共查询到20条相似文献，搜索用时 31 毫秒
1.
A. J. W. DUIJNDAM 《Geophysical Prospecting》1988,36(8):899-918
A parameter estimation or inversion procedure is incomplete without an analysis of uncertainties in the results. In the fundamental approach of Bayesian parameter estimation, discussed in Part I of this paper, the a posteriori probability density function (pdf) is the solution to the inverse problem. It is the product of the a priori pdf, containing a priori information on the parameters, and the likelihood function, which represents the information from the data. The maximum of the a posteriori pdf is usually taken as a point estimate of the parameters. The shape of this pdf, however, gives the full picture of uncertainty in the parameters. Uncertainty analysis is strictly a problem of information reduction. This can be achieved in several stages. Standard deviations can be computed as overall uncertainty measures of the parameters, when the shape of the a posteriori pdf is not too far from Gaussian. Covariance and related matrices give more detailed information. An eigenvalue or principle component analysis allows the inspection of essential linear combinations of the parameters. The relative contributions of a priori information and data to the solution can be elegantly studied. Results in this paper are especially worked out for the non-linear Gaussian case. Comparisons with other approaches are given. The procedures are illustrated with a simple two-parameter inverse problem. 相似文献
2.
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
3.
The normal-to-shear weakness ratio is commonly used as a fracture fluid indicator, but it depends not only on the fluid types but also on the fracture intensity and internal architecture. Amplitude variation with offset and azimuth is commonly used to perform the fluid identification and fracture characterization in fractured porous rocks. We demonstrate a direct inversion approach to utilize the observable azimuthal data to estimate the decoupled fluid (fluid/porosity term) and fracture (normal and shear weaknesses) parameters instead of the calculation of normal-to-shear weakness ratio to help reduce the uncertainties in fracture characterization and fluid identification of a gas-saturated porous medium permeated by a single set of parallel vertical fractures. Based on the anisotropic poroelasticity and perturbation theory, we first derive a linearized amplitude versus offset and azimuth approximation using the scattering function to decouple the fluid indicator and fracture parameters. Incorporating Bayes formula and convolution theory, we propose a feasible direct inversion approach in a Bayesian framework to obtain the direct estimations of model parameters, in which Cauchy and Gaussian distribution are used for the a priori information of model parameters and the likelihood function, respectively. We finally use the non-linear iteratively reweighted least squares to solve the maximum a posteriori solutions of model parameters. The synthetic examples containing a moderate noise demonstrate the feasibility of the proposed approach, and the real data illustrates the stabilities of estimated fluid indicator and dry fracture parameters in gas-saturated fractured porous rocks. 相似文献
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5.
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
6.
A. Ghorbani C. Camerlynck N. Florsch P. Cosenza A. Revil 《Geophysical Prospecting》2007,55(4):589-605
The inversion of induced‐polarization parameters is important in the characterization of the frequency electrical response of porous rocks. A Bayesian approach is developed to invert these parameters assuming the electrical response is described by a Cole–Cole model in the time or frequency domain. We show that the Bayesian approach provides a better analysis of the uncertainty associated with the parameters of the Cole–Cole model compared with more conventional methods based on the minimization of a cost function using the least‐squares criterion. This is due to the strong non‐linearity of the inverse problem and non‐uniqueness of the solution in the time domain. The Bayesian approach consists of propagating the information provided by the measurements through the model and combining this information with a priori knowledge of the data. Our analysis demonstrates that the uncertainty in estimating the Cole–Cole model parameters from induced‐polarization data is much higher for measurements performed in the time domain than in the frequency domain. Our conclusion is that it is very difficult, if not impossible, to retrieve the correct value of the Cole–Cole parameters from time‐domain induced‐polarization data using standard least‐squares methods. In contrast, the Cole–Cole parameters can be more correctly inverted in the frequency domain. These results are also valid for other models describing the induced‐polarization spectral response, such as the Cole–Davidson or power law models. 相似文献
7.
