Statistical Inversion Based on Nonlinear Weighted Anisotropic Total Variational Model and Its Application in Electrical Impedance Tomography

Electrical impedance tomography (EIT) aims to reconstruct the conductivity distribution using the boundary measured voltage potential. Traditional regularization based method would suffer from error propagation due to the iteration process. The statistical inverse problem method uses statistical inference to estimate unknown parameters. In this article, we develop a nonlinear weighted anisotropic total variation (NWATV) prior density function based on the recently proposed NWATV regularization method. We calculate the corresponding posterior density function, i.e., the solution of the EIT inverse problem in the statistical sense, via a modified Markov chain Monte Carlo (MCMC) sampling. We do numerical experiment to validate the proposed approach.

Model-constrained and data-driven double-supervision acoustic impedance inversion

Seismic impedance inversion is an important technique for structure identification and reservoir prediction.Model-based and data-driven impedance inversion are the commonly used inversion methods.In practice,the geophysical inversion problem is essentially an ill-posedness problem,which means that there are many solutions corresponding to the same seismic data.Therefore,regularization schemes,which can provide stable and unique inversion results to some extent,have been introduced into the objective function as constrain terms.Among them,given a low-frequency initial impedance model is the most commonly used regularization method,which can provide a smooth and stable solution.However,this model-based inversion method relies heavily on the initial model and the inversion result is band limited to the effective frequency bandwidth of seismic data,which cannot effectively improve the seismic vertical resolution and is difficult to be applied to complex structural regions.Therefore,we propose a data-driven approach for high-resolution impedance inversion based on the bidirectional long short-term memory recurrent neural network,which regards seismic data as time-series rather than image-like patches.Compared with the model-based inversion method,the data-driven approach provides higher resolution inversion results,which demonstrates the effectiveness of the data-driven method for recovering the high-frequency components.However,judging from the inversion results for characterization the spatial distribution of thin-layer sands,the accuracy of high-frequency components is difficult to guarantee.Therefore,we add the model constraint to the objective function to overcome the shortages of relying only on the data-driven schemes.First,constructing the supervisor1 based on the bidirectional long short-term memory recurrent neural network,which provides the predicted impedance with higher resolution.Then,convolution constraint as supervisor2 is introduced into the objective function to guarantee the reliability and accuracy of the inversion results,which makes the synthetic seismic data obtained from the inversion result consistent with the input data.Finally,we test the proposed scheme based on the synthetic and field seismic data.Compared to model-based and purely data-driven impedance inversion methods,the proposed approach provides more accurate and reliable inversion results while with higher vertical resolution and better spatial continuity.The inversion results accurately characterize the spatial distribution relationship of thin sands.The model tests demonstrate that the model-constrained and data-driven impedance inversion scheme can effectively improve the thin-layer structure characterization based on the seismic data.Moreover,tests on the oil field data indicate the practicality and adaptability of the proposed method.

Fast pre-stack multi-channel inversion constrained by seismic reflection features

Classical multi-channel technology can significantly reduce the pre-stack seismic inversion uncertainty, especially for complex geology such as high dipping structures. However, due to the consideration of complex structure or reflection features, the existing multi-channel inversion methods have to adopt the highly time-consuming strategy of arranging seismic data trace-by-trace, limiting its wide application in pre-stack inversion. A fast pre-stack multi-channel inversion constrained by seismic reflection features has been proposed to address this issue. The key to our method is to re-characterize the reflection features to directly constrain the pre-stack inversion through a Hadamard product operator without rearranging the seismic data. The seismic reflection features can reflect the distribution of the stratum reflection interface, and we obtained them from the post-stack profile by searching the shortest local Euclidean distance between adjacent seismic traces. Instead of directly constructing a large-size reflection features constraint operator advocated by the conventional methods, through decomposing the reflection features along the vertical and horizontal direction at a particular sampling point, we have constructed a computationally well-behaved constraint operator represented by the vertical and horizontal partial derivatives. Based on the Alternating Direction Method of Multipliers (ADMM) optimization, we have derived a fast algorithm for solving the objective function, including Hadamard product operators. Compared with the conventional reflection features constrained inversion, the proposed method is more efficient and accurate, proved on the Overthrust model and a field data set.

