A comparison of smooth and blocky inversion methods in 2-D electrical imaging surveys
TL;DR: In this paper, the L 2 norm based least squares optimisation method is used to map moderately complex structures with arbitrary resistivity distributions, and the blocky or L 1 norm optimization method can be used for such situations.
Abstract: Two-dimensional electrical imaging surveys are now widely used in engineering and environmental surveys to map moderately complex structures. In order to adequately resolve such structures with arbitrary resistivity distributions, the regularised least-squares optimisation method with a cell-based model is frequently used in the inversion of the electrical imaging data. The L 2 norm based least-squares optimisation method that attempts to minimise the sum of squares of the spatial changes in the model resistivity is often used. The resulting inversion model has a smooth variation in the resistivity values. In cases where the true subsurface resistivity consists of several regions that are approximately homogenous internally and separated by sharp boundaries, the result obtained by the smooth inversion method is not optimal. It tends to smear out the boundaries and give resistivity values that are too low or too high. The blocky or L 1 norm optimisation method can be used for such situations. This method attempts to minimise the sum of the absolute values of the spatial changes in the model resistivity. It tends to produce models with regions that are piecewise constant and separated by sharp boundaries. Results from tests of the smooth and blocky inversion methods with several synthetic and field data sets highlight the strengths and weaknesses of both methods. The smooth inversion method gives better results for areas where the subsurface resistivity changes in a gradual manner, while the blocky inversion method gives significantly better results where there are sharp boundaries. While fast computers and software have made the task of interpreting data from electrical imaging surveys much easier, it remains the responsibility of the interpreter to choose the appropriate tool for the task based on the available geological information.
Figures (9)
Fig. 1. (a) A typical field arrangement for 2D electrical imaging surveys. (b) A cell-based model used for 2D resistivity inversion. 
Fig. 2. (a) Wenner (alpha) array apparent resistivity pseudosection for a 100 Ω⋅m rectangular prism embedded in a 10 Ω⋅m medium. Models produced by the (b) smooth and (c) blocky inversion methods. The outline of the prism is shown in (b) and (c) for comparison. 
Fig. 3. (a) A plot of the resistivity of the row of model cells between depths of 1.4 and 1.8 metres for the prism model with a sharp boundary of Figure 2. (b) A similar plot for the prism model with a boundary layer of Figure 6. 
Fig. 4. (a) Wenner (alpha) array apparent resistivity pseudosection for a 100 Ω⋅m rectangular prism embedded in a 10 Ω⋅m medium with 3% random noise added. Models produced by the (b) smooth and (c) blocky inversion methods. 
Fig. 5. (a) Wenner (alpha) array apparent resistivity pseudosection for a 100 Ω⋅m rectangular prism embedded in a 10 Ω⋅m medium with 10% random noise added. Models produced by the (b) smooth and (c) blocky inversion methods. 
Fig. 6. (a) Apparent resistivity pseudosection for a 100 Ω⋅m rectangular prism embedded in a 10 Ω⋅m medium with a 33 Ω⋅m boundary layer. Models produced by the (b) smooth and (c) blocky inversion methods. The outline of the prism and boundary layer is shown in (b) and (c) for comparison. 
Fig. 7. Results from a survey using the Wenner gamma array in the Bauchi area, Nigeria. (a) Apparent resistivity pseudosection. The inversion model obtained with the (b) smooth and (c) blocky inversion methods. Note the location of the borehole at the 175-metre ark. 
Fig. 8. Results from a survey using the Wenner alpha array across the Odarslöv dyke, Sweden. (a) Apparent resistivity pseudosection. The inversion model obtained with the (b) smooth and (c) blocky inversion methods. 
Fig. 9. The Magusi River (Canada) survey with (a) the apparent resistivity pseudosection and the (b) smooth and (c) blocky inversion models. (d) The apparent I.P. pseudosection and the (e) smooth and (f) blocky inversion models.
Citations
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01 Jan 2001
904 citations
Cites methods from "A comparison of smooth and blocky i..."
...This method is also known as an l1-norm or robust or blocky inversion method (Loke et al. 2003), whereas the conventional smoothnessconstrained least-squares method as given in equation (4....
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TL;DR: In this article, numerical simulations are used to compare the resolution and efficiency of 2D resistivity imaging surveys for 10 electrode arrays, including pole-pole (PP), pole-dipole (PD), half-Wenner (HW), Wenner-α (WN), Schlumberger (SC), dipole-dipsole (DD), WenNER-β (WB), γ -array (GM), multiple or moving gradient array (GD) and midpoint-potential-referred measurement (MPR) arrays.
