Fuzzy constrained Lp-norm inversion of direct current resistivity data

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. E11-E24 ◽  
Author(s):  
Anand Singh ◽  
Shashi Prakash Sharma ◽  
İrfan Akca ◽  
Vikas Chand Baranwal
Keyword(s):  
Direct Current ◽  
Cluster Center ◽  
Resistivity Data ◽  
Data Inversion ◽  
Lp Norm ◽  
L2 Norm ◽  
Resistivity Model ◽  
Minimization Procedure ◽  
Fuzzy C Means Clustering ◽  
2D Resistivity

We evaluate the use of a fuzzy c-means clustering procedure to improve an inverted 2D resistivity model within the iterative error minimization procedure. The algorithm is coded in MATLAB language for the Lp-norm inversion of 2D direct current resistivity data and is referred to as fuzzy constrained inversion (FCI). Two additional input parameters are required to be provided by the interpreter: (1) the number of geologic units in the model (i.e., the number of clusters) and (2) the mean resistivity values of each geologic unit (i.e., cluster center values of the geologic units). The efficacy of our approach is evaluated by tests carried on the synthetic and field electrical resistivity tomography (ERT) data. Inversion results from the FCI algorithm are presented for conventional L1- and L2-norm minimization techniques. FCI indicates improvement over conventional inversion approaches in differentiating the geologic units if a proper number of the geologic units is provided to the algorithm. Inappropriate clustering information will affect the resulting resistivity models, particularly conductive geologic units existing in the model. We also determine that FCI is only effective when the observed ERT data can recognize the particular geologic units.

Unstructured Grid based Fuzzy Cooperative Resistivity Tomography for Electrical and Electromagnetic data

2020 ◽  
Author(s):  
Anand Singh
Keyword(s):  
Learning Algorithm ◽  
A Priori ◽  
Cluster Center ◽  
Resistivity Tomography ◽  
Study Region ◽  
Separation Result ◽  
Resistivity Model ◽  
Fuzzy C Means Clustering ◽  
Membership Value ◽  
Direct Outcome

<p>There are many inversion algorithms that have been developed in the literature to obtain the resistivity distribution of the subsurface. Recovered resistivity values are usually lower/higher than the actual resistivity as a consequence of the inversion algorithms. As a consequence, Identification of geologic units based on resistivity distribution can be done on a relative scale. In general, identification of different geologic units is a post step inversion process based on resistivity distribution in the study region.  I have presented a technique to enhance the resistivity image using cooperative inversion (named as fuzzy cooperative resistivity tomography) where two additional input parameters are added as the number of geologic units in the model (i.e. number of cluster) and the cluster center values of the geologic units (mean resistivity value of each geologic unit). Fuzzy cooperative resistivity tomography fulfills three needs: (1) to obtain a resistivity model which will satisfy the fitting between measured and modeled data, (2) the recovered resistivity model will be guided by additional a priori parametric information, and (3) resistivity distribution and geologic separation will be accomplished simultaneously (i.e. no post inversion step will be needed). Fuzzy cooperative resistivity tomography is based on fuzzy c-means clustering technique which is an unsupervised machine learning algorithm. The highest membership value which is a direct outcome from the FCRT corresponds to a geology separation result. To obtain a geology separation result, I adopted the defuzzification method to assign a single geologic unit for each model cell based on the membership values. Various synthetic and field example data show that FCRT is an effective approach to differentiate between various geologic units. However, I have also noted that this approach is only effective when measured data sets are sensitive to particular geologic units. This is the limitation of the presented FCRT.</p>

scholarly journals Electrical resistivity tomography applied to geologic, hydrogeologic, and engineering investigations at a former waste-disposal site

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. B231-B239 ◽  
Author(s):  
Jonathan E. Chambers ◽  
Oliver Kuras ◽  
Philip I. Meldrum ◽  
Richard D. Ogilvy ◽  
Jonathan Hollands
Keyword(s):  
Electrical Resistivity ◽  
Electrical Resistivity Tomography ◽  
Site Investigation ◽  
High Resistivity ◽  
Potential Zone ◽  
Resistivity Tomography ◽  
L2 Norm ◽  
Resistivity Model ◽  
2D And 3D ◽  
2D Resistivity

