Bayesian Approach to Retrieving a Vertical Ozone Profile from Radiometric Measurement Data

We propose a technique for retrieving a vertical profile of the atmospheric ozone number density from ground-based radiometry data. The technique is based on the Bayesian approach to solving inverse problems and permits one, by allowing for measurement noise and using certain a priori information on the retrieved profile, to construct the probability distribution of the ozone number density in the entire altitude range being monitored. Using the proposed technique, we compare the retrieval results obtained for various (both well-known and suggested for the first time in this paper) methods of approximation and regularization of retrieved profiles. Model examples demonstrate that the proposed technique is capable of retrieving ozone-profile disturbances which accompany the formation of ozone holes.     

An efficient inversion algorithm for atmospheric remote sensing with application to UV limb observations

In this paper we present a retrieval algorithm for atmospheric remote sensing. The algorithm combines Tikhonov regularization and the iteratively regularized Gauss-Newton method and is devoted to the solution of multi-parameter inverse problems with simple bounds on the variables. The basic features of the algorithm: the solution of the bound-constrained minimization problem, the selection of the optimal regularization parameter, the derivation of the global regularization matrix and the characterization of the solution (error analysis) are discussed in detailed. The inversion algorithm is applied to ozone retrieval from SCIAMACHY limb scatter measurements in the ultraviolet spectral range.     

Acoustical inverse problems regularization: Direct definition of filter factors using Signal-to-Noise Ratio

Acoustic imaging aims at localization and characterization of sound sources using microphone arrays. In this paper a new regularization method for acoustic imaging by inverse approach is proposed. The method first relies on the singular value decomposition of the plant matrix and on the projection of the measured data on the corresponding singular vectors. In place of regularization using classical methods such as truncated singular value decomposition and Tikhonov regularization, the proposed method involves the direct definition of the filter factors on the basis of a thresholding operation, defined from the estimated measurement noise. The thresholding operation is achieved using modified filter functions. The originality of the approach is to propose the definition of a filter factor which provides more damping to the singular components dominated by noise than that given by the Tikhonov filter. This has the advantage of potentially simplifying the selection of the best regularization amount in inverse problems. Theoretical results show that this method is comparatively more accurate than Tikhonov regularization and truncated singular value decomposition.     

Sound source reconstruction using inverse boundary element calculations

Whereas standard boundary element calculations focus on the forward problem of computing the radiated acoustic field from a vibrating structure, the aim in this work is to reverse the process, i.e., to determine vibration from acoustic field data. This inverse problem is brought on a form suited for solution by means of an inverse boundary element method. Since the numerical treatment of the inverse source reconstruction results in a discrete ill-posed problem, regularization is imposed to avoid unstable solutions dominated by errors. In the present work the emphasis is on Tikhonov regularization and parameter-choice methods not requiring an error-norm estimate for choosing the right amount of regularization. Several parameter-choice strategies have been presented lately, but it still remains to be seen how well these can handle industrial applications with real measurement data. In the present work it is demonstrated that the L-curve criterion is robust with respect to the errors in a real measurement situation. In particular, it is shown that the L-curve criterion is superior to the more conventional generalized cross-validation (GCV) approach for the present tire noise studies.     

含噪动态光散射测量数据反演中正则化算法与Chahine算法的比较

采用两种常用的粒度反演方法——正则化和Chahine算法,对90nm与250nm单峰分布、50nm与200nm双峰分布、100nm与300nm双峰分布的模拟动态光散射数据,以及105nm、300nm标准颗粒的实测动态光散射数据进行了反演分析.结果表明:噪声水平的高低是影响粒度分布反演准确性的关键因素之一,反演结果的准确性随噪声水平的增加而降低,噪声水平超过某一阈值后,将无法得到有意义的反演结果;不同反演方法具有不同的抗噪能力,在低噪声水平下反演结果无显著差别,随着噪声水平的增加,反演结果表现出很大差异;正则化方法通过正则参数的选择可以有效抑制噪声影响,表现出强于Chahine算法的抗噪能力;与Chahine算法相比,正则化方法不需要假定初始分布,因此,在噪声较大的实验或生产过程中进行颗粒分布测量时,宜采用正则化方法.     

Analytical determination of the regularization parameter in the retrieval of atmospheric vertical profiles

In the retrieval of vertical profiles of atmospheric constituents, regularization methods are frequently used to improve the conditioning of the solution. The regularization reduces the retrieval errors and causes the vertical resolution to deteriorate. One obtains a trade-off by tuning the strength of the regularization by way of a regularization parameter. A new analytical method for determining the regularization parameter is presented. This method is suitable for operational retrievals, for which an unattended procedure is required. The performance of the new method is compared with that of the L-curve method, and the results show that a better trade-off between retrieval errors and vertical resolution is obtained.     

