共查询到20条相似文献，搜索用时 46 毫秒
1.
Despite their apparent high dimensionality, spatially distributed hydraulic properties of geologic formations can often be compactly (sparsely) described in a properly designed basis. Hence, the estimation of high-dimensional subsurface flow properties from dynamic performance and monitoring data can be formulated and solved as a sparse reconstruction inverse problem. Recent advances in statistical signal processing, formalized under the compressed sensing paradigm, provide important guidelines on formulating and solving sparse inverse problems, primarily for linear models and using a deterministic framework. Given the uncertainty in describing subsurface physical properties, even after integration of the dynamic data, it is important to develop a practical sparse Bayesian inversion approach to enable uncertainty quantification. In this paper, we use sparse geologic dictionaries to compactly represent uncertain subsurface flow properties and develop a practical sparse Bayesian method for effective data integration and uncertainty quantification. The multi-Gaussian assumption that is widely used in classical probabilistic inverse theory is not appropriate for representing sparse prior models. Following the results presented by the compressed sensing paradigm, the Laplace (or double exponential) probability distribution is found to be more suitable for representing sparse parameters. However, combining Laplace priors with the frequently used Gaussian likelihood functions leads to neither a Laplace nor a Gaussian posterior distribution, which complicates the analytical characterization of the posterior. Here, we first express the form of the Maximum A-Posteriori (MAP) estimate for Laplace priors and then use the Monte-Carlo-based Randomize Maximum Likelihood (RML) method to generate approximate samples from the posterior distribution. The proposed Sparse RML (SpRML) approximate sampling approach can be used to assess the uncertainty in the calibrated model with a relatively modest computational complexity. We demonstrate the suitability and effectiveness of the SpRML formulation using a series of numerical experiments of two-phase flow systems in both Gaussian and non-Gaussian property distributions in petroleum reservoirs and successfully apply the method to an adapted version of the PUNQ-S3 benchmark reservoir model. 相似文献
2.
压缩感知技术通常利用地震信号在某一变换域内的稀疏性质,将随机缺失的地震数据重建问题转化为L1正则化问题.本文首先通过Shearlet变换获得地震信号的稀疏性质,再将广义全变分（TGV）约束引入L1正则化模型,构建了基于Shearlet变换的双正则化模型用于重建地下介质的图像.与传统L1正则化方法相比,基于Shearlet变换的双正则化方法不仅考虑了信号的稀疏性,同时兼顾了地下介质结构的复杂性,可以较好的重建地下结构体的图像.最后采用交替方向乘子法（ADMM）求解所建模型,每个子问题均可得到显式解.数值实验对比了基于小波变换、Shearlet变换的L1正则化方法和TGV正则化方法,结果表明基于Shearlet变换的双正则化方法对于随机采样50%数据的情况具有较好的重建结果,同时对于有限范围的连续缺失数据的重建亦具有一定的有效性. 相似文献
3.
A. J. W. DUIJNDAM 《Geophysical Prospecting》1988,36(8):878-898
This paper gives a review of Bayesian parameter estimation. The Bayesian approach is fundamental and applicable to all kinds of inverse problems. Its basic formulation is probabilistic. Information from data is combined with a priori information on model parameters. The result is called the a posteriori probability density function and it is the solution to the inverse problem. In practice an estimate of the parameters is obtained by taking its maximum. Well-known estimation procedures like least-squares inversion or l1 norm inversion result, depending on the type of noise and a priori information given. Due to the a priori information the maximum will be unique and the estimation procedures will be stable except (in theory) for the most pathological problems which are very unlikely to occur in practice. The approach of Tarantola and Valette can be derived within classical probability theory. The Bayesian approach allows a full resolution and uncertainty analysis which is discussed in Part II of the paper. 相似文献
4.
目前瑞雷波多阶模式频散曲线反演中仅考虑数据的拟合,缺乏对模型的约束,不能很好地刻画地层间断面的问题,针对此问题,研究了瑞雷波多阶模式频散曲线稀疏正则化反演方法.正演模拟基于广义反射-透射系数法,数值计算上采用一种快速求根方法,与二等分方法相比,能够在很短的时间内达到最优的收敛效果;反演建模时采用L1范数正则化方法对模型... 相似文献
5.
Michael J. Tompkins Juan Luis Fernández Martínez Zulima Fernández Muñiz 《Geophysical Prospecting》2013,61(1):28-41
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. 相似文献
6.
