Smooth Inversion Research Articles

The priority of microgravity focusing inversion in 3D modeling of subsurface voids

Geophysical methods can be used to detect voids, cavities, and tunnels. Inversion of microgravity data provides the distribution of the subsurface density. It can be effective to reconstruct concealed cavities and other underground openings, as they are often at different depths that have a low density. We have smooth and focusing inversion methods that apply the minimum norm and minimum support stabilizers. In this study, we used smooth and focusing inversion methods by reweighted regularized conjugate gradient (RRCG) method for inverting microgravity data. Here, these methods are utilized to inversion of microgravity data from two synthetic models and a microgravity data set over a tunnel located in Yucca Flat, Nevada. The results demonstrated smooth inversion cannot detect sharp boundaries and overestimates the bottom depths of the underground openings, but the focusing inversion method recovers models with sharp boundaries and does not have the problems of the previous method. The results indicate the priority of the focusing inversion method for the detection of voids by microgravity data.

Lp Norm Smooth Inversion of Magnetic Anomaly Based on Improved Adaptive Differential Evolution

Due to the approved applicability of differential evolution (DE) in geophysical problems, the algorithm has been widely concerned. The DE algorithms are mostly applied to solve the geophysical parametric estimation based on specific models, but they are rarely used in solving the physical property inverse problem of geophysical data. In this paper, an improved adaptive differential evolution is proposed to solve the lp norm magnetic inversion of 2D data, in which the perturbation direction in the mutation strategy is smoothed by using the moving average technique. Besides, a new way of updating the regularization coefficient is introduced to balance the effect of the model constraint adaptively. The inversion results of synthetic models demonstrate that the presented method can obtain a smoother solution and delineate the distributions of abnormal bodies better. In the field example of Zaohuoxi iron ore deposits in China, the reconstructed magnetic source distribution is in good agreement with the one inferred from drilling information. The result shows that the proposed method offers a valuable tool for magnetic anomaly inversion.

3D Gravity Inversion on Unstructured Grids

Compared with structured grids, unstructured grids are more flexible to model arbitrarily shaped structures. However, based on unstructured grids, gravity inversion results would be discontinuous and hollow because of cell volume and depth variations. To solve this problem, we first analyzed the gradient of objective function in gradient-based inversion methods, and a new gradient scheme of objective function is developed, which is a derivative with respect to weighted model parameters. The new gradient scheme can more effectively solve the problem with lacking depth resolution than the traditional inversions, and the improvement is not affected by the regularization parameters. Besides, an improved fuzzy c-means clustering combined with spatial constraints is developed to measure property distribution of inverted models in both spatial domain and parameter domain simultaneously. The new inversion method can yield a more internal continuous model, as it encourages cells and their adjacent cells to tend to the same property value. At last, the smooth constraint inversion, the focusing inversion, and the improved fuzzy c-means clustering inversion on unstructured grids are tested on synthetic and measured gravity data to compare and demonstrate the algorithms proposed in this paper.

Ground-wire Source TEM 3D Full Time Multinary Inversion Using Adaptive Regulation

Ground-wire source transient electromagnetic method (GTEM) provides better investigation ability than loop source TEM at a given noise level and decay time. However, at the present time, the method still stays in the one-dimensional inversion interpretation stage. Since actual geological structures are three-dimensionally distributed, the three-dimensional electromagnetic forward and inversion are crucial for understanding the electromagnetic responses of complex geological structures. Moreover, the traditional 3D smooth inversions of geophysical data have been found to inaccurately reflect small-scale and isolated anomalies. In this study, a multinary inversion method was introduced and applied to GTEM inversions. It was found that the proposed method had the ability to enable GTEM to more accurately delineate anomalous bodies when applied to detect high-resistivity target. Then, for the purpose of avoiding the need for multiple inversion tests to determine the regularization factors, a self-adaptive scheme was proposed based on the differences between the data fitting functional and the model functional during each iteration step. It was observed that by introducing the multinary inversion with adaptive regulation, more stable and accurate inversion results were obtained. In the current study, the numerical simulation results had successfully verified that the proposed multinary inversion method had provided better resolution than the traditional inversion methods.

