Smoothness-constrained Inversion Research Articles

Illuminating traces of an Achaemenid's monumental complex in the Southern Caucasus by electrical resistivity tomography

The Achaemenid Empire controlled the Western Asian landmass between ca. 550–300 BCE. By some estimates, its subject peoples made up almost 44% of the world population. In recent years, studies have moved from a consideration of its core region, located in the province of Fars, Iran, to the investigation of imperial rule in the dependent regions. Since Achaemenid centers were often built of mudbrick and thus not easy to recognize by traditional archaeological methods, geophysical methods play an important role in this research. This paper reports the results of an innovative Electrical Resistivity Tomography (ERT) study undertaken at one such center, Saaklemo, in the Kakheti Province, Republic of Georgia and shows the wide impact of the geophysical methods in the detection of the archaeological sites.In three areas of this ancient site between 2020 and 2021, following archaeological survey and magnetometer prospection, several three-dimensional (3D) ERT measurements, composed of dense parallel two-dimensional (2D) tomographies with dipole-dipole configuration, were carried out. To reconstruct the resistivity distribution of the subsurface, we used the smoothness-constrained inversion method, especially as the substructure was made of mudbrick and thus resistivity values of these anomalies were close to the background anomaly. Therefore, with smoothness-constrained inversion, we could detect the slight resistivity differences, which were partly due to density differences, and hence this method was capable of detecting these anomalies.In area 1, for the first time with ERT method, we were successful in discovering an extensive mudbrick (partly burnt) substructure at a depth of 40 cm to 1 m. After analyzing the magnetic data and a 3D model of this substructure based on the ERT data, a large excavation was conducted and a mudbrick substructure with six in-situ bell-shaped column bases was unearthed, which was destroyed by conflagration. The size and the architectural techniques used for this building prove that this excavated structure was an administrative building and served for representative purposes. Further presumed mudbrick structures were detected in the other two areas with the ERT method, but they have not yet been excavated.

Imaging Islamic architecture in the multileveled archaeological settlement of Tell Dibgou, Northeast Nile Delta, Egypt

Abstract Tell Dibgou, northeastern Nile Delta (Egypt), is a typical type of complex archaeological site that contains ruins from different chronological periods (Pharaonic, Byzantine, Greece-Roman and Islamic), as evidenced by the findings of pottery kilns, pottery shards, fired brick and colored glass. This study aims to image the topmost archaeological layer of Islamic architecture using electrical resistivity tomography and land-based magnetic surveys. Two nearby areas in the northern part of Tell Dibgou were investigated; the first was surveyed using resistivity tomography and magnetics, and the second used total-field and vertical magnetic gradients. Several electrode arrays (dipole–dipole, Wenner Beta, Wenner Schlumberger and pole–dipole) were tested to select the one that provides better resolution. Two inversion schemes were applied where the smoothness-constrained inversion smeared and blurred the archaeological boundaries, and the robust inversion produced models with well-defined and sharp edges. The vertical extent of the topmost archaeological layer of Islamic settlers was imaged using the electrical resistivity tomography; however, the presence of slag debris, collapsed bricks, and pottery shards at shallow depths complicated the discrimination of the closest walls. The interpretation of magnetic data (total and vertical gradients) in terms of depth and geometry was executed using semi-automated techniques such as three-dimensional, Euler deconvolution and tilt angle derivative. The geometry of archaeological targets is interpreted to be closed rooms, water tanks and corridors constructed from fired brick walls filled with clay sediments. The depths yielded from both techniques are comparable and range from >1 m to up to 2 m.

