利用共轭梯度法的电阻率三维反演研究

利用共轭梯度（ＣＧ）迭代技术,实现了直流电阻率测量数据的三维最小构造反演。 首先,运用共轭梯度迭代算法解反问题的线性方程组,只需求Ｊａｃｏｂｉａｎ矩阵Ｇ与任一向量ｘ的 乘积Ｇｘ及Ｇ^Ｔ与任何一向量ｙ的乘积Ｇ^Ｔｙ,再引入Ｇ的Ｒｏｄｉ算法,则Ｇｘ及Ｇ^Ｔｙ均可在每次反 演迭代中的一次正演计算后一并求得．因此,每次反演迭代仅需一次正演计算,大大加快了 计算速度;而且避免了直接求 Ｇ以及 Ｇ^ＴＧ的逆矩阵,也避免了存储 Ｇ和 Ｇ^ＴＧ所需庞大的存储 量。另外,由于反演参数太多,求模型光滑约束的最小构造反演能够有效地消除多余构造信 息,得到可靠的反演结果。将这３种方法和技术融合于三维反演中,取得了好的反演效果．为 改进传统最小构造反演收敛慢的问题,还提出了一种新的反演迭代技术,仅需１０次左右甚至 更少迭代即可收敛。     

电阻率层析成像的原理与初步应用

电阻率层析成像的基本原理是根据研究区域边界处的电压和电流值确定区域内部电阻率的分布,它同声波和电磁波层析成像的原理有着本质的不同。根据电阻率层析成像的数学原理,从积分变换、线性迭代反演和非线性迭代反演３种基本的反演途径系统地分析和归纳了当前几种主要的电阻率图像重建方法,并讨论了雅可比偏导数矩阵的求解。已开展的电阻率层析成像的野外观测试验,给出了实测的井间和地表两种观测系统的成像结果,分析了技术应用中的特点与限制     

一种块迭代的快速代数重建算法

常用的计算机层析成像的重建算法可分为:变换重建法、代数重建法和其它算法几大类.变换重建算法中最为常用的为"卷积反投影”算法,该算法重建速度较快,重建效果较好.但该算法也存在一些不足,它通常要求完全的、等间隔的平行采样数据.在天文、物探、地震成像等领域采样数据通常是不完全的和非等间隔的.代数重建算法简单,适用于不同格式的采样数据,对不完全数据亦可重建图像.还可以结合一些先验知识进行求解.可应用于工业检测、物探成像、天文成像等领域.其缺点主要是计算量大,收敛速度慢,难以重建大的图像. 计算机层析成像的重建问题,可离散化为线性方程组AF=P的求解问题,其中P是被采集的投影数据向量,A是投影系数矩阵,F是图像基函数.假设有M个投影数据,且重建的图像有N×N像素,则A为M行、N×N列矩阵.即使重建较小的图像,系数矩阵也是很大的,需要M×N×N个浮点数.A为大型稀疏矩阵,其非零元的个数约为2×M×N个浮点数.因此,想用代数重建算法重建中等或大的图像,必须寻找一种快速的投影系数矩阵实时计算方法. 其次,代数重建算法中迭代的收敛速度也是要解决的主要难点.初值的选取对收敛速度影响是很大的.如果选取的初值与原物体的密度分布较接近,迭代就容易满足收敛条件.传统的代数重建算法中,初值常选为零和某种平均值.在每次循环中都对N×N个图像值,进行逐线或逐点迭代修正.因此,需要大量计算时间,且收敛速度甚慢. 本文提出一种基于分块迭代的快速代数重建算法,其基本思想是采用对图像逐级分块,通过迭代使图像逐步细化,最终逼近于重建的图像.算法的实现过程如下:1.将重建图像按不同级别分块;2.根据块的大小,抽取投影数据,实时计算投影系数矩阵的非零元;3.对给定级图像块赋值,根据投影系数矩阵的非零元和阀值确定对哪些图像块的值进行修正:4.对给定级的图像块经一次循环迭代修正后,判断前后两次的图像是否满足该级迭代结束条件,满足时进入下一级块的迭代;最后一级块迭代满足条件后,块迭代结束.在每一级块迭代过程中,我们设计了求解系数矩阵非零元的快速计算方法,使得所需的系数矩阵的非零元可实时计算,而不必存贮. 利用X射线工业CT实采数据,我们对块迭代代数重建算法的测试结果表明:该方法重建速度快,重建图像精度高、伪影轻,并有较高的密度分辨率和空间分辨率.     

