Series Expansion-Based Genetic Inversion of Wireline Logging Data

An evolutionary approach is applied to solve the nonlinear well logging inverse problem. In the framework of the proposed interval inversion method, nuclear, sonic, and laterolog resistivity data measured at an arbitrary depth interval are jointly inverted, where the depth variation of porosity, water saturation, and shale volume is expanded into series using Legendre polynomials as basis functions. In the interval inversion procedure, the series expansion coefficients are estimated by using an adaptive float-encoded genetic algorithm. Since the solution of the inverse problem using traditional linear optimization tools highly depends on the selection of the initial model, a heuristic search is necessary to reduce the initial model dependence of the interval inversion procedure. The genetic inversion strategy used in interval inversion seeks the global extreme of the objective function and provides an estimate of the vertical distribution of petrophysical parameters, even starting the inversion procedure from extremely high distances from the optimum. For a faster computational process, after a couple of thousand generations, the genetic algorithm is replaced by some linear optimization steps. The added advantage of using the Marquardt algorithm is the possibility to characterize the accuracy of the series expansion coefficients and derived petrophysical properties. A Hungarian oil field example demonstrates the feasibility and stability of the improved interval inversion method. As a significance, the genetic inversion method does not require prior knowledge or strong restrictions on the values of petrophysical properties and gives highly reliable estimation results practically independent of the initial model and core information.

     

Inverse solution to the wave equation using the Laguerre transform

An inverse solution to the 1D wave equation is obtained using the spectral Laguerre transform to find the distribution of wave velocities at some point of the medium. The problem is solved as optimization in which the function of Laguerre harmonics is minimized by the conjugate gradient or Newton’s algorithms.Reported are velocities of a wave defined by a stepwise constant function. The accuracy of the inverse solution for the Laguerre harmonics is investigated against the approximation accuracy in the boundary problem. The accuracy and efficiency of the Laguerre method are compared to those in the Fourier method.     

位场广义线性综合反演系统的建立

以BG理论为基础,首先在重磁异常线性反演中将该理论与吉洪诺夫正则化方法相结合求解地下密度源(或磁源)分布及质心(或磁质心)位置;在重磁非线性反演中结合应用正则化方法和马奎特思想给出一种确定地下密度(或磁性)界面的稳定迭代算法:正则—马奎特法其次研究一种综合多和磁两种异常联合反演既是磁界面也是密度界面的方法。最后,以判断重磁异常是否同源的对应分析为先导,将上述方法综合于一起建立了重磁异常广义线性人机交互式综合反演系统。实际地区的应用已取得较好效果。     

三维电阻率成像新方法及应用

三维电阻率成像法包括近似三维反演因子和综合子空间解释器,三维反演因子用于正向线性电阻率近似解译,综合子空间法是共轭梯度法的变换,是求解大型最优化问题的有效方法.三维成像是线性反演问题,不需正演模拟或敏感度修改.在非线性三维反演中,三维电阻率成像技术可直接获取三维电阻率的分布信息,或提供中间过程的模式修改.野外资料验证表明,三维成像技术可提供电阻率空间的分布信息.     

Optimal lower bound of passive earth pressure using finite elements and non-linear programming

The present study pertains to the finding of the lower bound solution, formulating it as a non-linear programming problem using the generalized method developed by Lysmer with certain variations to incorporate the non-linear no-yield condition constraints directly in the analysis. The method considers the family of plane stress fields having the property that all stresses vary linearly within each triangular element of some mesh which covers the soil mass under study. For this type of stress field it is possible to express all equilibrium conditions as a set of linear constraints and the no-yield as a set of non-linear constraints. The boundary condition constraints may be of linear equality or inequality type. By expressing some of the design variables in terms of the remaining variables the linear equality constraints are implicity satisfied. Such a technique minimizes the complexity of the problem by eliminating the equality constraints and reduces the dimensionality of the problem, saving much, computational effort. The optimal lower bound is isolated by formulating it as a non-linear programming (NLP) problem subjected to both linear and non-linear inequality constraints. The sequential unconstrained minimization technique using the extended penalty function method as suggested by Kavlie has been used to isolate the optimal lower bound. The method has successfully been applied to the passive earth pressure and bearing capacity problem. Numerical results are obtained and compared with Lysmer's solution to show the effectiveness of the present approach.     

