Computations of secondary potential for 3D DC resistivity modelling using an incomplete Choleski conjugate‐gradient method

Quick clay, which is the main cause of landslides that occur in the northern countries, liquefies easily, and its presence implies an increased risk of landslide. Geophysical methods have been increasingly used in landslide investigations. Three-dimensional electric resistivity tomography, radio magnetotelluric (RMT), controlled-source RMT (CSRMT), and high-resolution reflection seismic data were acquired at a quick-clay landslide site in the southwest of Sweden. The main objectives were to evaluate the capability of each method in delineating different subsurface geologic structures that controlled a peculiar and hazardous retrogressive-type landslide in the study area. A 3D resistivity model from the inversion of CSRMT data showed the best correlation with the reflection seismic data and borehole information, thanks to the broad frequency range of the data set. It better imaged the resistive crystalline bedrock underlying the marine conductive clays and showed considerable correlations with the 3D reflection seismic data in resolving a coarse-grained layer that was interpreted to act as a conduit directing freshwater into the clays under a confined pressure, leaching their salt and forming quick clays. The 3D CSRMT resistivity model and 3D reflection seismic data showed that the coarse-grained layer has a varying thickness. At some locations, it was too thin to be resolved by the methods used here. Combination of the CSRMT model, reflection seismic data, and the borehole data suggested that a layer with thickness of approximately 5 m and resistivity between [Formula: see text] was potentially quick clay, which probably extended laterally in the entire study area. These observations suggested that future developments should focus on joint inversion of such 3D data sets incorporating sharp boundaries as constraints in the inversion and particularly when quick clays were studied.

A transient electromagnetic vector finite-element forward-modeling method based on 3D anisotropic resistivity constrained by well-logging and seismic data is developed. This method integrates detailed stratigraphic division provided by well-logging data and structural information from seismic interpretation to establish a structural model that reflects the actual geologic conditions. Based on the structural model, a kriging interpolation method is applied to the logging resistivity data to obtain the anisotropic resistivity distribution information of each layer in the reservoir, thereby establishing a 3D anisotropic resistivity model. To ensure the smoothness of the model, we use a variogram function based on a spherical model to fit the discrete interpolation points. After the resistivity model is spatially discretized by locally refined unstructured tetrahedral meshes and temporally discretized by a nonuniform time step, the large sparse matrix equation is solved using the PARDISO direct solver to obtain the forward response. We apply this method to layered models of isotropic and vertical transversely isotropic (VTI) media and verify the accuracy of the forward simulation method by comparative analysis with quasi-analytical solutions. Doubling the nonuniform time step in our forward simulations does not affect the accuracy of the solution. Applying our method to China’s D oilfield, the electromagnetic response at observation points generally matches traditional methods well. Comparative analysis of the forward results between our method and the model without geologic data or homogeneous resistivity shows that we achieve better fitting with the field measurement data and provide more detailed electrical properties.

For better manage the groundwater reservoir, it is crucial to delineate the potential area that can serve as the artificial recharge. We first conducted over 60 Transient electromagnetic (TEM) measurement in the upper-fan area of the Chuoshuichi alluvial. We used borehole logs as well as the vertical resistivity sounding (VES) conducted previously to verify the TEM results. We then construct a three-dimensional(3D) resistivity model by Krigging the TEM results. Finally, we conducted the hierarchical accumulative clustering in machine learning to characterize the different resistivity clusters. We attempt to build up the 3D geological model by comparing the clusters to the limited geological logs/records, and converting the aforementioned 3D resistivity model directly into the geological model accordingly.

Geoelectrical resistivity method is popular technique for subsurface mapping of groundwater, mining exploration, and structural geology. For case study in Tidar Plateau which characterized by hard rock of volcanic breccia, lava, tuff, and conglomerate, the groundwater system occurs only by the secondary porosity (faulting, facturing, and weathering). Therefore it becomes difficult task for determine position of good well site in Tidar Plateau. Geoelectrical technique by Schlumberger configuration was choosen to investigated the ground water system in this area. Total 5 Vertical Electrical Sounding (VES) acqusition point (400 m length measurement; spacing 10 m) for data measurement with coverage area about 400 x 400 m2. For data processing was used IP2Win, Progress3, and RockWork 16. The data processing results showed high accuration with avarage Root Mean Square (RMS) error about 0.3 – 8.3 %. According to resistivity results, the study area has resistivity between 0.3– 2.5x104 . The rocks which have resistivity between 0.3 – 12.52 interpreted as sandy-clay; 12.52 – 50 interpreted as wet sandstone (aquifer); 129 – 178 interpreted as conglomerat; 700 – 1300 interpreted as tuff, sand, and dry gravel; 2600 – 2.5x104 interpreted as volcanic breccia. This geoelectrical results has succesful finding the fresh water at VES 5 after drilled at depth 100 m at wet sandstone formation. To findout distrubution of aquifer layer, a three-dimensional (3D) resistivity model was developed. The 3D resistivity model shows the aquifer layer image look the dome-like structure and image the fracture regions which becomes the pathway of groundwater system.