根据贝叶斯理论给出了对含噪声地球物理数据处理的具体流程和方法,主要包括似然函数估计和后验概率计算.我们将数据向量的概念扩展为数据向量的集合,通过引入数据空间内的信赖度,把数据噪声转移到模型空间的概率密度函数上,即获得了反映数据本身的不确定性的似然函数.该方法由于避免了处理阶段数据空间内的人工干预,因而可以保证模型空间中的概率密度单纯反映数据噪声,具有信息保真度高、保留可行解的优点.为了得到加入先验信息的后验分布,本文提出了使用加权矩阵的概率分析法,该方法在模型空间直接引入地质信息,对噪声引起的反演多解性有很强的约束效果.整个处理流程均以大地电磁反演为例进行了展示. 相似文献
8.
1D joint multi‐offset inversion of time‐domain marine controlled source electromagnetic data 
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下载免费PDF全文 The accurate estimation of sub‐seafloor resistivity features from marine controlled source electromagnetic data using inverse modelling is hindered due to the limitations of the inversion routines. The most commonly used one‐dimensional inversion techniques for resolving subsurface resistivity structures are gradient‐based methods, namely Occam and Marquardt. The first approach relies on the smoothness of the model and is recommended when there are no sharp resistivity boundaries. The Marquardt routine is relevant for many electromagnetic applications with sharp resistivity contrasts but subject to the appropriate choice of a starting model. In this paper, we explore the ability of different 1D inversion schemes to derive sub‐seafloor resistivity structures from time domain marine controlled source electromagnetic data measured along an 8‐km‐long profile in the German North Sea. Seismic reflection data reveal a dipping shallow amplitude anomaly that was the target of the controleld source electromagnetic survey. We tested four inversion schemes to find suitable starting models for the final Marquardt inversion. In this respect, as a first scenario, Occam inversion results are considered a starting model for the subsequent Marquardt inversion (Occam–Marquardt). As a second scenario, we employ a global method called Differential Evolution Adaptive Metropolis and sequentially incorporate it with Marquardt inversion. The third approach corresponds to Marquardt inversion introducing lateral constraints. Finally, we include the lateral constraints in Differential Evolution Adaptive Metropolis optimization, and the results are sequentially utilized by Marquardt inversion. Occam–Marquardt may provide accurate estimation of the subsurface features, but it is dependent on the appropriate conversion of different multi‐layered Occam model to an acceptable starting model for Marquardt inversion, which is not straightforward. Employing parameter spaces, the Differential Evolution Adaptive Metropolis approach can be pertinent to determine Marquardt a priori information; nevertheless, the uncertainties in Differential Evolution Adaptive Metropolis optimization will introduce some inaccuracies in Marquardt inversion results. Laterally constrained Marquardt may be promising to resolve sub‐seafloor features, but it is not stable if there are significant lateral changes of the sub‐seafloor structure due to the dependence of the method to the starting model. Including the lateral constraints in Differential Evolution Adaptive Metropolis approach allows for faster convergence of the routine with consistent results, furnishing more accurate estimation of a priori models for the subsequent Marquardt inversion. 相似文献
9.
Allan D. Woodbury Tadeusz J. Ulrych 《Stochastic Environmental Research and Risk Assessment (SERRA)》1998,12(5):317-358
The similarity between maximum entropy (MaxEnt) and minimum relative entropy (MRE) allows recent advances in probabilistic
inversion to obviate some of the shortcomings in the former method. The purpose of this paper is to review and extend the
theory and practice of minimum relative entropy. In this regard, we illustrate important philosophies on inversion and the
similarly and differences between maximum entropy, minimum relative entropy, classical smallest model (SVD) and Bayesian solutions
for inverse problems. MaxEnt is applicable when we are determining a function that can be regarded as a probability distribution.
The approach can be extended to the case of the general linear problem and is interpreted as the model which fits all the
constraints and is the one model which has the greatest multiplicity or “spreadout” that can be realized in the greatest number
of ways. The MRE solution to the inverse problem differs from the maximum entropy viewpoint as noted above. The relative entropy
formulation provides the advantage of allowing for non-positive models, a prior bias in the estimated pdf and `hard' bounds
if desired. We outline how MRE can be used as a measure of resolution in linear inversion and show that MRE provides us with
a method to explore the limits of model space. The Bayesian methodology readily lends itself to the problem of updating prior
probabilities based on uncertain field measurements, and whose truth follows from the theorems of total and compound probabilities.