A Frequency Matching Method: Solving Inverse Problems by Use of Geologically Realistic Prior Information

The frequency matching method defines a closed form expression for a complex prior that quantifies the higher order statistics of a proposed solution model to an inverse problem. While existing solution methods to inverse problems are capable of sampling the solution space while taking into account arbitrarily complex a priori information defined by sample algorithms, it is not possible to directly compute the maximum a posteriori model, as the prior probability of a solution model cannot be expressed. We demonstrate how the frequency matching method enables us to compute the maximum a posteriori solution model to an inverse problem by using a priori information based on multiple point statistics learned from training images. We demonstrate the applicability of the suggested method on a synthetic tomographic crosshole inverse problem.     

Isotherms for the sorption of zinc and copper onto kaolinite: comparison of various error functions

In this research, the equilibrium sorption of Zn(II) and Cu(II) by kaolinite was explained using the Freundlich, Langmuir and Redlich–Peterson isotherms, via both linear and non-linear regression analyses. In the case of non-linear regression method, the best-fitting model was evaluated using six different error functions, namely coefficient of determination (r ²), hybrid fractional error function (HYBRID), Marquardt’s percent standard deviation (MPSD), average relative error (ARE), sum of the errors squared (SSE) and sum of the absolute errors (EABS). The examination of error estimation methods showed that the Langmuir model provides the best fit for the experimental equilibrium data for both linear and non-linear regression analyses. The SSE function was found to be a better option to minimize the error distribution between the experimental equilibrium data and predicted two-parameter isotherms. In the case of three-parameter isotherm, HYBRID was found to be the best error function to minimize the error distribution structure between experimental equilibrium data and theoretical isotherms. Non-linear method was found to be more appropriate method for estimating the isotherm parameters.     

高密度电阻率法比值参数基于阻尼最小二乘反演

三电位电极系中装置的探测精度和数据处理方法的研究一直是地球物理工作者研究的一个热点,同时也存在很大的争议。利用正演模拟结果讨论了岩溶地区几种可能存在的地质条件下三种装置的探测精度,发现β装置和γ装置的探测效果明显优于α装置;然后利用正演计算得到的数据合成比值参数(T),对合成数据T进行最小二乘反演,发现T值反演结果和视电阻率反演结果一致,并在噪声影响较大的区域,T值反演结果优于T值等值线图,可作为判断异常体特征的一个依据,也可验证视电阻率的反演结果,弥补由于噪声对某种单一装置探测效果的影响。以义马某地的水文地质勘察为例,T值最小二乘反演结果表明,在含水低阻区域T值也呈现小值异常,且显示的异常体边界准确,结构特征明显。利用T值反演对数据处理具有重要的意义,应予以重视。     

Evaluation of monotonic and non-monotonic dissipation test results

The dissipation test evaluation method presented here employs: (i) a point-symmetric, linear, coupled consolidation model with a new boundary condition and a new time factor, (ii) an automatic and mathematically precise, non-linear, inverse problem solver which includes some reliability testing methods and a kind of regularization technique, and (iii) a method to identify the initial condition.The evaluation method is equally applicable to pore water pressure data displaying monotonic or non-monotonic time variation, with the only difference that the initial condition is identified differently during the inverse problem solution.The necessary testing time is very short provided that the pore water pressure is measured well above the tip. This is attributed to the ability of the one-dimensional, linear consolidation model to account for the geometry and the unloading effects that occur when steady penetration of the static cone penetrometer ceases.     

