3D electrical resistivity inversion using prior spatial shape constraints

To minimize the number of solutions in 3D resistivity inversion, an inherent problem in inversion, the amount of data considered have to be large and prior constraints need to be applied. Geological and geophysical data regarding the extent of a geological anomaly are important prior information. We propose the use of shape constraints in 3D electrical resistivity inversion, Three weighted orthogonal vectors （a normal and two tangent vectors） were used to control the resistivity differences at the boundaries of the anomaly. The spatial shape of the anomaly and the constraints on the boundaries of the anomaly are thus established. We incorporated the spatial shape constraints in the objective function of the 3D resistivity inversion and constructed the 3D resistivity inversion equation with spatial shape constraints. Subsequently, we used numerical modeling based on prior spatial shape data to constrain the direction vectors and weights of the 3D resistivity inversion. We established a reasonable range between the direction vectors and weights, and verified the feasibility and effectiveness of using spatial shape prior constraints in reducing excessive structures and the number of solutions. We applied the prior spatially shape-constrained inversion method to locate the aquifer at the Guangzhou subway. The spatial shape constraints were taken from ground penetrating radar data. The inversion results for the location and shape of the aquifer agree well with drilling data, and the number of inversion solutions is significantly reduced.     

利用大地电磁三维反演方法获得二维剖面附近三维电阻率结构的可行性

大地电磁野外实测数据目前大多为二维剖面数据.如何反演这些二维剖面数据获得较为接近实际地电情况的结果,是多数大地电磁工作者关心的问题.我们通过对理论模型的三维响应进行分析和对合成数据及实测资料的反演结果进行对比研究,讨论了利用三维反演的方法来获得大地电磁二维剖面附近三维电阻率结构的可行性.结果表明:可用三维反演的方法来解释二维剖面数据.对大地电磁二维剖面的张量数据进行三维反演,不仅可以沿剖面获得较好的二维断面结果,还能够得到二维反演所不能获得的剖面附近的三维电阻率结构信息.合成数据的反演算例表明:对二维剖面数据进行三维反演时,对角元素对于圈定剖面附近三维异常体的空间分布具有独特作用,应尽量反演所有的张量元素.     

电阻率二维神经网络反演

由于非线性特性地球物理反演一直以来都是一个比较困难的问题. 近十年来,非线性反演方法如人工神经网络、遗传算法在地球物理数据解释中得到越来越多的应用,但目前基本仍限于一维反演问题. 对于二维反问题,反演参数较多,神经网络反演运用较少. 本文利用BP神经网络优化方法,实现了电阻率二维非线性反演. 与传统线性化的迭代反演比较,神经网络反演能够克服传统方法的不足、获得更好的反演结果.     

基于非结构网格的电阻率三维带地形反演

地表起伏地形在野外矿产资源勘察中不可避免, 其对直流电阻率法勘探影响巨大.近年来, 电阻率三维正演取得诸多进展, 特别是应用非结构网格我们能够进行任意复杂地形和几何模型的电阻率三维数值模拟, 但面向实际应用的起伏地形下电阻率三维反演依然困难.本文基于非结构化四面体网格, 并考虑到应用GPS/GNSS时, 区域地球物理调查中可非规则布设测网的实际特点, 实现了任意地形(平坦或起伏)条件下、任意布设的偶极-偶极视电阻率数据的不完全Gauss-Newton三维反演.合成数据的反演结果表明了方法的有效性, 可应用于复杂野外环境下的三维电法勘探.     