In the Bayesian approach information is complete and Bayes' theorem gives a unique posterior pdf. In comparing the results
of the classical, MaxEnt, MRE and Bayesian approaches we notice that the approaches produce different results. In␣comparing
MaxEnt with MRE for Jayne's die problem we see excellent comparisons between the results. We compare MaxEnt, smallest model
and MRE approaches for the density distribution of an equivalent spherically-symmetric earth and for the contaminant plume-source
problem. Theoretical comparisons between MRE and Bayesian solutions for the case of the linear model and Gaussian priors may
show different results. The Bayesian expected-value solution approaches that of MRE and that of the smallest model as the
prior distribution becomes uniform, but the Bayesian maximum aposteriori (MAP) solution may not exist for an underdetermined
case with a uniform prior. 相似文献
10.
11.
The similarity between maximum entropy (MaxEnt) and minimum relative entropy (MRE) allows recent advances in probabilistic
inversion to obviate some of the shortcomings in the former method. The purpose of this paper is to review and extend the
theory and practice of minimum relative entropy. In this regard, we illustrate important philosophies on inversion and the
similarly and differences between maximum entropy, minimum relative entropy, classical smallest model (SVD) and Bayesian solutions
for inverse problems. MaxEnt is applicable when we are determining a function that can be regarded as a probability distribution.
The approach can be extended to the case of the general linear problem and is interpreted as the model which fits all the
constraints and is the one model which has the greatest multiplicity or “spreadout” that can be realized in the greatest number
of ways. The MRE solution to the inverse problem differs from the maximum entropy viewpoint as noted above. The relative entropy
formulation provides the advantage of allowing for non-positive models, a prior bias in the estimated pdf and `hard' bounds
if desired. We outline how MRE can be used as a measure of resolution in linear inversion and show that MRE provides us with
a method to explore the limits of model space. The Bayesian methodology readily lends itself to the problem of updating prior
probabilities based on uncertain field measurements, and whose truth follows from the theorems of total and compound probabilities.
In the Bayesian approach information is complete and Bayes' theorem gives a unique posterior pdf. In comparing the results
of the classical, MaxEnt, MRE and Bayesian approaches we notice that the approaches produce different results. In␣comparing
MaxEnt with MRE for Jayne's die problem we see excellent comparisons between the results. We compare MaxEnt, smallest model
and MRE approaches for the density distribution of an equivalent spherically-symmetric earth and for the contaminant plume-source
problem. Theoretical comparisons between MRE and Bayesian solutions for the case of the linear model and Gaussian priors may
show different results. The Bayesian expected-value solution approaches that of MRE and that of the smallest model as the
prior distribution becomes uniform, but the Bayesian maximum aposteriori (MAP) solution may not exist for an underdetermined
case with a uniform prior. 相似文献
12.
Prestack amplitude versus angle inversion for Young's modulus and Poisson's ratio based on the exact Zoeppritz equations 下载免费PDF全文
下载免费PDF全文 Elastic parameters such as Young's modulus, Poisson's ratio, and density are very important characteristic parameters that are required to properly characterise shale gas reservoir rock brittleness, evaluate gas characteristics of reservoirs, and directly interpret lithology and oil‐bearing properties. Therefore, it is significant to obtain accurate information of these elastic parameters. Conventionally, they are indirectly calculated by the rock physics method or estimated by approximate formula inversion. The cumulative errors caused by the indirect calculation and low calculation accuracy of the approximate Zoeppritz equations make accurate estimation of Young's modulus, Poisson's ratio, and density difficult in the conventional method. In this paper, based on the assumption of isotropy, we perform several substitutions to convert the Zoeppritz equations from the classical form to a new form containing the chosen elastic constants of Young's modulus, Poisson's ratio, and density. The inversion objective function is then constructed by utilising Bayesian theory. Meanwhile, the Cauchy distribution is introduced as a priori information. We then combine the idea of generalised linear inversion with an iterative reweighed least squares algorithm in order to solve the problem. Finally, we obtain the iterative updating formula of the three elastic parameters and achieve the direct inversion of these elastic parameters based on the exact Zoeppritz equations. Both synthetic and field data examples show that the new method is not only able to obtain the two elastic parameters of Young's modulus and Poisson's ratio stably and reasonably from prestack seismic data but also able to provide an accurate estimation of density information, which demonstrates the feasibility and effectiveness of the proposed method. The proposed method offers an efficient seismic method to identify a “sweet spot” within a shale gas reservoir. 相似文献
13.
