共查询到20条相似文献，搜索用时 48 毫秒
1.
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
2.
The pioneering work of E. T. Jaynes in the field of Bayesian/Maximum Entropy methods has been successfully explored in a number of disciplines. The principle of maximum entropy (PME) is remarkably powerful and versatile and leads to results which are devoid of spurious structure. Minimum relative entropy (MRE) is a method which has all the important attributes of the maximum-entropy (ME) approach with the advantage that prior information may be easily included. These ‘soft’ prior constraints play a fundamental role in the solution of underdetermined problems. The MRE approach, like ME, has achieved considerable success in the field of spectral analysis where the spectrum is estimated from incomplete autocorrelations. In this paper we apply the MRE philosophy to 1D inverse problems where the model is not necessarily positive, and thus we show that MRE is a general method of tackling linear, underdetermined, inverse problems. We illustrate our discussion with examples which deal with the famous die problem introduced by Jaynes, the question of aliasing, determination of interval velocities from stacking velocities and, finally, the universal problem of band-limited extrapolation. It is found that the MRE solution for the interval velocities, when a uniform prior velocity is assumed, is exactly the Dix formulation which is generally used in the seismic industry. 相似文献
3.
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
4.
To reduce the dependence of EM inversion on the choice of initial model and to obtain the global minimum, we apply transdimensional Bayesian inversion to time-domain airborne electromagnetic data. The transdimensional Bayesian inversion uses the Monte Carlo method to search the model space and yields models that simultaneously satisfy the acceptance probability and data fitting requirements. Finally, we obtain the probability distribution and uncertainty of the model parameters as well as the maximum probability. Because it is difficult to know the height of the transmitting source during flight, we consider a fixed and a variable flight height. Furthermore, we introduce weights into the prior probability density function of the resistivity and adjust the constraint strength in the inversion model by changing the weighing coefficients. This effectively solves the problem of unsatisfactory inversion results in the middle high-resistivity layer. We validate the proposed method by inverting synthetic data with 3% Gaussian noise and field survey data. 相似文献
5.
This paper develops and applies the minimum relative entropy (MRE) theory with spectral power as a random variable for streamflow forecasting. The MRE theory consists of (1) hypothesizing a prior probability distribution for the random variable, (2) determining the spectral power distribution, (3) extending the autocorrelation function, and (4) doing forecasting. The MRE theory was verified using streamflow data from the Mississippi River watershed. The exponential distribution was chosen as a prior probability in applying the MRE theory by evaluating the historical data of the Mississippi River. If no prior information is given, the MRE theory is equivalent to the Burg entropy (BE) theory. The spectral density obtained by the MRE theory led to higher resolution than did the BE theory. The MRE theory did not miss the largest peak at 1/12th frequency, which is the main periodicity of streamflow of the Mississippi River, but the BE theory sometimes did. The MRE theory was found to be capable of forecasting monthly streamflow with a lead time from 12 to 48 months. The coefficient of determination (r 2) between observed and forecasted stream flows was 0.912 for Upper Mississippi River and was 0.855 for Lower Mississippi River. Both MRE and BE theories were generally more reliable and had longer forecasting lead times than the autoregressive (AR) method. The forecasting lead time for MRE and BE could be as long as 48–60 months, while it was less than 48 months for the AR method. However, BE was comparable to MRE only when observations fitted the AR process well. The MRE theory provided more reliable forecasts than did the BE theory, and the advantage of using MRE is more significant for downstream flows with irregular flow patterns or where the periodicity information is limited. The reliability of monthly streamflow forecasting was the highest for MRE, followed by BE followed by AR. 相似文献
6.
Categorical data play an important role in a wide variety of spatial applications, while modeling and predicting this type of statistical variable has proved to be complex in many cases. Among other possible approaches, the Bayesian maximum entropy methodology has been developed and advocated for this goal and has been successfully applied in various spatial prediction problems. This approach aims at building a multivariate probability table from bivariate probability functions used as constraints that need to be fulfilled, in order to compute a posterior conditional distribution that accounts for hard or soft information sources. In this paper, our goal is to generalize further the theoretical results in order to account for a much wider type of information source, such as probability inequalities. We first show how the maximum entropy principle can be implemented efficiently using a linear iterative approximation based on a minimum norm criterion, where the minimum norm solution is obtained at each step from simple matrix operations that converges to the requested maximum entropy solution. Based on this result, we show then how the maximum entropy problem can be related to the more general minimum divergence problem, which might involve equality and inequality constraints and which can be solved based on iterated minimum norm solutions. This allows us to account for a much larger panel of information types, where more qualitative information, such as probability inequalities can be used. When combined with a Bayesian data fusion approach, this approach deals with the case of potentially conflicting information that is available. Although the theoretical results presented in this paper can be applied to any study (spatial or non-spatial) involving categorical data in general, the results are illustrated in a spatial context where the goal is to predict at best the occurrence of cultivated land in Ethiopia based on crowdsourced information. The results emphasize the benefit of the methodology, which integrates conflicting information and provides a spatially exhaustive map of these occurrence classes over the whole country. 相似文献
7.
