Implicit structural inversion of gravity data using linear programming,a validation study*

In this study, a regional scale gravity data set has been inverted to infer the structure (topography) of the top of the basement underlying sub‐horizontal strata. We apply our method to this real data set for further proof of concept, validation and benchmarking against results from an earlier forward modelling done elsewhere. Our aim is to carry out implicit structural inversion, i.e., to obtain a geologically reasonable model, without specifically solving for structure. The 2.5D volume of interest is parametrized with homogeneous horizontal prisms and a two‐lithology medium is assumed. A possible regional linear trend and a general floating reference are also inverted for. Using a gridded parametrization, linear programming is used to minimize the L₁ ‐norm of the data misfit, relative to a floating reference. Given a known density contrast between the lithologies, an inversion using linear programming has the intrinsic advantage that a relatively sharp image of the sub‐surface is retrieved instead of a smooth one. The model recovered is almost bi‐modal and its general features seem to be robust with respect to several parametrization scenarios investigated. The floating reference and a linear trend in the data were also retrieved simultaneously. The inversion results, indicating two depressions in the basement, are robust and agree with those obtained earlier based upon detailed 2D forward modelling using many narrow, near‐vertical prisms.     

基于交叉梯度约束的电阻率法和背景噪声法三维联合反演研究

电阻率法和背景噪声法是通过获得地下介质的电阻率和速度参数的分布来探究地球内部物质分布的非均匀性特征,联合反演可以有效地发挥两种方法的优势,减小反演多解性.本文采用有限内存拟牛顿（L-BFGS）算法实现了电阻率法和背景噪声法的单方法三维反演,然后基于电阻率和速度模型结构耦合约束,采用交替迭代方式实现了电阻率法和背景噪声法的三维联合反演.通过设计规则体组合模型和不规则体组合模型进行理论模型合成数据三维反演试算,结果表明：联合反演可以获得同时满足数据拟合差和结构更为相似的速度-电阻率模型;联合反演可以压制背景噪声单方法反演出现的假异常,获得更精细的速度结构;联合反演获得的电阻率模型对倾斜异常体、高阻覆层下方异常体、圈闭的高/低阻体等边界信息有明显的提升,有效克服电阻率法单方法反演的局限,提高深部电阻率的分辨率.     

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.     

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.     

An assessment of inversion methods for AEM data applied to environmental studies

Airborne electromagnetic (AEM) surveys are currently being flown over populated areas and applied to detailed problems using high flight line densities. Interpretation information is supplied through a model of the subsurface resistivity distribution. Theoretical and survey data are used here to study the character and reliability of such models. Although the survey data were obtained using a fixed-wing system, the corresponding associations with helicopter, towed-bird systems are discussed. Both Fraser half-space and 1D inversion techniques are considered in relation to their ability to distinguish geological, cultural and environmental influences on the survey data. Fraser half-space modelling provides the dual interpretation parameters of apparent resistivity and apparent depth at each operational frequency. The apparent resistivity was found to be a remarkably stable parameter and appears robust to the presence of a variety of at-surface cultural features. Such features provide both incorrect altitude data and multidimensional influences. Their influences are observed most strongly in the joint estimate of apparent depth and this accounts for the stability of the apparent resistivity. Positive apparent depths, in the example data, result from underestimated altitude measurements. It is demonstrated that increasingly negative apparent depths are associated with increasing misfits between a 1D model and the data. Centroid depth calculations, which are a transform of the Fraser half-space parameters, provide an example of the detection of non-1D influences on data obtained above a populated area. 1D inversion of both theoretical and survey data is examined. The simplest use of the 1D inversion method is in providing an estimate of a half-space resistivity. This can be undertaken prior to multilayer inversion as an initial assessment. Underestimated altitude measurements also enter the problem and, in keeping with the Fraser pseudo-layer concept, an at-surface highly resistive layer of variable thickness can be usefully introduced as a constrained parameter. It is clearly difficult to ascribe levels of significance to a ‘measure’ of misfit contained in a negative apparent depth with the dimensions of metres. The reliability of 1D models is better assessed using a formal misfit parameter. With the misfit parameter in place, the example data suggest that the 1D inversion methods provide reliable apparent resistivity values with a higher resolution than the equivalent information from the Fraser half-space estimates.     