A former dolerite quarry and landfill site was investigated using 2D and 3D electrical resistivity tomography (ERT), with the aims of determining buried quarry geometry, mapping bedrock contamination arising from the landfill, and characterizing site geology. Resistivity data were collected from a network of intersecting survey lines using a Wenner-based array configuration. Inversion of the data was carried out using 2D and 3D regularized least-squares optimization methods with robust (L1-norm) model constraints. For this site, where high resistivity contrasts were present, robust model constraints produced a more accurate recovery of subsurface structures when compared to the use of smooth (L2-norm) constraints. Integrated 3D spatial analysis of the ERT and conventional site investigation data proved in this case a highly effective means of characterizing the landfill and its environs. The 3D resistivity model was successfully used to confirm the position of the landfill boundaries, which appeared as electrically well-defined features that corresponded extremely closely to both historic maps and intrusive site investigation data. A potential zone of leachate migration from the landfill was identified from the electrical models; the location of this zone was consistent with the predicted direction of groundwater flow across the site. Unquarried areas of a dolerite sill were imaged as a resistive sheet-like feature, while the fault zone appeared in the 2D resistivity model as a dipping structure defined by contrasting bedrock resistivities.

Uncertainties Analysis of Electrical Resistivity Tomography

2021 ◽  
Author(s):  
Yonatan Garkebo Doyoro ◽  
Chang Ping-Yu ◽  
Jordi Mahardika Puntu
Keyword(s):  
Model Resolution ◽  
Resistivity Data ◽  
Resistivity Method ◽  
Resistivity Model ◽  
2D Resistivity Imaging ◽  
Model Set ◽  
Different Depths ◽  
Measured Resistivity ◽  
2D Resistivity ◽  
Subsurface Cavities

<p>We examined the uncertainty of the resistivity method in cavity studies using a synthetic cavity model set at six-different depths. Conceptual models were simulated to generate synthetic resistivity data for dipole-dipole, pole-dipole, Wenner-Schlumberger, and pole-pole arrays. The 2D geoelectric models were recovered from the inversion of the synthetically measured resistivity data. The highest anomaly effect (1.46) and variance (24400) in resistivity data were recovered by dipole-dipole array, while the pole-pole array obtained the lowest anomaly effect (0.60) and variance (2401) for the target cavity T<sub>1</sub>. The anomaly effect and variance were linearly associated with the quality of the inverted models. The steeper anomaly gradient of resistivity indicated more distinct cavity boundaries, while the gentler gradient prevents the inference of the cavity boundaries. The recovered model zone above the depth of investigation index of 0.1 has shown relatively higher sensitivity. Modeling for dipole-dipole array provided the highest model resolution and anomaly gradient that shows a relatively distinct geometry of the cavity anomalies. On the contrary, the pole-dipole and Wenner-Schlumberger arrays recovered good model resolutions and moderate anomaly gradient but determining the anomaly geometries is relatively challenging. Whereas, the pole-pole array depicted the lowest model resolution and anomaly gradient with less clear geometry of the cavity anomalies. At deeper depths, the inverted models showed a reduction in model resolutions, overestimation in anomaly sizes, and deviation in anomaly positions, which can create ambiguity in resistivity model interpretations. Despite these uncertainties, our modeling specified that the 2D resistivity imaging is a potential technique to study subsurface cavities.</p>

scholarly journals Uncertainty of the 2D Resistivity Survey on the Subsurface Cavities

2021 ◽  
Vol 11 (7) ◽  
pp. 3143
Author(s):  
Yonatan Garkebo Doyoro ◽  
Ping-Yu Chang ◽  
Jordi Mahardika Puntu
Keyword(s):  
High Sensitivity ◽  
Two Dimensional ◽  
Model Resolution ◽  
Resistivity Data ◽  
Resistivity Method ◽  
Resistivity Model ◽  
Measured Resistivity ◽  
2D Resistivity ◽  
Subsurface Cavities