Satellite remote sensing deals with a complex system coupling atmosphere and surface. Any physical model with reasonable precision needs several to tens of parameters. Without a priori knowledge of these parameters, Proposition 3 of Verstraete et al. requires the number of independent observations to be greater than the number of unknown parameters. This requirement can hardly be satisfied even in the coming EOS era. As Tarantola pointed out, the inversion problems in geoscience are always underdetermined in some sense. In order to make good use of every kind of a priori knowledge for effectively extracting information from remote sensing observations, the right question to set is as follows:Given an imperfect model and a certain amount ofa priori information on model parameters, in which sense should one modify thea priori information, given the actual observation with noise?A priori knowledge of physical parameters can be presented in different ways such as physical limits, global statistical means and variance fora certain landcover type, or previous statistics and temporal variation of a specific target. When sucha priori knowledge can be expressed as joint probability density. Bayessian theorem can be used in the inversion to obtain posterior probability densities of parameters using newly acquired observations. There is no prerequirement on how many independent observations must be made, and the knowledge gained merely depends on the information content of the new observations. Some specific problems about knowledge accumulation and renewal are also discussed. 相似文献
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15.
A two‐and‐half dimensional model‐based inversion algorithm for the reconstruction of geometry and conductivity of unknown regions using marine controlled‐source electromagnetic (CSEM) data is presented. In the model‐based inversion, the inversion domain is described by the so‐called regional conductivity model and both geometry and material parameters associated with this model are reconstructed in the inversion process. This method has the advantage of using a priori information such as the background conductivity distribution, structural information extracted from seismic and/or gravity measurements, and/or inversion results a priori derived from a pixel‐based inversion method. By incorporating this a priori information, the number of unknown parameters to be retrieved becomes significantly reduced. The inversion method is the regularized Gauss‐Newton minimization scheme. The robustness of the inversion is enhanced by adopting nonlinear constraints and applying a quadratic line search algorithm to the optimization process. We also introduce the adjoint formulation to calculate the Jacobian matrix with respect to the geometrical parameters. The model‐based inversion method is validated by using several numerical examples including the inversion of the Troll field data. These results show that the model‐based inversion method can quantitatively reconstruct the shapes and conductivities of reservoirs. 相似文献
16.
岩相信息能够反映储层岩性及流体特征,在地震储层预测中具有重要作用.常规方法主要利用与岩相信息关系密切的弹性参数定性或定量地转化为岩相信息.在实际应用中,弹性参数的获取主要基于叠前地震反演技术.而不同弹性参数的叠前地震反演精度间存在着差异,势必影响岩相的整体预测精度.本文提出对弹性参数进行加权统计来预测岩相.首先,基于贝叶斯理论,引入权重系数来调节弹性参数信息的采用量,构建出最终的目标反演函数;其次,考虑到勘探初期缺少明确的测井岩相信息,提出利用高斯混合分布函数来自动估算岩相先验概率;最后,根据输入弹性参数的取值,计算每类岩相对应的后验概率密度,将目标反演函数取最大后验概率密度时对应的岩相类别作为最终预测的岩相.新方法旨在减少弹性参数精度间的精度差异对岩相预测结果的影响,以期提高地震岩相的预测精度.模型与实际资料测试均表明该方法可行、有效且预测精度较高. 相似文献
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18.