The use of asymmetric time constraints in 4-D ERT inversion

Time-lapse resistivity surveys are commonly used to monitor temporal changes in the subsurface. In certain cases, it is known from other information that the resistivity will only decrease or increase with time. The 4-D resistivity smoothness-constrained inversion method reduces artifacts due to noise by including a temporal roughness filter constraint that ensures the temporal changes vary in a smooth manner. A least-squares optimization method is used to find a solution by attempting to locate the minimum of an objective function that consists of the data misfit and model (spatial and temporal) roughness. In some cases, the 4-D time-lapse inverse models show an increase in the resistivity with time in parts of the subsurface where it is only expected to decrease (or vice versa). We compare two methods, the barrier function and transformation methods, that attempt to minimize or eliminate these artifacts. We incorporate the barrier function constraint into the 4-D inverse method by using a modified difference matrix as a temporal roughness filter. The barrier function constraint includes an additional term that increases the objective function value greatly if the model values cross the allowed thresholds. This greatly minimizes the artifacts but does not completely eliminate them. It has the advantage that there are minimal changes in the objective function in regions of model space that are not close to the imposed thresholds. The method of transformations changes the model parameter such that the additional positivity or negativity constraints are implicitly included in the transformed model parameter. It has the advantage that it can completely eliminate the artifacts. However, it modifies the entire objective function which could be a disadvantage in some cases. We also explore a combination of the two methods, using the barrier function method to generate an initial model that minimizes the artifacts followed by the transformation method. This hybrid technique completely removes the residual artifacts left by the barrier method, and produces an inverse model which is closer to the true model for a synthetic data set. We also describe a post-inversion modification of the L-curve method to determine the optimum model that takes into account the non-linear nature of the inverse problem and the forward modelling method. The technique gave an estimate of the noise level for a field data set and produced a model which is consistent with independent hydrological measurements at the test site.

Efficient multiscale imaging of subsurface resistivity with uncertainty quantification using ensemble Kalman inversion

SUMMARYElectrical resistivity tomography (ERT) is widely used to image the Earth’s subsurface and has proven to be an extremely useful tool in application to hydrological problems. Conventional smoothness-constrained inversion of ERT data is efficient and robust, and consequently very popular. However, it does not resolve well sharp interfaces of a resistivity field and tends to reduce and smooth resistivity variations. These issues can be problematic in a range of hydrological or near-surface studies, for example mapping regolith-bedrock interfaces. While fully Bayesian approaches, such as those using Markov chain Monte Carlo sampling, can address the above issues, their very high computation cost makes them impractical for many applications. Ensemble Kalman inversion (EKI) offers a computationally efficient alternative by approximating the Bayesian posterior distribution in a derivative-free manner, which means only a relatively small number of ‘black-box’ model runs are required. Although common limitations for ensemble Kalman filter-type methods apply to EKI, it is both efficient and generally captures uncertainty patterns correctly. We propose the use of a new EKI-based framework for ERT which estimates a resistivity model and its uncertainty at a modest computational cost. Our EKI framework uses a level-set parametrization of the unknown resistivity to allow efficient estimation of discontinuous resistivity fields. Instead of estimating level-set parameters directly, we introduce a second step to characterize the spatial variability of the resistivity field and infer length scale hyperparameters directly. We demonstrate these features by applying the method to a series of synthetic and field examples. We also benchmark our results by comparing them to those obtained from standard smoothness-constrained inversion. Resultant resistivity images from EKI successfully capture arbitrarily shaped interfaces between resistivity zones and the inverted resistivities are close to the true values in synthetic cases. We highlight its readiness and applicability to similar problems in geophysics.

Basis-constrained Bayesian Markov-chain Monte Carlo difference inversion for geoelectrical monitoring of hydrogeologic processes

Bayesian Markov-chain Monte Carlo (McMC) techniques are increasingly being used in geophysical estimation of hydrogeologic processes due to their ability to produce multiple estimates that enable comprehensive assessment of uncertainty. Standard McMC sampling methods can, however, become computationally intractable for spatially distributed, high-dimensional problems. We have developed a novel basis-constrained Bayesian McMC difference inversion framework for time-lapse geophysical imaging. The strategy parameterizes the Bayesian inversion model space in terms of sparse, hydrologic-process-tuned bases, leading to dimensionality reduction while accounting for the physics of the target hydrologic process. We evaluate the algorithm on cross-borehole electrical resistivity tomography (ERT) field data acquired during a heat-tracer experiment. We validate the ERT-estimated temperatures with direct temperature measurements at two locations on the ERT plane. We also perform the inversions using the conventional smoothness-constrained inversion (SCI). Our approach estimates the heat plumes without excessive smoothing in contrast with the SCI thermograms. We capture most of the validation temperatures within the 90% confidence interval of the mean. Accounting for the physics of the target process allows the detection of small temperature changes that are undetectable by the SCI. Performing the inversion in the reduced-dimensional model space results in significant gains in computational cost.