A method of time-dependent analysis using elastic solutions for nonlinear materials

The formulation of viscoelastic solutions from elastic equations using the ‘correspondence principle’ and an inverse Laplace transform has been discussed extensively in the literature. Because this method has been developed, many time-dependent solutions can be obtained from closed form elastic solutions and conditions have been delineated in which the ‘quasi-elastic’ approximation of the viscoelastic solution is within acceptable tolerance. This communication shows the feasibility of the application of these methods to formulate approximate nonlinear viscoelastic solutions with nonlinear stress-strain materials, and for want of a specific nonlinear model to demonstrate this, the hyperbolic model was selected. The ‘power law’ is used to model the relaxation modulus of the viscoelastic materials. There are five related development that are discussed here using a simple numerical example to illustrate each of them and they are: (1) a linear elastic solution, (2) a linear viscoelastic solution, (3) a nonlinear elastic solution, (4) a nonlinear viscoelastic solution and finally, (5) a ‘regression’ approximation of the nonlinear viscoelastic solution which is suggested by the series form of the elastic solution. All of these are related to one another and each provides an acceptably accurate solution of the problem it addresses. The latter is of particular practical interest since it can be used to provide answers to problems involving nonlinear viscoelastic materials while requiring only very small calculation times. The problem used as an example is the calculation of the displacement of a circular hole in an infinite plate made of a material with a nonlinear time-dependent stress-strain relationship. The nonlinear elastic form of the solution was developed by matching results from nonlinear finite element analysis.     

Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem

Probabilistic domain decomposition is proposed as a novel method for solving the two-dimensional Maxwell’s equations as used in the magnetotelluric method. The domain is split into non-overlapping sub-domains and the solution on the sub-domain boundaries is obtained by evaluating the stochastic form of the exact solution of Maxwell’s equations by a Monte-Carlo approach. These sub-domains can be naturally chosen by splitting the sub-surface domain into regions of constant (or at least continuous) conductivity. The solution over each sub-domain is obtained by solving Maxwell’s equations in the strong form. The sub-domain solver used for this purpose is a meshless method resting on radial basis function-based finite differences. The method is demonstrated by solving a number of classical magnetotelluric problems, including the quarter-space problem, the block-in-half-space problem and the triangle-in-half-space problem.     

激发极化法测深曲线反演方法及对比分析

在电阻率反演的基础上,运用等效电阻率进行激发极化法测深曲线的反演.通过对线性反演法、精确形式法、非线性反演法三种极化率反演方法的计算与分析,编制程序对水平层状模型进行反演,对比其在耗费时间和反演效果等方面的差别,认为非线性反演是理论上最好的方法,而在实际生产中建议采用线性反演方法.     

利用一维正则化反演进行大地电磁测深数据拟二维反演解释

大地电磁测深的反演问题是不适定的,其反演结果不稳定,且具有非唯一性。通过在目标函数中采用正则化方法,可以使得不适定反演问题具有稳定的反演结果,并改善解的稳定性和非唯一性问题。为了提高野外大地电磁测深数据的处理效率和初步解释的精度,提出了大地电磁测深数据的一维正则化反演进行拟二维反演解释方法。这里所述的大地电磁测深一维反演解释,与以往的解释方法不同,其思路首先用Bostick反演的深度来控制层参数,使反演计算的模型参数仅存在电阻率;最后采用阻尼高斯-牛顿算法进行反演计算,并将Bostick反演结果作为反演计算的初始模型。通过模型试算,结果表明其处理速度快、解释直观,对野外大地电磁测深数据进行初步反演解释是可行的。     

On Mathematical Aspects of a Combined Inversion of Gravity and Normal Mode Variations by a Spline Method

This paper provides a brief overview of two linear inverse problems concerned with the determination of the Earth’s interior: inverse gravimetry and normal mode tomography. Moreover, a vector spline method is proposed for a combined solution of both problems. This method uses localised basis functions, which are based on reproducing kernels, and is related to approaches which have been successfully applied to the inverse gravimetric problem and the seismic traveltime tomography separately.     

