The inversion of data from very large three-dimensional electrical resistivity tomography mobile surveys

New developments in mobile resistivity meter instrumentation have made it possible to survey large areas with dense data coverage. The mobile system usually has a limited number of electrodes attached to a cable that is pulled along behind an operator so that a large area can be covered within a short time. Such surveys can produce three-dimensional datasets with hundreds of thousands of electrodes positions and data points. Similarly, the inverse model used to interpret the data can have several hundred thousand cells. It is impractical to model such large datasets within a reasonable time on microcomputers used by many small companies employing standard inversion techniques. We describe a model segmentation technique that subdivides the finite-element mesh used to calculate the apparent resistivity and Jacobian matrix values into a number of smaller meshes. A fast technique that optimizes the calculation of the Jacobian matrix values for multi-channel systems was also developed. A one-dimensional wavelet transform method was then used to compress the storage of the Jacobian matrix, in turn reducing the computer time and memory required to solve the least-squares optimization equation to determine the inverse model resistivity values. The new techniques reduce the calculation time and memory required by more than 80% while producing models that differ by less than 1% from that obtained using the standard inversion technique with a single mesh. We present results using a synthetic model and a field dataset that illustrates the effectiveness of the proposed techniques.     

2D data modelling by electrical resistivity tomography for complex subsurface geology

A new tool for two‐dimensional apparent‐resistivity data modelling and inversion is presented. The study is developed according to the idea that the best way to deal with ill‐posedness of geoelectrical inverse problems lies in constructing algorithms which allow a flexible control of the physical and mathematical elements involved in the resolution. The forward problem is solved through a finite‐difference algorithm, whose main features are a versatile user‐defined discretization of the domain and a new approach to the solution of the inverse Fourier transform. The inversion procedure is based on an iterative smoothness‐constrained least‐squares algorithm. As mentioned, the code is constructed to ensure flexibility in resolution. This is first achieved by starting the inversion from an arbitrarily defined model. In our approach, a Jacobian matrix is calculated at each iteration, using a generalization of Cohn's network sensitivity theorem. Another versatile feature is the issue of introducing a priori information about the solution. Regions of the domain can be constrained to vary between two limits (the lower and upper bounds) by using inequality constraints. A second possibility is to include the starting model in the objective function used to determine an improved estimate of the unknown parameters and to constrain the solution to the above model. Furthermore, the possibility either of defining a discretization of the domain that exactly fits the underground structures or of refining the mesh of the grid certainly leads to more accurate solutions. Control on the mathematical elements in the inversion algorithm is also allowed. The smoothness matrix can be modified in order to penalize roughness in any one direction. An empirical way of assigning the regularization parameter (damping) is defined, but the user can also decide to assign it manually at each iteration. An appropriate tool was constructed with the purpose of handling the inversion results, for example to correct reconstructed models and to check the effects of such changes on the calculated apparent resistivity. Tests on synthetic and real data, in particular in handling indeterminate cases, show that the flexible approach is a good way to build a detailed picture of the prospected area.     

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.     

An algorithm for fast 3D inversion of surface electrical resistivity tomography data: application on imaging buried antiquities

In this work a new algorithm for the fast and efficient 3D inversion of conventional 2D surface electrical resistivity tomography lines is presented. The proposed approach lies on the assumption that for every surface measurement there is a large number of 3D parameters with very small absolute Jacobian matrix values, which can be excluded in advance from the Jacobian matrix calculation, as they do not contribute significant information in the inversion procedure. A sensitivity analysis for both homogeneous and inhomogeneous earth models showed that each measurement has a specific region of influence, which can be limited to parameters in a critical rectangular prism volume. Application of the proposed algorithm accelerated almost three times the Jacobian (sensitivity) matrix calculation for the data sets tested in this work. Moreover, application of the least squares regression iterative inversion technique, resulted in a new 3D resistivity inversion algorithm more than 2.7 times faster and with computer memory requirements less than half compared to the original algorithm. The efficiency and accuracy of the algorithm was verified using synthetic models representing typical archaeological structures, as well as field data collected from two archaeological sites in Greece, employing different electrode configurations. The applicability of the presented approach is demonstrated for archaeological investigations and the basic idea of the proposed algorithm can be easily extended for the inversion of other geophysical data.     