Simulation of fine-scale electrical conductivity fields using resolution-limited tomograms and area-to-point kriging

SUMMARY Deterministic geophysical inversion approaches yield tomographic images with strong imprints of the regularization terms required to solve otherwise ill-posed inverse problems. While such tomograms enable an adequate assessment of the larger-scale features of the probed subsurface, the finer-scale details tend to be unresolved. Yet, representing these fine-scale structural details is generally desirable and for some applications even mandatory. To address this problem, we have developed a two-step methodology based on area-to-point kriging to generate fine-scale multi-Gaussian realizations from smooth tomographic images. Specifically, we use a co-kriging system in which the smooth, low-resolution tomogram is related to the fine-scale heterogeneity through a linear mapping operation. This mapping is based on the model resolution and the posterior covariance matrices computed using a linearization around the final tomographic model. This, in turn, allows us for analytical computations of covariance and cross-covariance models. The methodology is tested on a heterogeneous synthetic 2-D distribution of electrical conductivity that is probed with a surface-based electrical resistivity tomography (ERT) survey. The results demonstrate the ability of this technique to reproduce a known geostatistical model characterizing the fine-scale structure, while simultaneously preserving the large-scale structures identified by the smoothness-constrained tomographic inversion. Small discrepancies between the geophysical forward responses of the realizations and the reference synthetic data are attributed to the underlying linearization. Overall, the method provides an effective and fast alternative to more comprehensive, but computationally more expensive approaches, such as, for example, Markov chain Monte Carlo techniques. Moreover, the proposed method can be used to generate fine-scale multivariate Gaussian realizations from virtually any smoothness-constrained inversion results given the corresponding resolution and posterior covariance matrices.

Using impressed current cathodic protection systems of pipelines for electromagnetic exploration

Low-frequency electromagnetic (EM) signals generated by networks of technical infrastructure such as power-lines, pipelines, or railways may provide a cheap and efficient means to perform EM depth sounding of the upper few kilometers of the earth. We attempt to use the signals emitted by an impressed current cathodic protection (ICCP) system of a 35 km long gas pipeline segment in northwestern Germany. The installed ICCP system uses a periodical 12 s on/3 s off current switching scheme, which resembles current waveforms used in controlled-source electromagnetics (CSEM). In contrast to CSEM, where a grounded electrical dipole is used as the source, the current flow in pipelines is not constant along its legs. Our efforts are therefore concentrated toward the determination of the temporal and spatial behavior of the electrical current within the investigated pipeline segment. Although the time dependency of the current can be measured directly at the injection point, the spatial distribution is only accessible through indirect observations. We use fluxgate magnetic field measurements at multiple locations directly above the pipeline to infer the local source current and its frequency-dependency and phase lag. We observe that the current decays roughly exponentially away from the injection point, exhibits a position-dependent frequency dependency, and experiences a phase shift that accumulates to more than 30° at the ends of the segment. These effects can be consistently explained with a transmission line model. Having determined the current distribution, we can represent the pipeline as an EM source superposed of point dipoles. The estimated source model allows us to predict the electric (and magnetic) fields at remote locations. To verify our approach, we deploy an array of telluric recorders in the vicinity of the pipeline, estimate the frequency-domain transfer functions, and invert the data into a 3D electrical conductivity model using smoothness-constrained inversion techniques.