Assessment of the reliability of 2D inversion of apparent resistivity data

The reliability of inversion of apparent resistivity pseudosection data to determine accurately the true resistivity distribution over 2D structures has been investigated, using a common inversion scheme based on a smoothness‐constrained non‐linear least‐squares optimization, for the Wenner array. This involved calculation of synthetic apparent resistivity pseudosection data, which were then inverted and the model estimated from the inversion was compared with the original 2D model. The models examined include (i) horizontal layering, (ii) a vertical fault, (iii) a low‐resistivity fill within a high‐resistivity basement, and (iv) an upfaulted basement block beneath a conductive overburden. Over vertical structures, the resistivity models obtained from inversion are usually much sharper than the measured data. However, the inverted resistivities can be smaller than the lowest, or greater than the highest, true model resistivity. The substantial reduction generally recorded in the data misfit during the least‐squares inversion of 2D apparent resistivity data is not always accompanied by any noticeable reduction in the model misfit. Conversely, the model misfit may, for all practical purposes, remain invariant for successive iterations. It can also increase with the iteration number, especially where the resistivity contrast at the bedrock interface exceeds a factor of about 10; in such instances, the optimum model estimated from inversion is attained at a very low iteration number. The largest model misfit is encountered in the zone adjacent to a contact where there is a large change in the resistivity contrast. It is concluded that smooth inversion can provide only an approximate guide to the true geometry and true formation resistivity.     

基于印模法的地面磁电阻率数据三维非线性共轭梯度反演

本文提出了能提高异常体分辨能力,同时得到绝对电导率的地面磁电阻率数据三维反演方法.磁电阻率响应用准直流的低频磁场代替;数值模拟由频率域电场满足的Helmholtz方程出发,采用三维交错网格有限差分法;长直导线源作为发射源,其中源的计算包含在背景场中;结合地面磁电阻率数据各分量的特点,选择y分量进行反演研究;反演采用三维非线性共轭梯度反演技术,为了提高异常体的深度分辨能力,进行迭代重构反演;用印模法对初始模型进行重构,采用的是辅模型在浅部,元模型在深部的组合方式.从合成数据和实际数据的反演结果可以得到以下的认识：（1）由频率域麦克斯韦方程组出发,低频磁场数据反演可以直接得到电导率,而不是相对电导率之比;（2）采用印模法组合初始模型,进行迭代重构反演,可以提高地面磁电阻率数据反演对异常体的分辨能力,确定埋深位置,同时不会丧失对于浅部异常体的分辨能力;（3）在结合印模法的地面磁电阻率数据三维反演中,深部异常体的分辨能力受地表不均匀导电体影响较小;（4）确定印模深度可以采用上一次重构反演结束时的模型变化量,通过相邻两次重构反演结束时的模型变化量之差来确定迭代重构是否终止.因为静磁场与重力场在数学上的相似性,本文的反演方法可以被运用到重力场等位场的地面数据的反演中.

     

井地三维电阻率反演研究.

本文首先讨论了垂直有限线源三维电阻率反演方法并利用阻尼最小二乘法技术实现了垂直有限线源三维电阻率反演,然后研究了反演中的几个相关问题.模型试验结果表明：初值选择越合理,反演结果越接近真值;地表的观测范围越大,包含深部电阻率的信息越丰富,反演效果越好.     

1D inversion of DC resistivity data using a quality-based truncated SVD

Many DC resistivity inversion schemes use a combination of standard iterative least-squares and truncated singular value decomposition (SVD) to optimize the solution to the inverse problem. However, until quite recently, the truncation was done arbitrarily or by a trial-and-error procedure, due to the lack of workable guidance criteria for discarding small singular values. In this paper we present an inversion scheme which adopts a truncation criterion based on the optimization of the total model variance. This consists of two terms: (i) the term associated with the variance of statistically significant principal components, i.e. the standard model estimate variance, and (ii) the term associated with statistically insignificant principal components of the solution, i.e. the variance of the bias term. As an initial model for the start of iterations, we use a multilayered homogeneous half-space whose layer thicknesses increase logarithmically with depth to take into account the decrease of the resolution of the DC resistivity technique with depth. The present inversion scheme has been tested on synthetic and field data. The results of the tests show that the procedure works well and the convergence process is stable even in the most complicated cases. The fact that the truncation level in the SVD is determined intrinsically in the course of inversion proves to be a major advantage over other inversion schemes where it is set by the user.     