Magnetotelluric exploration of the Wagner Basin, Gulf of California, Mexico: Evidence for an axial magma chamber and hydrothermal circulation

Different geological and geophysical disciplines have suggested the Wagner Basin as a promising location of geothermal resources. As a reconnaissance tool Magnetotelluric (MT) data were measured at 10 sites along a profile over this basin in the northern Gulf of California. The subsurface distribution of electric resistivity was estimated with two-dimensional smooth inversion using the apparent resistivities and phases from both field polarizations. We found a deep conductor underneath the center of the basin, interpreted as a region of partial melt feeding the accretion zone of this incipient oceanic crust. From six MT sites with close-by heat flow measurements, we also found a positive correlation between the interpreted shallow (first 100 m) low resistivities and high heat flows. This shows that the MT method can be used to map submarine heat sources.

Active Vibration Suppression of Uncertain Hose and Drogue Systems in the Presence of Actuator Nonlinearities

This paper investigates vibration suppression of uncertain hose and drogue systems in the presence of actuator nonlinearities. Firstly, a previously presented model of the hose and drogue systems is extended to describe how the hose and drogue systems restrain the vibration, while the accompanying unknown aerodynamic coefficients are estimated by invoking the parameter projection method. Subsequently, for the actuator nonlinearities of dead-zone and saturation, a smooth dead-zone approximate function is constructed to design the dead-zone compensation method, based upon which the proposed control scheme can handle actuator dead-zone and saturation simultaneously while improving the output efficiency of the actuator. Next, for the actuator nonlinearities of backlash and saturation, a smooth backlash inverse is constructed based upon which the presented control scheme can cope with the both actuator nonlinearities simultaneously. Finally, by utilizing backstepping method and hyperbolic tangent function, the proposed control schemes can also achieve the control objectives of vibration suppression and external disturbance attenuation. Simulation examples are included to demonstrate the validity of the proposed control schemes.

Magnetotelluric extremum boundary inversion based on different stabilizers and its application in a high radioactive waste repository site selection

Geophysical inversion under different stabilizers has different descriptions of the target body boundary, especially in complex geological structures. In this paper, we present an extremum boundary inversion algorithm based on different stabilizers for electrical interface recognition. Firstly, we use the smoothest and minimum-support stabilizing functional to study the applicability of adaptive regularization inversion algorithm. Then, an electrical interface recognition method based on different stabilizers is developed by introducing extremum boundary inversion algorithm. The testing shows that the adaptive regularization inversion method does work for different stabilizers and has a low dependence on the initial models. The ratio of the smooth and focusing upper and lower boundaries obtained using the extremum boundary inversion algorithm can clearly demarcate electrical interfaces. We apply the inversion algorithm to the magnetotelluric (MT) data collected from a preselected area of a high-level-waste clay-rock repository site in the Tamusu area. We recognized regional structures with smooth inversion and the local details with focusing inversion and determined the thickness of the target layer combined with the geological and drilling information, which meets the requirement for the site of the high-level waste clay-rock repository.

Application of Refraction Seismic Tomography Method Using Wavepath Eikonal Traveltime Inversion for Modelling the Subsurface

Refraction seismic tomography is a geophysical method that is able to image shallow subsurface velocity variation. Common application of this method is for Civil Engineering include near-surface problem such as weathering zone and bedrock. In this study, refraction seismic tomography was using Wavepath Eikonal Traveltime (WET) inversion. This method is performed by software RayfractTM. The initial model is generated by Delta-t-V method and smoothing inversion. The Delta-t-V method obtain a 2D initial model, while the smoothing inversion obtain a 1D initial model. The forward modelling is done by finite-difference solution using Eikonal equation to obtain the ray tracing. Then WET inversion will obtain the subsurface model. The imaging result is able to show the thickness of weathering layer and the depth of the bedrock. The tomographic interpretation indicates the thickness of weathering layer is 2-10 meter, while the bedrock presence at depth 7-35 meter from the surface.

