fuzzy model trained by a locally linear model tree algorithm resulted in more accurate porosity prediction than standard neural networks, particularly in the case where irregularities increase in the data set. The production data have been also used to verify the reliability of the porosity model. The porosity sections through the two wells demonstrate that the porosity model conforms to the production rate of wells. Comparison of the locally linear neuro‐fuzzy model performance on different wells indicates that there is a distinct discrepancy in the performance of this model compared with the other techniques. This discrepancy in the performance is a function of the correlation between the model inputs and output. In the case where the strength of the relationship between seismic attributes and effective porosity decreases, the neuro‐fuzzy model results in more accurate prediction than regular neural networks, whereas the neuro‐fuzzy model has a close performance to neural networks if there is a strong relationship between seismic attributes and effective porosity. The effective porosity map, presented as the output of the method, shows a high‐porosity area in the centre of zone 2 of the Ilam reservoir. Furthermore, there is an extensive high‐porosity area in zone 4 of Sarvak that extends from the centre to the east of the reservoir.     

Permeability inversion from low‐frequency seismoelectric logs in fluid‐saturated porous formations

In this paper, a method is proposed to invert permeability from seismoelectric logs in fluid‐saturated porous formations. From the analysis of both the amplitude and the phase of simulated seismoelectric logs, we find that the Stoneley wave amplitude of the ratio of the converted electric field to the pressure (REP) is sensitive to porosity rather than permeability while the tangent of the REP's phase is sensitive to permeability. The REP's phase reflects the phase discrepancy between the electric field and the pressure at the same location in the borehole. We theoretically derive the frequency‐dependent expression of the REP of the low‐frequency Stoneley wave and find that the tangent of the REP's argument is approximately in inverse proportion to permeability. We obtain an inversion formula and present the permeability inversion method by using the tangent of the REP's phase. To test this method, the permeabilities of different sandstones are inverted from the synthetic full‐waveform data of seismoelectric logs. A modified inversion process is proposed based on the analysis of inversion errors, by which the relative errors are controlled below 25% and they are smaller than those of the permeability inversion from the Stoneley wave of acoustic logs.     

Applying Neural Network Classification to Obtain Mangrove Landscape Characteristics for Monitoring the Travel Environment Quality on the Beihai Coast of Guangxi,P. R. China

The spectral characteristics of mangroves on the Beihai Coast of Guangxi, P. R. China are acquired on the basis of spectral data from field measurements. Following this, the 3‐layer reverse‐conversing neural networks (NN) classification technology is used to analyze the Landsat TM5 image obtained on January 8, 2003. It is detailed enough to facilitate the introduction of the algorithm principle and trains project of the neural network. Neural network algorithms have characteristics including large‐scale data handling and distributing information storage. This research firstly analyzes the necessity and complexity of this translation system, and then introduces the strong points of the neural network. Processing mangrove landscape characteristics by using neural network is an important innovation, with great theoretical and practical significance. This kind of neural network can greatly improve the classification accuracy. The spatial resolution of Landsat TM5 is high enough to facilitate the research, and the false color composite from 3‐, 4‐, and 5‐bands has a clear boundary and provides a significant quantity of information and effective images. On the basis of a field survey, the exported layers are defined as mangrove, vegetation, bare land, wetlands and shrimp pool. TM satellite images are applied to false color composites by using 3‐, 4‐, and 5‐bands, and then a supervised classification model is used to classify the image. The processing method of hyper‐spectrum remote sensing allows the spectral characteristics of the mangrove to be determined, and integrates the result with the NN classification for the false color composite by using 3‐, 4‐, and 5‐bands. The network model consists of three layers, i. e., the input layer, the hidden layer, and the output layer. The input layer number of classification is defined as 3, and the hidden layers are defined as 5 according to the function operation. The control threshold is 0.9. The training ratio is 0.2. The maximum permit error is 0.08. The classification precision reaches 86.86%. This is higher than the precision of maximal parallel classification (50.79%) and the spectrum angle classification (75.39%). The results include the uniformity ratio (1.7789), the assembly ratio (0.6854), the dominance ratio (–1.5850), and the fragmentation ratio (0.0325).     