In this paper finite element method for 3D DC resistivity modeling accelerated using multi-GPU (Graphics Processing Unit). Solution of the large system of linear equations is the most expensive computation in finite element method performed in GPUs to reduce the computational time. Conjugate gradient solver used to solve large system of linear equations. We developed kernel for conjugate gradient solver that exploit data vectorization and written in PTX assembly form. We perform test on GTX 750Ti GPU and Tesla C2050 GPU. Our kernel have better sparse matrix-vector performance than CUSPARSE library in the first GPU, but lower performance in the latter GPU. The performance comparisons to the library are about 1.4 times and 0.7 times respectively. Our multi-GPU implementation achieved about 1.9 times performance of single GPU by using 2 identical GPUs. In comparison to the serial CPU implementation, about 10 times speedup could be achieved by using 2 GPUs.

A program generator for the incomplete cholesky conjugate gradient (ICCG) method with a symmetrizing preprocessor

The effectiveness of renumbering for the incomplete Cholesky conjugate gradient (ICCG) solver, which is usually applied to direct solvers, is examined quantitatively by analyzing 3D standard benchmark models. On an acceleration factor which is introduced to obtain convergence quickly, indices for determining the optimum value of the acceleration factor, which minimizes the number of iterations, are discussed. It is found that the renumbering is effective to use with the ICCG solver, and the solver using the acceleration factor gives a good convergence characteristic even in the case when the conventional solver fails to provide convergent solutions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The effects of initial solution and order of observations on the solving of mixed model equations (MME) by traditional iterative algorithms-Gauss-Seidel, successive over-relaxation (SOR), and second-order Jacobi-and by preconditioned conjugate gradient algorithms-scaled conjugate gradient (SCG) and incomplete Cholesky conjugate gradient (ICCG) simulation were investigated by using data in a computer. With SOR, the use of adjusted initial solutions based on phenotype and heritability of the trait and order of the animal within the trait in the MME was effective for fast convergence. Adjustment of initial solutions was not an advantage in preconditioned conjugate gradient algorithms. Of the five algorithms, ICCG required the lowest number of iterations until convergence. However, ICCG needed the largest central processing unit (CPU) time to calculate one round of iteration. The SCG algorithm was an attractive alternative for solving MME because it required the second fewest rounds of iteration and had the shortest CPU time per round of iteration until convergence.

Solving the 3D resistivity problem is fundamental to understand the complex response in most environmental contexts. The finite-elements method is a classical tool to solve the dc equations (Coggon 1971 , Pridmore et al. 1981, Holcombe et al. 1984). The forward modelling is often performed in a limited domain, and requires to specify potential boundary conditions. Since the latter are physically unrealistic, they lead to unacceptable errors, especially when dealing with small domains. This paper presents the improvements which can be obtained by using infinite elements in the outer part of the main studied domain, as they permit to discard the Dirichlet condition to infinity.

The Arabian–Nubian Shield (ANS) hosts numerous volcanogenic massive sulphide (VMS) deposits formed in submarine volcanic settings and enriched by hydrothermal processes, making it a critical region for mineral exploration due to the types of deposits it hosts and its geological complexity. The Wadi Bidah Mineral Belt (WBMB), located within the Arabian Shield, contains over 30 polymetallic VMS occurrences associated with an island arc system active between 950 and 800 million years ago. Despite its mineral potential, the WBMB still needs to be explored, with limited geophysical studies to support resource evaluation. This study focuses on the Hawiah area, a prominent VMS site within the WBMB, to delineate subsurface mineralization using transient electromagnetic (TEM) methods. TEM surveys were conducted to characterize the conductivity structure and identify potential zones of sulphide mineralization. Data were processed and inverted to generate 1D, 2D, and 3D resistivity models, providing critical insights into the depth, geometry, and continuity of the mineralized zones based on the final 3D resistivity distribution. The results revealed distinct conductive (very low resistivity) anomalies, correlating with known surface gossans and inferred sulphide-rich layers, and extended these features into the subsurface. The integration of TEM results with geological and geochemical data highlights the effectiveness of this approach in detecting and mapping concealed mineral deposits in complex geological environments. This study advances the understanding of VMS systems in the WBMB and demonstrates the potential of TEM surveys as a key tool for mineral exploration in the Arabian Shield.

