Hidden equivalence and efficiency of gravity data interpretation

Analytical properties of the solutions of the inverse problem of gravimetry are studied in the context of the approximative approach and method of linear integral representations. A new effect of the inheritance of specific analytical properties in the solutions, which is named the effect of hidden equivalence, is revealed and investigated. This effect significantly influences the interpretation informativity in the complex models of a medium, and it should be taken into account in the geological interpretation of gravity data. Hidden equivalence is studied for both linear and nonlinear inverse problems. As an example of a nonlinear problem, the inverse problem of structural gravimetry is analyzed. The correlation between the boundary equations and boundary values of the harmonic functions is demonstrated. Methods are suggested to allow for the effects that occur by expanding the approximative approach for complex conditions (the models of the media with spatially distributed parameters) by the dynamical and criterion principles.     

On the new approach to solving the inverse problem of gravimetry

The results of the studies within the new approach to solving the inverse problem of gravimetry are considered. This approach consists in direct (analytical) continuation of the anomalous gravitational field specified on the Earth’s surface into the lower half-space with the use of the method of discrete approximations. The solution of the problem of analytical continuation is demonstrated by the model example. In the solution of the problem of analytical continuation, the developed algorithms and computer programs were implemented in two program packages which are used both in the model computations and in practice.     

Minimax approach to inverse problems of geophysics

A new approach is suggested for solving the inverse problems that arise in the different fields of applied geophysics (gravity, magnetic, and electrical prospecting, geothermy) and require assessing the spatial region occupied by the anomaly-generating masses in the presence of different types of a priori information. The interpretation which provides the maximum guaranteed proximity of the model field sources to the real perturbing object is treated as the best interpretation. In some fields of science (game theory, economics, operations research), the decision-making principle that lies in minimizing the probable losses which cannot be prevented if the situation develops by the worst-case scenario is referred to as minimax. The minimax criterion of choice is interesting as, instead of being confined to the indirect (and sometimes doubtful) signs of the “optimal” solution, it relies on the actual properties of the information in the results of a particular interpretation. In the hierarchy of the approaches to the solution of the inverse problems of geophysics ordered by the volume and quality of the retrieved information about the sources of the field, the minimax approach should take special place.     

声波方程逆散射反演的近似方法

我们在文献[1]里研究了介质参考波速沿某个方向线性变化时的三维声散射理论,导出了声波方程逆散射反演问题解的解析表达式.考虑到应用时的实际条件,本文根据上述反演方法导出2.5维模型的声波方程逆散射反演的波速扰动计算公式,给出该方法在“高频”近似条件下的波速扰动反演计算公式,从而使我们提出的“参考波速线性变化时的声波方程逆散射反演”理论更接近实际应用条件.本文给出的这些反演公式仍然具有原方法的优点,即不但可以使Born近似的假定在大多数情况下能得以满足,而且可以利用快速Fourier变换来快速实现介质波速扰动的反演成象.     

声波方程逆散射反演的近似方法

我们在文献[1]里研究了介质参考波速沿某个方向线性变化时的三维声散射理论,导出了声波方程逆散射反演问题解的解析表达式.考虑到应用时的实际条件,本文根据上述反演方法导出2.5维模型的声波方程逆散射反演的波速扰动计算公式,给出该方法在“高频”近似条件下的波速扰动反演计算公式,从而使我们提出的“参考波速线性变化时的声波方程逆散射反演”理论更接近实际应用条件.本文给出的这些反演公式仍然具有原方法的优点,即不但可以使Born近似的假定在大多数情况下能得以满足,而且可以利用快速Fourier变换来快速实现介质波速扰动的反演成象.     

Numerical solution of the two-dimensional inverse geomagnetic induction problem

Summary An effective numerical approach to the solution of the two-dimensional inverse geomagnetic induction problem using the linearization method is presented. The numerical realization of the inversion is based on Marquardt's algorithm, for which the solution of the direct problem and the partial derivatives of this solution with respect to the electrical parameters of the medium are computed by the finite difference method. Theoretical models are studied and numerical results are presented.     