We focus on the Bayesian estimation of strongly heterogeneous transmissivity fields conditional on data sampled at a set of locations in an aquifer. Log-transmissivity, Y, is modeled as a stochastic Gaussian process, parameterized through a truncated Karhunen–Loève (KL) expansion. We consider Y fields characterized by a short correlation scale as compared to the size of the observed domain. These systems are associated with a KL decomposition which still requires a high number of parameters, thus hampering the efficiency of the Bayesian estimation of the underlying stochastic field. The distinctive aim of this work is to present an efficient approach for the stochastic inverse modeling of fully saturated groundwater flow in these types of strongly heterogeneous domains. The methodology is grounded on the construction of an optimal sparse KL decomposition which is achieved by retaining only a limited set of modes in the expansion. Mode selection is driven by model selection criteria and is conditional on available data of hydraulic heads and (optionally) Y. Bayesian inversion of the optimal sparse KLE is then inferred using Markov Chain Monte Carlo (MCMC) samplers. As a test bed, we illustrate our approach by way of a suite of computational examples where noisy head and Y values are sampled from a given randomly generated system. Our findings suggest that the proposed methodology yields a globally satisfactory inversion of the stochastic head and Y fields. Comparison of reference values against the corresponding MCMC predictive distributions suggests that observed values are well reproduced in a probabilistic sense. In a few cases, reference values at some unsampled locations (typically far from measurements) are not captured by the posterior probability distributions. In these cases, the quality of the estimation could be improved, e.g., by increasing the number of measurements and/or the threshold for the selection of KL modes. 相似文献
7.
The technique of seismic amplitude-versus-angle inversion has been widely used to estimate lithology and fluid properties in seismic exploration. The amplitude-versus-angle inversion problem is intrinsically ill-posed and generally stabilized by the use of L2-norm regularization methods but with drawback of smoothing important boundaries between adjacent layers. In this study, we propose a sparse Bayesian linearized solution for amplitude-versus-angle inversion problem to preserve the sharp geological interfaces. In this regard, a priori constraint term with two regularization functions is presented: the sparse constraint regularization and the low-frequency model information. In addition, to obtain high-resolution reflectivity estimation, the model parameters decorrelation technique combined with dipole decomposition method is employed. We validate the applicability of the presented method by both synthetic and real seismic data from the Gulf of Mexico. The accuracy improvement of the presented method is also confirmed by comparing the results with the commonly used Bayesian linearized amplitude-versus-angle inversion. 相似文献
8.
Anyone working on inverse problems is aware of their ill-posed character. In the case of inverse problems, this concept (ill-posed) proposed by J. Hadamard in 1902, admits revision since it is somehow related to their ill-conditioning and the use of local optimization methods to find their solution. A more general and interesting approach regarding risk analysis and epistemological decision making would consist in analyzing the existence of families of equivalent model parameters that are compatible with the prior information and predict the observed data within the same error bounds. Otherwise said, the ill-posed character of discrete inverse problems (ill-conditioning) originates that their solution is uncertain. Traditionally nonlinear inverse problems in discrete form have been solved via local optimization methods with regularization, but linear analysis techniques failed to account for the uncertainty in the solution that it is adopted. As a result of this fact uncertainty analysis in nonlinear inverse problems has been approached in a probabilistic framework (Bayesian approach), but these methods are hindered by the curse of dimensionality and by the high computational cost needed to solve the corresponding forward problems. Global optimization techniques are very attractive, but most of the times are heuristic and have the same limitations than Monte Carlo methods. New research is needed to provide uncertainty estimates, especially in the case of high dimensional nonlinear inverse problems with very costly forward problems. After the discredit of deterministic methods and some initial years of Bayesian fever, now the pendulum seems to return back, because practitioners are aware that the uncertainty analysis in high dimensional nonlinear inverse problems cannot (and should not be) solved via random sampling methodologies. The main reason is that the uncertainty “space” of nonlinear inverse problems has a mathematical structure that is embedded in the forward physics and also in the observed data. Thus, problems with structure should be approached via linear algebra and optimization techniques. This paper provides new insights to understand uncertainty from a deterministic point of view, which is a necessary step to design more efficient methods to sample the uncertainty region(s) of equivalent solutions. 相似文献
9.
Regularization methods are used to recover a unique and stable solution in ill-posed geophysical inverse problems. Due to the connection of homogeneous operators that arise in many geophysical inverse problems to the Fourier basis, for these operators classical regularization methods possess some limitations that one may try to circumvent by wavelet techniques.