Practical techniques for 3D resistivity surveys and data inversion1

Techniques to reduce the time needed to carry out 3D resistivity surveys with a moderate number (25 to 100) of electrodes and the computing time required to interpret the data have been developed. The electrodes in a 3D survey are normally arranged in a square grid and the pole-pole array is used to make the potential measurements. The number of measurements required can be reduced to about one-third of the maximum possible number without seriously degrading the resolution of the resulting inversion model by making measurements along the horizontal, vertical and 45° diagonal rows of electrodes passing through the current electrode. The smoothness-constrained least-squares inversion method is used for the data interpretation. The computing time required by this technique can be greatly reduced by using a homogeneous half-space as the starting model so that the Jacobian matrix of partial derivatives can be calculated analytically. A quasi-Newton updating method is then used to estimate the partial derivatives for subsequent iterations. This inversion technique has been tested on synthetic and field data where a satisfactory model is obtained using a modest amount of computer time. On an 80486DX2/66 microcomputer, it takes about 20 minutes to invert the data from a 7 by 7 electrode survey grid. using the techniques described below, 3D resistivity surveys and data inversion can be carried out using commercially available field equipment and an inexpensive microcomputer.     

4D active time constrained resistivity inversion

Techniques that incorporate regularization in space and time have been proposed to reduce inversion artifacts that may lead a misinterpretation of geophysical monitoring data. Applying this time regularization, however, may result in a model too smoothly carrying in the time domain. To alleviate this problem, we propose an algorithm for inverting time-lapse resistivity monitoring data.Here the time regularization is not considered to be constant between different time steps but is now allowed to vary depending on the degree of spatial resistivity changes occurring between different monitoring stages. Two methods are proposed to assign different time Lagrangian values, one based on a pre-estimation during execution time, and one using a-priori information. Both methods require a threshold to characterize the significance of the observed resistivity changes with time. We performed numerous numerical experiments using synthetic data to provide reasonable threshold values. Synthetic data tests illustrate that the new algorithm, named 4D Active Time Constrained (4D-ATC), produces in most cases improved time-lapse images when compared with existing techniques. Further the applicability of the new scheme is demonstrated with real data. Overall, the new algorithm is shown to be a useful tool for processing time-lapse resistivity data, which can be used with minor modifications to other types of time-lapse geophysical data.     

Optimizing a layered and laterally constrained 2D inversion of resistivity data using Broyden's update and 1D derivatives

Dense profile-oriented resistivity data allows for 2D and 3D inversions. However, huge amounts of data make it practically impossible to do full 2D or 3D inversions on a routine basis. Therefore, a number of approximations have been suggested over the years to speed up computations. We suggest using a combination of Broyden's update on the Jacobian matrix with derivatives calculated using a 1D formulation on a parameterized 2D model of locally 1D layered models. The approximations bring down the effective number of 2D forward responses to a minimum, which again gives us the ability to invert very large sections. Broyden's update is not as useful with a parameterized problem as is the case with a smooth minimum structure problem that has been the usual application. 1D derivatives, however, seem to be very effective when initiating a full 2D solution with Broyden's update. We compare the different methods using two different kinds of data on two synthetic models and on two field examples. The most effective and reliable optimization combines 1D derivatives with a full 2D solution and Broyden's update. When using Broyden's update the Jacobian matrix needs to be reset every once in a while. We do this whenever the difference in data residual from the previous iteration is less than 5%. This combined inversion method reduces the computation time approximately a factor of 3 without losing model resolution.     

Possibilities of electric field measurements in the audio frequency range using ungrounded electric sensors

Possibilities of electric field measurements in the audio frequency range using underground (capacitive) electric sensors are considered. Features of parameters of electric sensors and input circuits of the measurement equipment, robust methods, and software tools for processing obtained data are analyzed. Examples of experimental studies by the audiomagnetotelluric sounding (AMTS) method, demonstrating possibilities of implementing ungrounded electric field measurements in a frequency range of 7?C300 Hz, are presented.     