Abstract: Numerical simulations are used to compare the resolution and efficiency of 2D resistivity imaging surveys for 10 electrode arrays. The arrays analysed include polepole (PP), pole-dipole (PD), half-Wenner (HW), Wenner-α (WN), Schlumberger (SC), dipole-dipole (DD), Wenner-β (WB), γ -array (GM), multiple or moving gradient array (GD) and midpoint-potential-referred measurement (MPR) arrays. Five synthetic geological models, simulating a buried channel, a narrow conductive dike, a narrow resistive dike, dipping blocks and covered waste ponds, were used to examine the surveying efficiency (anomaly effects, signal-to-noise ratios) and the imaging capabilities of these arrays. The responses to variations in the data density and noise sensitivities of these electrode configurations were also investigated using robust (L1-norm) inversion and smoothness-constrained least-squares (L2-norm) inversion for the five synthetic models. The results show the following. (i) GM and WN are less contaminated by noise than the other electrode arrays. (ii) The relative anomaly effects for the different arrays vary with the geological models. However, the relatively high anomaly effects of PP, GM and WB surveys do not always give a high-resolution image. PD, DD and GD can yield better resolution images than GM, PP, WN and WB, although they are more susceptible to noise contamination. SC is also a strong candidate but is expected to give more edge effects. (iii) The imaging quality of these arrays is relatively robust with respect to reductions in the data density of a multi-electrode layout within the tested ranges. (iv) The robust inversion generally gives better imaging results than the L2-norm inversion, especially with noisy data, except for the dipping block structure presented here. (v) GD and MPR are well suited to multichannel surveying and GD may produce images that are comparable to those obtained with DD and PD. Accordingly, the GD, PD, DD and SC arrays are strongly recommended for 2D resistivity imaging, where the final choice will be determined by the expected geology, the purpose of the survey and logistical considerations.
731 citations
TL;DR: There have been major improvements in instrumentation, field survey design and data inversion techniques for the geoelectrical method over the past 25 years as mentioned in this paper, which has made it possible to conduct large 2D, 3D and even 4D surveys efficiently to resolve complex geological structures that were not possible with traditional 1-D surveys.
702 citations
Cites background or methods from "A comparison of smooth and blocky i..."
...In addition to robust (L1-norm) inversion (Loke et al., 2003), a range of other interface detection approaches have been applied....
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...Regularization is used, often in the form of a smoothing matrix (Loke et al., 2003), to enforce uniqueness without...
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...Regularization is used, often in the form of a smoothing matrix (Loke et al., 2003), to enforce uniqueness without sacrificing too much resolution....
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...An L1-norm criterion can be used to produce ‘blocky’ models for regions that are piecewise constant and separated by sharp boundaries (Farquharson and Oldenburg, 1998; Loke et al., 2003)....
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01 Jan 2005
TL;DR: In this paper, the relationship between direct current resistivity and hydrological properties, such as porosity and moisture content, is investigated. But the applications of induced polarization methods in hydrogeophysics have been limited.
Abstract: Direct current (DC) resistivity (here referred to as resistivity) and induced polarization (IP) methods allow, respectively, the determination of the spatial distribution of the low-frequency resistive and capacitive characteristics of soil. Since both properties are affected by lithology, pore fluid chemistry, and water content (see Chapter 4 of this volume), these methods have significant potential for hydrogeophysical applications. The methods can be applied at a wide range of laboratory and field scales, and surveys may be made in arbitrary geometrical configurations (e.g., on the soil surface and down boreholes). In fact, resistivity methods are one of the most widely used sets of geophysical techniques in hydrogeophysics. These surveys are relatively easy to carry out, instrumentation is inexpensive, data processing tools are widely available, and the relationships between resistivity and hydrological properties, such as porosity and moisture content, are reasonably well established. In contrast, applications of induced polarization methods in hydrogeophysics have been limited. As noted by Slater and Lesmes (2002), this is partly because of the more complex procedure for data acquisition, but also because the physicochemical interpretation of induced polarization parameters is not fully understood.
618 citations
TL;DR: In this article, a 2D inversion scheme with lateral constraints and sharp boundaries (LCI) is presented for continuous resistivity data, where all data and models are inverted as one system, producing layered solutions with laterally smooth transitions.