In this paper, we introduce a two-step wavelet-based regularization method that combines classical regularization methods with wavelet transform to solve ill-posed linear inverse problems in geophysics. The power of the two-step wavelet-based regularization for linear inversion is twofold. First, regularization parameter choice is straightforward; it is obtained from a priori estimate of data variance. Second, in two-step wavelet-based regularization the basis can simultaneously diagonalize both the operator and the prior information about the model to be recovered. The latter is performed by wavelet-vaguelette decomposition using orthogonal symmetric fractional B-spline wavelets.
In the two-step wavelet-based regularization method, at the first step where fully classical tools are used, data is inverted for the Moore-Penrose solution of the problem, which is subsequently used as a preliminary input model for the second step. Also in this step, a model-independent estimate of data variance is made using nonparametric estimation and L-curve analysis. At the second step, wavelet-based regularization is used to partially recover the smoothness properties of the exact model from the oscillatory preliminary model.
We illustrated the efficiency of the method by applying on a synthetic vertical seismic profiling data. The results indicate that a simple non-linear operation of weighting and thresholding of wavelet coefficients can consistently outperform classical linear inverse methods. 相似文献
In this paper, we introduce a two-step wavelet-based regularization method that combines classical regularization methods with wavelet transform to solve ill-posed linear inverse problems in geophysics. The power of the two-step wavelet-based regularization for linear inversion is twofold. First, regularization parameter choice is straightforward; it is obtained from a priori estimate of data variance. Second, in two-step wavelet-based regularization the basis can simultaneously diagonalize both the operator and the prior information about the model to be recovered. The latter is performed by wavelet-vaguelette decomposition using orthogonal symmetric fractional B-spline wavelets.
In the two-step wavelet-based regularization method, at the first step where fully classical tools are used, data is inverted for the Moore-Penrose solution of the problem, which is subsequently used as a preliminary input model for the second step. Also in this step, a model-independent estimate of data variance is made using nonparametric estimation and L-curve analysis. At the second step, wavelet-based regularization is used to partially recover the smoothness properties of the exact model from the oscillatory preliminary model.
We illustrated the efficiency of the method by applying on a synthetic vertical seismic profiling data. The results indicate that a simple non-linear operation of weighting and thresholding of wavelet coefficients can consistently outperform classical linear inverse methods. 相似文献
10.
为解决介质圆柱体逆散射问题，提出一种新的在线逆散射方法,通过支持向量机将逆散射问题转化成一个回归估计问题. 该方法可应用于各种逆散射方面, 尤其是目标的几何与电磁参数重构和埋地目标探测. 文中首次将支持向量机方法应用到该领域，设置多个散射场的观测点，通过提取散射场的不同信息作为样本信息训练支持向量机, 建立了介质圆柱体的逆散射模型, 利用该模型重构了介质圆柱体的电磁参数，同时探测了埋地位置. 数值结果显示了该方法的有效性和准确性,为目标的实时逆散射研究提供了一种有效方法. 相似文献
11.
We introduce a nonlinear orthogonal matching pursuit (NOMP) for sparse calibration of subsurface flow models. Sparse calibration is a challenging problem as the unknowns are both the non-zero components of the solution and their associated weights. NOMP is a greedy algorithm that discovers at each iteration the most correlated basis function with the residual from a large pool of basis functions. The discovered basis (aka support) is augmented across the nonlinear iterations. Once a set of basis functions are selected, the solution is obtained by applying Tikhonov regularization. The proposed algorithm relies on stochastically approximated gradient using an iterative stochastic ensemble method (ISEM). In the current study, the search space is parameterized using an overcomplete dictionary of basis functions built using the K-SVD algorithm. The proposed algorithm is the first ensemble based algorithm that tackels the sparse nonlinear parameter estimation problem. 相似文献
12.
Antoine Guitton 《Geophysical Prospecting》2012,60(5):870-884
With ill‐posed inverse problems such as Full‐Waveform Inversion, regularization schemes are needed to constrain the solution. Whereas many regularization schemes end up smoothing the model, an undesirable effect with FWI where high‐resolution maps are sought, blocky regularization does not: it identifies and preserves strong velocity contrasts leading to step‐like functions. These models might be needed for imaging with wave‐equation based techniques such as Reverse Time Migration or for reservoir characterization. Enforcing blockiness in the model space amounts to enforcing a sparse representation of discontinuities in the model. Sparseness can be obtained using the ?1 norm or Cauchy function which are related to long‐tailed probability density functions. Detecting these discontinuities with vertical and horizontal gradient operators helps constraining the model in both directions. Blocky regularization can also help recovering higher wavenumbers that the data used for inversion would allow, thus helping controlling the cost of FWI. While the Cauchy function yields blockier models, both ?1 and Cauchy attenuate illumination and inversion artifacts. 相似文献
13.