Regularization,Bayesian Inference,and Machine Learning Methods for Inverse Problems

Classical methods for inverse problems are mainly based on regularization theory, in particular those, that are based on optimization of a criterion with two parts: a data-model matching and a regularization term. Different choices for these two terms and a great number of optimization algorithms have been proposed. When these two terms are distance or divergence measures, they can have a Bayesian Maximum A Posteriori (MAP) interpretation where these two terms correspond to the likelihood and prior-probability models, respectively. The Bayesian approach gives more flexibility in choosing these terms and, in particular, the prior term via hierarchical models and hidden variables. However, the Bayesian computations can become very heavy computationally. The machine learning (ML) methods such as classification, clustering, segmentation, and regression, based on neural networks (NN) and particularly convolutional NN, deep NN, physics-informed neural networks, etc. can become helpful to obtain approximate practical solutions to inverse problems. In this tutorial article, particular examples of image denoising, image restoration, and computed-tomography (CT) image reconstruction will illustrate this cooperation between ML and inversion.     

Bayesian approach to recovery of the vertical profile of the stratosphere temperature from the ground-based measurements of solar radiation in millimeter absorption lines of molecular oxygen

We propose a method for recovery of the altitude profile of the stratosphere temperature using ground-based radiometric measurements. The method is based on the Bayesian approach to solving inverse problems and involves determining the probability distribution for the temperature in the entire range of the sounded altitudes. In this case, we use some assumptions on noise in the experimental data and available a priori information about the recovered profile. Using the proposed approach, we compare the recovery results for two methods of approximation and regularization of the recovered profile. It is shown that using the ground-based observations of solar radiation in the oxygen absorption line 27_, it is possible to recover the temperature profile in the altitude range 20–55 km.     

Vacuum polarization in the field of a multidimensional global monopole

An approximate expression for the Euclidean Green function of a massless scalar field in the spacetime of a multidimensional global monopole has been derived. Expressions for the vacuum expectation values 〈?²〉_ren and 〈T ₀₀〉_ren have been derived by the dimensional regularization method. Comparison with the results obtained by alternative regularization methods is made.     

Hierarchical Bayesian inference for Ill-posed problems via variational method

This paper investigates a novel approximate Bayesian inference procedure for numerically solving inverse problems. A hierarchical formulation which determines automatically the regularization parameter and the noise level together with the inverse solution is adopted. The framework is of variational type, and it can deliver the inverse solution and regularization parameter together with their uncertainties calibrated. It approximates the posteriori probability distribution by separable distributions based on Kullback–Leibler divergence. Two approximations are derived within the framework, and some theoretical properties, e.g. variance estimate and consistency, are also provided. Algorithms for their efficient numerical realization are described, and their convergence properties are also discussed. Extensions to nonquadratic regularization/nonlinear forward models are also briefly studied. Numerical results for linear and nonlinear Cauchy-type problems arising in heat conduction with both smooth and nonsmooth solutions are presented for the proposed method, and compared with that by Markov chain Monte Carlo. The results illustrate that the variational method can faithfully capture the posteriori distribution in a computationally efficient way.     

Estimation of Electrical Source Localization Using the Temporal Correlation between Magnetic Field Measurements

It is well known that an inverse problem of biomagnetic imaging is generally ill-posed. Recently several attempts using biocoherence have been reported, but the ill-condition is not considered in these previous methods. If the conventional reconstruction formula is under the ill-condition, the reconstruction problem is also ill-posed and the ill-posedness cannot be improved. In this paper, we propose an approach to estimate the location of current sources under the condition that sources of different positions have little or weak correlation. Our method is based on a well-known regularization method, the truncated SVD method. Compared with the conventional method, our technique allows accurate determination of the location of current sources because it improves the ill-posedness of the inverse problem. We carried out computer simulations using the half-space conducting model and confirmed the effectiveness of our method.     

Atmospheric trace gases profile retrievals using the nonlinear regularized total least squares method

We discuss the regularized total least squares (RTLS) method for the nonlinear overdetermined ill-posed problem of atmospheric trace gas retrieval. The RTLS method is used for the automatic determination of the regularization strength of the ill-conditioned matrix in an iterative process, and a mixed quadratic and cubic line search method is used for the nonlinear retrievals. Additional retrieval metrics, such as the model resolution matrix and the degrees of freedom in the retrieval, which characterize the vertical resolution of the retrievals, are also derived. Simulated retrievals as well as the retrieval from the data of a balloon-based spectroscopic measurement will be discussed. Retrieval results obtained using O₃ and CH₄ as test cases will be presented.     