Hamid Salahshoor Alexey Lyubushin Elham Shabani Javad Kazemian 《Journal of Seismology》2018,22(6):1515-1527
Bayesian probability theory is an appropriate and useful method for estimating parameters in seismic hazard analysis. The analysis in Bayesian approaches is based on a posterior belief, also their special ability is to take into account the uncertainty of parameters in probabilistic relations and a priori knowledge. In this study, we benefited the Bayesian approach in order to estimate maximum values of peak ground acceleration (Amax) also quantiles of the relevant probabilistic distributions are figured out in a desired future interval time in Iran. The main assumptions are Poissonian character of the seismic events flow and properties of the Gutenberg-Richter distribution law. The map of maximum possible values of Amax and also map of 90% quantile of distribution of maximum values of Amax on a future interval time 100 years is presented. According to the results, the maximum value of the Amax is estimated for Bandar Abbas as 0.3g and the minimum one is attributed to Esfahan as 0.03g. Finally, the estimated values in Bayesian approach are compared with what was presented applying probabilistic seismic hazard (PSH) methods based on the conventional Cornel (1968) method. The distribution function of Amax for future time intervals of 100 and 475 years are calculated for confidence limit of probability level of 90%. 相似文献
19.
On the geostatistical approach to the inverse problem 总被引：5，自引：0，他引：5
Peter K. Kitanidis 《Advances in water resources》1996,19(6):333-342
The geostatistical approach to the inverse problem is discussed with emphasis on the importance of structural analysis. Although the geostatistical approach is occasionally misconstrued as mere cokriging, in fact it consists of two steps: estimation of statistical parameters (“structural analysis”) followed by estimation of the distributed parameter conditional on the observations (“cokriging” or “weighted least squares”). It is argued that in inverse problems, which are algebraically undetermined, the challenge is not so much to reproduce the data as to select an algorithm with the prospect of giving good estimates where there are no observations. The essence of the geostatistical approach is that instead of adjusting a grid-dependent and potentially large number of block conductivities (or other distributed parameters), a small number of structural parameters are fitted to the data. Once this fitting is accomplished, the estimation of block conductivities ensues in a predetermined fashion without fitting of additional parameters. Also, the methodology is compared with a straightforward maximum a posteriori probability estimation method. It is shown that the fundamental differences between the two approaches are: (a) they use different principles to separate the estimation of covariance parameters from the estimation of the spatial variable; (b) the method for covariance parameter estimation in the geostatistical approach produces statistically unbiased estimates of the parameters that are not strongly dependent on the discretization, while the other method is biased and its bias becomes worse by refining the discretization into zones with different conductivity. 相似文献
20.
2D data modelling by electrical resistivity tomography for complex subsurface geology 总被引：3，自引：1，他引：3
A new tool for two‐dimensional apparent‐resistivity data modelling and inversion is presented. The study is developed according to the idea that the best way to deal with ill‐posedness of geoelectrical inverse problems lies in constructing algorithms which allow a flexible control of the physical and mathematical elements involved in the resolution. The forward problem is solved through a finite‐difference algorithm, whose main features are a versatile user‐defined discretization of the domain and a new approach to the solution of the inverse Fourier transform. The inversion procedure is based on an iterative smoothness‐constrained least‐squares algorithm. As mentioned, the code is constructed to ensure flexibility in resolution. This is first achieved by starting the inversion from an arbitrarily defined model. In our approach, a Jacobian matrix is calculated at each iteration, using a generalization of Cohn's network sensitivity theorem. Another versatile feature is the issue of introducing a priori information about the solution. Regions of the domain can be constrained to vary between two limits (the lower and upper bounds) by using inequality constraints. A second possibility is to include the starting model in the objective function used to determine an improved estimate of the unknown parameters and to constrain the solution to the above model. Furthermore, the possibility either of defining a discretization of the domain that exactly fits the underground structures or of refining the mesh of the grid certainly leads to more accurate solutions. Control on the mathematical elements in the inversion algorithm is also allowed. The smoothness matrix can be modified in order to penalize roughness in any one direction. An empirical way of assigning the regularization parameter (damping) is defined, but the user can also decide to assign it manually at each iteration. An appropriate tool was constructed with the purpose of handling the inversion results, for example to correct reconstructed models and to check the effects of such changes on the calculated apparent resistivity. Tests on synthetic and real data, in particular in handling indeterminate cases, show that the flexible approach is a good way to build a detailed picture of the prospected area. 相似文献