Simultaneous determination of aquifer parameters and zone structures with fuzzy c-means clustering and meta-heuristic harmony search algorithm

This study proposes an inverse solution algorithm through which both the aquifer parameters and the zone structure of these parameters can be determined based on a given set of observations on piezometric heads. In the zone structure identification problem fuzzy c-means (FCM) clustering method is used. The association of the zone structure with the transmissivity distribution is accomplished through an optimization model. The meta-heuristic harmony search (HS) algorithm, which is conceptualized using the musical process of searching for a perfect state of harmony, is used as an optimization technique. The optimum parameter zone structure is identified based on three criteria which are the residual error, parameter uncertainty, and structure discrimination. A numerical example given in the literature is solved to demonstrate the performance of the proposed algorithm. Also, a sensitivity analysis is performed to test the performance of the HS algorithm for different sets of solution parameters. Results indicate that the proposed solution algorithm is an effective way in the simultaneous identification of aquifer parameters and their corresponding zone structures.     

Global surface wave inversion with model constraints‡

Practical applications of surface wave inversion demand reliable inverted shear‐wave profiles and a rigorous assessment of the uncertainty associated to the inverted parameters. As a matter of fact, the surface wave inverse problem is severely affected by solution non‐uniqueness: the degree of non‐uniqueness is closely related to the complexity of the observed dispersion pattern and to the experimental inaccuracies in dispersion measurements. Moreover, inversion pitfalls may be connected to specific problems such as inadequate model parametrization and incorrect identification of the surface wave modes. Consequently, it is essential to tune the inversion problem to the specific dataset under examination to avoid unnecessary computations and possible misinterpretations. In the heuristic inversion algorithm presented in this paper, different types of model constraints can be easily introduced to bias constructively the solution towards realistic estimates of the 1D shear‐wave profile. This approach merges the advantages of global inversion, like the extended exploration of the parameter space and a theoretically rigorous assessment of the uncertainties on the inverted parameters, with the practical approach of Lagrange multipliers, which is often used in deterministic inversion, which helps inversion to converge towards models with desired properties (e.g., ‘smooth’ or ‘minimum norm' models). In addition, two different forward kernels can be alternatively selected for direct‐problem computations: either the conventional modal inversion or, instead, the direct minimization of the secular function, which allows the interpreter to avoid mode identification. A rigorous uncertainty assessment of the model parameters is performed by posterior covariance analysis on the accepted solutions and the modal superposition associated to the inverted models is investigated by full‐waveform modelling. This way, the interpreter has several tools to address the more probable sources of inversion pitfalls within the framework of a rigorous and well‐tested global inversion algorithm. The effectiveness and the versatility of this approach, as well as the impact of the interpreter's choices on the final solution and on its posterior uncertainty, are illustrated using both synthetic and real data. In the latter case, the inverted shear velocity profiles are blind compared with borehole data.     

捷联式航空重力测量算法比较

捷联式航空重力测量系统与平台式系统相比具有体积小、重量轻、功耗低等许多优点,近些年来取得了显著的研究进展.本文给出了捷联式航空重力测量的两种算法模型:捷联式惯性标量重力测量(SISG)和旋转不变式标量重力测量(RISG)模型,并对其误差模型作了初步讨论.利用我国首套捷联式航空重力仪SGA-WZ01在某海域的部分试验数据,对两种算法模型进行了比较分析,表明其差值之标准差对于200s的滤波长度小于0.5mGal.同时,利用两组重复测线数据估算了不同滤波尺度下的两种算法的内符合精度,表明SISG算法略优于RISG算法.对于200s和300s的滤波长度,SISG的内符合精度分别为1.06mGal和0.80mGal.     

Development and application of a coupled-process parameter inversion model based on the maximum likelihood estimation method

The coupled flow-mass transport inverse problem is formulated using the maximum likelihood estimation concept. An evolutionary computational algorithm, the genetic algorithm, is applied to search for a global or near-global solution. The resulting inverse model allows for flow and transport parameter estimation, based on inversion of spatial and temporal distributions of head and concentration measurements. Numerical experiments using a subset of the three-dimensional tracer tests conducted at the Columbus, Mississippi site are presented to test the model's ability to identify a wide range of parameters and parametrization schemes. The results indicate that the model can be applied to identify zoned parameters of hydraulic conductivity, geostatistical parameters of the hydraulic conductivity field, angle of hydraulic conductivity anisotropy, solute hydrodynamic dispersivity, and sorption parameters. The identification criterion, or objective function residual, is shown to decrease significantly as the complexity of the hydraulic conductivity parametrization is increased. Predictive modeling using the estimated parameters indicated that the geostatistical hydraulic conductivity distribution scheme produced good agreement between simulated and observed heads and concentrations. The genetic algorithm, while providing apparently robust solutions, is found to be considerably less efficient computationally than a quasi-Newton algorithm.     