Analysis of the ambiguity of log-constrained seismic impedance inversion

Although the ambiguity of seismic inversion is widely recognized in both theory and practice, so far as a concrete inversion example is concerned, there is not any objective, controllable method or any standard for how to evaluate and determine its ambiguity and reliability, especially for the high frequency components beyond the effective seismic frequency band. Taking log-constrained impedance inversion as an example, a new appraisal method is proposed on the basis of analyzing a simple geological model. Firstly, the inverted impedance model is transformed to a reflection coefficient series. Secondly, the maximum effective frequency of the real seismic data is chosen as a cutoff point and the reflection coefficient series is decomposed into two components by low-pass and high-pass filters. Thirdly, the geometrical reflection characteristics of the high-frequency components and that of the real seismic data are compared and analyzed. Then, the reliability of the inverted impedance model is appraised according to the similarity of geometrical characteristics between the high-frequency components and the real seismic data. The new method avoids some subjectivity in appraising the inverted result, and helps to enhance the reliability of reservoir prediction by impedance inversion technology.

Two-dimensional regularized inversion of AMT data based on rotation invariant of Central impedance tensor

Considering the uncertainty of the electrical axis for two-dimensional audo-magnetotelluric(AMT) data processing, an AMT inversion method with the Central impedance tensor was presented. First, we present a calculation expression of the Central impedance tensor in AMT, which can be considered as the arithmetic mean of TE-polarization mode and TM-polarization mode in the twodimensional geo-electrical model. Second, a least-squares iterative inversion algorithm is established, based on a smoothnessconstrained model, and an improved L-curve method is adopted to determine the best regularization parameters. We then test the above inversion method with synthetic data and field data. The test results show that this two-dimensional AMT inversion scheme for the responses of Central impedance is effective and can reconstruct reasonable two-dimensional subsurface resistivity structures. We conclude that the Central impedance tensor is a useful tool for two-dimensional inversion of AMT data.

A comparison of deep learning methods for seismic impedance inversion

Deep learning is widely used for seismic impedance inversion,but few work provides in-depth research and analysis on designing the architectures of deep neural networks and choosing the network hyperparameters.This paper is dedicated to comprehensively studying on the significant aspects of deep neural networks that affect the inversion results.We experimentally reveal how network hyperparameters and architectures affect the inversion performance,and develop a series of methods which are proven to be effective in reconstructing high-frequency information in the estimated impedance model.Experiments demonstrate that the proposed multi-scale architecture is helpful to reconstruct more high-frequency details than a conventional network.Besides,the reconstruction of high-frequency information can be further promoted by introducing a perceptual loss and a generative adversarial network from the computer vision perspective.More importantly,the experimental results provide valuable references for designing proper network architectures in the seismic inversion problem.

Lamé parameters inversion based on elastic impedance and its application

Connolly (1999 ) 有弹性的阻抗(EI ) 方程是 P 波浪速度， S 波浪速度，密度，和发生角度的功能。常规倒置方法能仅仅直接基于这个方程提取 P 速度， S 速度，和密度数据，在不同发生角度的有弹性的阻抗不在一样的规模，它使比较困难。我们基于灰色的等建议一个新有弹性的阻抗方程。(1999 ) 地址的 Zoeppritz 近似使用 Lam é参数常规倒置方法的缺乏。这个方程被使正常化在不同角度统一有弹性的阻抗尺寸并且为倒置使用了。Lam é参数能从用在 Lam é参数和有弹性的阻抗之间的线性关系从倒置获得的有弹性的阻抗数据直接被提取。有弹性的参数用这个新方法提取了的申请例子表演更稳定、正确并且能恢复水库信息很好。新方法是常规方法上的改进基于 Connolly 的方程。

Model-data-driven P-wave impedance inversion using ResNets and the normalized zero-lag cross-correlation objective function

Model-driven and data-driven inversions are two prominent methods for obtaining P-wave impedance,which is significant in reservoir description and identification.Based on proper initial models,most model-driven methods primarily use the limited frequency bandwidth information of seismic data and can invert P-wave impedance with high accuracy,but not high resolution.Conventional data-driven methods mainly employ the information from well-log data and can provide high-accuracy and highresolution P-wave impedance owing to the superior nonlinear curve fitting capacity of neural networks.However,these methods require a significant number of training samples,which are frequently insufficient.To obtain P-wave impedance with both high accuracy and high resolution,we propose a model-data-driven inversion method using Res Nets and the normalized zero-lag cross-correlation objective function which is effective for avoiding local minima and suppressing random noise.By using initial models and training samples,the proposed model-data-driven method can invert P-wave impedance with satisfactory accuracy and resolution.Tests on synthetic and field data demonstrate the proposed method’s efficacy and practicability.