Abstract: In a sedimentary environment, quasi-layered models often can represent the actual geology more accurately than smooth minimum-structure models. We present a 2D inversion scheme with lateral constraints and sharp boundaries (LCI) for continuous resistivity data. All data and models are inverted as one system, producing layered solutions with laterally smooth transitions. The models are regularized through lateral constraints that tie interface depths or thicknesses and resistivities of adjacent layers. A priori information, used to resolve ambiguities and to add, for example, geological information, can be added at any point of the profile and migrates through the lateral constraints to parameters at adjacent sites. Similarly, information from areas with well-resolved parameters migrates through the constraints to help resolve areas with poorly constrained parameters. The estimated model is complemented by a full sensitivity analysis of the model parameters supporting quantitative evaluation of the inversion result. A simple synthetic model proves the need for a quasilayered, 2D inversion when compared with a traditional 2D minimum-structure inversion. A 2D minimum-structure inversion produces models with spatially smooth resistivity transitions, making identification of layer boundaries difficult. A continuous vertical electrical sounding field example from Sweden with a depression in the depth to bedrock supports the conclusions drawn from the synthetic example. A till layer on top of the bedrock, hidden in the traditional inversion result, is identified using the 2D LCI scheme. Furthermore, the depth to the bedrock surface is easily identified for most of the profile with the 2D LCI model, which is not the case with the model from the traditional minimumstructure inversion.
379 citations
Cites background or methods from "A comparison of smooth and blocky i..."
...A robust inversion scheme (L1 norm) tends to give a more blocky appearance of the model section (Loke et al., 2001), but layer boundaries are still smeared....
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...When applying an L1 norm misfit criterion as in Figure 4c instead of the usual L2 norm misfit criterion, the model appearance becomes more blocky, with sharp resistivity transitions both laterally and vertically (Loke et al., 2001; Farquharson and Oldenburg, 2003)....
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References
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Book•
01 Jan 1992
4,247 citations
"A comparison of smooth and blocky i..." refers methods in this paper
...In order to study the effects of noise on the inversion results, Gaussian random noise (Press et al., 1992) with an amplitude of 3% was added to the apparent resistivity data....
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...According to Farquharson and Oldenburg (1998), and other authors (Claerbout and Muir, 1973; Menke, 1989; Press et al., 1992; Parker, 1994), this method is less sensitive to outliers in the data particularly when used with the regularised least-squares optimisation method....
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Book•
01 Jan 1989
TL;DR: In this article, the authors describe a number of different types of inverse problems, such as the least squares problem, the purely underdetermined problem, and the Mixed*b1Determined problem.
Abstract: Preface. Introduction. DESCRIBING INVERSE PROBLEMS Formulating Inverse Problems. The Linear Inverse Problem. Examples of Formulating Inverse Problems. Solutions to Inverse Problems. SOME COMMENTS ON PROBABILITY THEORY Noise and Random Variables. Correlated Data. Functions of Random Variables. Gaussian Distributions. Testing the Assumption of Gaussian Statistics Confidence Intervals. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1:THE LENGTH METHOD The Lengths of Estimates. Measures of Length. Least Squares for a Straight Line. The Least Squares Solution of the Linear Inverse Problem. Some Examples. The Existence of the Least Squares Solution. The Purely Underdetermined Problem. Mixed*b1Determined Problems. Weighted Measures of Length as a Type of A Priori Information. Other Types of A Priori Information. The Variance of the Model Parameter Estimates. Variance and Prediction Error of the Least Squares Solution. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 2: GENERALIZED INVERSES Solutions versus Operators. The Data Resolution Matrix. The Model Resolution Matrix. The Unit Covariance Matrix. Resolution and Covariance of Some Generalized Inverses. Measures of Goodness of Resolution and Covariance. Generalized Inverses with Good Resolution and Covariance. Sidelobes and the Backus-Gilbert Spread Function. The Backus-Gilbert Generalized Inverse for the Underdetermined Problem. Including the Covariance Size. The Trade-off of Resolution and Variance. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 3: MAXIMUM LIKELIHOOD METHODS The Mean of a Group of Measurements. Maximum Likelihood Solution of the Linear Inverse Problem. A Priori Distributions. Maximum Likelihood for an Exact Theory. Inexact Theories. The Simple Gaussian Case with a Linear Theory. The General Linear, Gaussian Case. Equivalence of the Three Viewpoints. The F Test of Error Improvement Significance. Derivation of the Formulas of Section 5.7. NONUNIQUENESS AND LOCALIZED AVERAGES Null Vectors and Nonuniqueness. Null Vectors of a Simple Inverse Problem. Localized Averages of Model Parameters. Relationship to the Resolution Matrix. Averages versus Estimates. Nonunique Averaging Vectors and A Priori Information. APPLICATIONS OF VECTOR SPACES Model and Data Spaces. Householder Transformations. Designing Householder Transformations. Transformations That Do Not Preserve Length. The Solution of the Mixed-Determined Problem. Singular-Value Decomposition and the Natural Generalized Inverse. Derivation of the Singular-Value Decomposition. Simplifying Linear Equality and Inequality Constraints. Inequality Constraints. LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONS L1 Norms and Exponential Distributions.