Modelling elastic properties of impure chalk from South Arne Field, North Sea

In impure chalk, the elastic moduli are not only controlled by porosity but also by contact‐cementation, resulting in relatively large moduli for a given porosity, and by admixtures of clay and fine silica, which results in relatively small moduli for a given porosity. Based on a concept of solids suspended in pore fluids as well as composing the rock frame, we model P‐wave and S‐wave moduli of dry and wet plug samples by an effective‐medium Hashin–Shtrikman model, using chemical, mineralogical and textural input. For a given porosity, the elastic moduli correspond to a part of the solid (the iso‐frame value) forming the frame of an Upper Hashin–Shtrikman bound, whereas the remaining solid is modelled as suspended in the pore fluid. The iso‐frame model is thus a measure of the pore‐stiffness or degree of cementation of the chalk. The textural and mineralogical data may be assessed from logging data on spectral gamma radiation, density, sonic velocity and water saturation in a hydrocarbon zone, whereas the iso‐frame value of a chalk may be assessed from the density and acoustic P‐wave logs alone. The iso‐frame concept may thus be directly used in conventional log‐analysis and is a way of incorporating sonic‐logging data. The Rigs‐1 and Rigs‐2 wells in the South Arne field penetrate the chalk at the same depth but differ in porosity and in water saturation although almost the entire chalk interval has irreducible water saturation. Our model, combined with petrographic data, indicates that the difference in porosity is caused by a higher degree of pore‐filling cementation in Rigs‐1. Petrographic data indicate that the difference in water saturation is caused by a higher content of smectite in the pores of Rigs‐1. In both wells, we find submicron‐size diagenetic quartz.     

The potential of different ANN techniques in evapotranspiration modelling

The potential of three different artificial neural network (ANN) techniques, the multi‐layer perceptrons (MLPs), radial basis neural networks (RBNNs) and generalized regression neural networks (GRNNs), in modelling of reference evapotranspiration (ET₀) is investigated in this paper. Various daily climatic data, that is, solar radiation, air temperature, relative humidity and wind speed from two stations, Pomona and Santa Monica, in Los Angeles, USA, are used as inputs to the ANN techniques so as to estimate ET₀ obtained using the FAO‐56 Penman–Monteith (PM) equation. In the first part of the study, a comparison is made between the estimates provided by the MLP, RBNN and GRNN and those of the following empirical models: The California Irrigation Management Information System (CIMIS) Penman (1985), Hargreaves (1985) and Ritchie (1990). In this part of the study, the empirical models are calibrated using the standard FAO‐56 PM ET₀ values. The estimates of the ANN techniques are also compared with those of the calibrated empirical models. Mean square errors, mean absolute errors and determination coefficient statistics are used as comparing criteria for the evaluation of the models' performances. Based on the comparisons, it is found that the MLP and RBNN techniques could be employed successfully in modelling the ET₀ process. In the second part of the study, the potential of ANN techniques and the empirical methods in ET₀ estimation using nearby station data is investigated. Among the models, the calibrated Hargreaves model is found to perform better than the others. Copyright © 2007 John Wiley & Sons, Ltd.     

Deep Artificial Neural Networks as a Tool for the Analysis of Seismic Data

The number of seismological studies based on artificial neural networks has been increasing. However, neural networks with one hidden layer have almost reached the limit of their capabilities. In the last few years, there has been a new boom in neuroinformatics associated with the development of third-generation networks, deep neural networks. These networks operate with data at a higher level. Unlabeled data can be used to pretrain the network, i.e., there is no need for an expert to determine in advance the phenomenon to which these data correspond. Final training requires a small amount of labeled data. Deep networks have a higher level of abstraction and produce fewer errors. The same network can be used to solve several tasks at the same time, or it is easy to retrain it from one task to another. The paper discusses the possibility of applying deep networks in seismology. We have described what deep networks are, their advantages, how they are trained, how to adapt them to the features of seismic data, and what prospects are opening up in connection with their use.     

An ANN‐based model for spatiotemporal groundwater level forecasting

This paper evaluates the feasibility of using an artificial neural network (ANN) methodology for estimating the groundwater levels in some piezometers placed in an aquifer in north‐western Iran. This aquifer is multilayer and has a high groundwater level in urban areas. Spatiotemporal groundwater level simulation in a multilayer aquifer is regarded as difficult in hydrogeology due to the complexity of the different aquifer materials. In the present research the performance of different neural networks for groundwater level forecasting is examined in order to identify an optimal ANN architecture that can simulate the piezometers water levels. Six different types of network architectures and training algorithms are investigated and compared in terms of model prediction efficiency and accuracy. The results of different experiments show that accurate predictions can be achieved with a standard feedforward neural network trained usung the Levenberg–Marquardt algorithm. The structure and spatial regressions of the ANN parameters (weights and biases) are then used for spatiotemporal model presentation. The efficiency of the spatio‐temporal ANN (STANN) model is compared with two hybrid neural‐geostatistics (NG) and multivariate time series‐geostatistics (TSG) models. It is found in this study that the ANNs provide the most accurate predictions in comparison with the other models. Based on the nonlinear intrinsic ANN approach, the developed STANN model gives acceptable results for the Tabriz multilayer aquifer. Copyright © 2008 John Wiley & Sons, Ltd.     