常规AVO三参数反演是通过Zoeppritz方程的近似公式来建立AVO正演模拟的过程,然而在P波入射角过临界角和弹性参数在纵向上变化剧烈的情况下,Zoeppritz方程近似公式精度有限.针对这种情况,可以使用精确的Zoeppritz方程来构建反演目标函数,由于精确Zoeppritz方程中P波反射系数和弹性参数之间是一种复杂的非线性关系,通常解决途径是利用非线性的优化算法来进行数值计算,但是非线性优化算法的缺点是计算量过大;另外一种途径是利用广义线性反演的方法,通过泰勒一阶展开式将P波反射振幅展开后,用线性关系近似表达非线性关系,经过几次迭代后,在理论上可以达到很高的精度,但是广义线性反演算法的核心部分--Jacobian矩阵由于矩阵条件数过大,往往会造成反演算法的不稳定,其应用范围得到了限制.贝叶斯反演方法是通过引入模型参数的先验分布结合噪声的似然函数,生成模型参数的后验分布,通过求取模型参数的最大后验概率分布来得到模型参数的反演解,由于引入模型参数的先验分布信息,可以有效的降低反演的不适定问题.本文将两种反演算法的思想相结合,利用广义线性反演算法的思想,构建AVO正演模拟的过程来提高大角度地震数据反演的精度,同时结合贝叶斯理论,通过引入模型参数的先验分布信息构建反演目标函数的正则化项,可以有效降低由于Jacob矩阵条件数过大带来的反演不适定问题,该算法假设模型参数服从三变量柯西分布. 相似文献
8.
A key point in the application of multi‐model Bayesian averaging techniques to assess the predictive uncertainty in groundwater modelling applications is the definition of prior model probabilities, which reflect the prior perception about the plausibility of alternative models. In this work the influence of prior knowledge and prior model probabilities on posterior model probabilities, multi‐model predictions, and conceptual model uncertainty estimations is analysed. The sensitivity to prior model probabilities is assessed using an extensive numerical analysis in which the prior probability space of a set of plausible conceptualizations is discretized to obtain a large ensemble of possible combinations of prior model probabilities. Additionally, the value of prior knowledge about alternative models in reducing conceptual model uncertainty is assessed by considering three example knowledge states, expressed as quantitative relations among the alternative models. A constrained maximum entropy approach is used to find the set of prior model probabilities that correspond to the different prior knowledge states. For illustrative purposes, a three‐dimensional hypothetical setup approximated by seven alternative conceptual models is employed. Results show that posterior model probabilities, leading moments of the predictive distributions and estimations of conceptual model uncertainty are very sensitive to prior model probabilities, indicating the relevance of selecting proper prior probabilities. Additionally, including proper prior knowledge improves the predictive performance of the multi‐model approach, expressed by reductions of the multi‐model prediction variances by up to 60% compared with a non‐informative case. However, the ratio between‐model to total variance does not substantially decrease. This suggests that the contribution of conceptual model uncertainty to the total variance cannot be further reduced based only on prior knowledge about the plausibility of alternative models. These results advocate including proper prior knowledge about alternative conceptualizations in combination with extra conditioning data to further reduce conceptual model uncertainty in groundwater modelling predictions. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
9.