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.     

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.     

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.     

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.     

A priori models and inversion of gravity gradient data in hilly terrain

The increased popularity of airborne measurements of the gravity gradient tensor for resource studies and geological mapping has resulted in a new awareness of the importance of terrain effects. In these measurements, the terrain effect often overwhelms that of the underlying crust and it becomes important to formulate a strategy for taking it into account when presenting the data and when inverting the data into density models. Using newly acquired data from Northern Sweden, we first attempted to estimate a variable terrain density model by inverting the data using a terrain model with a laterally varying density. Using data weights related to the topography variations, we find the best estimate of the lateral variation of the terrain density. We translate this model into a full three-dimensional model such that all columns have the same vertical centre of mass as estimated from inspecting the radially averaged power spectrum of the area. This then defines a reference model for subsequent three-dimensional inversion of the gravity gradient tensor dataset. We tested this approach first on synthetic data calculated from the measured topography including two density anomalies before we applied it to the measured data. The result is a model in which the surface density variations are propagated downwards in a systematic manner now in better agreement with measured densities of rock samples in the area.     

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.     

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.

     

基于三维高密度电法的地质BIM模型应用研究

地质体三维可视化BIM(Building Information Model)技术具有直观呈现地质体的三维赋存情况、实现地质体的空间计算与分析等众多功能,为工程设计和施工提供可视化的基础操作平台.目前,地质BIM模型的构建主要依赖大量钻孔资料和地质测绘成果.而由这些点状信息构建的三维地质模型易出现孔间盲区,且大量钻孔费时费力、破坏三维地质体结构,难以满足精细化、快捷、无损的地质建模需求.针对此问题,本文以豫北某拟建水库大坝工程为依托,利用三维高密度电法对坝址区左岸杂填土进行探测,物探解译成果结合少量钻孔资料快速精确构建工区地质模型.在此基础上,将大坝输水洞设计参数引入模型,直观展示沿洞线地质体与输水洞的空间关系.通过该方法计算施工开挖土方量比传统地质断面法多6000 m3,系孔间盲区基岩面下降形成基岩凹槽所致.研究表明,三维高密度电法成果结合少量钻孔资料可快速实现三维地质BIM精细化建模,为工程后期预算、施工地质预报以及构筑物优化设计提供可靠地质依据,也为精细三维地质BIM建模提供了一种新思路.     

3D resistivity inversion using 2D measurements of the electric field

Field and 'noisy' synthetic measurements of electric-field components have been inverted into 3D resistivities by smoothness-constrained inversion. Values of electrical field can incorporate changes in polarity of the measured potential differences seen when 2D electrode arrays are used with heterogeneous 'geology', without utilizing negative apparent resistivities or singular geometrical factors. Using both the X - and Y -components of the electric field as measurements resulted in faster convergence of the smoothness-constrained inversion compared with using one component alone. Geological structure and resistivity were reconstructed as well as, or better than, comparable published examples based on traditional measurement types. A 2D electrode grid (20 × 10), incorporating 12 current-source electrodes, was used for both the practical and numerical experiments; this resulted in 366 measurements being made for each current-electrode configuration. Consequently, when using this array for practical field surveys, 366 measurements could be acquired simultaneously, making the upper limit on the speed of acquisition an order of magnitude faster than a comparable conventional pole–dipole survey. Other practical advantages accrue from the closely spaced potential dipoles being insensitive to common-mode noise (e.g. telluric) and only 7% of the electrodes (i.e. those used as current sources) being susceptible to recently reported electrode charge-up effects.     

SOTEM数据一维OCCAM反演及其应用于三维模型的效果

本文基于垂直磁场分量研究了SOTEM数据的一维OCCAM反演方法,并将其应用于理论三维数据及野外实测数据的反演.对于大部分一维模型,OCCAM反演可取得较好的反演效果,且反演结果不依赖于偏移距;噪声对SOTEM数据的OCCAM反演具有较大影响,但当信号含噪水平不超过5%时,反演结果仍具有较好的准确性;若浅层存在较厚的低阻层,OCCAM反演结果对下部地层的分辨能力下降,仅能获得具有平均效应的电阻率.将一维算法应用于SOTEM三维数据的反演,会产生较大的误差,尤其是在异常体边缘地带影响最为严重.该影响程度与异常体和背景电阻率之间的差异有关,对于大多数电性近似呈连续变化的真实大地而言,一维OCCAM反演算法仍可获得较好的效果.最后通过陕西某煤田深部富水性调查的实测SOTEM数据反演验证了本文的研究成果.