Rapid inversion of data from 2D resistivity surveys with electrode displacements

Resistivity monitoring surveys are used to detect temporal changes in the subsurface using repeated measurements over the same site. The positions of the electrodes are typically measured at the start of the survey program and possibly at occasional later times. In areas with unstable ground, such as landslide‐prone slopes, the positions of the electrodes can be displaced by ground movements. If this occurs at times when the positions of the electrodes are not directly measured, they have to be estimated. This can be done by interpolation or, as in recent developments, from the resistivity data using new inverse methods. The smoothness‐constrained least squares optimisation method can be modified to include the electrode positions as additional unknown parameters. The Jacobian matrices with the sensitivity of the apparent resistivity measurements to changes in the electrode positions are then required by the optimisation method. In this paper, a fast adjoint‐equation method is used to calculate the Jacobian matrices required by the least squares method to reduce the calculation time. In areas with large near‐surface resistivity contrasts, the inversion routine sometimes cannot accurately distinguish between electrode displacements and subsurface resistivity variations. To overcome this problem, the model for the initial time‐lapse dataset (with accurately known electrode positions) is used as the starting model for the inversion of the later‐time dataset. This greatly improves the accuracy of the estimated electrode positions compared to the use of a homogeneous half‐space starting model. In areas where the movement of the electrodes is expected to occur in a fixed direction, the method of transformations can be used to include this information as an additional constraint in the optimisation routine.     

基于遗传算法和统计学的电阻率测深二维反演研究

本文介绍了一种以遗传算法和统计学为基础的反演方法,该方法利用一维反演结果的统计学特征确定地层电性结构并设计初始模型,使用遗传算法修改模型,经过二维有限单元法正演达到拟合误差最小,从而完成电阻率测深数据的二维反演.这个方法的优点在于不需要解大型线性方程组,降低计算量,先判断地层电性结构,反演后给出用地电断面表示的直观的解释结果.理论模型试算和模型实验证明该方法效果较好.     

基于不等式约束的三维电阻率探测混合反演方法

三维电阻率探测的线性反演和非线性反演中均存在着多解性的固有难题.电阻率线性反演方法的效率较高,但反演结果对初始模型的依赖性较强,易陷入局部极小;而非线性反演方法不依赖初始模型,但搜索效率极低,尚未见到关于三维电阻率非线性反演的文献.针对上述问题,融合线性与非线性反演方法的互补优势,提出了最小二乘法（线性方法）与改进遗传算法（非线性方法）相结合的混合反演方法的概念和思想.首先,提出了将介质电阻率变化范围作为不等式约束引入反演方程的思路,以实现压制多解性、提高可靠性的目标.提出了宽松不等式约束和基于钻孔推断的局部严格不等式约束的获取及定义方法.在此基础上,分别提出了基于不等式约束的最小二乘线性反演方法和遗传算法非线性反演方法.其次,对于遗传算法在变异搜索方向控制、初始群体产生等方面进行了改进,优化了其搜索方向和初始群体多样性.然后,提出了混合反演方法及其实现方案,利用改进遗传算法进行第一阶段反演,发挥其对初始模型的依赖程度低的优势,搜索到最优解附近的空间,输出当前最优个体;利用最小二乘法进行第二阶段反演,将遗传算法得到的当前最优个体作为初始模型,在最优解附近空间执行高效率的局部线性搜索,最终实现地电结构的三维成像.最后,开展了合成数据与实际工程算例验证,与传统最小二乘方法进行了对比,发现混合反演方法在压制多解性、摆脱初始模型依赖和提高反演效果方面有较好效果.

     

Piecewise 1D laterally constrained inversion of resistivity data

In a sedimentary environment, layered models are often capable of representing the actual geology more accurately than smooth minimum structure models. Furthermore, interval thicknesses and resistivities are often the parameters to which non‐geophysicist experts can relate and base decisions on when using them in waste site remediation, groundwater modelling and physical planning. We present a laterally constrained inversion scheme for continuous resistivity data based on a layered earth model (1D). All 1D data sets and models are inverted as one system, producing layered sections with lateral smooth transitions. The models are regularized through laterally equal constraints that tie interface depths and resistivities of adjacent layers. Prior information, e.g. originating from electric logs, migrates through the lateral constraints to the adjacent models, making resolution of equivalences possible to some extent. Information from areas with well‐resolved parameters will migrate through the constraints in a similar way to help resolve the poorly constrained parameters. The estimated model is complemented by a full sensitivity analysis of the model parameters, supporting quantitative evaluation of the inversion result. Examples from synthetic 2D models show that the model recognition of a sublayered 2D wedge model is improved using the laterally constrained inversion approach when compared with a section of combined 1D models and when compared with a 2D minimum structure inversion. Case histories with data from two different continuous DC systems support the conclusions drawn from the synthetic example.     