Leachate detection via statistical analysis of electrical resistivity and induced polarization data at a waste disposal site (Northern Israel)

Landfill site is an intensely heterogeneous medium, in which application of 2-D geophysical techniques frequently does not enable its accurate examination due to the lack of correlation between different geophysical profiles. The study was aimed to eliminate such difficulty. Hereafter, the leachate detection methodology is presented including 2-D smoothness-constrained inversion of geoelectrical raw data collected in situ, the inverted data normalization and statistical examination. To eradicate influence of high heterogeneity of the landfill body, the generalized statistical model of the entire site was created including the likelihood analysis of the entire normalized data set. Three geoelectrical parameters were involved in the analysis: inversed electrical resistivity, inversed chargeability (IP) and the ratio of inversed chargeability-to-inversed electrical resistivity (IP/Res). The study was combined with the synthetic modeling, the analysis of bulk versus fluid electrical resistivity relationship and test borehole drillings. Such generalized analysis ensures independency of the results from the locality of sampling/measurements. It was shown that leachate detection can be correctly performed utilizing the IP/Res ratio while the anomaly’s likelihood is higher than 90%.

Comparison of smoothness-constrained and geostatistically based cross-borehole electrical resistivity tomography for characterization of solute tracer plumes

Experiments using electrical resistivity tomography (ERT) have shown promising results in reducing the uncertainty of solute plume characteristics related to estimates based on the analysis of local point measurements only. To explore the similarities and differences between two cross-borehole ERT inversion approaches for characterizing salt tracer plumes, namely the classical smoothness-constrained inversion and a geostatistically based approach, we performed two-dimensional synthetic experiments. Simplifying assumptions about the solute transport model and the electrical forward and inverse model allowed us to study the sensitivity of the ERT inversion approaches towards a variety of basic conditions, including the number of boreholes, measurement schemes, contrast between the plume and background electrical conductivity, use of a priori knowledge, and point conditioning. The results show that geostatistically based and smoothness-constrained inversions of electrical resistance data yield plume characteristics of similar quality, which can be further improved when point measurements are incorporated and advantageous measurement schemes are chosen. As expected, an increased number of boreholes included in the ERT measurement layout can highly improve the quality of inferred plume characteristics, while in this case the benefits of point conditioning and advantageous measurement schemes diminish. Both ERT inversion approaches are similarly sensitive to the noise level of the data and the contrast between the solute plume and background electrical conductivity, and robust with regard to biased input parameters, such as mean concentration, variance, and correlation length of the plume. Although sophisticated inversion schemes have recently become available, in which flow and transport as well as electrical forward models are coupled, these schemes effectively rely on a relatively simple geometrical parameterization of the hydrogeological model. Therefore, we believe that standard uncoupled ERT inverse approaches, like the ones discussed and assessed in this paper, will continue to be important to the imaging and characterization of solute plumes in many real-world applications.

Covariance-constrained difference inversion of time-lapse electrical resistivity tomography data

Hydrogeophysics has become a major field of research in the past two decades, and time-lapse electrical resistivity tomography (ERT) is one of the most popular techniques to monitor passive and active processes in shallow subsurface reservoirs. Time-lapse inversion schemes have been developed to refine inversion results, but they mostly still rely on a spatial regularization procedure based on the standard smoothness constraint. We have applied a covariance-based regularization operator to the time-lapse ERT inverse problem. We first evaluated the method for surface and crosshole ERT with two synthetic cases and compared the results with the smoothness-constrained inversion (SCI). These tests showed that the covariance-constrained inversion (CCI) better images the target in terms of shape and amplitude. Although more important in low-sensitivity zones, we have observed improvements everywhere in the tomograms. Those synthetic examples also show that an error made in the range or in the type of the variogram model had a limited impact on the resulting image, which still remained better than SCI. We then applied the method to cross-borehole ERT field data from a heat-tracing experiment, in which the comparison with direct measurements showed a strong improvement of the breakthrough curves retrieved from ERT. This method could be extended to the time dimension, which would allow the use of CCI in 4D inversion schemes.