重力反演是恢复地下密度空间分布的有效工具,而选择合理的密度模型约束方法是提升重力反演分辨率和可靠性的关键.常规约束方法大多是从剖分网格空间中的密度模型出发,通过调整光滑或稀疏约束权重来匹配反演目标,但当地质体类型多样、异常分离不准确及网格剖分方案不合理时,模型约束的合理性与灵活性难以得到有效保证.为此,本文提出了一种基于密度模型稀疏表征的重力反演方法.首先假设待反演的密度模型表征为模型特征矩阵和稀疏分解系数的线性组合,之后重新推导了重力反演目标函数,并给出了分解系数的稀疏求解过程.相比现有重力反演方法,用于构建模型特征矩阵的特征模型可包含不同类型地质体的先验几何信息,分解系数的稀疏性保证了待反演目标来自于最典型的地质模式组合.最后,通过模型试验及实际资料验证了基于密度模型稀疏表征的重力反演方法的有效性. 相似文献
14.
不规则采样地震数据会对地震数据的多道处理造成严重影响,将非均匀Fourier变换和贝叶斯参数反演方法相结合,对不规则空间带限地震数据进行反演重建.对每一个频率依据最小视速度确定出重建数据的带宽,然后从不规则地震数据中估计出重建数据的空间Fourier系数.将不规则地震数据重建视为信息重建的地球物理反演问题,运用贝叶斯参数反演理论来估计Fourier系数.在反演求解时,使用共轭梯度算法,以保证求解的稳定性,加快解的收敛速度.理论模型和实际资料处理验证了本方法的有效性和实用性. 相似文献
15.
随着地震勘探目标复杂化和精细化程度的提高以及"两宽一高"等采集技术的广泛应用,当前地震数据采集的时间越来越长、成本越来越高.针对此问题,本文基于压缩感知理论开展了地震数据高效采集方法的改进和探索研究.根据波动方程解的一般表示式,从波场传播的角度给出了地震数据具有稀疏性的数学物理依据及寻找适应地震数据稀疏变换的一般方案;在稀疏性先验信息的指导下,发展了具有"蓝色噪声"频谱特征的改进的分段采样方法,并基于最优化理论提出了地震数据重建方法.地震数据的稀疏性理论、稀疏约束下的高效采集方法以及地震数据的重建方法构成了相对完善的地震数据高效采集理论.把该理论用于指导地震数据采集,即利用稀疏约束的随机采样方法改变常规规则密集测网中炮点和检波点(或二者之一)的分布,设计了三种随机且均匀的高效采集测网,提出了利用相应测网获取的地震数据重建为常规规则密集测网地震数据的针对性方案,并使用重建精度、高效采集数据的直接成像和重建后再成像的结果对比证明了上述重建方案的有效性.基于Marmousi模型的高效采集试验检验了本文构建的基于稀疏约束的地震数据高效采集方法理论框架在提高当前地震数据采集效率、降低勘探成本上的优势以及方法的有效性和可行性. 相似文献
16.
为了提高二维大地电磁反演对异常体边界的刻画能力,我们引入曲波变换建立一种新的稀疏正则化反演方法.与传统的在空间域中对模型电阻率参数求解的方式不同,我们借助曲波变换将二维电阻率模型转换为曲波系数,并采用L1范数约束以保证系数的稀疏性.曲波变换是一种多尺度分析方法,其系数分为粗尺度系数和精细尺度系数,粗尺度的系数代表电阻率模型的整体概貌,而精细尺度中较大系数代表目标体的边缘细节.此外,曲波变换的窗函数满足各向异性尺度关系,并具有多方向性,因此曲波变换可以近似最佳地提取目标体的边缘特征信息,这为我们在反演中恢复边界提供有利条件.通过对大地电磁的理论模型合成数据和实测数据反演,验证了基于曲波变换稀疏正则化反演对异常体边界的刻画能力优于常规的L2范数和L1范数反演方法. 相似文献
17.
