Combined straightforward inversion of resistivity and induced polarization (time-domain) sounding data

It is proposed that the Straightforward Inversion Scheme (SIS) developed by the authors for 1D inversion of resistivity sounding and magneto-telluric sounding data can also be used in similar fashion for time-domain induced polarization sounding data. The necessary formulations based on dynamic dipole theory are presented. It is shown that by using induced polarization potential, measured at the instant when steady state current is switched off, an equation can be developed for apparent ‘chargeability–resistivity’ which is similar to the one for apparent resistivity. The two data sets of apparent resistivity and apparent chargeability–resistivity can be inverted in a combined manner, using SIS for a common uniform thickness layer earth model to estimate the respective subsurface distributions of resistivity and chargeability–resistivity. The quotient of the two profiles will give the sought after chargeability profile. A brief outline of SIS is provided for completeness. Three theoretical models are included to confirm the efficacy of SIS software by inverting only the synthetic resistivity sounding data. Then one synthetic data set based on a geological model and three field data sets (combination of resistivity and IP soundings) from diverse geological and geographical regions are included as validation of the proposal. It is hoped that the proposed scheme would complement the resistivity interpretation with special reference to shaly sand formations.     

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.     

3D inversion of DC data using artificial neural networks

In this paper, we investigate the applicability of artificial neural networks in inverting three-dimensional DC resistivity imaging data. The model used to produce synthetic data for training the artificial neural network (ANN) system was a homogeneous medium of resistivity 100 Ωm with an embedded anomalous body of resistivity 1000 Ωm. The different sizes for anomalous body were selected and their location was changed to different positions within the homogeneous model mesh elements. The 3D data set was generated using a finite element forward modeling code through standard 3D modeling software. We investigated different learning paradigms in the training process of the neural network. Resilient propagation was more efficient than any other paradigm. We studied the effect of the data type used on neural network inversion and found that the use of location and the apparent resistivity of data points as the input and corresponding true resistivity as the output of networks produces satisfactory results. We also investigated the effect of the training data pool volume on the inversion properties. We created several synthetic data sets to study the interpolation and extrapolation properties of the ANN. The range of 100–1000 Ωm was divided into six resistivity values as the background resistivity and different resistivity values were also used for the anomalous body. Results from numerous neural network tests indicate that the neural network possesses sufficient interpolation and extrapolation abilities with the selected volume of training data. The trained network was also applied on a real field dataset, collected by a pole-pole array using a square grid (8 ×8) with a 2-m electrode spacing. The inversion results demonstrate that the trained network was able to invert three-dimensional electrical resistivity imaging data. The interpreted results of neural network also agree with the known information about the investigation area.     

基于非结构双网格的2D RMT双参数同步反演研究

Radio-magnetotelluric(RMT)是以无线电发射机为信号源的一种地球物理勘探方法,近年来被广泛应用于数米至数十米内的近地表工程和环境地球物理勘探.目前,各类电磁资料的反演均是以寻求满足目标拟合差的地下介质电阻率分布为目的.然而,对于勘探频率为10~300kHz的RMT数据,由介电常数所引起的波动场在总场中的比例可达20%以上,在这种情况下,忽略介电常数,仅通过电阻率参数的反演来进行数据拟合势必降低反演资料解释的准确性.为解决这一问题,本文研究了基于电阻率-介电常数的双参数同步反演算法.构建了一个全新的双参数目标函数,并推导了双参数反演迭代方程组;通过灵敏度分析,研究了电阻率和介电常数对正演响应的影响,并据此提出相对电导率的概念,统一了反演参数的灵敏度;通过理论模型分析了参考频率、双参数正则化因子对反演结果的影响,并给出了一般性的参数优选方案.此外,为了能够灵活处理复杂地形,本文采用非结构的正反演双网格进行模型离散,并通过局部加密技术保证反演的速度和精度.最后,对一带地形的理论模型分别进行了单参数和双参数反演,结果表明单参数反演无法正确反映出地电信息,而双参数反演能够准确得到异常的分布,验证了本文所开发的双参数反演程序的有效性.     