Acoustic impedance inversion of zero-offset VSP data

高度精确的声学的阻抗倒置是为预先钻的一种关键技术由 VSP 数据的预言。在这篇论文，基于 VSP 数据有高分辨率的事实，高发信号到噪音比率，并且下面去并且在去上面，波浪能精确地被分开，我们建议用 VSP 数据在位前面在地上凿穿下面预言阻抗的一个方法。首先，非线性的反复的倒置的方法被采用用 VSP 走廊栈为阻抗转换。然后，由在重复并且用 preconditioned 修改抑制因素结合坡度方法解决倒置结果的方程，稳定性和集中能被提高。理论模型和实际数据的结果证明方法为预先钻是有效的预言使用 VSP 数据。

Seismic fluid identification using a nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo method

Elastic impedance inversion with high efficiency and high stability has become one of the main directions of seismic pre-stack inversion. The nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo （MCMC） method is proposed in this paper, combining conventional MCMC method based on global optimization with a preconditioned conjugate gradient （PCG） algorithm based on local optimization, so this method does not depend strongly on the initial model. It converges to the global optimum quickly and efficiently on the condition that effi- ciency and stability of inversion are both taken into consid- eration at the same time. The test data verify the feasibility and robustness of the method, and based on this method, we extract the effective pore-fluid bulk modulus, which is applied to reservoir fluid identification and detection, and consequently, a better result has been achieved.

Robust elastic impedance inversion using L1-norm misfit function and constraint regularization

The classical elastic impedance(EI) inversion method,however,is based on the L2-norm misfit function and considerably sensitive to outliers,assuming the noise of the seismic data to be the Guassian-distribution.So we have developed a more robust elastic impedance inversion based on the L1-norm misfit function,and the noise is assumed to be non-Gaussian.Meanwhile,some regularization methods including the sparse constraint regularization and elastic impedance point constraint regularization are incorporated to improve the ill-posed characteristics of the seismic inversion problem.Firstly,we create the Ll-norm misfit objective function of pre-stack inversion problem based on the Bayesian scheme within the sparse constraint regularization and elastic impedance point constraint regularization.And then,we obtain more robust elastic impedances of different angles which are less sensitive to outliers in seismic data by using the IRLS strategy.Finally,we extract the P-wave and S-wave velocity and density by using the more stable parameter extraction method.Tests on synthetic data show that the P-wave and S-wave velocity and density parameters are still estimated reasonable with moderate noise.A test on the real data set shows that compared to the results of the classical elastic impedance inversion method,the estimated results using the proposed method can get better lateral continuity and more distinct show of the gas,verifying the feasibility and stability of the method.

Pre-stack inversion for caved carbonate reservoir prediction:A case study from Tarim Basin,China

The major storage space types in the carbonate reservoir in the Ordovician in the TZ45 area are secondary dissolution caves.For the prediction of caved carbonate reservoir,post-stack methods are commonly used in the oilfield at present since pre-stack inversion is always limited by poor seismic data quality and insufficient logging data.In this paper,based on amplitude preserved seismic data processing and rock-physics analysis,pre-stack inversion is employed to predict the caved carbonate reservoir in TZ45 area by seriously controlling the quality of inversion procedures.These procedures mainly include angle-gather conversion,partial stack,wavelet estimation,low-frequency model building and inversion residual analysis.The amplitude-preserved data processing method can achieve high quality data based on the principle that they are very consistent with the synthetics.Besides,the foundation of pre-stack inversion and reservoir prediction criterion can be established by the connection between reservoir property and seismic reflection through rock-physics analysis.Finally,the inversion result is consistent with drilling wells in most cases.It is concluded that integrated with amplitude-preserved processing and rock-physics,pre-stack inversion can be effectively applied in the caved carbonate reservoir prediction.