高斯混合模型(Gaussian Mixture Model, GMM)可以用来描述储层性质的多峰分布特性,多峰特性主要是由于它们在不同离散变量内的变化而引起的.在高斯混合模型中,高斯分量的权值代表离散变量的概率.然而,基于高斯混合模型的贝叶斯线性反演可能会对某些点的离散变量错误地分类,进而影响连续变量的反演结果,尤其存在强噪声的时候.在本文中,我们考虑了离散变量的空间变化性,并将高斯混合模型与序贯指示模拟(Sequential Indicator Simulation, SIS)相结合来确定离散变量的后验条件权值,形成了结合序贯指示模拟的贝叶斯高斯混合线性反演方法.该方法能够准确地对离散变量进行归类,且具有良好的抗噪性.通过模型试算,我们证明了这种方法的可行性,并在实际资料中取得了较好的结果. 相似文献
10.
Cauchy priori distribution-based Bayesian AVO reflectivity inversion may lead to sparse estimates that are sensitive to large reflectivities. For the inversion, the computation of the covariance matrix and regularized terms requires prior estimation of model parameters, which makes the iterative inversion weakly nonlinear. At the same time, the relations among the model parameters are assumed linear. Furthermore, the reflectivities, the results of the inversion, or the elastic parameters with cumulative error recovered by integrating reflectivities are not well suited for detecting hydrocarbons and fuids. In contrast, in Bayesian linear AVO inversion, the elastic parameters can be directly extracted from prestack seismic data without linear assumptions for the model parameters. Considering the advantages of the abovementioned methods, the Bayesian AVO reflectivity inversion process is modified and Cauchy distribution is explored as a prior probability distribution and the time-variant covariance is also considered. Finally, we propose a new method for the weakly nonlinear AVO waveform inversion. Furthermore, the linear assumptions are abandoned and elastic parameters, such as P-wave velocity, S-wave velocity, and density, can be directly recovered from seismic data especially for interfaces with large reflectivities. Numerical analysis demonstrates that all the elastic parameters can be estimated from prestack seismic data even when the signal-to-noise ratio of the seismic data is low. 相似文献
11.
Stacy L. Tantum Quan Zhu Peter A. Torrione Leslie M. Collins 《Stochastic Environmental Research and Risk Assessment (SERRA)》2009,23(2):155-167
Buried unexploded ordnance (UXO) continues to be a difficult remediation problem from both a sensing and a discrimination
point of view. Modern approaches to both the sensing and discrimination problems utilize high bandwidth electromagnetic induction
(EMI) sensors to collect geo-referenced data which is then inverted, or fit, using a forward model in order to obtain features
that can be directly interpreted using the physics associated with electromagnetic induction-based sensing. These features
are then used in a variety of classification architectures. One aspect of this process that has captured recent interest is
that uncertainty in the positions at which data was collected can degrade the inversion performance and thus the subsequent
classification. Several mechanisms to address this issue have been explored that range from filtering and prediction of actual
positions to exploiting Bayesian approaches for uncertainty mitigation. In the Bayesian approach, a statistical model of the
position errors is used as a prior for integrating over the uncertainty in the inversion process. In this study, we demonstrate
that errors in the statistical priors used in this process can negatively impact subsequent classification performance, thus
highlighting the need for an accurate statistical model for the position errors. Next, we propose a mechanism by which to
obtain such models. Specifically, we utilize a Goff–Jordan rough surface model and simulate the sensor data collection system
motion over the simulated ground or ocean surfaces to calculate errors and generate statistical models. Our results suggest
that this approach can be used to develop the statistical models necessary for mitigating uncertain position information. 相似文献
12.
Rafi Baker George Christakos 《Stochastic Environmental Research and Risk Assessment (SERRA)》2007,21(4):435-446
The well-known “Maximum Entropy Formalism” offers a powerful framework for deriving probability density functions given a
relevant knowledge base and an adequate prior. The majority of results based on this approach have been derived assuming a
flat uninformative prior, but this assumption is to a large extent arbitrary (any one-to-one transformation of the random
variable will change the flat uninformative prior into some non-constant function). In a companion paper we introduced the
notion of a natural reference point for dimensional physical variables, and used this notion to derive a class of physical
priors that are form-invariant to changes in the system of dimensional units. The present paper studies effects of these priors
on the probability density functions derived using the maximum entropy formalism. Analysis of real data shows that when the
maximum entropy formalism uses the physical prior it yields significantly better results than when it is based on the commonly
used flat uninformative prior. This improvement reflects the significance of the incorporating additional information (contained
in physical priors), which is ignored when flat priors are used in the standard form of the maximum entropy formalism. A potentially
serious limitation of the maximum entropy formalism is the assumption that sample moments are available. This is not the case
in many macroscopic real-world problems, where the knowledge base available is a finite sample rather than population moments.