Rapid least-squares inversion of apparent resistivity pseudosections by a quasi-Newton method1

A fast inversion technique for the interpretation of data from resistivity tomography surveys has been developed for operation on a microcomputer. This technique is based on the smoothness-constrained least-squares method and it produces a two-dimensional subsurface model from the apparent resistivity pseudosection. In the first iteration, a homogeneous earth model is used as the starting model for which the apparent resistivity partial derivative values can be calculated analytically. For subsequent iterations, a quasi-Newton method is used to estimate the partial derivatives which reduces the computer time and memory space required by about eight and twelve times, respectively, compared to the conventional least-squares method. Tests with a variety of computer models and data from field surveys show that this technique is insensitive to random noise and converges rapidly. This technique takes about one minute to invert a single data set on an 80486DX microcomputer.     

Application of the 3D magnetotelluric inversion code in a geologically complex area

The WSINV3DMT code makes the implementation of 3D inversion of magnetotelluric data feasible using a single PC. Audio‐magnetotelluric data were collected along two profiles in a Cu‐Ni mining area in Xinjiang, China, where the apparent resistivity and phase curves, the phase tensors and the magnetic induction vectors indicate a complex 3D conductivity structure. 3D inversions were carried out to reveal the electrical structure of the area. The final 3D model is selected from the inversion results using different initial Lagrange values and steps. The relatively low root‐mean‐square (rms) misfit and model norm indicate a reliable electrical model. The final model includes four types of low resistivity areas, the first ones coincide with the known location of an orebody and further forward modelling indicates that they are not in full connectivity to form a low resistivity zone. The second ones are not controlled by magnetotelluric sites and embody little information of the observed data, they are considered as tedious structures. The third one is near to the regional Kangguer fault and should be treated carefully considering the effect of the fault. The last ones are isolated and existing at a limited level as the first ones, they should be paid more attention to.     

Field test to compare 3D imaging capabilities of three arrays in a site with high resistivity contrast regions

The applicability of three kinds of electrode configurations used to delineate a buried horizontal pipe was studied. A 3D resistivity imaging survey was carried out along eight parallel lines using pole-pole, pole-dipole, and dipole-dipole arrays with 1m minimum electrode spacings. Roll-along measurements were carried out to cover a rectangular grid. The 2D and 3D least squares algorithms based on the robust inversion method were used in the inversion of the apparent resistivity data sets. The 2D inversion of data sets could not delineate the orientation and dimension of the subsurface anomalies clearly. To obtain more accurate results, a 3D joint inversion of the pole-pole and pole-dipole data sets was performed, as well as of pole-pole and dipole-dipole data sets. In this case, both horizontal and vertical dimensions of subsurface structures were resolved. The resulting model obtained from each array was compared to those of joint inversion method. The result showed that the horizontal resolution does not improve so much as that in the vertical direction when joint inversion is applied.     

The use of reference models from a priori data to guide 2D inversion of electrical resistivity tomography data

Clay-rich till plains cover much of the UK. Such sites are attractive locations for landfills, since clay aquitards lower the risk of landfill leachate entering groundwater. However, such tills often contain sand and gravel bodies that can act as leachate flow routes. Such bodies may not be detected by conventional site investigation techniques such as drilling boreholes and trial pitting. A method of guided inversion, where a priori data are used to construct structural reference models for use in inverting electrical resistivity tomography data, was proposed as a tool to improve the detection of sand and gravel bodies within clay-rich till sequences.
Following a successful 2D guided inversion synthetic modelling study, a field study was undertaken. Wenner 2D electrical resistivity tomography lines, resistivity cone penetrometry bores and electromagnetic induction ground resistivity data were collected over a site on the East Yorkshire coast, England, where sand and gravel lenses were known to exist from cliff exposures. A number of equally valid geoelectrical models were constructed using the electromagnetic and resistivity cone data. These were used as structural reference models in the inversion of the resistivity tomography data. Blind inversion using an homogenous reference model was also carried out for comparison.
It was shown for the first time that the best solution model produced by 2D inversion of one data set with a range of structural reference models could be determined by using the l ₂ model misfit between the solution models and associated reference models (reference misfit) as a proxy for the l ₂ misfit between the solution models and the synthetic model or 'best-guess' geoelectrical model (true misfit). The 2D methodology developed here is applicable in clay-rich till plains containing sand and gravel bodies throughout the UK.     