有限角度CT图像重建算法综述

本文主要介绍了处理有限角度CT图像重建的思路和方法。有限角度CT图像重建属于不完全数据重建范畴,由于不满足数据完备性条件,因此不能精确重建。其处理方法大致可以分为两类:基于变换的迭代-解析重建算法和基于级数展开的迭代-代数/统计重建算法。同时,有限角度重建等价于病态矩阵求逆问题,适当的约束条件、先验知识以及正则化因子对提高重建图像质量非常重要。     

多准则迭代卷积重建理论及实现

多准则迭代卷积重建是多目标决策下的Ｒａｄｏｎ变换求逆过程。我们从图像场的本质出发，设立反映图像场平滑性测度－熵函数、反映图像场的能量一平方范数及反映重建精度测试－投影响与重建图象的再投影之间的平方误差之和作为重建的三个性能测试。基于这些准则的多准则正则化方法是非适定图象重建一个有效的工具，从德国西门子公司实际ＣＴ投影数据的重建结果来看，多准则迭代卷积重建优于ＣＴ机用的卷积算法。     

三维重力反演新方法——调和密度成像理论与模型测试

三维重力反演问题具有多解性,通常要施加一些先验信息来约束反演结果,得到最优解,但是通常要获取满足约束条件的先验信息具有一定困难.本文总结了调和密度直接成像理论的基础和发展现状,并给出模型测试结果.研究表明:调和重力反演方法不需要先验信息约束,可以直接计算出三维空间的等效源密度结构,结果可以为三维物性反演提供初始解,也可以通过约束来构建迭代运算得到合理的三维连续密度特征,本文基于该方法提出了一种等效模型置换法的反演思路.     

地震波场反演的BG－逆散射方法

本文讨论利用三维反射地震数据进行波场反演的一种方法，旨在取得高分辨率的地球模型．这种方法用Ｂａｃｋｕｓ－Ｇｉｌｂｅｒｔ的理论构造波动方程非线性反问题的逐次线性化迭代格式，用逆散射原理导出泛函的Ｆｒｅｃｈｅｔ导数，并用最佳折衷准则求解线性化后的方程组．根据迭代过程中不断提高分辨率的思想和减少计算成本的原则，设计了可供实用的反演算法流程．     

地震波场反演的BG－逆散射方法

本文讨论利用三维反射地震数据进行波场反演的一种方法，旨在取得高分辨率的地球模型．这种方法用Ｂａｃｋｕｓ－Ｇｉｌｂｅｒｔ的理论构造波动方程非线性反问题的逐次线性化迭代格式，用逆散射原理导出泛函的Ｆｒｅｃｈｅｔ导数，并用最佳折衷准则求解线性化后的方程组．根据迭代过程中不断提高分辨率的思想和减少计算成本的原则，设计了可供实用的反演算法流程．     

井间电阻率层析成象的几个问题研究

井间电阻率层析成象是一种较新的高精度电法勘探方法，尚有许多问题需要探讨。本文针对平滑度约束反演算法，研究了平滑因子及迭代次数对成象结果的影响，给出了约束条件的施加方法。     

电阻率层析成像用于水文地质勘探

电阻率层析成像是根据阵列电极的电位测量重建电阻率图象的一种新技术，与波动CT问题有着本质的不同，图象重建的关键在于快速完成对稳定电流场的泊松方程求解和反演迭代．本文以岛裕雅的级联算法为例，简介了数学方法的基本步骤．针对山东汶南煤矿的矿井涌水问题，实施了地表层析探测，3条剖面总长5 832 m，最大成像深度120 m．层析图象清晰地揭示了地层结构、煤矿采空区、破碎带和裂隙(或管道状)漏水区的分布．所得结果经钻孔检验正确，已成功地应用于继而展开的堵漏截流治水工程．文中针对技术方法和存在的问题进行了讨论．      