As a result, the maximum entropy formalism generates a family of “nested models” parameterized by the unknown values of the
population parameters. In this work we combine this formalism with a model selection scheme based on Akaike’s information
criterion to derive the maximum entropy model that is most consistent with the available sample. This combination establishes
a general inference framework of wide applicability in scientific/engineering problems. 相似文献
13.
Non-stationarity in statistical properties of the subsurface is often ignored. In a classical linear Bayesian inversion setting of seismic data, the prior distribution of physical parameters is often assumed to be stationary. Here we propose a new method of handling non-stationarity in the variance of physical parameters in seismic data. We propose to infer the model variance prior to inversion using maximum likelihood estimators in a sliding window approach. A traditional, and a localized shrinkage estimator is defined for inferring the prior model variance. The estimators are assessed in a synthetic base case with heterogeneous variance of the acoustic impedance in a zero-offset seismic cross section. Subsequently, this data is inverted for acoustic impedance using a non-stationary model set up with the inferred variances. Results indicate that prediction as well as posterior resolution is greatly improved using the non-stationary model compared with a common prior model with stationary variance. The localized shrinkage predictor is shown to be slightly more robust than the traditional estimator in terms of amplitude differences in the variance of acoustic impedance and size of local neighbourhood. Finally, we apply the methodology to a real data set from the North Sea basin. Inversion results show a more realistic posterior model than using a conventional approach with stationary variance. 相似文献
14.
Surface wave methods have received much attention due to their efficient, flexible and convenient characteristics. However, there are still critical issues regarding a key step in surface wave inversion. In most existing methods, the number of layers is assumed to be known prior to the process of inversion. However, improper assignment of this parameter leads to erroneous inversion results. A Bayesian nonparametric method for Rayleigh wave inversion is proposed herein to address this problem. In this method, each model class represents a particular number of layers with unknown S-wave velocity and thickness of each layer. As a result, determination of the number of layers is equivalent to selection of the most applicable model class. Regarding each model class, the optimization search of S-wave velocity and thickness of each layer is implemented by using a genetic algorithm. Then, each model class is assessed in view of its efficiency under the Bayesian framework and the most efficient class is selected. Simulated and actual examples verify that the proposed Bayesian nonparametric approach is reliable and efficient for Rayleigh wave inversion, especially for its capability to determine the number of layers.
相似文献15.
We consider a Bayesian model for inversion of observed amplitude variation with offset data into lithology/fluid classes, and study in particular how the choice of prior distribution for the lithology/fluid classes influences the inversion results. Two distinct prior distributions are considered, a simple manually specified Markov random field prior with a first-order neighbourhood and a Markov mesh model with a much larger neighbourhood estimated from a training image. They are chosen to model both horizontal connectivity and vertical thickness distribution of the lithology/fluid classes, and are compared on an offshore clastic oil reservoir in the North Sea. We combine both priors with the same linearized Gaussian likelihood function based on a convolved linearized Zoeppritz relation and estimate properties of the resulting two posterior distributions by simulating from these distributions with the Metropolis–Hastings algorithm. The influence of the prior on the marginal posterior probabilities for the lithology/fluid classes is clearly observable, but modest. The importance of the prior on the connectivity properties in the posterior realizations, however, is much stronger. The larger neighbourhood of the Markov mesh prior enables it to identify and model connectivity and curvature much better than what can be done by the first-order neighbourhood Markov random field prior. As a result, we conclude that the posterior realizations based on the Markov mesh prior appear with much higher lateral connectivity, which is geologically plausible. 相似文献
16.
Huijuan Cui Vijay P. Singh 《Stochastic Environmental Research and Risk Assessment (SERRA)》2016,30(6):1545-1563
This paper develops a minimum relative entropy theory with frequency as a random variable, called MREF henceforth, for streamflow forecasting. The MREF theory consists of three main components: (1) determination of spectral density (2) determination of parameters by cepstrum analysis, and (3) extension of autocorrelation function. MREF is robust at determining the main periodicity, and provides higher resolution spectral density. The theory is evaluated using monthly streamflow observed at 20 stations in the Mississippi River basin, where forecasted monthly streamflows show the coefficient of determination (r 2) of 0.876, which is slightly higher in the Upper Mississippi (r 2 = 0.932) than in the Lower Mississippi (r 2 = 0.806). Comparison of different priors shows that the prior with the background spectral density with a peak at 1/12 frequency provides satisfactory accuracy, and can be used to forecast monthly streamflow with limited information. Four different entropy theories are compared, and it is found that the minimum relative entropy theory has an advantage over maximum entropy (ME) for both spectral estimation and streamflow forecasting, if additional information as a prior is given. Besides, MREF is found to be more convenient to estimate parameters with cepstrum analysis than minimum relative entropy with spectral power as random variable (MRES), and less information is needed to assume the prior. In general, the reliability of monthly streamflow forecasting from the highest to the lowest is for MREF, MRES, configuration entropy (CE), Burg entropy (BE), and then autoregressive method (AR), respectively. 相似文献
17.
