An approximation method for solving the inverse MTS problem with the use of neural networks

The work develops the approximation approach to solving the inverse MTS problem with the use of neural networks. The inverse problem is considered in model classes of parametrized geoelectric structures, whose electric conductivity is controlled by a few hundreds of macroparameters (N ∼ 300). An approximate inverse operator of the problem is constructed for each model class as a neural network, whose coefficients are determined in the process of training on a representative sample of standard examples of forward problem solutions. The problem of determination of the model class of geolectric structures corresponding to the presented input MT data is solved with the use of the neural network classifier constructed for the available set of model classes of structures. Regularizing factors and errors of the neural network method are analyzed. The operation of the algorithm is illustrated by examples of the 2-D inversion of synthetic MT data.     

Quasi-one-dimensional method for solving the inverse magnetotelluric problem

We consider the iterative numerical method for solving two-dimensional (2D) inverse problems of magnetotelluric sounding, which significantly reduces the computational burden of the inverse problem solution in the class of quasi-layered models. The idea of the method is to replace the operator of the direct 2D problem of calculating the low-frequency electromagnetic field in a quasi-layered medium by a quasi-one dimensional operator at each observation point. The method is applicable for solving the inverse problems of magnetotellurics with either the E- and H-polarized fields and in the case when the inverse problem is simultaneously solved using the impedance values for the fields with both polarizations. We describe the numerical method and present the examples of its application to the numerical solution of a number of model inverse problems of magnetotelluric sounding.     

地震波传播的三维伪谱和高阶有限差分混合方法并行模拟

基于交错网格伪谱法和高阶精度有限差分方法,发展了模拟非均匀介质地震波传播的三维伪谱和有限差分混合算法.该方法在两个水平方向利用交错网格伪谱算子计算空间微分,保留了该方法高效、高精度的优势,在垂直方向采用交错网格高阶精度有限差分算子实现空间微分计算．利用有限差分方法的局部性特征,将三维计算区域在垂直方向上划分为一系列子区域,并分配给不同的处理器,实现了在并行计算机集群上的三维并行计算．通过模拟算例,与离散波数法比较,检验了该算法的精度．为了检验该方法的实用性,在64个处理器上,对三维沉积盆地模型进行了67108864个网格点的并行计算,模拟的波场主频率为1.25Hz,讨论了沉积盆地深度对三维沉积盆地地面运动的影响．      

TTI介质的交错网格伪P波正演方法

研究了三维弱各向异性近似下,利用伪P波(伪纵波)模拟弹性波场P分量在倾斜对称轴的横向各向同性(TTI)介质中的传播过程,并对比了分别基于弹性Hooke定律、弹性波投影和运动学色散方程所建立的三种二阶差分伪P波方程的正演特点.目前这些伪P波方程数值计算主要采用规则网格差分,但是规则网格在TTI模拟中有低效率、低精度以及不稳定的缺点.为了提高计算的精度,本文构建出相应方程的交错网格有限差分格式.通过对比伪P波方程在三维TTI介质中不同的数值模拟的表达形式,本文认为基于色散方程所建立的伪P波方程在模拟弹性波中P波传播的过程中具有最小的噪声.本文分析不同的各向同性对称轴空间角度的频散特征,并引入适当的横波速度维持计算的稳定.二维模型算例表明,本文提出的交错网格正演算法可以得到稳定光滑的伪P波正演波场.使用本文交错网格算法对二维BP TTI模型的逆时偏移也具有较稳定的偏移结果.     

Neural network inversion of data in classes of parametrized geoelectric sections

A method of approximate magnetotelluric sounding (MTS) data inversion is developed on the basis of the representation of the inverse operator by an artificial neural network in classes of geoelectric structures. A methodology of the neural network inversion of magnetotelluric data is proposed for a family of classes of geoelectric structures and the uncertainty of the inferred results is estimated. A neural network algorithm of MTS data inversion is tested using synthetic 2-D data.     