We examined the uncertainty of the two-dimensional (2D) resistivity method using conceptual cavity models. The experimental cavity study was conducted to validate numerical model results. Spatial resolution and sensitivity to resistivity perturbations were also assessed using checkerboard tests. Conceptual models were simulated to generate synthetic resistivity data for dipole-dipole (DD), pole-dipole (PD), Wenner–Schlumberger (WS), and pole-pole (PP) arrays. The synthetically measured resistivity data were inverted to obtain the geoelectric models. The highest anomaly effect (1.46) and variance (24,400 Ω·m) in resistivity data were recovered by the DD array, whereas the PP array obtained the lowest anomaly effect (0.60) and variance (2401 Ω·m) for the shallowest target cavity set at 2.2 m depth. The anomaly effect and variance showed direct dependency on the quality of the inverted models. The DD array provided the highest model resolution that shows relatively distinct anomaly geometries. In contrast, the PD and WS arrays recovered good resolutions, but it is challenging to determine the correct anomaly geometries with them. The PP array reproduced the lowest resolution with less precise anomaly geometries. Moreover, all the tested arrays showed high sensitivity to the resistivity contrasts at shallow depth. The DD and WS arrays displayed the higher sensitivity to the resistivity perturbations compared to the PD and PP arrays. The inverted models showed a reduction in sensitivity, model resolution, and accuracy at deeper depths, creating ambiguity in resistivity model interpretations. Despite these uncertainties, our modeling specified that two-dimensional resistivity imaging is a potential technique to study subsurface cavities. We inferred that the DD array is the most appropriate for cavity surveys. The PD and WS arrays are adequate, while the PP array is the least suitable for cavity studies.

scholarly journals Geoelectrical Data Inversion by Clustering Techniques of Fuzzy Logic to Estimate the Subsurface Layer Model

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
A. Stanley Raj ◽  
D. Hudson Oliver ◽  
Y. Srinivas
Keyword(s):  
Fuzzy Logic ◽  
Soft Computing ◽  
Subsurface Layer ◽  
Subtractive Clustering ◽  
Model Parameters ◽  
Layer Model ◽  
Inference System ◽  
Resistivity Data ◽  
Data Inversion ◽  
Geoelectrical Resistivity

Soft computing based geoelectrical data inversion differs from conventional computing in fixing the uncertainty problems. It is tractable, robust, efficient, and inexpensive. In this paper, fuzzy logic clustering methods are used in the inversion of geoelectrical resistivity data. In order to characterize the subsurface features of the earth one should rely on the true field oriented data validation. This paper supports the field data obtained from the published results and also plays a crucial role in making an interdisciplinary approach to solve complex problems. Three clustering algorithms of fuzzy logic, namely, fuzzyC-means clustering, fuzzyK-means clustering, and fuzzy subtractive clustering, were analyzed with the help of fuzzy inference system (FIS) training on synthetic data. Here in this approach, graphical user interface (GUI) was developed with the integration of three algorithms and the input data (AB/2 and apparent resistivity), while importing will process each algorithm and interpret the layer model parameters (true resistivity and depth). A complete overview on the three above said algorithms is presented in the text. It is understood from the results that fuzzy logic subtractive clustering algorithm gives more reliable results and shows efficacy of soft computing tools in the inversion of geoelectrical resistivity data.