Advances in the combined interpretation of seismics with magnetotellurics1

Five examples, obtained during exploration for hydrocarbons in the Pannonian Basin of Hungary, are used to show how the interpretation of seismic sections can be usefully complemented by results from MT surveys. Selection of the most appropriate MT quantities, considered to be proper ‘MT attributes’ for the purpose of visualization as well as recognition of the subsurface structures and the different inversions of MT data is essential for practical integration of seismic and MT surveys. A new technique providing a semiquantitive MT-attribute pseudosection for the purpose of visualization of the subsurface structures is proposed. The procedure utilizes derivative functions of the phase of MT impedance for visualization and derives estimated depths from the Bostick transformation of Cagniard apparent resistivities. On the basis of the MT-attribute pseudosections, constructed from the phase derivatives and transformed resistivity data, depths are estimated for interfaces between geological formations with significant resistivity contrast. In particular examples, the interface between the Tertiary sediments and the older basement rocks as well as tectonic fracture zones with decreased resistivity can be resolved.     

一维垂直各向异性介质频率域海洋可控源电磁资料反演方法

本文提出了一维垂直各向异性(VTI)介质倾斜偶极源频率域海洋可控源电磁(CSEM)资料高斯-牛顿反演方法.在电阻率各向异性介质水平偶极源和垂直偶极源海洋CSEM正演算法的基础上,利用欧拉旋转方法,实现了各向异性介质倾斜偶极源海洋CSEM正演算法.海洋可控源电磁场关于地下介质横向电阻率(ρ_h)和垂向电阻率(ρ_v)的偏导数(即灵敏度矩阵)是解析计算的,结合垂直各向异性介质横向电阻率与垂向电阻率的关系,将各向异性率融入到正则化因子选择中,实现了正则化因子的自适应选择.理论模型合成数据和实测资料反演算例表明,我们提出的反演方法能够较准确的重构海底围岩和基岩的各向异性电阻率以及高阻薄层的埋藏深度、厚度和垂向电阻率.     

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.     

Field test of sub‐basalt hydrocarbon exploration with marine controlled source electromagnetic and magnetotelluric data

The recent use of marine electromagnetic technology for exploration geophysics has primarily focused on applying the controlled source electromagnetic method for hydrocarbon mapping. However, this technology also has potential for structural mapping applications, particularly when the relative higher frequency controlled source electromagnetic data are combined with the lower frequencies of naturally occurring magnetotelluric data. This paper reports on an extensive test using data from 84 marine controlled source electromagnetic and magnetotelluric stations for imaging volcanic sections and underlying sediments on a 128‐km‐long profile. The profile extends across the trough between the Faroe and Shetland Islands in the North Sea. Here, we focus on how 2.5D inversion can best recover the volcanic and sedimentary sections. A synthetic test carried out with 3D anisotropic model responses shows that vertically transverse isotropy 2.5D inversion using controlled source electromagnetic and magnetotelluric data provides the most accurate prediction of the resistivity in both volcanic and sedimentary sections. We find the 2.5D inversion works well despite moderate 3D structure in the synthetic model. Triaxial inversion using the combination of controlled source electromagnetic and magnetotelluric data provided a constant resistivity contour that most closely matched the true base of the volcanic flows. For the field survey data, triaxial inversion of controlled source electromagnetic and magnetotelluric data provides the best overall tie to well logs with vertically transverse isotropy inversion of controlled source electromagnetic and magnetotelluric data a close second. Vertical transverse isotropy inversion of controlled source electromagnetic and magnetotelluric data provided the best interpreted base of the volcanic horizon when compared with our best seismic interpretation. The structural boundaries estimated by the 20‐Ω·m contour of the vertical resistivity obtained by vertical transverse isotropy inversion of controlled source electromagnetic and magnetotelluric data gives a maximum geometric location error of 11% with a mean error of 1.2% compared with the interpreted base of the volcanic horizon. Both the model study and field data interpretation indicate that marine electromagnetic technology has the potential to discriminate between low‐resistivity prospective siliciclastic sediments and higher resistivity non‐prospective volcaniclastic sediments beneath the volcanic section.     