Water and graphite in the Earth's crust —An approach to interpretation of conductivity models

In the present discussion on the origin of highly conducting layers in the middle and lower crust, water (brines) and graphite are widely accepted as the two most promising explanations. In this review, petrophysical and petrological aspects are summarized aiming at an understanding of how the indispensable interconnectedness of these highly conducting phases within a normally insulating rock matrix may be established and maintained under conditions of the upper and the lower crust. Accordingly, interpretation of conductivity models will be conclusive in cases of conductance values reaching several thousand S for layers at deeper crustal level since they can not be explained other than by the existence of metaanthracitic or graphitic rocks. In cases of low to moderate conductance values as often observed, e.g., at the transition zone from the upper to the lower crust, interpretation may remain ambiguous and may need further multidisciplinary investigations.     

Truncated multiGaussian fields and effective conductance of binary media

Truncated Gaussian fields provide a flexible model for defining binary media with dispersed (as opposed to layered) inclusions. General properties of excursion sets on these truncated fields are coupled with a distance-based upscaling algorithm and approximations of point process theory to develop an estimation approach for effective conductivity in two-dimensions. Estimation of effective conductivity is derived directly from knowledge of the kernel size used to create the multiGaussian field, defined as the full-width at half maximum (FWHM), the truncation threshold and conductance values of the two modes. Therefore, instantiation of the multiGaussian field is not necessary for estimation of the effective conductance. The critical component of the effective medium approximation developed here is the mean distance between high conductivity inclusions. This mean distance is characterized as a function of the FWHM, the truncation threshold and the ratio of the two modal conductivities. Sensitivity of the resulting effective conductivity to this mean distance is examined for two levels of contrast in the modal conductances and different FWHM sizes. Results demonstrate that the FWHM is a robust measure of mean travel distance in the background medium. The resulting effective conductivities are accurate when compared to numerical results and results obtained from effective media theory, distance-based upscaling and numerical simulation.     

Impedance spectroscopy analysis on electrical properties of serpentine at high pressure and high temperature

The electrical conductivity of serpentine is measured and the microscopic conductance mechanisms are investigated with impedance spectroscopy at 2.5–4.0 GPa and 220–780°C. The results show that the electrical conductivity is strongly dependent on the frequencies used, and that only arc I, which reflects grain interior conductance, occurs and dominates the whole conductance processes over 12-10⁵Hz at high pressure before dehydration. The arc II, which indicates the grain boundary process, begins to occur at the initial stage of dehydration. After dehydration, due to the presence of highly conductive networks of free water, the electrical conductivity is not dependent on frequencies any longer and the total electrical conductivity is dominated by process of ionic conductance of free water in interconnected networks. Dehydration of serpentine enhances pronouncedly the total electrical conductivity, through which highly conductive layers (HCL) may be formed in the earth’s interior.     

层状介质中地震面波频散函数和体波广义反射系数的计算

本文在B、P、C坐标中给出了弹性波在横向均匀介质中的传播矩阵,并将其表示为五个形式简单的矩阵乘积,其中有四个矩阵是与频率无关的。我们用传播矩阵的分离方式,将Abo-Zena（1979）算法大量简化。用本文的算法计算综合地震图时,计算量要比文献[2,7,8]的算法少一半以上。 文中还给出了地震面波频散函数和体波广义反射系数的快速计算步骤。     