Petrophysical parameters inversion for heavy oil reservoir based on a laboratory-calibrated frequency-variant rock-physics model

Heavy oil has high density and viscosity, and exhibits viscoelasticity. Gassmann's theory is not suitable for materials saturated with viscoelastic fluids. Directly applying such model leads to unreliable results for seismic inversion of heavy oil reservoir. To describe the viscoelastic behavior of heavy oil, we modeled the elastic properties of heavy oil with varying viscosity and frequency using the Cole-Cole-Maxwell (CCM) model. Then, we used a CCoherent Potential Approximation (CPA) instead of the Gassmann equations to account for the fluid effect, by extending the single-phase fluid condition to two-phase fluid (heavy oil and water) condition, so that partial saturation of heavy oil can be considered. This rock physics model establishes the relationship between the elastic modulus of reservoir rock and viscosity, frequency and saturation. The viscosity of the heavy oil and the elastic moduli and porosity of typical reservoir rock samples were measured in laboratory, which were used for calibration of the rock physics model. The well-calibrated frequency-variant CPA model was applied to the prediction of the P- and S-wave velocities in the seismic frequency range (1–100 Hz) and the inversion of petrophysical parameters for a heavy oil reservoir. The pre-stack inversion results of elastic parameters are improved compared with those results using the CPA model in the sonic logging frequency (∼10 kHz), or conventional rock physics model such as the Xu-Payne model. In addition, the inversion of the porosity of the reservoir was conducted with the simulated annealing method, and the result fits reasonably well with the logging curve and depicts the location of the heavy oil reservoir on the time slice. The application of the laboratory-calibrated CPA model provides better results with the velocity dispersion correction, suggesting the important role of accurate frequency dependent rock physics models in the seismic prediction of heavy oil reservoirs.

Nonlinear Electrical Impedance Tomography Method Using a Complete Electrode Model for the Characterization of Heterogeneous Domains

This paper presents an electrical impedance tomography(EIT)method using a partial-differential-equationconstrained optimization approach.The forward problem in the inversion framework is described by a complete electrodemodel(CEM),which seeks the electric potential within the domain and at surface electrodes considering the contact impedance between them.The finite element solution of the electric potential has been validated using a commercial code.The inverse medium problem for reconstructing the unknown electrical conductivity profile is formulated as an optimization problem constrained by the CEM.The method seeks the optimal solution of the domain’s electrical conductivity to minimize a Lagrangian functional consisting of a least-squares objective functional and a regularization term.Enforcing the stationarity of the Lagrangian leads to state,adjoint,and control problems,which constitute the Karush-Kuhn-Tucker(KKT)first-order optimality conditions.Subsequently,the electrical conductivity profile of the domain is iteratively updated by solving the KKT conditions in the reduced space of the control variable.Numerical results show that the relative error of the measured and calculated electric potentials after the inversion is less than 1%,demonstrating the successful reconstruction of heterogeneous electrical conductivity profiles using the proposed EIT method.This method thus represents an application framework for nondestructive evaluation of structures and geotechnical site characterization.

Pre-stack seismic density inversion in marine shale reservoirs in the southern Jiaoshiba area, Sichuan Basin, China

For a typical marine shale reservoir in the Jiaoshiba area, Sichuan Basin of China, P-impedance is sensitive for identifying lithology but not suitable for indicating good shale reservoirs. In comparison, density is an important quantity, which is sensitive for identifying the organic-rich mud shale from non-organic-rich mud shale. Due to the poor data quality and incidence angle range, density cannot be easily inverted by directly solving the ill-posed pre-stack seismic inversion in this area. Meanwhile, the traditional density regularizations implemented by directly using the more robust P-impedance inversion tend to be inaccurate for recovering density for this shale reservoir. In this paper, we combine the P-impedance and the minus uranium to construct the pseudo-P-impedance（PIp） at well locations. The PIp is observed to be sensitive for identifying organic-rich mud shale and has a good correlation with density in this area. We employ the PIp–density relation into the pre-stack inversion framework to estimate density. Three types of regularization are tested on both numerical and field data： These are no regularization, traditional regularization and the proposed approach. It is observed that the proposed method is better for recovering the density of organic-rich mud shale in the Jiaoshiba area.

Pre-stack AVO inversion with adaptive edge preserving smooth filter regularization based on Aki-Richard approximation

With the development of exploration of oil and gas resources,the requirements for seismic inversion results are getting more accurate.In particular,it is hoped that the distribution patterns of oil and gas reservoirs can be finely characterized,and the seismic inversion results can clearly characterize the location of stratigraphic boundaries and meet the needs of accurate geological description.Specifically,for pre-stack AVO inversion,it is required to be able to distinguish smaller geological targets in the depth or time domain,and clearly depict the vertical boundaries of the geological objects.In response to the above requirements,we introduce the preprocessing regularization of the adaptive edge-preserving smooth filter into the pre-stack AVO elastic parameter inversion to clearly invert the position of layer boundary and improve the accuracy of the inversion results.