3,592 citations
"A comparison of smooth and blocky i..." refers methods in this paper
...According to Farquharson and Oldenburg (1998), and other authors (Claerbout and Muir, 1973; Menke, 1989; Press et al., 1992; Parker, 1994), this method is less sensitive to outliers in the data particularly when used with the regularised least-squares optimisation method....
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TL;DR: In this paper, the authors propose an extension of the existing 1-D algorithm, Occam's inversion, to smooth 2-D models using an extension to the existing Occam inversion.
Abstract: Magnetotelluric (MT) data are inverted for smooth 2-D models using an extension of the existing 1-D algorithm, Occam’s inversion. Since an MT data set consists of a finite number of imprecise data, an infinity of solutions to the inverse problem exists. Fitting field or synthetic electromagnetic data as closely as possible results in theoretical models with a maximum amount of roughness, or structure. However, by relaxing the misfit criterion only a small amount, models which are maximally smooth may be generated. Smooth models are less likely to result in overinterpretation of the data and reflect the true resolving power of the MT method. The models are composed of a large number of rectangular prisms, each having a constant conductivity. Apriori information, in the form of boundary locations only or both boundary locations and conductivity, may be included, providing a powerful tool for improving the resolving power of the data. Joint inversion of TE and TM synthetic data generated from known models al...
1,411 citations
"A comparison of smooth and blocky i..." refers methods in this paper
...A first-order finitedifference operator (deGroot-Hedlin and Constable, 1990) is used for the roughness filter W....
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...The L 2 norm or smoothness-constrained least-squares optimisation equation (deGroot-Hedlin and Constable, 1990; Ellis and Oldenburg, 1994) is given by Ji T Ji + λiW T W ∆ri = Ji Tgi − λiW T Wr i−1 where g i is the data misfit vector containing the difference between the logarithms of the measured…...
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...A commonly used inversion technique for 2D and 3D resistivity inversion is the regularised least-squares optimisation method (Sasaki, 1989; deGroot-Hedlin and Constable, 1990; Oldenburg and Li, 1994; Loke and Barker, 1996; Li and Oldenburg, 2000)....
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...One commonly used version of the regularised least-squares optimisation method is the smoothness-constrained or L 2 norm method (deGroot-Hedlin and Constable, 1990)....
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TL;DR: In this article, a nonlinear conjugate gradients (NLCG) algorithm was proposed to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity.
Abstract: We investigate a new algorithm for computing regularized solutions of the 2-D magnetotelluric inverse problem. The algorithm employs a nonlinear conjugate gradients (NLCG) scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity. We compare this algorithm theoretically and numerically to two previous algorithms for constructing such “minimum‐structure” models: the Gauss‐Newton method, which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics, and an algorithm due to Mackie and Madden, which solves a sequence of linearized inverse problems incompletely using a (linear) conjugate gradients technique. Numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients (NLCG and Mackie‐Madden) are more efficient than the Gauss‐Newton algorithm in terms of both computer memory requirements and CPU time needed to find accurate solutio...
1,185 citations
"A comparison of smooth and blocky i..." refers background in this paper
...These include the least-squares (Inman, 1975), conjugategradient (Rodi and Mackie, 2001), maximum entropy (Bassrei and 1 School of Physics, Universiti Sains Malaysia, 11800 Penang, Malaysia Tel: 60 4 6574525 Fax: 60 4 6579150 Email: mhloke@tm.net.my 2 Water Research Laboratory, School of Civil and…...
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Book•
16 May 1994
TL;DR: In this article, the Dilogarithm function is used for 1-norm Misfits in linear problems with exact and uncertain data and nonlinear problems with uncertain data.
Abstract: PrefaceCh. 1Mathematical PrecursorCh. 2Linear Problems with Exact DataCh. 3Linear Problems with Uncertain DataCh. 4Resolution and InferenceCh. 5Nonlinear ProblemsAppendix A: The Dilogarithm FunctionAppendix B: Table for 1-norm MisfitsReferencesIndex
871 citations
"A comparison of smooth and blocky i..." refers methods in this paper
...According to Farquharson and Oldenburg (1998), and other authors (Claerbout and Muir, 1973; Menke, 1989; Press et al., 1992; Parker, 1994), this method is less sensitive to outliers in the data particularly when used with the regularised least-squares optimisation method....
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