Inversion of controlled‐source electromagnetic data using a model‐based approach

A two‐and‐half dimensional model‐based inversion algorithm for the reconstruction of geometry and conductivity of unknown regions using marine controlled‐source electromagnetic (CSEM) data is presented. In the model‐based inversion, the inversion domain is described by the so‐called regional conductivity model and both geometry and material parameters associated with this model are reconstructed in the inversion process. This method has the advantage of using a priori information such as the background conductivity distribution, structural information extracted from seismic and/or gravity measurements, and/or inversion results a priori derived from a pixel‐based inversion method. By incorporating this a priori information, the number of unknown parameters to be retrieved becomes significantly reduced. The inversion method is the regularized Gauss‐Newton minimization scheme. The robustness of the inversion is enhanced by adopting nonlinear constraints and applying a quadratic line search algorithm to the optimization process. We also introduce the adjoint formulation to calculate the Jacobian matrix with respect to the geometrical parameters. The model‐based inversion method is validated by using several numerical examples including the inversion of the Troll field data. These results show that the model‐based inversion method can quantitatively reconstruct the shapes and conductivities of reservoirs.     

An iterative scheme for use in MT 2D forward modelling — to speed up production of the Jacobian and to improve accuracy of the inverse matrix

An important stage in two-dimensional magnetotelluric modelling is the calculation of the Earth's response functions for an assumed conductivity model and the calculation of the associated Jacobian relating those response functions to the model parameters. The efficiency of the calculation of the Jacobian will affect the efficiency of the inversion modelling. Rodi (1976) produced all the Jacobian elements by inverting a single matrix and using an approximate first-order algorithm. Since only one inverse matrix required calculation the procedure speeded up the inversion. An iterative scheme to improve the approximation to the Jacobian information is presented in this paper. While this scheme takes a little longer than Rodi's algorithm, it enables a more accurate determination of the Jacobian information. It is found that the Jacobian elements can be produced in 10% of the time required to calculate an inverse matrix or to calculate a 2D starting model. A modification of the algorithm can further be used to improve the accuracy of the original inverse matrix calculated in a 2D finite difference program and hence the solution this program produces. The convergence of the iteration scheme is found to be related both to the originally calculated inverse matrix and to the change in the newly formed matrix arising from perturbation of the model parameter. A ridge regression inverse algorithm is used in conjunction with the iterative scheme for forward modelling described in this paper to produce a 2D conductivity section from field data.     

Extrapolating EISCAT Pedersen conductances to other parts of the sky using ground-based TV auroral images

Ionospheric conductivity is not very easily measured directly. Incoherent scatter radars perhaps offer the best method but can only measure at one point in the sky at any one time and are limited in their time resolution. Statistical models of average conductivity are available but these may not be applied to individual case studies such as substorms. There are many instances where a real-time estimate of ionospheric conductivity over a large field-of-view is highly desirable at a high temporal and spatial resolution. We show that it is possible to make a reasonable estimate of the noctural height-integrated Pedersen conductivity, or conductance, with a single all-sky TV camera operating at 557.7 nm. This is not so in the case of the Hall conductance where at least two auroral wavelengths should be imaged in order to estimate additionally the energy of the precipitating particles.     

Conductivity model for pyrite-bearing laminated and dispersed shaly sands based on a differential equation and the generalized Archie equation

The conductance of pyrite-bearing laminated and dispersed shaly sands is not well understood and resistivity models for pyrite-bearing shaly sands are nonexistent. Thus, we first synthesize clean pyrite-matrix samples, and quartz-matrix samples with variable laminated shale, dispersed shale, and pyrite content and then perform petrophysics experiments to assess the effect of pyrite content on the conductivity of pyrite-bearing shaly sands. Second, based on the differences in conductivity and conduction pathways and geometries because of the variable composition of the pyrite-bearing laminated and dispersed shaly sands, we divide the shaly sands into their components, i.e., laminated shale, quartz grains, pyrite grains, hydrocarbon, dispersed shale, microscopic capillary water, and mobile water. A generalized resistivity model is proposed to describe the conductivity of pyrite-bearing laminated and dispersed shaly sands, based on the combined conductivity differential equation and generalized Archie equation. In the generalized resistivity model, the conductivity differential equation is used to describe the conductivity of dispersed inclusions in a host, whereas the generalized Archie equation is used to describe the conductivity of two conducting phases. Moreover, parallel conductance theory is used to describe the conductivity of dispersed shaly sands and laminated shale. Theoretical analysis suggests that the proposed model satisfies the physical constraints and the model and experimental results agree. The resistivity and resistivity index of shaly sands decrease with increasing conductivity and pyrite. Finally, the accuracy of the resistivity model is assessed based on experimental data from 46 synthetic core samples with different oil saturation. The model can describe the conductivity of clean pyrite-matrix samples, and quartz-matrix samples with different volumes of laminated shale, dispersed shale, and pyrite. An accurate saturation model of pyrite-bearing laminated and dispersed shaly sands is thus obtained and the log data interpretation in complex shaly sands can improve with the proposed model.     