     

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.     

The use of non-conventional CPTe data in determination of 3-D electrical resistivity distribution

The spatial distribution of the electrical resistivity data provides useful information for investigating and modeling the fluid transport processes. 3D electrical resistivity distribution provides information about water flow and changes in electrical resistivity of the pore fluid.Therefore, to assist in understanding and modeling of the fluid transport process, 3D spatial distribution of the electrical resistivity data with the corresponded 3D geological section were mapped and interpreted in the test site located in western Germany. A process of deriving electrical resistivity values from the mechanical and radioactive parameters of cone penetration tests (CPT) and geological information of boreholes was presented. A reliable method which gives accurate resistivity values in cases of near surface sediments was introduced. Then a field test was executed where the calculated resistivity values were compared with the measured CPTe resistivity data. The CPTe (cone penetration test with electrical extension) data were also used in correlating to the ERT (electrical resistivity tomography) data. Consequently, obtained dense CPT surveys give us the possibility to determine a high resolution resistivity distribution of the investigated area.     

Sensitivity of transversal apparent resistivity to the changes in electrical resistivity of the elements of a 2D geoelectrical structure

The sensitivity of transversal apparent resistivity to the changes in the electrical resistivity of elements of a two-dimensional (2D) geoelectrical structure is studied by numerical modeling. This sensitivity is found to have a series of specific features, due to which the monitoring of transversal apparent resistivity can serve as a helpful addition to the monitoring of longitudinal apparent resistivity to trace the dynamics of subsurface and shallow crustal elements of the medium. It is shown that the method previously suggested for the inversion of relative changes in apparent resistivity into relative changes in electrical resistivities of the elements of the geoelectrical structure is applicable to the transversal electrical resistivity.     

Compiling and verifying 3D models of 2D induced polarization and resistivity data by geostatistical methods

A 3D model of collected time-domain induced polarization (IP) and electrical resistivity tomography (ERT) data is compiled by geostatistical methods as well as studying spatial correlation among the database. Mesgaran copper deposit, located in Birjand eastern Iran, was chosen to compile and verify the model, leading to five parallel surveyed IP and ERT profiles with dipole–dipole arrays. The collected data were inverted, and then 2D models of IP and ER were prepared; also 3D inversion was done. Afterward, the 3D model has been built by geostatistical methods. Correspondingly, the anomalies threshold was detected by fractal methods and the estimation variance and Kriging efficiency were calculated to validate the modeling. The mineralization zones were determined according to the classified anomalies and those with the lowest error. Results indicated a high correlation between anomalies identified from the model and mineralization. The results made it possible to construct 3D models from surveyed 2D data with acceptable error level.     

On the use of statistical methods to interpret electrical resistivity data from the Eumsung basin (Cretaceous), Korea

Electrical resistivity mapping was conducted to delineate boundaries and architecture of the Eumsung Basin Cretaceous. Basin boundaries are effectively clarified in electrical dipole–dipole resistivity sections as high-resistivity contrast bands. High resistivities most likely originate from the basement of Jurassic granite and Precambrian gneiss, contrasting with the lower resistivities from infilled sedimentary rocks. The electrical properties of basin-margin boundaries are compatible with the results of vertical electrical soundings and very-low-frequency electromagnetic surveys. A statistical analysis of the resistivity sections is tested in terms of standard deviation and is found to be an effective scheme for the subsurface reconstruction of basin architecture as well as the surface demarcation of basin-margin faults and brittle fracture zones, characterized by much higher standard deviation. Pseudo three-dimensional architecture of the basin is delineated by integrating the composite resistivity structure information from two cross-basin E–W magnetotelluric lines and dipole–dipole resistivity lines. Based on statistical analysis, the maximum depth of the basin varies from about 1 km in the northern part to 3 km or more in the middle part. This strong variation supports the view that the basin experienced pull-apart opening with rapid subsidence of the central blocks and asymmetric cross-basinal extension.     

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.