A Trade-Off Solution between Model Resolution and Covariance in Surface-Wave Inversion

Regularization is necessary for inversion of ill-posed geophysical problems. Appraisal of inverse models is essential for meaningful interpretation of these models. Because uncertainties are associated with regularization parameters, extra conditions are usually required to determine proper parameters for assessing inverse models. Commonly used techniques for assessment of a geophysical inverse model derived (generally iteratively) from a linear system are based on calculating the model resolution and the model covariance matrices. Because the model resolution and the model covariance matrices of the regularized solutions are controlled by the regularization parameter, direct assessment of inverse models using only the covariance matrix may provide incorrect results. To assess an inverted model, we use the concept of a trade-off between model resolution and covariance to find a proper regularization parameter with singular values calculated in the last iteration. We plot the singular values from large to small to form a singular value plot. A proper regularization parameter is normally the first singular value that approaches zero in the plot. With this regularization parameter, we obtain a trade-off solution between model resolution and model covariance in the vicinity of a regularized solution. The unit covariance matrix can then be used to calculate error bars of the inverse model at a resolution level determined by the regularization parameter. We demonstrate this approach with both synthetic and real surface-wave data.     

Controversial points in the methodology of modern interpretation theory of gravity anomalies and the ways to resolving contradictions

The fact that most of the basic points in the methodology of interpretation theory of gravity anomalies are inadequate to the targeted intention—to extract the maximal volume of reliable information about the sources of the field—has barely been noticed so far. It is established that objectivation of the real possibilities of gravimetry in the particular conditions of interpretation is mainly blocked by the contradiction between the objective existence of a variety of a priori equivalent variants of the interpretation, on one hand, and the desire to describe its structure by a single element, the “optimal” solution of the inverse problem, on the other hand. A complete, to any extent, solution of the problem of reliability of the information retrieved from the gravimetry data can only be obtained in the case if the results of the interpretation contain the information about the properties of the studied object that is consistent with all the admissible solutions of the inverse problem. The informativity of conventional representations of the results of interpretation is assessed in comparison to the informativity of the alternative representations.     

Basin-scale transmissivity and storativity estimation using hydraulic tomography

While tomographic inversion has been successfully applied to laboratory- and field-scale tests, here we address the new issue of scale that arises when extending the method to a basin. Specifically, we apply the hydraulic tomography (HT) concept to jointly interpret four multiwell aquifer tests in a synthetic basin to illustrate the superiority of this approach to a more traditional Theis analysis of the same tests. Transmissivity and storativity are estimated for each element of a regional numerical model using the geostatistically based sequential successive linear estimator (SSLE) inverse solution method. We find that HT inversion is an effective strategy for incorporating data from potentially disparate aquifer tests into a basin-wide aquifer property estimate. The robustness of the SSLE algorithm is investigated by considering the effects of noisy observations, changing the variance of the true aquifer parameters, and supplying incorrect initial and boundary conditions to the inverse model. Ground water flow velocities and total confined storage are used as metrics to compare true and estimated parameter fields; they quantify the effectiveness of HT and SSLE compared to a Theis solution methodology. We discuss alternative software that can be used for implementing tomography inversion.     

基于逆算子估计的AVO反演方法研究

传统反演算法以优化算法为主,而基于逆算子估计的AVO反演算法则利用了直接求逆的思路.算法的关键在于寻找存在逆函数的子域,进而可以在子域内直接求逆,这种解决反问题的思路不同于一般的优化类算法所采用的直接搜索解的方式,具有更高的效率.AVO反演利用了振幅随着偏移距的变化特征,反演的精度受到地震资料质量的影响,通过加入L1范数约束以及合理的初始模型有助于提高反演的稳定性以及准确度.模型测算和实际应用表明,基于逆算子估计的AVO反演方法具有较高的精确程度和可靠性.     

The method of linear integral representations for solving 3D structural inverse problems of potential theory

The algorithm for the solution of a three-dimensional (3D) structural inverse problem in potential theory is described. The algorithm is based on the method of linear integral representations and on the method of extending compacts. The proposed approach is verified on model examples for an anomalous gravity field.     