Correcting the distortion of apparent resistivity pseudosection produced by 3D effect of the embankment geometry

Two-dimensional electrical resistivity tomography data acquired on the crest of the embankment are distorted by the 3D effect of the embankment geometry, reservoir water level and abutment. The distortion affects seriously the final solution of the 2D inverse problem. By comparing the apparent resistivity pseudosections from a 3D and 2D electrical resistivity model of the embankment, the distortion degrees of the apparent resistivity pseudosections along the axis on the crest were estimated for the cases of reservoir and which does not contain water. The obtained results indicate that the distortion degree acquired in the case of a reservoir that contains water was much less than that in the case of the reservoir that does not contain water. In the case of reservoir that contains water, the apparent resistivity pseudosections of the P–P and ED–ED arrays had the largest distortion degree and of D–D, W–S and P–D arrays had the smallest distortion degree. In the case of the reservoir that does not contain water, the apparent resistivity pseudosection of P–P array had the smallest distortion among all arrays. Through modeling investigation, a correction process to reduce the distortion of the apparent resistivity pseudosection was proposed. The correction process was tested in the embankment model, and two field works were carried out in the To Lich River in Hanoi and Khuan Cat embankment in Lang Son province, Vietnam. It is possible to bring the distortion of the apparent resistivity pseudosection down to 2.8–13.9%, depending on the type of electrode arrays and the type of reservoirs, containing or does not contain water. The distortion correction of the apparent resistivity pseudosection is strongly recommended before doing the 2D inverse interpretation.     

硫酸侵蚀下水泥土的电阻率特性研究

通过室内试验模拟了水泥土试块受不同浓度硫酸（H2SO4）溶液侵蚀的过程,采用电阻率法研究了电阻率与各参数、无侧限抗压强度之间的关系。结果表明,受H2SO4侵蚀后,水泥土电阻率、孔隙水电阻率、水泥土抗压强度均随着溶液浓度的增加而减小,而水泥土的结构因子和重度均随着溶液浓度的增加而增加,水泥土抗压强度与水泥土电阻率成正比。在试验的基础上,考虑侵蚀因素对水泥土电阻率的影响,通过对Archie法则进行修正,得到了H2SO4侵蚀下水泥土电阻率模型。通过对无侧限抗压强度试验结果的分析,并结合试块表观变化,提出了利用电阻率来评价侵蚀程度的方法。     

Joint hydrogeophysical inversion: state estimation for seawater intrusion models in 3D

Seawater intrusion (SWI) is a complex process, where 3D modeling is often necessary in order to monitor and manage the affected aquifers. Here, we present a synthetic study to test a joint hydrogeophysical inversion approach aimed at solving the inverse problem of estimating initial and current saltwater distribution. First, we use a 3D groundwater model for variable density flow based on discretized flow and solute mass balance equations. In addition to the groundwater model, a 3D geophysical model was developed for direct current resistivity imaging and inversion. The objective function of the coupled problem consists of data misfit and regularization terms as well as a coupling term that relates groundwater and geophysical states. We present a novel approach to solve the inverse problem using an alternating direction method of multipliers (ADMM) to minimize this coupled objective function. ADMM enables to treat the groundwater and geophysical part separately and thus use the existing software with minor changes. To further reduce the computational cost, the sensitivities are derived analytically for the discretized system of equations, which allows us to efficiently compute the gradients in the minimization procedure. The method was tested on different synthetic scenarios with groundwater and geophysical data represented by solute mass fraction data and direct current resistivity data. With the ADMM approach, we were able to obtain better estimates for the solute distribution compared to just considering each data separately, solving the problem with a simple coupled approach or by a direct substitution of the coupling constraint.     

L1-L2范数联合约束稀疏脉冲反演的应用

稀疏脉冲反演实际上就是利用反褶积原理, 从带有噪声的地震道中计算出具有稀疏分布特征的反射系数的振幅和时间.稀疏脉冲反演是非线性优化问题, 通常都是把非线性优化问题转化为线性优化问题, 然后用线性优化算法求解.以范数约束为基础, 提出L1-L2范数联合约束求解的方法.该算法采用了目前国际流行的内点算法, 与传统的优化算法相比, 这种算法具有精度高, 速度快的优点.通过研究人工模型和南海某油田实际数据, 表明该算法的计算结果和测井记录吻合好, 提高了地震分辨率, 目的层段分辨率优于8m.利用反射系数剖面预测的储层厚度和开发井吻合很好, 大大地降低了资源量计算的风险和油田开发的不确定性.