Numerical modelling of airborne electromagnetic anomalies originating from low-conductivity 3D bodies1

Responses of a multifrequency, multicoil airborne electromagnetic (AEM) system were modelled numerically for 3D electrical conductors embedded in a resistive bedrock and overlain by an overburden of low to moderate conductivity. The results cover a horizontal coplanar coil configuration and two frequencies, 7837 Hz and 51 250 Hz. The models studied are single or multiple, poor conductors (conductance lower than 0.1 S) embedded in a host rock of high but finite resistivity (5000 Ωm) and overlain by a layer of overburden with finite thickness and low to moderate conductivity (conductance up to 2 S). On the basis of the modelling results, limits of detectability for poor conductors have been studied for the various model structures. The results indicate that the anomaly from a steeply dipping, plate-like conductor will decrease significantly when the conductor is embedded in a weakly conductive host rock and is overlain by a conductive overburden. However, an anomaly is obtained, and its magnitude can even increase with increasing overburden conductivity or frequency. The plate anomaly remains practically constant when only the overburden thickness is varied. Changes in overburden conductivity will cause the plate-anomaly values to change markedly. If the plate conductance is less than that of the overburden, a local anomaly opposite in sign to the normal type of anomaly will be recorded. Another major consequence is that conductors interpreted with free-space models will be heavily overestimated in depth or underestimated in conductance, if in reality induction and current channelling in the host rock and overburden make even a slight contribution to the anomalous EM field. The lateral resolution for the horizontal coplanar coil system was found to be about 1.7 times the sensor altitude. Similarly, the lateral extension of a horizontal conductive ribbon, required to reach the semi-infinite (half-space) behaviour, is more than three times the sensor altitude. Finally, screening of a steeply dipping plate, caused by a small, conductive horizontal ribbon, is much more severe than screening of the same plate by an extensive horizontal layer.     

Magnetotelluric soundings over the northeast Pacific may reveal spatial dependence of depth and conductance of the asthenosphere

A seafloor electrical conductivity profile resulting from a more thorough analysis of magnetotelluric data from station S.F. Revisited than previously presented is compared to an earlier profile at Farewell to Aggy, station III. Both stations are located over the same interfracture zone segment of the Pacific plate, the first roughly 700 km off the coast of California (position 31°18′N, 128°20′W, water depth 4.5 km, plate age 30 m.y. estimated from nearby magnetic reversal number 12), the second approximately 800 km to the NNE of the mainland of Hawaii (position 26°32′N, 151°20′W, depth 5.3 km, age 72 m.y. estimated from adjacent magnetic reversal 30–31).The seafloor impedances at S.F. Revisited are only mildly polarized and their interpretation in terms of an isotropic, horizontally layered structure suggests the occurrence at about 85 km depth of a highly conducting layer with a conductance exceeding by roughly 4 × 10³ S, an otherwise monotonically increasing conductivity trend. The implied dependence of conductivity with depth is therefore similar to that found earlier for station III, however with the following differences: the high-conductivity layer at station III occurs at a greater depth (140 km), it appears to have a slightly reduced excess conductance over the background, 3.5 × 10³ S although this evidence should be used with caution, and the lithospheric conductivity at station III, surprisingly seems to be somewhat higher, an effect possibly related to the proximity of the Hawaiian chain and to its generic processes.