Implementation and model uniqueness of Particle Swarm Optimization method with a 2D smooth modeling approach for Radio-Magnetotelluric data

We present a simple workflow for 2D smooth modeling of Radio-Magnetotelluric (RMT) data using Particle Swarm Optimization (PSO). The workflow aims for the general problems of the Global Optimization (GO) methods, encountered during multidimensional geophysical modeling; time consumption, and difficulties in solving large amount of model parameters. In our implementation of PSO, starting models are used and an optimum parameter change vector (Δm) is searched by the algorithm. Our initial studies showed, implementing a smooth modeling approach is necessary for modeling large amount of model parameters using PSO. To obtain smooth models and avoiding multi-objective optimization we used an operator to smooth the model parameters to the desired level. As result, we used the PSO to search for the Δm minimizing the functional when the model parameters are smoothed. The smoothness is gradually decreased when rougher models are wanted. As the smoothness decrease, artefacts are found to be emerging; especially at the low sensitivity zones. To prevent these artefacts, we added Minimum Gradient Support (MGS) to our functional with a multiplier. Since the value of this multiplier is necessarily small, the algorithm doesn't search for a balance between these objectives, and the contribution from the MGS acts as a threshold value to introduce a structure to the model. We tested this approach using a synthetic and a field dataset. In all cases the algorithm successfully minimized the functional and obtained geologically meaningful models. We also compared the uniqueness of the models recovered from the field dataset by multiple modeling realizations using smooth inversion and PSO. Our results suggest, when higher smoothness and minimum structures are aimed, the uniqueness of the models is significantly higher, however, for a fixed set of modeling parameters with moderate values, the non-uniqueness is observed to be higher than that suggested by the smooth inversions.

AMT survey in the Outokumpu ore belt, eastern Finland

ABSTRACTThe Outokumpu belt in eastern Finland is known for numerous polymetallic (Cu–Co–Zn–Ni–Ag–Au) sulphide ore deposits and it is a potential area for new discoveries. Audiomagnetotelluric (AMT) data have recently been collected to image the subsurface conductivity structure of the belt. The data were acquired along five profiles aiming to explore deep conductors, study their geometry and investigate the deep conductivity structure of the belt.The acquired AMT data were mostly adequate for 2D inversion, as the survey area is characterised by thin, laterally extensive conductors indicated by airborne electromagnetic data, deep drillings and regional strike analysis of the impedance tensor data. 2D smooth inversion was carried out jointly for transverse electric and transverse magnetic data. The results show southeast-dipping and sub-horizontal continuous conductors southeast of the Outokumpu old mine, suggesting the presence of prospective Outokumpu assemblage rocks. One ∼ 1 km deep sub-horizontal conductor is verified by a deep drill-hole located ∼ 8 km southeast of the mine. The results demonstrate a similar southeast-dipping structure in the Vuonos mine area, which is located ∼ 7 km northeast of the Outokumpu old mine. An eastwards-dipping conductor detected in the Miihkali serpentinite area indicates deep exploration potential in this area. Besides the smooth 2D inversion, a sharp-boundary 2D inversion was performed and the resulting model was assessed with the existing geological interpretation of the Miihkali area. In the area of the Sotkuma gneiss inlier, conductors are absent from the uppermost ∼ 6 km. This feature may represent an uplifted fault block rather than a thin thrust sheet of Archaean basement rocks.