Use of VFSA for resolution, sensitivity and uncertainty analysis in 1D DC resistivity and IP inversion

We present results from the resolution and sensitivity analysis of 1D DC resistivity and IP sounding data using a non-linear inversion. The inversion scheme uses a theoretically correct Metropolis–Gibbs' sampling technique and an approximate method using numerous models sampled by a global optimization algorithm called very fast simulated annealing (VFSA). VFSA has recently been found to be computationally efficient in several geophysical parameter estimation problems. Unlike conventional simulated annealing (SA), in VFSA the perturbations are generated from the model parameters according to a Cauchy-like distribution whose shape changes with each iteration. This results in an algorithm that converges much faster than a standard SA. In the course of finding the optimal solution, VFSA samples several models from the search space. All these models can be used to obtain estimates of uncertainty in the derived solution. This method makes no assumptions about the shape of an a posteriori probability density function in the model space. Here, we carry out a VFSA-based sensitivity analysis with several synthetic and field sounding data sets for resistivity and IP. The resolution capability of the VFSA algorithm as seen from the sensitivity analysis is satisfactory. The interpretation of VES and IP sounding data by VFSA, incorporating resolution, sensitivity and uncertainty of layer parameters, would generally be more useful than the conventional best-fit techniques.     

时间域航空电磁2.5维非线性共轭梯度反演

对于时间域航空电磁法二维和三维反演来说,最大的困难在于有效的算法和大的计算量需求.本文利用非线性共轭梯度法实现了时间域航空电磁法2.5维反演方法,着重解决了迭代反演过程中灵敏度矩阵计算、最佳迭代步长计算、初始模型选取等问题.在正演计算中,我们采用有限元法求解拉式傅氏域中的电磁场偏微分方程,再通过逆拉氏和逆傅氏变换高精度数值算法得到时间域电磁响应.在灵敏度矩阵计算中,采用了基于拉式傅氏双变换的伴随方程法,时间消耗只需计算两次正演,从而节约了大量计算时间.对于最佳步长计算,二次插值向后追踪法能够保证反演迭代的稳定性.设计两个理论模型,检验反演算法的有效性,并讨论了选择不同初始模型对反演结果的影响.模型算例表明:非线性共轭梯度方法应用于时间域航空电磁2.5维反演中稳定可靠,反演结果能够有效地反映地下真实电性结构.当选择的初始模型电阻率值与真实背景电阻率值接近时,能得到较好的反演结果,当初始模型电阻率远大于或远小于真实背景电阻率值时反演效果就会变差.     

Three-dimensional imaging of subsurface structures using resistivity data

We have developed a three-dimensional inverse scheme for carrying out DC resistivity surveys, incorporating complicated topography as well as arbitrary electrode arrays. The algorithm is based on the finite-element approximation to the forward problem, so that the effect of topographic variation on the resistivity data is effectively evaluated and incorporated in the inversion. Furthermore, we have enhanced the resolving power of the inversion using the active constraint balancing method. Numerical verifications show that a correct earth image can be derived even when complicated topographic variation exists. By inverting the real field data acquired at a site for an underground sewage disposal plant, we obtained a reasonable image of the subsurface structures, which correlates well with the surface geology and drill log data.     

Electrical Imaging of Impact Structures

Electrical imaging provides important subsurface information for the construction of hypervelocity impact models. We here provide an overview and evaluation of the current electrical imaging methods used in impact cratering studies. Although apparent resistivity models are commonly used in the geoelectrical imaging of impact structures, the reliability of these models has not hitherto been determined. In order to assess these imaging approaches in impact cratering, we investigate for the first time the discrepancies between the apparent resistivity and true resistivity models of an impact structure. To this end, we present (1) a new true resistivity model of the Araguainha impact structure in central Brazil by applying L2-norm inversion to previously published data, (2) apparent resistivity model of the impact structure, and (3) models obtained from different stages of the iterative tomographic inversions. Our results show that changes in vertical resistivity gradient are significantly better defined in the true resistivity models than in the apparent resistivity model. On the basis of these results, we outline a new approach that true resistivity models can be effectively assessed by applying both L1- and L2-norm inversion schemes together with the monitoring of intermediate models from iterative inversion. The results of our study highlight the importance of tomographic inversion of resistivity data in impact cratering studies, and they provide a data modeling framework and foundation for cost-effective subsurface imaging of impact structures in the future.     