Constrained regularization methods for ozone profile retrieval from UV/VIS nadir spectrometers

In this paper we present several constrained regularization methods for ozone profile retrieval from UV/VIS nadir sounding instruments such as GOME, SCIAMACHY, OMI and GOME-2. These methods extend the Tikhonov regularization and the iteratively regularized Gauss-Newton method with equality and inequality constraints imposed on the vertical column. It will be shown that this type of information, which is delivered by an independent algorithm like DOAS or GODFIT, significantly improves the accuracy and stability of the profile retrieval.     

遗传算法结合正则化方法反演海洋大气波导

针对正则化方法在解决实际反演问题时既能克服问题的不适定性又可以很大程度上抑制噪声和误差的传播, 本文提出了利用遗传算法结合正则化方法的新算法, 在遗传算法适应度函数中引入正则化项来反演波导参数; 然后对算法进行仿真试验, 结果表明新算法与传统遗传算法相比具有较高的反演精度, 并指出当噪声误差小于10%时, 算法具有较强的“去噪”性能; 最后利用机载雷达在Wallops岛探测的海表面处局部回波资料进行反演试验, 将反演结果与实测大气折射率廓线进行比较, 说明该算法的有效性. 新方法为海洋大气波导反演研究提供了一种新思路.     

Using Regularization Methods for Image Reconstruction of Electrical Capacitance Tomography

Image reconstruction of electrical capacitance tomography (ECT) is a typical inverse problem owing to non‐linearity and ill‐posedness. At the same time, progress towards the solution of this kind of problem has been made at good speed as a branch of mathematics in the past three decades. In this paper, most of the regularization tools developed for the inverse problem are applied to the reconstruction of various simulated images by ECT. The results show promise for ECT image reconstruction by regularization methods. The non‐linearity of the sensitivity matrix seems to be the major problem.     

Particle Size Distribution Retrieval from Elastic Light Scattering Measurements by a Modified Regularization Method

Retrieval of polymeric emulsions Particle Size Distribution (PSD) from Elastic Light Scattering (ELS) measurements requires the solution of an ill conditioned inverse problem. In this paper we compare the performance of classical regularization methods such as Truncated Singular Value Decomposition and Tikhonov's, to a regularization technique due to Pike that includes positivity constraints on the PSD. We also propose a new method that combines Tikhonov's approach with Pike's method. Simulated ELS measurements for two polymeric materials were generated under different experimental noise assumptions. Uncertainties on the polymer refractive index value were also considered. The PSD estimations obtained show that the implementation of the proposed more sophisticated algorithm is convenient for this very ill‐conditioned problem.     

大气廓线参数反演中基于大气可反演指数的正 则化参数选择方法

以大气可反演指数为指标, 采用Nelder-Mead单纯形法求解方法, 提出了一种卫星遥感资料反演大气廓线算法中动态的正则化参数选择方案. 同时针对四条典型区域廓线, 进行了选择试验. 从选择结果看, 不同廓线所对应的最优正则化参数变化幅度非常小, 因此在实际反演过程中, 为提高计算效率, 可近似取正则化参数为固定值. 这为利用卫星遥感资料进行大气廓线反演问题中正则化参数的选择提供了一种可借鉴的方法.     

Design of an optimal wave-vector filter for enhancing the resolution of reconstructed source field by near-field acoustical holography (NAH)

In near-field acoustical holography using the boundary element method, the reconstructed field often diverges due to the presence of small measurement errors. In order to handle this instability in the inverse problem, the reconstruction process should include some form of regularization for enhancing the resolution of source images. The usual method of regularization has been the truncation of wave vectors associated with small singular values, although the determination of an optimal truncation order is difficult. In this article, an iterative inverse solution technique is suggested in which the mean-square error prediction is used. A statistical estimation of the minimum mean-square error between measured pressures and the model solution is required for yielding the optimal number of iterations. The continuous curve of an optimal wave-vector filter is designed, for suppressing the high-order modes that can produce large reconstruction errors. Experimental results from a baffled radiator reveal that the reconstruction errors can be reduced by this form of regularization, by at least 48% compared to those without any regularization. In comparison to results using the optimal truncation method of regularization, the new scheme is shown to give further reductions of truncation error of between 7% and 39%, for the example in this article.