Improved 1D inversions for sea ice thickness and conductivity from electromagnetic induction data: Inclusion of nonlinearities caused by passive bucking

The porosity of sea ice is a fundamental physical parameter that governs the mechanical strength of sea ice and the mobility of gases and nutrients for biological processes and biogeochemical cycles in the sea ice layer. However, little is known about the spatial distribution of the sea ice porosity and its variability between different sea ice types; an efficient and nondestructive method to measure this property is currently missing. Sea ice porosity is linked to the bulk electrical conductivity of sea ice, a parameter routinely used to discriminate between sea ice and seawater by electromagnetic (EM) induction sensors. Here, we have evaluated the prospect of porosity retrieval of sea ice by means of bulk conductivity estimates using 1D multifrequency EM inversion schemes. We have focused on two inversion algorithms, a smoothness-constrained inversion and a Marquardt-Levenberg inversion, which we modified for the nonlinear signal bias caused by a passive bucking coil operated in such a highly conductive environment. Using synthetic modeling studies, 1D inversion algorithms and multiple frequencies, we found that we can resolve the sea ice conductivity within [Formula: see text]. Using standard assumptions for the conductivity-porosity relation of sea ice, we were able to estimate porosity with an uncertainty of [Formula: see text], which enables efficient and nondestructive surveys of the internal state of the sea ice cover.

Structure-based geoelectrical models derived from genetic algorithms: A case study for hydrogeological investigations along Elbe River coastal area, Germany

Abstract Vertical electrical sounding (VES) and electrical resistivity tomography (ERT) surveys are performed to assess the hydrogeological conditions along Elbe River coastal area, Germany. Because the interpretation of actual resistivity data still has a degree of non-uniqueness and ill-conditioning, linear and non-linear inversion methods have been applied in this paper for optimal interpretation of the measured data. The 1D model generation using hybrid genetic algorithms (GA) represents an accurate and quick solution to image the subsurface resistivity distributions; freshwater aquifer and two highly conductive zones of perched saltwater and seawater intrusion. The longitudinal conductance of the interpreted layers above the water table is calculated to explain why the vulnerable zone to the perched saltwater concentrates at the central and southern parts of the area investigated. Two-dimensional layered-earth models consisting of undulating interfaces are generated by the 2D hybrid GA. These structure-based models are then compared to the cell-models derived from the conventional smoothness-constrained inversion in view of available borehole data. A finite element forward 2D modeling scheme is used for the calculation of theoretical data. This survey example demonstrates that the combined use of the GA with structure-based model and conventional derivative-based inversion with cell-model provides valuable information to constrain the number of interfaces to interpret the DC resistivity measurements for hydrogeological investigations. The limitations of conventional inversion methods under the presence of conductive layers can be overcome by the use of GA with a structure-based model. However, the structure-based parameterization is not practical in the case of significant and sudden discontinuities along the lateral direction.

Evaluation of known-boundary and resistivity constraints for improving cross-borehole DC electrical resistivity imaging of discrete fractures

There is a need to better characterize discrete fractures in contaminated hard rock aquifers to determine the fate of remediation injections away from boreholes and also to evaluate hydraulic fracturing performance. A synthetic cross-borehole electrical resistivity study was conducted assuming a discrete fracture model of an existing contaminated site with known fracture locations. Four boreholes and two discrete fracture zones, assumed to be the dominant electrical and hydraulically conductive pathways, were explicitly modeled within an unstructured tetrahedral mesh. We first evaluated different regularization constraints starting with an uninformed smoothness-constrained inversion, to which a priori information was incrementally added. We found major improvements when (1) smoothness regularization constraints were relaxed (or disconnected) along boreholes and fractures, (2) a homogeneous conductivity was assumed along boreholes, and (3) borehole conductivity constraints that could be determined from a specific conductance log were applied. We also evaluated the effect of including borehole packers on fracture zone model recovery. We found that the fracture zone conductivities with the inclusion of packers were comparable to similar trials excluding the use of packers regardless of electrical potential changes. The misplacement of fracture regularization disconnects (FRDs) can easily be misinterpreted as actual fracture locations. Conductivities within these misplaced disconnects were near the starting model value, and removing smoothing between boreholes and assumed fracture locations helped in identifying incorrectly located FRDs. We found that structural constraints used after careful evaluation of a priori information are critical to improve imaging of fracture electrical conductivities, locations, and orientations.