The <Emphasis Type="Italic">T</Emphasis>-matrix approach for the mathematical modeling of the effective elastic properties of hydrocarbon reservoirs

Various methods of the effective medium theory (EMT)—the T-matrix approach (optical potential approximation and coherent potential approximation), Mori–Tanaka method, generalized singular approximation, etc.—for modeling the effective elastic properties of hydrocarbon reservoir rocks are considered. The relationship of the different approaches (perturbation theory, self-consistent methods, and variational principles) is demonstrated. The classification of the methods by the degree of complexity of solving the inverse problem is suggested. An example of the theoretical modeling of an oil-, gas-, and water-saturated rock with the oriented axis of the fractures is presented. Conclusions about the applicability of different methods for rock modeling are made. In this study we review and compare different methods of the EMT and present the guidelines for determining the effective properties of the medium in rock physics modeling.     

Inverse data-space multiple elimination with 3D curvelet sparsity promotion

This paper describes an effective implementation of the inverse data-space multiple elimination method via the three-dimensional (3D) curvelet domain. The method can separate the surface-related operator (A) and primaries (P ₀) through seismic data matrix inversion. A 3D curvelet transform is introduced to sparsely represent the seismic data in the inverse data space. Hence, this approach is suitable for obtaining an accurate solution because of its multiscale and multidirectional analysis properties. The L1 norm is used to promote sparseness in the transform domain. Then, a high-fidelity separation of the operator (A) and the primaries (P ₀) is realized. The proposed method is applied to synthetic data from a model containing a salt structure. We compare the results with that of the traditional inverse data-space multiple elimination method and also with that of two-dimensional surface-related multiple elimination. The findings fully demonstrate the superiority of the proposed method over the traditional inverse method; moreover, the proposed method protects the primary energy more effectively than the SRME method.     

Inversion of DC resistivity data using neural networks

The inversion of geoelectrical resistivity data is a difficult task due to its non-linear nature. In this work, the neural network (NN) approach is studied to solve both 1D and 2D resistivity inverse problems. The efficiency of a widespread, supervised training network, the back-propagation technique and its applicability to the resistivity problem, is investigated. Several NN paradigms have been tried on a basis of trial-and-error for two types of data set. In the 1D problem, the batch back-propagation paradigm was efficient while another paradigm, called resilient propagation, was used in the 2D problem. The network was trained with synthetic examples and tested on another set of synthetic data as well as on the field data. The neural network gave a result highly correlated with that of conventional serial algorithms. It proved to be a fast, accurate and objective method for depth and resistivity estimation of both 1D and 2D DC resistivity data. The main advantage of using NN for resistivity inversion is that once the network has been trained it can perform the inversion of any vertical electrical sounding data set very rapidly.     

3D Neural Network Inversion of Field Geoelectric Data with Calculating Posterior Estimates

Izvestiya, Physics of the Solid Earth - Abstract—The approximation neural network (ANN) method for solving the inverse geoelectric problem in the piecewise constant classes of media with the...     

1D and 2D Cole-Cole-inversion of time-domain induced-polarization data

A new method for the 2D inversion of induced polarization (IP) data in the time domain has been developed. The entire IP transients were observed and inverted into 2D Cole-Cole earth models, including resistivity, chargeability, relaxation time and the frequency constant. Firstly, a modified 1D time-domain electromagnetic algorithm was used to calculate the response of a layered polarizable ground. The transient signals were then inverted using the Marquardt method to derive the Cole-Cole parameters of each layer. However, model calculations showed that the EM effects could be neglected for the time range (>1 ms) and for the transmitter–receiver distances (<50 m) used in this study. Therefore, the induction effects were not considered for the solution of the 2D inverse problem and a DC solution was applied. An approximative forward algorithm was introduced in order to calculate the IP transients directly in the time domain and in order to speed up the inverse procedure. The approximation is highly accurate, and this is demonstrated by comparing the approximations with their exact solutions up to 3D. The inverse algorithm presented consists of two steps. The transient voltages of an array data set were inverted separately into a two-dimensional resistivity model for each time channel. The time-dependent resistivity of each cell was then interpreted as the response of a homogeneous half-space. In the 2D inversion algorithm, a 3D DC algorithm was used as a forward operator. The method only requires a standard 2D DC inversion and a homogenous half-space Cole-Cole inversion. The developed algorithm has been successfully applied to synthetic data sets and to a field data set obtained from a waste site situated close to Düren in Germany.     

