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.     

Using informative priors in facies inversion: The case of C-ISR method

Inverse problems involving the characterization of hydraulic properties of groundwater flow systems by conditioning on observations of the state variables are mathematically ill-posed because they have multiple solutions and are sensitive to small changes in the data. In the framework of McMC methods for nonlinear optimization and under an iterative spatial resampling transition kernel, we present an algorithm for narrowing the prior and thus producing improved proposal realizations. To achieve this goal, we cosimulate the facies distribution conditionally to facies observations and normal scores transformed hydrologic response measurements, assuming a linear coregionalization model. The approach works by creating an importance sampling effect that steers the process to selected areas of the prior. The effectiveness of our approach is demonstrated by an example application on a synthetic underdetermined inverse problem in aquifer characterization.     

Solving nonlinear water management models using a combined genetic algorithm and linear programming approach

Gradient-based nonlinear programming (NLP) methods can solve problems with smooth nonlinear objectives and constraints. However, in large and highly nonlinear models, these algorithms can fail to find feasible solutions, or converge to local solutions which are not global. Evolutionary search procedures in general, and genetic algorithms (GAs) specifically, are less susceptible to the presence of local solutions. However, they often exhibit slow convergence, especially when there are many variables, and have problems finding feasible solutions in constrained problems with “narrow” feasible regions. In this paper, we describe strategies for solving large nonlinear water resources models management, which combine GAs with linear programming. The key idea is to identify a set of complicating variables in the model which, when fixed, render the problem linear in the remaining variables. The complicating variables are then varied by a GA. This GA&LP approach is applied to two nonlinear models: a reservoir operation model with nonlinear hydropower generation equations and nonlinear reservoir topologic equations, and a long-term dynamic river basin planning model with a large number of nonlinear relationships. For smaller instances of the reservoir model, the CONOPT2 nonlinear solver is more accurate and faster, but for larger instances, the GA&LP approach finds solutions with significantly better objective values. The multiperiod river basin model is much too large to be solved in its entirety. The complicating variables are chosen here so that, when they are fixed, each period's model is linear, and these models can be solved sequentially. This approach allows sufficient model detail to be retained so that long-term sustainability issues can be explored.     

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.     

反演模型分辨率的估算方法

反演问题的时空间分辨率或称时空分辨长度是评估模型精细程度的重要参数,决定了该模型应用的范围和价值,但是分辨长度估算却是比反演更复杂和麻烦的数学问题。除了层析成像中广泛利用理论模型恢复试验定性提取空间分辨长度外,通过求解分辨率矩阵可定量获得分辨长度。通过矩阵操作给出的分辨率矩阵包括三类：直接分辨率矩阵、正则化分辨率矩阵和混合分辨率矩阵。这三类矩阵包含了反演本身不同侧面的信息,因此在一个反演应用中,同时提供这三类分辨率矩阵可更全面地评估反演模型分辨率分布。最近An（2012）提出了从大量随机理论模型及其解中统计出分辨率矩阵的方法。这种分辨率矩阵是从模拟真实反演实验的输入和输出模型中通过反演得到的,因此这种分辨率矩阵更能反映整个反演所涉及到的更多因素和过程;同时由于这种分辨率矩阵计算过程无需进行矩阵操作且不依赖于具体正演和反演方法,因此可以被应用于更普遍的反演问题。实际应用证明统计分辨率分析方法适用于对二维和三维层析成像反演模型进行分辨率分析。     

Uncertainty analysis and probabilistic segmentation of electrical resistivity images: the 2D inverse problem

In this paper, we present the uncertainty analysis of the 2D electrical tomography inverse problem using model reduction and performing the sampling via an explorative member of the Particle Swarm Optimization family, called the Regressive‐Regressive Particle Swarm Optimization. The procedure begins with a local inversion to find a good resistivity model located in the nonlinear equivalence region of the set of plausible solutions. The dimension of this geophysical model is then reduced using spectral decomposition, and the uncertainty space is explored via Particle Swarm Optimization. Using this approach, we show that it is possible to sample the uncertainty space of the electrical tomography inverse problem. We illustrate this methodology with the application to a synthetic and a real dataset coming from a karstic geological set‐up. By computing the uncertainty of the inverse solution, it is possible to perform the segmentation of the resistivity images issued from inversion. This segmentation is based on the set of equivalent models that have been sampled, and makes it possible to answer geophysical questions in a probabilistic way, performing risk analysis.     

电导率随深度分层线性变化的大地电磁响应

本文在电导率随深度分层线性变化的地球模型上,研究了大地电磁的正演问题.由Airy微分方程导出了电场和磁场水平分量的解、地面阻抗的递推表达式和视电阻率的计算公式;并证明了分层线性模型与均匀层状模型一样,也存在高阻薄层的H等值性和低阻薄层的S等值性;总结了电导率随深度的变化率α,对大地电磁测深曲线的影响规律.     