Detection of sinkholes using 2D electrical resistivity imaging

Sinkholes in dolomitic areas are notoriously difficult geophysical targets, and selecting an appropriate geophysical solution is not straightforward. Electrical resistivity imaging, or tomography (RESTOM) is well suited to mapping sinkholes because of the ability of the technique for detecting resistive features and discriminating subtle resistivity variations. RESTOM surveys were conducted at two sinkhole sites near Pretoria, South Africa. The survey areas are located in the dolomites of the Lyttelton Formation, which forms part of the Malamani Subgroup and Chuniespoort Group of the Transvaal Supergroup. The survey results suggest that RESTOM is an ideal geophysical tool to aid in the detection and monitoring of sinkholes and other subsurface cavities.     

随钻超深前视电阻率测井二次场耦合势三维有限体积法高效数值仿真

随钻超深前视电阻率测井是一种新型井中电磁测量技术,能够实现复杂储层探测和开发中井眼轨迹优化与地质导向.本文将基于二次场耦合势三维有限体积法建立一套非均质各向异性地层中随钻超深前视多分量电阻率测井高效数值模拟算法.首先,利用Yee氏交错网格、非均质网格上的等效电导率标准化技术以及三维有限体积法实现二次场耦合势Helmholtz方程离散,并采用传输线法建立背景电磁场并矢Green函数库,同时结合二维Newton插值技术确定背景电磁场,提高异常体内散射电流场计算效率.在此基础上,针对仪器移动测量方式的特点,通过并行直接稀疏求解器PARDISO求解离散方程,结合并行计算同时确定不同位置发射源产生的电磁场.最后,数值结果验证了算法的有效性,显示了二次场方法具有更高的计算精度;建库插值方法显著提高了背景场计算效率,拓展了二次场方法的应用价值.

     

2D and 3D resistivity inversion of Schlumberger vertical electrical soundings in Wadi El Natrun,Egypt: A case study

The Wadi El Natrun area is characterized by a very complicated geological and hydrogeological system. 45 vertical electrical soundings (Schlumberger array) were measured in the study area to elucidate the peculiarity of this unique regime, specifically the nature of waterless area. 2D and 3D resistivity inversion based on the finite element technique and regularization method were applied on the data set. 2D and 3D model resolution was investigated through the use of the Depth and Volume of Investigation Indexes. A very good matching was found between the zones of high resistivity, the waterless area, and the non-productive wells. The low resistivity zones (corresponding to Lower Pliocene clay) were also identified. The middle resistivity fresh water aquifer zones were recognized. Available results can assist in the aquifer management by selecting the most productive zone of groundwater.     

Large‐scale 3D inversion of potential field data

Inversion of gravity and/or magnetic data attempts to recover the density and/or magnetic susceptibility distribution in a 3D earth model for subsequent geological interpretation. This is a challenging problem for a number of reasons. First, airborne gravity and magnetic surveys are characterized by very large data volumes. Second, the 3D modelling of data from large‐scale surveys is a computationally challenging problem. Third, gravity and magnetic data are finite and noisy and their inversion is ill posed so regularization must be introduced for the recovery of the most geologically plausible solutions from an infinite number of mathematically equivalent solutions. These difficulties and how they can be addressed in terms of large‐scale 3D potential field inversion are discussed in this paper. Since potential fields are linear, they lend themselves to full parallelization with near‐linear scaling on modern parallel computers. Moreover, we exploit the fact that an instrument’s sensitivity (or footprint) is considerably smaller than the survey area. As multiple footprints superimpose themselves over the same 3D earth model, the sensitivity matrix for the entire earth model is constructed. We use the re‐weighted regularized conjugate gradient method for minimizing the objective functional and incorporate a wide variety of regularization options. We demonstrate our approach with the 3D inversion of 1743 line km of FALCON gravity gradiometry and magnetic data acquired over the Timmins district in Ontario, Canada. Our results are shown to be in good agreement with independent interpretations of the same data.