Robust inversion of vertical electrical sounding data using a multiple reweighted least-squares method

The root cause of the instability problem of the least-squares (LS) solution of the resistivity inverse problem is the ill-conditioning of the sensitivity matrix. To circumvent this problem a new LS approach has been investigated in this paper. At each iteration, the sensitivity matrix is weighted in multiple ways generating a set of systems of linear equations. By solving each system, several candidate models are obtained. As a consequence, the space of models is explored in a more extensive and effective way resulting in a more robust and stable LS approach to solving the resistivity inverse problem. This new approach is called the multiple reweighted LS method (MRLS). The problems encountered when using the L ₁- or L ₂-norm are discussed and the advantages of working with the MRLS method are highlighted. A five-layer earth model which generates an ill-conditioned matrix due to equivalence is used to generate a synthetic data set for the Schlumberger configuration. The data are randomly corrupted by noise and then inverted by using L ₂, L ₁ and the MRLS algorithm. The stabilized solutions, even though blurred, could only be obtained by using a heavy ridge regression parameter in L ₂- and L ₁-norms. On the other hand, the MRLS solution is stable without regression factors and is superior and clearer. For a better appraisal the same initial model was used in all cases. The MRLS algorithm is also demonstrated for a field data set: a stable solution is obtained.     

改进蜂群算法及其在面波频散曲线反演中的应用

应用改进蜂群算法反演面波频散曲线以获得近地表横波速度剖面.蜂群算法属于群智能算法中的一种,灵感来源于蜜蜂群体特定的觅食行为,在该算法的基础上结合粒子群算法中的全局最优解引导思想,同时引入遗传算法中交叉运算操作,即采用基于交叉操作的全局人工蜂群算法对面波频散曲线进行反演研究.改进蜂群算法在继承传统算法精于探索特性的同时,针对其疏于开发的缺陷着重加强了算法对全局的探索能力.使用理论和实测瑞雷波数据,本文研究了改进蜂群算法在推导近地表横波速度分布的有效性和适用性.在反演中,目标函数的收敛性好,改进算法在迭代的过程中能够快速收敛到全局最优;模型参数的概率分布高,即在寻找到全局最优解的同时,能够确保解中每个参数同时达到最优,保证了反演的结果可靠度,使其能有效地应用于瑞雷波频散曲线的反演和解释中.     

A parallel evolutionary strategy based simulation–optimization approach for solving groundwater source identification problems

Groundwater characterization involves the resolution of unknown system characteristics from observation data, and is often classified as an inverse problem. Inverse problems are difficult to solve due to natural ill-posedness and computational intractability. Here we adopt the use of a simulation–optimization approach that couples a numerical pollutant-transport simulation model with evolutionary search algorithms for solution of the inverse problem. In this approach, the numerical transport model is solved iteratively during the evolutionary search. This process can be computationally intensive since several hundreds to thousands of forward model evaluations are typically required for solution. Given the potential computational intractability of such a simulation–optimization approach, parallel computation is employed to ease and enable the solution of such problems. In this paper, several variations of a groundwater source identification problem is examined in terms of solution quality and computational performance. The computational experiments were performed on the TeraGrid cluster available at the National Center for Supercomputing Applications. The results demonstrate the performance of the parallel simulation–optimization approach in terms of solution quality and computational performance.     

基于背向散射脉冲传播时间统计特性的随机界面参数反演研究

研究了利用背向散射脉冲传播时间统计特性反演随机界面参数的问题.首先扩展了Fuks & Godin关于随机界面背向散射脉冲传播时间统计特性理论,给出了当随机界面高度-斜率统计相关情形下,最早返回的两个脉冲传播时间及其时延的概率密度函数（PDF）表达式.进而提出了一种基于背向散射脉冲传播时间统计特性的随机界面参数反演算法：采用遗传算法匹配前两个背向散射脉冲传播时间及其时延的PDF实现随机界面高度—斜率相关系数的绝对值|ρ|和无量纲参数T的反演.最后利用参数的后验概率分析反演结果的不确定性.数值模拟结果表明：该算法可有效反演随机界面的参数,参数|ρ|相对于T反演精度较高;|ρ|和T之间存在较强的参数耦合.与前人认为可忽略高度—斜率相关性（|ρ|）的观点不同：此类反演问题中应同时考虑|ρ|和T.