Delineating Lake Bottom Structure by TEM on Water Surface

Recently the development of TEM (Transient Electromagnetic) provides higher resolution and greater exploration depth than ever before and it is time to consider whether this advanced TEM technique can be applied to under-water mapping. About 250 TEM soundings have been measured within a lake of about 120m by 60m in a muddy deposits environments in southern Taiwan. For the TEM measurements the terrTEM unit (Monex GeoScope) was used with an inloop configuration, for which the receiver spacing is about 5 m and the transmitter loop is about the same size as the lake. For imaging the vertical and horizontal resistivity distribution within the deposit and the surrounding area, all sounding results were interpreted using a smooth inverse procedure. Some auger drilling and sounding were available for correlation. This led to a final model. Underneath the water bottom is mud/shale inter-bed with a depth greater than 50 m and resistivity ranging from 1 to 5 ohm-m. The depth of the clayey layer predicted by TEM data is comparable to that of drilled wells nearby, and the thick mud to the southeast suggests the lake dipping to the southeast. The success of the method depends upon the appropriate design of survey parameters; this new application is open to further refinement, both in instruments and in survey techniques.

Automatic identification of fresh–saline groundwater interfaces from airborne electromagnetic data in Zeeland, the Netherlands

ABSTRACTIn a setting of predominantly saline surface waters in Zeeland, the Netherlands, the only available shallow fresh groundwater resource is present in the form of freshwater lenses floating on top of saline groundwater. This fresh water is vital for agricultural, industrial, ecological, water conservation and drinking water functions. An essential first step for managing the usable water properly is to know the location of the fresh–saline groundwater interface. Traditional salinity mapping with ground‐based vertical electrical soundings, electrical cone penetration tests or chloride measurements from groundwater samples is time‐consuming and, therefore, expensive to cover large areas. Airborne electromagnetics, which is fast and can cover large areas in short time, is an efficient alternative. Therefore, a consortium of BGR, Deltares and TNO conducted the project FRESHEM Zeeland during 2014–2017. An area of more than 2000 km² was surveyed using BGR's helicopter‐borne geophysical system totalling 9640 line‐km. The helicopter‐borne electromagnetic data, after inversion to resistivity–depth models, served as baseline information for further interpretation. Without information on lithology, however, an accurate discrimination between fresh and saline groundwater applying fixed resistivity thresholds would fail if clayey sediments exist. Therefore, a probabilistic Monte Carlo approach was developed within the FRESHEM project. This approach combines helicopter‐borne electromagnetic resistivities, 3D geological model (GeoTOP), laboratory results (formation factor and surface conductivity) and local in situ groundwater measurements for the translation of resistivity data to chloride concentration. As such detailed information is generally not available, another approach, which uses only helicopter‐borne electromagnetic results to derive the thicknesses of the freshwater lenses from smooth inversion models, is presented in this paper. The corresponding fresh–saline groundwater interfaces are derived from steepest resistivity–depth (log–log) gradients appearing within a certain resistivity range. The bounds of this range are defined by resistivity values, which predominantly correlate with fresh or saline water, nearly independent of the lithology type. The results of this approach are checked using both synthetic and field data. The latter are compared with electrical cone penetration test measurements, the common threshold approach and the 3D chloride distribution of the FRESHEM results, particularly in an evaluation area where the transition of fresh to saline groundwater is relatively sharp. The fresh–saline groundwater interfaces derived by all methods are quite similar on average with deviations in the order of a metre. Locally, however, greater deviations occur, particularly close to the coast and along creek ridges or dunes, where the elevation of the fresh–saline groundwater interface varies notably.

Acceleration of 3D potential field data inversion using a BB iterative algorithm

As the big-data age arrives, the efficiency of three dimensional inversions of potential field data can be paid enough attention by scholars. A new inversion method is considered to deal with the large potential field data with a fast rate of convergence. Hence, in this paper, a fast inversion method is researched. And the gravity data is used as an example of potential field data to test the efficiency of our inversion method. To achieve this aim, the study region will be divided into huge amounts of rectangular prisms with unknown constant physical properties of rock. The traditional smooth inversion method is the main principle, and a new Barzilai-Borwein iterative algorithm is applied to ensure the rapid rate of convergence of the inversion method. To compare the rate of convergence of the BB (Barzilai-Borwein) iterative algorithm, the iterative gradient descent algorithm and the iterative conjugate gradient algorithm are used as the compared algorithms. To test high efficiency of the new fast developed inversion method, the large synthetic gravity data are performed. The contrast analysis results can easily reflect the high efficiency of our new inversion method. The great practical value of our inversion method is expounded by a real gravity data. Therefore, the new inversion method may have a great influence on the potential field data inversion.