Two-dimensional inversion of spectral induced polarization data using MPI parallel algorithm in data space

Traditional two-dimensional (2D) complex resistivity forward modeling is based on Poisson’s equation but spectral induced polarization (SIP) data are the coproducts of the induced polarization (IP) and the electromagnetic induction (EMI) effects. This is especially true under high frequencies, where the EMI effect can exceed the IP effect. 2D inversion that only considers the IP effect reduces the reliability of the inversion data. In this paper, we derive differential equations using Maxwell’s equations. With the introduction of the Cole–Cole model, we use the finite-element method to conduct 2D SIP forward modeling that considers the EMI and IP effects simultaneously. The data-space Occam method, in which different constraints to the model smoothness and parametric boundaries are introduced, is then used to simultaneously obtain the four parameters of the Cole—Cole model using multi-array electric field data. This approach not only improves the stability of the inversion but also significantly reduces the solution ambiguity. To improve the computational efficiency, message passing interface programming was used to accelerate the 2D SIP forward modeling and inversion. Synthetic datasets were tested using both serial and parallel algorithms, and the tests suggest that the proposed parallel algorithm is robust and efficient.     

Joint inversion of Rayleigh‐wave dispersion data and vertical electric sounding data: synthetic tests on characteristic sub‐surface models

In the traditional inversion of the Rayleigh dispersion curve, layer thickness, which is the second most sensitive parameter of modelling the Rayleigh dispersion curve, is usually assumed as correct and is used as fixed a priori information. Because the knowledge of the layer thickness is typically not precise, the use of such a priori information may result in the traditional Rayleigh dispersion curve inversions getting trapped in some local minima and may show results that are far from the real solution. In this study, we try to avoid this issue by using a joint inversion of the Rayleigh dispersion curve data with vertical electric sounding data, where we use the common‐layer thickness to couple the two methods. The key idea of the proposed joint inversion scheme is to combine methods in one joint Jacobian matrix and to invert for layer S‐wave velocity, resistivity, and layer thickness as an additional parameter, in contrast with a traditional Rayleigh dispersion curve inversion. The proposed joint inversion approach is tested with noise‐free and Gaussian noise data on six characteristic, synthetic sub‐surface models: a model with a typical dispersion; a low‐velocity, half‐space model; a model with particularly stiff and soft layers, respectively; and a model reproduced from the stiff and soft layers for different layer‐resistivity propagation. In the joint inversion process, the non‐linear damped least squares method is used together with the singular value decomposition approach to find a proper damping value for each iteration. The proposed joint inversion scheme tests many damping values, and it chooses the one that best approximates the observed data in the current iteration. The quality of the joint inversion is checked with the relative distance measure. In addition, a sensitivity analysis is performed for the typical dispersive sub‐surface model to illustrate the benefits of the proposed joint scheme. The results of synthetic models revealed that the combination of the Rayleigh dispersion curve and vertical electric sounding methods in a joint scheme allows to provide reliable sub‐surface models even in complex and challenging situations and without using any a priori information.     

1D and 2D Cole-Cole-inversion of time-domain induced-polarization data

A new method for the 2D inversion of induced polarization (IP) data in the time domain has been developed. The entire IP transients were observed and inverted into 2D Cole-Cole earth models, including resistivity, chargeability, relaxation time and the frequency constant. Firstly, a modified 1D time-domain electromagnetic algorithm was used to calculate the response of a layered polarizable ground. The transient signals were then inverted using the Marquardt method to derive the Cole-Cole parameters of each layer. However, model calculations showed that the EM effects could be neglected for the time range (>1 ms) and for the transmitter–receiver distances (<50 m) used in this study. Therefore, the induction effects were not considered for the solution of the 2D inverse problem and a DC solution was applied. An approximative forward algorithm was introduced in order to calculate the IP transients directly in the time domain and in order to speed up the inverse procedure. The approximation is highly accurate, and this is demonstrated by comparing the approximations with their exact solutions up to 3D. The inverse algorithm presented consists of two steps. The transient voltages of an array data set were inverted separately into a two-dimensional resistivity model for each time channel. The time-dependent resistivity of each cell was then interpreted as the response of a homogeneous half-space. In the 2D inversion algorithm, a 3D DC algorithm was used as a forward operator. The method only requires a standard 2D DC inversion and a homogenous half-space Cole-Cole inversion. The developed algorithm has been successfully applied to synthetic data sets and to a field data set obtained from a waste site situated close to Düren in Germany.     