ESTIMATION OF TEMPORAL CHANGES IN SOIL MOISTURE USING RESISTIVITY METHOD

The temporal variation in a soil moisture profile can be studied using resistivity sounding data acquired at different times. The layered earth model based estimation of soil moisture from apparent resistivity data is a two-step non-linear inversion. Firstly, the apparent resistivity data are inverted to derive the layer resistivity variations and thicknesses and, secondly, the moisture content is estimated from these layer resistivity variations using a calibration equation. The soil moisture–resistivity problem was studied using the one-dimensional formulation of resistivity problem. A generalized geoelectric earth model was considered to simulate the soil moisture distribution and its temporal variation in the unsaturated zone. An algorithm (RESMOS) for the interpretation of the apparent resistivity data in terms of soil moisture variations through this two-step inversion process is reported.     

裂陷盆地基底双界面模式二维重力反演

裂陷盆地基底的起伏表现为非光滑的几何形态,传统的重力反演结果并不能很好地反映这种特点.此外,大多数情况下,重力观测面并不位于盆地上界面,应为单独的起伏观测面,盆地应为上界面和基底组成的双界面模式.基于此,本文研究了起伏观测面上裂陷盆地基底双界面模式二维重力反演方法.研究中假设沉积盆地的沉积层与基底的密度差随深度按双曲线规律变化.将沉积盆地的沉积层剖分成相邻的垂直柱体,其水平尺寸是已知的,顶面与沉积层上界面重合,底面深度代表基底的深度,即为要反演的参数.反演中引入全变差函数作为盆地模型的约束,使得反演结果呈现非光滑形态,符合裂陷盆地基底特征.为减小反演多解性,引入已知深度点作为约束.建立由重力数据拟合、已知深度约束及全变差函数组成的目标函数,采用非线性共轭梯度算法使目标函数最小化.模型试算结果表明该方法可反演裂陷盆地基底起伏,并通过调整正则化参数的值可反演坳陷盆地基底起伏.将该反演方法用于珠江口盆地惠州凹陷和运城-临汾裂陷盆地实际资料处理,其结果较好地反映了裂陷盆地基底起伏特征,为研究盆地构造、油气勘探等提供重要参考.     

Feasibility study for an anisotropic full waveform inversion of cross‐well seismic data

Anisotropy is often observed due to the thin layering or aligned micro‐structures, like small fractures. At the scale of cross‐well tomography, the anisotropic effects cannot be neglected. In this paper, we propose a method of full‐wave inversion for transversely isotropic media and we test its robustness against structured noisy data. Optimization inversion techniques based on a least‐square formalism are used. In this framework, analytical expressions of the misfit function gradient, based on the adjoint technique in the time domain, allow one to solve the inverse problem with a high number of parameters and for a completely heterogeneous medium. The wave propagation equation for transversely isotropic media with vertical symmetry axis is solved using the finite difference method on the cylindrical system of coordinates. This system allows one to model the 3D propagation in a 2D medium with a revolution symmetry. In case of approximately horizontal layering, this approximation is sufficient. The full‐wave inversion method is applied to a crosswell synthetic 2‐component (radial and vertical) dataset generated using a 2D model with three different anisotropic regions. Complex noise has been added to these synthetic observed data. This noise is Gaussian and has the same amplitude f?k spectrum as the data. Part of the noise is localized as a coda of arrivals, the other part is not localized. Five parameter fields are estimated, (vertical) P‐wave velocity, (vertical) S‐wave velocity, volumetric mass and the Thomsen anisotropic parameters epsilon and delta. Horizontal exponential correlations have been used. The results show that the full‐wave inversion of cross‐well data is relatively robust for high‐level noise even for second‐order parameters such as Thomsen epsilon and delta anisotropic parameters.     