Why do we Need to Know the Electrical Resistivity Structure of Oceanic Lithosphere?

Regional-scale continental magnetotelluric (MT) programs such as AusLAMP are naturally bounded by the continental shelf and electrically-conducting seawater. Within a few hundred kilometres of the coastline, long-period MT data may be strongly influenced by induction in the seawater, a phenomenon known as the coast-effect. Thus, 3D inversion of gridded long-period MT data for continental lithosphere models requires good constraints on the resistivity of the seawater, oceanic crust and upper mantle, and into the asthenosphere.In this paper, we discuss the concept of a horizontal adjustment distance. This is the horizontal distance away from a major contrast in electrical conductance at which the anomalous electric fields are attenuated by a factor of 1/e from a 1D response, and is somewhat analogous to the more widely-known skin-depth concept. For seafloor MT, this adjustment distance can be thousands of kilometres. Inland, the effect depends on the conductance of the sedimentary cover, and the depth-integrated resistivity of the upper crust, and can vary from a few kilometres to hundreds of kilometres. We discuss the implications in terms of 3D smooth inversion that inherently minimises gradients in subsurface resistivity and suggest that the coast effect may be significantly underestimated in some continental models.

Discrete-valued gravity inversion using the guided fuzzy c-means clustering technique

The definition of boundaries, in a recovered model from an inversion, can be improved through the incorporation of known physical property values of a small number of geologic units. Directly imposing strict physical property values into a Tikhonov regularized inversion transforms it into an integer programming problem. Solving an integer programming problem can be prohibitively expensive for large problems in practical applications. We have developed a method to approximate a discrete-valued inverse problem by applying the guided fuzzy c-means clustering technique. This method enforces the discrete values to a high degree of approximation within the inversion by guiding the recovered model to cluster tightly around the known physical property values. Using this method, we are able to incorporate the uncertainty in our physical property information and solve the corresponding minimization problem with derivative-based minimization techniques, making this approach more efficient and broadly applicable. We applied the method to gravity inversions with two clusters, where the density contrast is restricted to be equal to either zero, for the background, or an anomalous value. We examine the method using synthetic and field data sets and determine that it recovers models with better distinguished density anomalies when compared with smooth inversion methods.

Adaptive backstepping control for air-breathing hypersonic vehicles with input nonlinearities

This paper addresses the control problem of air-breathing hypersonic vehicles subject to input nonlinearities, aerodynamic uncertainties and flexible modes. An adaptive backstepping controller and a dynamic inverse controller are developed for the altitude subsystem and the velocity subsystem, respectively, where the former eliminates the problem of “explosion of terms” inherent in backstepping control. Moreover, a modified smooth inverse of the dead-zone is proposed to compensate for the dead-zone effects and reduce the computational burden. Based on this smooth inverse, an input nonlinear pre-compensator is designed to handle input saturation and dead-zone nonlinearities, which leads to a simpler control design for the altitude subsystem subject to these two input nonlinearities. It is proved that the proposed controllers can guarantee that all closed-loop signals are bounded and the tracking errors converge to an arbitrarily small residual set. Simulation results are carried out to demonstrate the effectiveness of the proposed control scheme.

Santo Domingo’s Hispanic convent, Panama. A new appraisal from 3D resistivity survey

In 2012, a geophysical survey was conducted inside the open space of a Hispanic Dominican convent in Panama (Central America); its ruins form part of the Panama Viejo Historical Monument Complex, which was declared a UNESCO World Heritage site in 2003. The geophysical survey consisted of measuring the apparent electrical resistivity values along 24 parallel profiles inside the convent; the aim of this approach was focused on generating 3D electrical resistivity tomography in order to detect buried structures that could be associated with a possible cloister in this area of the convent. This dataset was inverted using a smooth inversion algorithm and shows important electrical anomalies on three horizontal maps at depths of 0.29, 0.90 and 1,60 m. The main electrical anomalies presented linear and isolated geometries that, in accordance with the archaeological excavations conducted at the site, are linked to a series of buried cultural features associated with the convent’s cloister, such as perimeter walls, pillar footings and bases made of stone as supports for the gallery of a second floor. This study was also able to identify other archaeological features that could be linked to the initial stages of the construction of this important building.