为了提高AVO(amplitude versus offset)反演结果的精度和横向连续性,本文提出了一种新的AVO反演约束方法,该方法结合贝叶斯原理和卡尔曼滤波算法实现了对反演参数纵向和横向的同时约束.文章首先结合反演参数的纵向贝叶斯先验概率约束和反演参数的横向连续性假设建立了与卡尔曼滤波算法对应的AVO反演系统的数学模型,然后将该数学模型代入卡尔曼滤波算法框架,利用卡尔曼滤波算法实现了双向约束AVO反演.二维模型测试和实际数据测试结果表明,相对于单纯的纵向贝叶斯先验概率约束,双向约束能更准确地刻画参数的横向变化,得到更准确、横向连续性更好的反演结果. 相似文献
18.
P. Bogaert 《Stochastic Environmental Research and Risk Assessment (SERRA)》2002,16(6):425-448
Being a non-linear method based on a rigorous formalism and an efficient processing of various information sources, the Bayesian
maximum entropy (BME) approach has proven to be a very powerful method in the context of continuous spatial random fields,
providing much more satisfactory estimates than those obtained from traditional linear geostatistics (i.e., the various kriging
techniques). This paper aims at presenting an extension of the BME formalism in the context of categorical spatial random
fields. In the first part of the paper, the indicator kriging and cokriging methods are briefly presented and discussed. A
special emphasis is put on their inherent limitations, both from the theoretical and practical point of view. The second part
aims at presenting the theoretical developments of the BME approach for the case of categorical variables. The three-stage
procedure is explained and the formulations for obtaining prior joint distributions and computing posterior conditional distributions
are given for various typical cases. The last part of the paper consists in a simulation study for assessing the performance
of BME over the traditional indicator (co)kriging techniques. The results of these simulations highlight the theoretical limitations
of the indicator approach (negative probability estimates, probability distributions that do not sum up to one, etc.) as well
as the much better performance of the BME approach. Estimates are very close to the theoretical conditional probabilities,
that can be computed according to the stated simulation hypotheses. 相似文献
19.
Viacheslav V. Spichak 《Acta Geophysica》2012,60(3):942-958
Feasibility of recovering the magma chamber’s parameters by 3D Bayesian statistical inversion of magnetotelluric data is estimated
for the simplified conductivity model of the Vesuvios volcano. The results indicate that in the lack of prior information
and data, the most efficient approach may consist in successive estimation of the geometry and the depth of the anomaly followed
by estimation of the electric conductivity distribution in it. The horizontal boundaries of the target could be outlined by
the high gradients of the impedance determinant phase pseudosections determined by the upward analytical continuation of the
anomalous electromagnetic fields from the relief surface to the artificial reference plane located above the summit of the
volcano. The vertical boundaries and the target extension as well as the electric conductivity could be estimated successively
by means of 3D Bayesian statistical inversion of the collected magnetotelluric data carried out in the domain delimited by
the estimated horizontal boundaries. 相似文献
20.
Conventional joint PP—PS inversion is based on approximations of the Zoeppritz equations and assumes constant VP/VS; therefore, the inversion precision and stability cannot satisfy current exploration requirements. We propose a joint PP—PS inversion method based on the exact Zoeppritz equations that combines Bayesian statistics and generalized linear inversion. A forward model based on the exact Zoeppritz equations is built to minimize the error of the approximations in the large-angle data, the prior distribution of the model parameters is added as a regularization item to decrease the ill-posed nature of the inversion, low-frequency constraints are introduced to stabilize the low-frequency data and improve robustness, and a fast algorithm is used to solve the objective function while minimizing the computational load. The proposed method has superior antinoising properties and well reproduces real data. 相似文献