Imaging artificial salt water infiltration using electrical resistivity tomography constrained by geostatistical data

► Surface electrical resistivity tomography to image salt water infiltration. ► Borehole-derived variogram used as inversion constraints. ► Significant improvement compared to standard smoothness-constrained imaging. ► Plume scale imaging (200 m). ► Validated results and more reliable TDS content estimates. Electrical resistivity tomography is a well-known technique to monitor fresh-salt water transitions. In such environments, boreholes are often used to validate geophysical results but rarely used to constrain the geoelectrical inversion. To estimate the extent of salt water infiltration in the dune area of a Natural Reserve (Westhoek, Belgium), electrical resistivity tomography profiles were carried out together with borehole electromagnetic measurements. The latter were used to calculate a vertical variogram, representative of the study site. Then, a geostatistical constraint, in the form of an a priori model covariance matrix based on the variogram, was imposed as regularization to solve the electrical inverse problem. Inversion results enabled to determine the extension of the salt water plume laterally and at depth, but also to estimate the total dissolved solid content within the plume. These results are in agreement with the hydrogeological data of the site. A comparison with borehole data showed that the inversion results with geostatistical constraints are much more representative of the seawater body (in terms of total dissolved solids, extension and height) than results using standard smoothness-constrained inversion. The field results obtained for the Westhoek site emphasize the need to go beyond standard smoothness-constrained images and to use available borehole data as prior information to constrain the inversion.

Inequality constraint in least-squares inversion of geophysical data

This paper presents a simple, generalized parameter constraint using a priori information to obtain a stable inverse of geophysical data. In the constraint the a priori information can be expressed by two limits: lower and upper bounds. This is a kind of inequality constraint, which is usually employed in linear programming. In this paper, we have derived this parameter constraint as a generalized version of positiveness constraint of parameter, which is routinely used in the inversion of electrical and EM data. However, the two bounds are not restricted to positive values. The width of two bounds reflects the reliability of ground information, which is obtained through well logging and surface geology survey. The effectiveness and convenience of this inequality constraint is demonstrated through the smoothness-constrained inversion of synthetic magnetotelluric data.

Smoothness-constrained inversion for two-dimensional electrical resistance tomography

Two-dimensional image reconstruction in electrical resistance tomography (ERT) involves determination of the electrical conductivities throughout the cross section of the region being imaged. Quantitative image reconstruction in ERT is an ill-conditioned problem and therefore some sort of regularization is required in order to provide meaningful images. Typically, this is achieved using the Marquardt - Levenberg regularization. In this work, a new data acquisition system developed at UMIST, the Mark 2a, is briefly described and a smoothness-constrained regularization is presented, which is shown to be superior to the widely used Marquardt - Levenberg regularization in the presence of Gaussian noise in the collected data.

A map of the horizontal flows in the solar convection zone

The first map of the horizontal flows as a function of depth and heliocentric position in the solar convection zone is presented. The map is inferred from a least-squares smoothness-constrained inversion of velocities measured from ring diagrams of the solar p-mode oscillations. The data provide information in four longitude regions at a latitude just south of the solar equator. The presence of several features is suggested by the results: (1) A prograde directed longitudinal jet between the hydrogen and first helium ionization zone, reversing direction to retrograde below the second helium ionization zone. (2) A equatorward directed latitudinal flow above the second helium ionization zone. (3) A poleward latitudinal flow below the second helium ionization zone, but only in one of the four longitude bands. A large active region was present in this band, but not in two of the other bands, suggesting a possible relationship between the activity and the different flow pattern.