Two-dimensional direct current (DC) resistivity inversion: Data space Occam's approach

A data space Occam's inversion algorithm for 2D DC resistivity data has been developed to seek the smoothest structure subject to an appropriate fit to the data. For traditional model space Gauss–Newton (GN) type inversion, the system of equations has the dimensions of M × M, where M is the number of model parameter, resulting in extensive computing time and memory storage. However, the system of equations can be mathematically transformed to the data space, resulting in a dramatic drop in its dimensions to N × N, where N is the number of data parameter, which is usually less than M. The transformation has helped to significantly reduce both computing time and memory storage. Numerical experiments with synthetic data and field data show that applying the data space technique to 2D DC resistivity data for various configurations is robust and accurate when compared with the results from the model space method and the commercial software RES2DINV.     

地电阻率多极距观测反演初步研究

地电阻率多极距观测的目的是借助对观测数据的反演获得地下介质中不同层位真电阻率的变化.本文以3层结构为例,对地电阻率多极距观测的一维反演的效果进行了初步地理论研究.首先研究了一维地电阻率结构下观测数据一维反演的模拟效果,得到了各层电阻率值,且与真电阻率值很接近,说明在地下电性结构为一维的情况下,电阻率多极距观测可以区分出不同地层的电阻率变化.其次,考虑到台址下电阻率结构的复杂性,研究了上两层界面存在起伏的情况下,多极距观测数据一维反演的效果,结果显示:当电阻率变化较小时,各层反演得到的电阻率的变化与真电阻率十分符合;当上两层介质的电阻率变化较大时,各层反演得到的电阻率出现畸变,与真实电阻率的变化情况存在一定差别,表明浅层电阻率变化达到一定程度后将会影响对深部电阻率变化情况的正确判断.一般情况下,观测的时间间隔越短,则连续两次观测时段内各层介质的电阻率变化越小,因此缩短多极距观测的时间间隔可能是避免出现上述畸变现象的有效观测手段.      

大地电磁一维连续介质反演的曲线对比法

根据电磁波的传播特性，把视电阻率随周期变化的曲线转化为电阻率随深度变化的曲线，并以此作为初始反演的地电模型．通过初始地电模型得到的视电阻率曲线与真实模型的视电阻率曲线的对比，对初始地电模型的电阻率值进行校正．校正后的反演模型的视电阻率曲线与真实模型的视电阻曲线的拟合程度有所提高．然后如此反复进行多次校正，获得与真实模型更为接近的反演模型，反演的拟合误差一般小于ｌ％．模型试验和实际例子表明，该方法的拟合程度优于Ｂｏｓｔｉｃｋ法．     

Inversion of band-limited TEM responses

It is shown that the electromagnetic (EM) spectrum is characterized by strong amplitude-modulated transmitters operating in the target bandwidth of transient electromagnetic (TEM) measurements. As these transmitters cause significant noise in TEM soundings, it is mandatory to band-limit the input signals to improve the signal-to-noise ratio and thereby the depth of exploration. Band-limitation will distort the TEM responses, which leads to erroneous inversion results if the applied low-pass filters are not accounted for in the inversion scheme. We incorporate the low-pass filters in the inversion scheme and test the inversion approach on theoretical and field data. Inversion of band-limited theoretical responses results in recovery of erroneous resistivity models if the filters are not included in the inversion scheme. By contrast, inversion of band-limited theoretical and field data, for which the applied low-pass filters are included in the inversion scheme, leads to recovery of similar resistivity models, independent of the applied cut-off frequencies.     