Inverse problems of gravimetry as retrieval of reliable information under uncertainty

The guaranteed approach to the solution of inverse problems of gravimetry, which is fundamentally different from the common solution, is presented. Instead of providing single (optimum) estimates of the model parameters, whose quality is, generally, random, the interpretation in the suggested approach yields that volume of reliable information on disturbing objects, which is contained in the field measurements, together with the a priori constraints. The method of solving the inverse problem of gravimetry is developed within this approach, where, in contrast to the conventional approach, both the geometrical parameters of the geological bodies and the densities of their composing rocks are treated as unknowns. The generalized assembly algorithm suggested by V.N. Strakhov is proposed as a basic working tool to implement this approach. The results of testing this algorithm and the guaranteed approach itself on the model and practical examples are discussed.     

Non-uniqueness in surface-wave inversion and consequences on seismic site response analyses

Surface-wave tests are based on the solution of an inverse problem for shear-wave velocity profile identification from the experimentally measured dispersion curve. The main criticisms for these testing methodologies are related to the inverse problem solution and arise from the possible equivalence of different shear-wave velocity profiles. In this paper, some implications of solution non-uniqueness for seismic response studies are investigated using both numerical simulations and experimental data. A Monte Carlo approach for the inversion problem has been used to obtain a set of equivalent shear-wave velocity models. This selection is based on a statistical test which takes into account both data uncertainty and model parameterization. This set of solutions (i.e., soil profiles) is then used to evaluate the seismic response with a conventional one-dimensional analysis. It is shown that equivalent profiles with respect to surface-wave testing are equivalent also with respect to site amplification, thus countering the criticism related to inversion uncertainty for the engineering use of surface-wave tests.     

Solutions of the inherent problem of the equivalence in direct current resistivity and electromagnetic methods through global optimization and joint inversion by successive refinement of model space

The problem of equivalence in direct current (DC) resistivity and electromagnetic methods for a thin resistive and conducting layer is well‐known. Attempts have been made in the past to resolve this problem through joint inversion. However, equivalence still remains an unresolved problem. In the present study, an effort is made to reduce non‐uniqueness due to equivalence using global optimization and joint inversion by successive refinement of the model space. A number of solutions derived for DC resistivity data using very fast simulated annealing global inversion that fits the observations equally well, follow the equivalence principle and show a definite trend. For a thin conductive layer, the quotient between resistivity and thickness is constant, while for a resistive one, the product between these magnitudes is constant. Three approaches to obtain very fast simulated annealing solutions are tested. In the first one, layer resistivities and thicknesses are optimized in a linear domain. In the second, layer resistivities are optimized in the logarithmic domain and thicknesses in the linear domain. Lastly, both layer resistivities and thicknesses are optimized in the logarithmic domain. Only model data from the mean models, corresponding to very fast simulated annealing solutions obtained for approach three, always fit the observations. The mean model defined by multiple very fast simulated annealing solutions shows extremely large uncertainty (almost 100%) in the final solution after inversion of individual DC resistivity or electromagnetic (EM) data sets. Uncertainty associated with the intermediate resistive and conducting layers after global optimization and joint inversion is still large. In order to reduce the large uncertainty associated with the intermediate layer, global optimization is performed over several iterations by reducing and redefining the search limits of model parameters according to the uncertainty in the solution. The new minimum and maximum limits are obtained from the uncertainty in the previous iteration. Though the misfit error reduces in the solution after successive refinement of the model space in individual inversion, it is observed that the mean model drifts away from the actual model. However, successive refinement of the model space using global optimization and joint inversion reduces uncertainty to a very low level in 4–5 iterations. This approach works very well in resolving the problem of equivalence for resistive as well as for conducting layers. The efficacy of the approach has been demonstrated using DC resistivity and EM data, however, it can be applied to any geophysical data to solve the inherent ambiguities in the interpretations.     

Closed-form solutions for unsaturated flow under variable flux boundary conditions

It is shown that from any solution of the linear diffusion equation, we may construct a solution of a realistic form of the Richards equation for unsaturated flow. Compared to the usual direct linearization method, our inverse approach involves a quite different sequence of transformations. This opens the possibility of exact solutions with a wider variety of continuously varying flux boundary conditions. Closed-form solutions are presented for two examples. In these, the varying water flux boundary conditions resemble (i) the passage of a peaking storm and (ii) the continuous opening of a valve preceding a steady water supply. Unlike earlier more systematic approaches to this problem, our method does not require the numerical solution of an integral equation.     