Adjustment of regularization in ill-posed linear inverse problems by the empirical Bayes approach 总被引：1，自引：0，他引：1
Yuji Mitsuhata 《Geophysical Prospecting》2004,52(3):213-239
Regularization is the most popular technique to overcome the null space of model parameters in geophysical inverse problems, and is implemented by including a constraint term as well as the data‐misfit term in the objective function being minimized. The weighting of the constraint term relative to the data‐fitting term is controlled by a regularization parameter, and its adjustment to obtain the best model has received much attention. The empirical Bayes approach discussed in this paper determines the optimum value of the regularization parameter from a given data set. The regularization term can be regarded as representing a priori information about the model parameters. The empirical Bayes approach and its more practical variant, Akaike's Bayesian Information Criterion, adjust the regularization parameter automatically in response to the level of data noise and to the suitability of the assumed a priori model information for the given data. When the noise level is high, the regularization parameter is made large, which means that the a priori information is emphasized. If the assumed a priori information is not suitable for the given data, the regularization parameter is made small. Both these behaviours are desirable characteristics for the regularized solutions of practical inverse problems. Four simple examples are presented to illustrate these characteristics for an underdetermined problem, a problem adopting an improper prior constraint and a problem having an unknown data variance, all frequently encountered geophysical inverse problems. Numerical experiments using Akaike's Bayesian Information Criterion for synthetic data provide results consistent with these characteristics. In addition, concerning the selection of an appropriate type of a priori model information, a comparison between four types of difference‐operator model – the zeroth‐, first‐, second‐ and third‐order difference‐operator models – suggests that the automatic determination of the optimum regularization parameter becomes more difficult with increasing order of the difference operators. Accordingly, taking the effect of data noise into account, it is better to employ the lower‐order difference‐operator models for inversions of noisy data. 相似文献
18.
地震数据的随机噪声去除是地震数据处理中的一项重要步骤,双稀疏字典提供了两层稀疏模型,比单层稀疏模型可以更好地去除噪声.该方法首先利用contourlet变换对地震数据进行稀疏表示,然后在contourlet域中使用快速迭代收缩阈值算法（fast iterative shrinkage-thresholding algorithm,FISTA）对初始字典系数进行更新,接着采用数据驱动紧标架（data-driven tight frame,DDTF）在contourlet域中得到DDTF字典并通过FISTA得到更新后的字典系数,最后通过DDTF字典和更新后的字典系数获得新的contourlet系数,并对新的contourlet系数进行硬阈值和contourlet反变换得到去噪后的数据.通过模拟数据和实际数据的实验证明：与固定基变换去噪方法相比,该方法可以自适应地对地震数据进行稀疏表示,在地震数据较为复杂时得到更高的信噪比;与字典学习去噪方法相比,该方法不仅拥有较快的去噪速度,而且克服了字典学习因为缺少先验约束造成瑕疵的缺点. 相似文献
19.
In most real-world hydrogeologic situations, natural heterogeneity and measurement errors introduce major sources of uncertainty in the solution of the inverse problem. The Bayesian Maximum Entropy (BME) method of modern geostatistics offers an efficient solution to the inverse problem by first assimilating various physical knowledge bases (hydrologic laws, water table elevation data, uncertain hydraulic resistivity measurements, etc.) and then producing robust estimates of the subsurface variables across space. We present specific methods for implementing the BME conceptual framework to solve an inverse problem involving Darcys law for subsurface flow. We illustrate one of these methods in the case of a synthetic one-dimensional case study concerned with the estimation of hydraulic resistivity conditioned on soft data and hydraulic head measurements. The BME framework processes the physical knowledge contained in Darcys law and generates accurate estimates of hydraulic resistivity across space. The optimal distribution of hard and soft data needed to minimize the associated estimation error at a specified sampling cost is determined.
This work was supported by grants from the National Institute of Environmental Health Sciences (Grant no. 5 P42 ES05948 and P30ES10126), the National Aeronautics and Space Administration (Grant no. 60-00RFQ041), the Army Research Office (Grant no. DAAG55-98-1-0289), and the National Science Foundation under Agreement No. DMS-0112069. 相似文献
20.
Apex shift hyperbolic Radon transform (ASHRT) is an extension of hyperbolic Radon transform (HRT). We have developed a novel sparsity-promoting framework for ASHRT by employing curvelet transform (CT) in the sparse inversion. RT-based seismic data processing can be considered as an optimization problem and a mixed norms inversion, therefore, objective function with CT can promote the sparsity of the transformed domain, which makes the sparse inversion more efficient. Compared with the conventional sparse inversion of ASHRT, the proposed method weights the sparse penalization, which indicates a sparser solution of ASHRT. We use synthetic and field data examples to demonstrate the performance of ASHRT. Compared to the conventional solution, the ours may lead to more accurately reconstructed results and have a better noise immunity. 相似文献