Appraisal of equivalence and suppression problems in 1D EM and DC measurements using global optimization and joint inversion*

Global optimization with very fast simulated annealing (VFSA) in association with joint inversion is performed for 1D earth structures. The inherent problems of equivalence and suppression in electromagnetic (EM) and direct current (DC) resistivity methods are studied. Synthetic phase data from multifrequency sounding using a horizontal coplanar coil system and synthetic apparent resistivity data from Schlumberger DC resistivity measurements are inverted individually and jointly over different types of layered earth structures. Noisy data are also inverted. The study reveals that global optimization of individual data sets cannot solve inherent equivalence or suppression problems. Joint inversion of EM and DC measurements can overcome the problem of equivalence very well. However, a suppression problem cannot be solved even after combination of data sets. This study reveals that the K-type earth structure is easiest to resolve while the A-type is the most difficult. We also conclude that the equivalence associated with a thin resistive layer can be resolved better than that for a thin conducting layer.     

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.     

Calibrating a Salt Water Intrusion Model with Time‐Domain Electromagnetic Data

Salt water intrusion models are commonly used to support groundwater resource management in coastal aquifers. Concentration data used for model calibration are often sparse and limited in spatial extent. With airborne and ground‐based electromagnetic surveys, electrical resistivity models can be obtained to provide high‐resolution three‐dimensional models of subsurface resistivity variations that can be related to geology and salt concentrations on a regional scale. Several previous studies have calibrated salt water intrusion models with geophysical data, but are typically limited to the use of the inverted electrical resistivity models without considering the measured geophysical data directly. This induces a number of errors related to inconsistent scales between the geophysical and hydrologic models and the applied regularization constraints in the geophysical inversion. To overcome these errors, we perform a coupled hydrogeophysical inversion (CHI) in which we use a salt water intrusion model to interpret the geophysical data and guide the geophysical inversion. We refer to this methodology as a Coupled Hydrogeophysical Inversion‐State (CHI‐S), in which simulated salt concentrations are transformed to an electrical resistivity model, after which a geophysical forward response is calculated and compared with the measured geophysical data. This approach was applied for a field site in Santa Cruz County, California, where a time‐domain electromagnetic (TDEM) dataset was collected. For this location, a simple two‐dimensional cross‐sectional salt water intrusion model was developed, for which we estimated five uniform aquifer properties, incorporating the porosity that was also part of the employed petrophysical relationship. In addition, one geophysical parameter was estimated. The six parameters could be resolved well by fitting more than 300 apparent resistivities that were comprised by the TDEM dataset. Except for three sounding locations, all the TDEM data could be fitted close to a root‐mean‐square error of 1. Possible explanations for the poor fit of these soundings are the assumption of spatial uniformity, fixed boundary conditions and the neglecting of 3D effects in the groundwater model and the TDEM forward responses.     

Deep resistivity structure of the Dikili-Bergama region, west Anatolia, revealed by two- dimensional inversion of vertical electrical sounding data

Within the framework of the National Marine Geological and Geophysical Program, we re‐examined deep vertical electrical sounding (VES) data. The data, measured in 1968 by the General Directorate of Mineral Research and Exploration (MTA) of Turkey with the aim of exploring the deep resistivity structure of the Dikili–Bergama region, focus on the geothermal potential. The geoelectrical resistivity survey was conducted using a Schlumberger array with a maximum electrode half‐spacing of 4.5 km. The two‐dimensional (2D) inversion was utilized to interpret the VES data that were collected along 15‐ to 30‐km profiles. The 2D resistivity–depth cross‐sections obtained show very low resistivity values near the Dikili and Kaynarca hot springs. The 2D inversion results also indicate the presence of fault zones striking nearly N–S and E–W, and fault‐bounded graben‐horst structures that show promising potential for geothermal field resources. The 2D gravity model, which is in good agreement with the density variation of the region, supports the resistivity structure revealed by 2D inversion. The lithology information obtained from the borehole near Kaynarca also confirms the results of the resistivity interpretation and the density model.