Simultaneous Waveform Inverse Modelling for Litho-Fluid Prediction in an Old Marginal, “Agbbo”Field, Onshore Niger Delta, Nigeria

Simultaneous waveform inversion was used to predict lithofacies and fluid type across the field. Very often, characterizing reservoirs in terms of lithology and fluid type using conventional methods is replete with uncertainties, especially in marginal fields. An approach is employed in this study that integrated rock physics and waveform inverse modelling for lithology and fluid-type characterization to appropriately identify potential hydrocarbon saturated zones and their corresponding lithology. Seismic and well-log data were analyzed using Hampson Russel software. The method adopted includes lithofacies and fluid content analysis using rock physics parameters and seismic simultaneous inverse modelling. Rock physics analysis identified 2 broad reservoirs namely: HDZ1 and HDZ2 reservoirs. Results from the inverse modelling showed that low values of acoustic impedance from 19,743 to 20,487 (ft/s)(g/cc) reflect hydrocarbon-bearing reservoirs while medium to high values shows brine and shale respectively, with brine zone ranging from 20,487 to 22,531 (ft/s)(g/cc) and shale above 22,531 (ft/s)(g/cc). Two lithofacies were identified from inversion analysis of Vp/Vs and Mu-Rho, namely: sand and shale with VpVs 1.95 values respectively. Mu-Rho > 12.29 (GPa)(g/cc) and <12.29 (GPa) (g/cc) represent sand and shale respectively. From 3D volume, it was observed that a high accumulation of hydrocarbon was observed to be saturated at the north to the eastern part of the field forming a meandering channel. Sands were mainly distributed around the northeastern to the southwestern part of the field, that tends to be away from Well 029. This was also validated by the volume of rigidity modulus (Mu-Rho) showing high values indicating sands fall within the northeastern part of the field.

Improved reservoir characterization by means of supervised machine learning and model-based seismic impedance inversion in the Penobscot field,Scotian Basin

The present research work attempted to delineate and characterize the reservoir facies from the Dawson Canyon Formation in the Penobscot field,Scotian Basin.An integrated study of instantaneous frequency,P-impedance,volume of clay and neutron-porosity attributes,and structural framework was done to unravel the Late Cretaceous depositional system and reservoir facies distribution patterns within the study area.Fault strikes were found in the EW and NEE-SWW directions indicating the dominant course of tectonic activities during the Late Cretaceous period in the region.P-impedance was estimated using model-based seismic inversion.Petrophysical properties such as the neutron porosity(NPHI)and volume of clay(VCL)were estimated using the multilayer perceptron neural network with high accuracy.Comparatively,a combination of low instantaneous frequency(15-30 Hz),moderate to high impedance(7000-9500 gm/cc*m/s),low neutron porosity(27%-40%)and low volume of clay(40%-60%),suggests fair-to-good sandstone development in the Dawson Canyon Formation.After calibration with the welllog data,it is found that further lowering in these attribute responses signifies the clean sandstone facies possibly containing hydrocarbons.The present study suggests that the shale lithofacies dominates the Late Cretaceous deposition(Dawson Canyon Formation)in the Penobscot field,Scotian Basin.Major faults and overlying shale facies provide structural and stratigraphic seals and act as a suitable hydrocarbon entrapment mechanism in the Dawson Canyon Formation's reservoirs.The present research advocates the integrated analysis of multi-attributes estimated using different methods to minimize the risk involved in hydrocarbon exploration.

A Method of Improving Seismic Data Resolution：Comprehensive Inversion of Well logging and Seismic Data

Comprehensive inversion of logging and seismic data presented in this paper is a method to improve seismic data resolution. It involves using ample high-frequency information and complete low-frequency information of known logging to make up for the lack of limited bandwidth of practical seismic recording, obtaining an approximate reflection coefficient sequence (or wave impedance) of high resolution by iterative inversion and providing more reliable seismic evidence for further lithologic interpretation and lateral tracking, correlation and prediction of thin reservoir. The comprehensive inversion can be realized in the following steps: (1) to establish an initial model of higher resolution; (2) to obtain wavelets, and (3) to constrain iterative inversion. The key to this inversion lies in building an initial model. It is assumed from our experience that when the initial model is properly given, iterative inversion can be quickly converged to the ideal result.