A three-dimensional lithospheric electrical resistivity model of the Archaean-Proterozoic Gawler Craton in southern Australia has been developed, to define the tectonic framework of the craton and identify craton margins under regolith cover. Knowledge of cratonic margins is important as upwards of 60% of known mineral wealth is located in palaeoconvergent zones between ancient cratons. The research was conducted in three phases. First, all previous magnetotelluric (MT) and geomagnetic depth sounding (GDS) data were compiled to establish a regional scale induction database. Almost 400 observation sites were found in the literature and from personal communication, collected over the last 30 years by various research groups. Of these, most measurements were GDS only, but recent 2D MT sites along major profiles provide further depth constraints. Second, eight long period MT sites were collected from central areas of the craton to provide a window into the deeper lithosphere and asthenosphere. Data were collected over a number of weeks to establish high quality MT responses in the bandwidth 10–104 s. Onedimensional modelling indicated that the lithosphere was resistive (> 100 Ω m), with a more conductive midmantle at 70 km and a decrease in resistivity at the transition zone. These MT responses agree with the long period (104−107 s) continental lithosphere induction responses of Olsen (N. Olsen: 'Long-period (30 days–1 year) electromagnetic sounding and the electrical conductivity of the lower mantle beneath Europe', Geophys. J. Int., 1999, 138, 179–187). Finally, a 3D resistivity model of the lithosphere was compiled from 2D inversions of MT profiles, with constraints from lithosphere asthenosphere soundings, and regional scale GDS data. The Gawler Craton is shown to have a resistive core of dimension 500 km, surrounded by more conductive terrains. The core is largely, but not exclusively, Archaean crust, surrounded by conductive Proterozoic crust. The edge of the core on the eastern side of the Gawler Craton correlates with the location of several iron oxide–copper–gold deposits, including the world class Olympic Dam deposit, potentially making mapping of large scale resistivity structures a useful tool for regional scale exploration.

Although most finite element programs have quite effective iterative solvers such as an incomplete Cholesky (IC) or symmetric successive overrelaxation (SSOR) preconditioned conjugate gradient (CG) method, the solution time may still become unacceptably long for very large systems. Convergence and thus total solution time can be shortened by using better preconditioners such as geometric multigrid methods. Algebraic multigrid methods have the supplementary advantage that no geometric information is needed and can thus be used as black box equation solvers. In the case of a finite element solution of a non-linear magnetostatic problem, the algebraic multigrid method reduces the overall computation time by a factor of 6 compared to a SSOR-CG solver.

The Nisa uranium deposit, located in Central Portugal, has been known since the late 1960s. Some areas were explored at that time. Today, a few open pits and dumps remain in place and are a concern to local authorities. To assess the geoenvironmental problems caused by the main mining exploration composed of an experimental rock blasting, 3D electrical conductivity and resistivity models were made to develop a hydrogeological model to understand the possibility of contaminants transportation, such as uranium, from the dumps towards a dam located nearby. These 3D models were the support to show alteration layer thickness variations and fault zones at depths controlling groundwater circulation. Spectrometric surveys were also carried out and correlated with geology and geoelectrical structure. All this information was used in the construction of the 3D steady state hydrogeological model of the experimental rock blasting of Nisa. In this model, groundwater flow and the contaminant pathways were simulated. Some areas have very high radioactive values resulting from the geological formation characteristics and old dumps. However, results of the environmental assessment using geophysical and hydrogeological methods point to a critical situation restricted only to the area of the experimental rock blasting of the Nisa uranium deposit and its dumps.

The A-phi formulation, which is widely used in electromagnetic analysis, leads to a redundant linear system of equations that includes a substantial number of redundant degrees of freedom (DOF). We can derive a redundancy-reduced linear system of equations by eliminating the redundant DOF, thereby decreasing the computation costs per iteration for iterative solvers, such as the incomplete Cholesky conjugate gradient (ICCG) solver. This does not, however, result in a reduction in total computation time, due to significant convergence deterioration. In this paper, we present a solution to this problem in the form of folded preconditioners. First, the theorem presented reveals that, for any preconditioned Krylov subspace method for the original redundant linear systems, we can derive the equivalent Krylov subspace method for the redundancy-reduced linear systems by using the corresponding folded preconditioner. As an uncomplicated example, the standard ICCG solver for the original redundant systems has exactly the same convergence property as the CG solver for the redundancy-reduced systems using the folded variant of the IC preconditioner (the folded IC preconditioner). Furthermore, we discuss efficient computational procedures for the folded preconditioners and the design of Krylov subspace algorithms using the preconditioners. A sample full-wave analysis demonstrates the good performance of a newly developed solver, the conjugate orthogonal conjugate gradient (COCG) method with the folded IC preconditioner. The new solver not only lowers the computation costs per iteration by reducing the number of DOF, but also completely avoids the convergence deterioration.

The profile of deep magnetotelluric sounding (MT) from Duc Trong - Tuy Phong has been carried out in Lam Dong and Binh Thuan provinces. The length of the Duc Trong - Tuy Phong profile is about 80 km with 15 stations and the distance between the stations measures about 5 km. Two-dimensional MT inversion was used to find a resistivity model that fits the data. The 2D resistivity model allows determining position and development formation of the Nha Trang - Tanh Linh fault. This is the deep fault, which is showed by the boundaries of remarkable change of resistivity. In the near surface of the Earth (from ground to the depth of 6 km), the angle of inclination of this fault is about 60o; in the next part, the direction of the Nha Trang - Tanh Linh faut is vertical. Geoelectrical section of the Nha Trang - Tanh Linh profile shows that the resistivity of mid-crust is higher than that of lower-crust and of upper-crust.