Improved Fuzzy FCM-LI Algorithm

2013 ◽  
Vol 765-767 ◽  
pp. 670-673
Author(s):  
Li Bo Hou
Keyword(s):  
Real Time ◽  
Clustering Algorithm ◽  
Feature Analysis ◽  
Cluster Center ◽  
High Dimensional ◽  
Fuzzy C Means ◽  
Sample Data ◽  
Fuzzy C Means Clustering ◽  
Fcm Clustering ◽  
Np Hard Problem

Fuzzy C-means (FCM) clustering algorithm is one of the widely applied algorithms in non-supervision of pattern recognition. However, FCM algorithm in the iterative process requires a lot of calculations, especially when feature vectors has high-dimensional, Use clustering algorithm to sub-heap, not only inefficient, but also may lead to "the curse of dimensionality." For the problem, This paper analyzes the fuzzy C-means clustering algorithm in high dimensional feature of the process, the problem of cluster center is an np-hard problem, In order to improve the effectiveness and Real-time of fuzzy C-means clustering algorithm in high dimensional feature analysis, Combination of landmark isometric (L-ISOMAP) algorithm, Proposed improved algorithm FCM-LI. Preliminary analysis of the samples, Use clustering results and the correlation of sample data, using landmark isometric (L-ISOMAP) algorithm to reduce the dimension, further analysis on the basis, obtained the final results. Finally, experimental results show that the effectiveness and Real-time of FCM-LI algorithm in high dimensional feature analysis.

Quality Control of LWD Resistivity Data Inversion Results

2019 ◽  
Author(s):  
Mikhail Sviridov ◽  
Yuriy Antonov ◽  
Sergey Martakov ◽  
Nikita Tropin ◽  
Henrik Andersson
Keyword(s):  
Quality Control ◽  
Resistivity Data ◽  
Data Inversion ◽  
Lwd Resistivity

Transient electromagnetic search engine for real-time imaging

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. E277-E285 ◽  
Author(s):  
Jide Nosakare Ogunbo ◽  
Jie Zhang ◽  
Xiong Zhang
Keyword(s):  
Real Time ◽  
Search Engine ◽  
Solution Space ◽  
Inversion Method ◽  
Model Parameters ◽  
Transient Electromagnetic ◽  
Model Solutions ◽  
Resistivity Model ◽  
2D Resistivity ◽  
Engine Method

To image the resistivity distribution of the subsurface, transient electromagnetic (TEM) surveying has been established as an effective geophysical method. Conventionally, an inversion method is applied to resolve the model parameters from the available measurements. However, significant time and effort are involved in preparing and executing an inversion and this prohibits its use as a real-time decision-making tool to optimize surveying in the field. We have developed a search engine method to find approximate 1D resistivity model solutions for circular central-loop configuration TEM data in real time. The search engine method is a concept used for query searches from large databases on the Internet. By extension, approximate solutions to any input TEM data can be found rapidly by searching a preestablished database. This database includes a large number of forward simulation results that represent the possible model solutions. The database size is optimized by the survey depth of investigation and the sensitivity analysis of the model layers. The fast-search speed is achieved by using the multiple randomized [Formula: see text]-dimensional tree method. In addition to its high speed in finding solutions, the search engine method provides a solution space that quantifies the resolutions and uncertainties of the results. We apply the search engine method to find 1D model solutions at different data points and then interpolate them to a pseudo-2D resistivity model. We tested the method with synthetic and real data.

THE INTERPRETATION OF DIRECT CURRENT RESISTIVITY MEASUREMENTS

Geophysics ◽  
1978 ◽  
Vol 43 (3) ◽  
pp. 610-625 ◽  
Author(s):  
D. W. Oldenburg
Keyword(s):  
Continuous Function ◽  
Direct Current ◽  
Inverse Theory ◽  
Iterative Technique ◽  
Electrode Arrays ◽  
Resistivity Measurements ◽  
The Earth ◽  
Resistivity Model ◽  
Surface Observations ◽  
Layered Half Space

The linearized inverse theory of Backus and Gilbert has been used to invert potential difference measurements obtained from direct current resistivity soundings. The resistivity is assumed to be a continuous function of depth, hence many of the difficulties encountered when assuming that the earth is a layered half‐space are avoided. An iterative technique is used to construct a resistivity model whose calculated responses agree with the observations, and the model is then appraised to find those features which are uniquely determined by the surface observations. Also, the existence of the Fréchet kernels allows direct comparisons of the resolution provided by various electrode geometries and thus the design of electrode arrays to enhance resolution becomes more feasible.

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