Diary of forthcoming events

Abstract

Currently, environmental modelling is frequently conducted with the aid of artificial neural networks (ANNs) in an effort to achieve greater accuracy in simulation and forecasting beyond that typically obtained when using solely linear models. For the design of an ANN, modellers must contend with two key issues: (a) the selection of model input and (b) the determination of the number of hidden neurons. A novel approach is introduced to address the optimal design of ANNs based on a multi-objective strategy that enables the user to find a set of feasible ANNs, determined as optimal trade-off solutions between model simplicity and accuracy. This is achieved in a multi-objective fashion by simultaneously minimizing three different cost functions: the model input dimension, the hidden neuron number and the generalization error computed on a validation set of data. The multi-objective approach is based on the Pareto dominance criterion and an evolutionary strategy has been employed to solve the combinatorial optimization problem. From a theoretical perspective, the choice of a multi-objective approach marks an attempt to account for, and overcome, the “curse of dimensionality” and to circumvent the drawbacks of “overfitting” that are inherent in ANNs. Moreover, it is demonstrated that the strategy renders the choice of the ANN more robust, as is evident by “unseen data” in the testing stage, since structure determination is not merely based on the statistical evaluation of the generalization performance. The methodology is tested and the results are reported in a case study relating groundwater level predictions to total monthly rainfall.     

Evaluation and forecasting of daily groundwater outflow in a small chalky watershed

The purpose of this research is to include expert knowledge as one part of the modelling system and therefore offer the chance to create a productive interaction system between expert, mathematical model (MMO8) and artificial neural networks (ANNs). In the present project, the first objective is to determine some parameters by the MMO8 model, introduced as ANN input parameters to forecast spring outflow. The second objective is first to investigate the effect of temporal information by taking current and past data sets and then to forecast spring outflow. The good results obtained reveal the merit of the ANNs–MMO8 combination, and specifically multilayer perceptron (MLP) models. This methodology, for a network with lower, lag and number hidden layer, consistently produced better performance. Copyright © 2003 John Wiley & Sons, Ltd.     

Artificial Neural Network Prediction of Tropospheric Ozone Concentrations in Istanbul,Turkey

Tropospheric (ground‐level) ozone has adverse effects on human health and environment. In this study, next day's maximum 1‐h average ozone concentrations in Istanbul were predicted using multi‐layer perceptron (MLP) type artificial neural networks (ANNs). Nine meteorological parameters and nine air pollutant concentrations were utilized as inputs. The total 578 datasets were divided into three groups: training, cross‐validation, and testing. When all the 18 inputs were used, the best performance was obtained with a network containing one hidden layer with 24 neurons. The transfer function was hyperbolic tangent. The correlation coefficient (R), mean absolute error (MAE), root mean squared error (RMSE), and index of agreement or Willmott's Index (d₂) for the testing data were 0.90, 8.78 µg/m³, 11.15 µg/m³, and 0.95, respectively. Sensitivity analysis has indicated that the persistence information (current day's maximum and average ozone concentrations), NO concentration, average temperature, PM₁₀, maximum temperature, sunshine time, wind direction, and solar radiation were the most important input parameters. The values of R, MAE, RMSE, and d₂ did not change considerably for the MLP model using only these nine inputs. The performances of the MLP models were compared with those of regression models (i.e., multiple linear regression and multiple non‐linear regression). It has been found that there was no significant difference between the ANN and regression modeling techniques for the forecasting of ozone concentrations in Istanbul.     

潜在地震灾害预测和评价神经网络系统及其若干问题

结合人工神经网络自身的特性和地震灾害预测研究的特点,本文应用神经网络模型,建立了潜在地震灾害预测和评价系统。针对网络模型参数设置、数据归一化、中间层神经元最优数目以及泛化分类评价指标等若干实际问题给出了实际可行的解决方案。通过大样本数据对网络的训练,形成了有识别和记忆功能的非线性预测和评价系统。对网络的测试和检验,论证了该系统在预测潜在地震灾害上的可行性和有效性。同时,从测试精度出发,探讨了这种预测网络存在的不足,并给出了相应的改进建议,为开展进一步的研究工作提供了参考。     

SHEAR-WAVE VELOCITY ESTIMATION IN POROUS ROCKS: THEORETICAL FORMULATION,PRELIMINARY VERIFICATION AND APPLICATIONS1

Shear-wave velocity logs are useful for various seismic interpretation applications, including bright spot analyses, amplitude-versus-offset analyses and multicomponent seismic interpretations. Measured shear-wave velocity logs are, however, often unavailable. We developed a general method to predict shear-wave velocity in porous rocks. If reliable compressional-wave velocity, lithology, porosity and water saturation data are available, the precision and accuracy of shear-wave velocity prediction are 9% and 3%, respectively. The success of our method depends on: (1) robust relationships between compressional- and shear-wave velocities for water-saturated, pure, porous lithologies; (2) nearly linear mixing laws for solid rock constituents; (3) first-order applicability of the Biot–Gassmann theory to real rocks. We verified these concepts with laboratory measurements and full waveform sonic logs. Shear-wave velocities estimated by our method can improve formation evaluation. Our method has been successfully tested with data from several locations.