Wasser — Sphinx unter den Flüssigkeiten

Water is a rather peculiar substance with respect to a wide variety of static and dynamic properties. It can be considered as a highly organized complex liquid with a complicated structure determined by extended hydrogen bondings. Mixture models and continuum models have been developed to explain the structural features of liquid water, both of them being approximative and hitherto not at all convincing. Aqueous solutions can be discussed in terms of structure breaking and structure making solutes. The most surprising and fascinating properties of water and its outstanding importance require further detailed and unusual investigations.     

复杂介质小波多尺度井间地震层析成像方法研究

复杂介质井间地震层析成像是一个很复杂的非线性反 演问题，常规的线性化反演方法无法得到好的解. 采用基于图形的弯曲射线追踪方法， 并将小波多尺度思想引入到井间层析成像，提出了小波多尺度井间地震层析成像方法，很好 地解决了非线性成像的难题，提高了图像的质量和分辨率. 物理模型实验结果表明， 该方法适合于复杂介质成像，并具有良好的实用性和效果.     

Inverse problems of gravity prospecting as a decision-making problem under uncertainty and risk

A new class of algorithms for solving the inverse problems of gravity prospecting is considered. The best interpretation is selected from the set Q of the admissible versions by the optimality criteria that are borrowed from the solution-making theory and adapted for the geophysical problems. The concept of retrieving the information about the sources of gravity anomalies, which treats the result of the interpretation as a set of locally optimal solutions of the inverse problem but not as a single globally optimal solution is discussed. The locally optimal solutions of the inverse problem are sort of singularity points of set Q. They are preferable to the other admissible solutions by a certain criterion formulated in terms of the geologically important information about the anomalous bodies. The admissible versions of the interpretation of the gravimetry data that meet the criteria of the decision-making theory are the primary candidates for the singularity points. The results of the numerical calculations are presented. The set of the admissible solutions from which the locally optimal versions of interpretation are selected is formed by the modifications of the assembly method developed by V.N. Strakhov.     

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.     

Nonlinear aspects of phreatic water filtration

Horizontal gravity filtration of groundwater in soil is considered. Under Boussinesq approximation, the problem is reduced to a one-dimensional nonlinear parabolic equation in phreatic water level. The problem of linearizing the original equation is discussed. The comparison of gravity-filtration problem solutions in the nonlinear and linearized formulations shows considerable discrepancies to exist between the solutions, especially, for boundary problems with mixed boundary conditions, when the value of the function is not fixed on the right boundary. An analytical solution is obtained for steady-state flow from a water body into the soil with subsequent leakage into underlying beds. Two regimes are shown to exist: one with an infinite exponential tail, and another in the form of a finite groundwater mound. A new approach is proposed to the linearization problem—quasilinearization with the use of the Burgers equation.     

Coupling basin- and site-scale inverse models of the Española aquifer

Large-scale models are frequently used to estimate fluxes to small-scale models. The uncertainty associated with these flux estimates, however, is rarely addressed. We present a case study from the Espa?ola Basin, northern New Mexico, where we use a basin-scale model coupled with a high-resolution, nested site-scale model. Both models are three-dimensional and are analyzed by codes FEHM and PEST. Using constrained nonlinear optimization, we examine the effect of parameter uncertainty in the basin-scale model on the nonlinear confidence limits of predicted fluxes to the site-scale model. We find that some of the fluxes are very well constrained, while for others there is fairly large uncertainty. Site-scale transport simulation results, however, are relatively insensitive to the estimated uncertainty in the fluxes. We also compare parameter estimates obtained by the basin- and site-scale inverse models. Differences in the model grid resolution (scale of parameter estimation) result in differing delineation of hydrostratigraphic units, so the two models produce different estimates for some units. The effect is similar to the observed scale effect in medium properties owing to differences in tested volume. More important, estimation uncertainty of model parameters is quite different at the two scales. Overall, the basin inverse model resulted in significantly lower estimates of uncertainty, because of the larger calibration dataset available. This suggests that the basin-scale model contributes not only important boundary condition information but also improved parameter identification for some units. Our results demonstrate that caution is warranted when applying parameter estimates inferred from a large-scale model to small-scale simulations, and vice versa.     

A parallel method of solving nonlinear inverse problems in gravimetry and magnetometry

A parallel iterative method of solving nonlinear inverse logarithmic potential problems and three-dimensional problems in gravimetry and magnetometry is proposed.     

A double-porosity model for a fractured aquifer with non-Darcian flow in fractures

Abstract

Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model.     

From Bayes to Tarantola: New insights to understand uncertainty in inverse problems

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.