Regularized magnetotelluric inversion based on a minimum support gradient stabilizing functional

Regularization is used to solve the ill-posed problem of magnetotelluric inversion usually by adding a stabilizing functional to the objective functional that allows us to obtain a stable solution. Among a number of possible stabilizing functionals, smoothing constraints are most commonly used, which produce spatially smooth inversion results. However, in some cases, the focused imaging of a sharp electrical boundary is necessary. Although past works have proposed functionals that may be suitable for the imaging of a sharp boundary, such as minimum support and minimum gradient support (MGS) functionals, they involve some difficulties and limitations in practice. In this paper, we propose a minimum support gradient (MSG) stabilizing functional as another possible choice of focusing stabilizer. In this approach, we calculate the gradient of the model stabilizing functional of the minimum support, which affects both the stability and the sharp boundary focus of the inversion. We then apply the discrete weighted matrix form of each stabilizing functional to build a unified form of the objective functional, allowing us to perform a regularized inversion with variety of stabilizing functionals in the same framework. By comparing the one-dimensional and two-dimensional synthetic inversion results obtained using the MSG stabilizing functional and those obtained using other stabilizing functionals, we demonstrate that the MSG results are not only capable of clearly imaging a sharp geoelectrical interface but also quite stable and robust. Overall good performance in terms of both data fitting and model recovery suggests that this stabilizing functional is effective and useful in practical applications.

Sparsity constraints using a Laplacian kernel to get geological structures from potential field data

ABSTRACTWe propose a new method for including sparsity constraints into potential field data inversion using a Laplacian kernel. The method obtains three‐dimensional density distributions, which are at best a proxy for geological structures. Compressive sensing has been used in many domains to recover the original data from the acquisition data. Compressive sensing is an inversion philosophy where sparsity constraints are applied, and thus, we use this principle into geophysical inversion. Here, we extend this algorithm to potential field data inversion. We introduced the Laplacian smoothed L0 norm into the stabiliser function as a sparse constraint, and the density constraint function has been introduced into our inversion method to guarantee the inverted density to be in the geological and physical meaning range. Compared to the traditional smooth inverse algorithm, our proposed method can obtain accurate geological structures with sharp boundary and sub‐surface “block” structures. This method permits reconstruction of (non‐smooth) density functions that represent a blocky geological structure. Our results using synthetic gravity data show that the Laplacian smoothed L0 norm inversion method with sparsity constraints predicts more focused and accurate depth and density anomalies than smooth inversion method. Application of this Laplacian smoothed L0 norm sparsity constraints method to the inversion of gravity data collected over the Humble salt dome, Harris County, near Houston, United States, leads to improved interpretation of geological structures. These results confirm the validity of the proposed method and its potential application for other potential field data inversions to explore geological structures.

A new method for 3-D magnetic data inversion with physical bound

Inversion of magnetic data is an important step towards interpretation of the practical data. Smooth inversion is a common technique for the inversion of data. Physical bound constraint can improve the solution to the magnetic inverse problem. However, how to introduce the bound constraint into the inversion procedure is important. Imposing bound constraint makes the magnetic data inversion a non-linear inverse problem. In this work, a new algorithm is developed for the 3D inversion of magnetic data, which uses an efficient penalization function for imposing the bound constraint and Gauss Newton method to achieve the solution. An adaptive regularization method is used in order to choose the regularization parameter in this inversion approach. The inversion results of synthetic data show that the new method can produce models that adequately match the real location and shape of the synthetic bodies. The test carried out on the field data from Mt. Milligan copper-gold porphyry deposit shows that the new inversion approach can produce the magnetic susceptibility models consistent with the true structures.