波动方程CT技术在地震数据处理中的应用

本文从畸变的Born近似的微扰技术出发,给出了利用广义Radon变换和Fourier积分算子的理论反演介质间断性的原理.将声学的广义Radon变换与经典Radon变换进行类比,近似地导出了声学广义Radon变换的反演公式. 本文对于反射地震学的情况,提出了一种拟线性化方法,考虑了成象点的一次散射场,从某种程度上减少了Born近似对弱散射的苛求. 利用同一模型的理论记录和物理实验记录的反演计算结果对提出的方法进行了验证,并讨论了进一步提高成象精度的方法.     

Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse‐grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse‐grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis‐Hastings. While both the sparse‐grid and Bayesian samplers produce compatible results, in both examples, the sparse‐grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse‐grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.     

渗流方程的三维粗化算法

本文提出了地下流体渗流问题的三维解粗化算法,在粗网格内流体压强分布用直接解法求解三维渗流方程,用这些解计算粗网格的等效渗透率,在流体流速大的区域仍采用精细网格的计算方法.用所得等效渗透率计算了粗化网格的渗流场的压强分布,结果表明渗流方程的三维粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了100多倍.     

弱各向异性介质弹性波的准各向同性近似正演模拟

准各向同性(QI)近似可用于弱各向异性介质的正演模拟.本文通过运用QI方法的零阶和一阶近似,计算了VTI介质模型的地震记录.得出的地震记录与标准各向同性射线理论(IRT)和基于伪谱法的三维地震正演模拟得出的地震记录作了比较,可以认为是精确的合成地震记录.     

Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels

We present a new approach to reservoir simulation that gives accurate resolution of both large-scale and fine-scale flow patterns. The method uses a mixed multiscale finite-element method (MMsFEM) to solve the pressure equation on a coarse grid and a streamline-based technique to solve the fluid transport on a fine-scale subgrid. The MMsFEM is based on the construction of special approximation velocity spaces that are adaptive to the local properties of the differential operator. As such, MMsFEM produces a detailed subgrid velocity field that reflects the impact of the fine-scale heterogeneous structures. By combining MMsFEM with rapid streamline simulation of the fluid transport, we aim towards a numerical scheme that facilitates routine reservoir simulation of large heterogeneous geomodels without upscaling. The new method is applied to two different test cases. The first test case consists of two (strongly) heterogeneous quarter five-spot problems in 2D. The second test case is a 3D upscaling benchmark taken from the 10th SPE Comparative Solution Project, a project whose purpose is to compare and validate upscaling techniques. The test cases demonstrate that the combination of multiscale methods and streamlines is a robust and viable alternative to traditional upscaling-based reservoir simulation.     

一种模拟非均匀介质中弹性波传播新的k-space方法

传统的伪谱(PS)方法,采用傅里叶变换(FT)计算空间导数具有很高的精度,每个波长仅需要两个采样点,而时间导数采用有限差分(FD)近似因而精度较低.当采用大时间步长时,由于时空精度不平衡,PS法存在不稳定性问题.原始的k-space方法可以有效地克服这些问题但是却无法适用于非均匀介质.为了提高原始k-space方法模拟非均匀介质波动方程的精度,我们提出了一种新的k-space算子族.它是用非均匀介质的变速度代替原k-space算子中的常数补偿速度构造得到,引入低秩近似可以高效求解.我们将构造的新的k-space算子应用于耦合的二阶位移波动方程,而不是交错网格一阶速度应力波动方程,使模拟弹性波的计算存储量减少.我们从数学上证明了基于二阶波动方程的k-space方法与基于一阶波动方程的k-space方法是等价的.数值模拟实验表明,与传统的PS、交错网格PS和原始的k-space方法相比,我们的新方法可以在时间和空间步长较大的均匀和非均匀介质中,为弹性波的传播提供更精确的数值解.在保持稳定性和精度的同时,采用较大的时空采样间隔,可以大大降低数值模拟的计算成本.