DAMPED LEAST-SQUARES INVERSION OF TIME-DOMAIN AIRBORNE EM DATA BASED ON SINGULAR VALUE DECOMPOSITION1

Airborne electromagnetic (AEM) methods are increasingly being used as tools of geological mapping, groundwater exploration and prospecting for coal and lignite. In such applications, quantitative interpretation is commonly based on the layered-earth model. A new approach, a damped least-squares inversion with singular value decomposition, is proposed for interpretation of time-domain, towed-bird AEM data. Studies using theoretical and field AEM data indicate that inversion techniques are dependable and provide fast converging solutions. An analysis has been made of the accuracy of model parameter determination, which depends on resistivity and thickness distribution. In the common case of conductive overburden, upper-layer resistivity and thickness are usually well determined, although situations exist where their separation becomes difficult. In the case of a resistive layer overlying a conductive basement, the layer thickness is the best-determined parameter. In both cases, estimates of basement resistivity are the least reliable. Field data obtained with the Chinese-made M-l AEM system in Dongling, Anhui Province, China, were processed using the described inversion algorithm. The survey area comprised fluvial Cenozoic clays and weathered Mesozoic sediments. Inversion of AEM data resulted in accurate depth-to-bedrock sections and realistic estimates of the resistivities of overburden and bedrock which agree with the results of drilling and resistivity sounding.     

EXAMPLES OF RESOLUTION IMPROVEMENT IN GEOELECTRICAL SOUNDINGS APPLIED TO GROUNDWATER INVESTIGATIONS1

One simulation and two field examples from New Jersey illustrate resolution improvement in geoelectrical soundings applied to groundwater exploration. Layered-earth parameter resolution is derived from data obtained with the commonly used methods of resistivity, induced polarization (IP) and transient electromagnetic (TEM) soundings. Resolution improvement is achieved by simultaneous inversion of two or more data sets and by constraining parameters of the inverse problem. A quantitative analysis showing the contribution of IP data to the resolution of geo-electric sections is presented. Simultaneous inversion of simple IP data with conventional resistivity and resistivity-TEM data sets resulted in improved parameter resolution. IP data improved resolution in three ways: (1) by decoupling correlated layered-earth parameters, (2) by adding information to a geological interpretation about a second physical property, namely chargeability, and (3) by increasing the electrical information available.     

大地电磁法三维快速松弛反演

实现大地电磁法快速三维反演的关键在于找到快速计算灵敏度矩阵的方案. 本文在对大地电磁三维张量阻抗表达式进行深入分析的基础上，获得了三维快速松弛反演算法的灵敏度函数表达式，解决了三维快速松弛反演的核心问题；为了减小反演解的多解性，针对三维问题定义了最小构造函数，实现了求最小构造的三维快速松弛反演算法. 对二维棱柱体、三维棱柱体理论模型的大地电磁合成数据进行了反演试算，反演结果与理论模型相吻合. 对日本Kayabe地区实测资料进行了XY模式反演，反演得到的地电模型较好地反映了地热资源的分布. 反演试算结果表明大地电磁三维快速松弛反演具有计算速度快、结果稳定可靠等特点；使三维快速松弛反演在普通微机上得以实现.     

A NONITERATIVE PROCEDURE FOR INVERTING PLANE-WAVE REFLECTION DATA AT SEVERAL ANGLES OF INCIDENCE USING THE RICCATI EQUATION*

Various exact methods of inverting the complete waveform of vertical seismic reflection data to produce acoustic impedance profiles have been suggested. These inverse methods generally remain valid for nonvertical, plane-wave data, provided total reflection does not occur. Thus, in principle, the “seismogram” at each ray parameter in a slant stack can be interpreted separately. Rather than invert each plane-wave seismogram separately, they can all be interpreted simultaneously and an “average” model thus obtained. Inversion for both the velocity and the density also becomes possible when two or more plane-wave seismograms are simultaneously inverted. The theory for a noniterative inversion method, based on the time-domain Riccati equation, is discussed. Numerical examples of inversions using this technique on synthetic data demonstrate its numerical stability and the advantage of simultaneous inversion of several seismograms to reduce the effect of noise in the data and increase the stability of the inversion process.