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Numerical inversion of geophysical data does not normally require user interaction apart from the selection of initial inversion parameters. However, such an inversion often returns a single solution based upon default parameters. While this solution will be geophysically correct, assuming convergence of the algorithm, it may not be the most geologically reasonable answer. It is necessary to incorporate human interaction in selecting inversion solutions, this being the most efficient method for adding qualitative geological constraints. An automatic system provides a user-directed search of the space of geophysical solutions. Rankings assigned to numerical inversion results guide a genetic algorithm in advancing towards a conceptual target. Our example uses resistivity and chargeability data from a pole-dipole induced polarisation survey collected during a mineral exploration program. We invert for specific geological features: a defined, conductive top layer, sharp geological boundaries in the resistivity, and greatest depth of resolution of the inversion algorithm. The interactive system is an organised way to investigate the solution space for valid inversion results that emphasise these geological possibilities.

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... Quantifying uncertainties of 3D geological models arising from sparse geological field measurements (e.g., dip and strike measurements) has been reported before (e.g., Wellmann and Regenauer-Lieb, 2012;Lindsay et al., 2012;Jessell et al., 2014;Pakyuz-Charrier et al., 2018;Stamm et al., 2019;Giraud et al., 2017Giraud et al., , 2019Giraud et al., , 2020. Wijns et al. (2003); Wijns and Kowalczyk (2007) use interactive evolutionary computation to incorporate human evaluations of model outputs and achieve a user-directed search of the model space. However, uncertainty analysis of geological units resulting from multiple geophysical measurements has not been attempted yet. ...

The physical property models obtained from geophysical inversions can be converted to a 3D quasi-geology model via a process termed geology differentiation. Recent works show that geology differentiation can help maximize the value of information contained in geophysical data. However, it remains largely unexplored as to how to quantify the uncertainties of a 3D quasi-geology model. We approach this problem by using a recently developed mixed Lp norm regularization and a priori physical property measurements. We use mixed Lp norm joint inversion to construct a large sequence of physical property models based on the Gzz component of the airborne gravity gradient and magnetic measurements. The available physical property measurements are used to determine which physical property models to accept. We then construct a sequence of 3D quasi-geology models by performing geology differentiation for all the accepted models, which allows us to compute the probabilities of our geology differentiation results. We apply our approach to a set of field data collected over the Decorah area located in northeast Iowa. We successfully quantify the uncertainties of the spatial extents for the identified geological units and compute probabilities of geologic units at any location in our study area. The proposed workflow has broad implications for 3D geological model building based on multiple geophysical and/or rock sample measurements.

... An additional interesting path combining geophysical inversion with geological knowledge is the concept of interactive inversion (Boschetti & Moresi, 2001;Wijns, Boschetti, & Moresi, 2003;Wijns & Kowalczyk, 2007), which aims at an explicit integration of expert knowledge in the inversion process. This is achieved through a system generating possible scenarios. ...

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... An additional interesting path combining geophysical inversion with geological knowledge is the concept of interactive inversion (Boschetti & Moresi, 2001;Wijns, Boschetti, & Moresi, 2003;Wijns & Kowalczyk, 2007), which aims at an explicit integration of expert knowledge in the inversion process. This is achieved through a system generating possible scenarios. ...

The Earth below ground is the subject of interest for many geophysical as well as geological investigations. Even though most practitioners would agree that all available information should be used in such an investigation, it is common practice that only a part of geological and geophysical information is actually integrated in structural geological models. We believe that some reasons for this omission are (a) an incomplete picture of available geological modeling methods, and (b) the problem of the perceived static picture of an inflexible geological representation in an image or geological model. With this work, we aim to contribute to the problem of subsurface interface detection through (a) the review of state-of-the-art geological modeling methods that allow the consideration of multiple aspects of geological realism in the form of observations, information, and knowledge, cast in geometric representations of subsurface structures, and (b) concepts and methods to analyze, quantify, and communicate related uncertainties in these models. We introduce a formulation for geological model representation and interpolation and uncertainty analysis methods with the aim to clarify similarities and differences in the diverse set of approaches that developed in recent years. We hope that this chapter provides an entry point to recent developments in geological modeling methods, helps researchers in the field to better consider uncertainties, and supports the integration of geological observations and knowledge in geophysical interpretation, modeling and inverse approaches.

... An additional interesting path combining geophysical inversion with geological knowledge is the concept of interactive inversion (Boschetti & Moresi, 2001;Wijns, Boschetti, & Moresi, 2003;Wijns & Kowalczyk, 2007), which aims at an explicit integration of expert knowledge in the inversion process. This is achieved through a system generating possible scenarios. ...

... The scientific literature is extraordinarily rich in algorithms concerning 2D and 3D potential field data inversion (e.g., Last and Kubik, 1983;Guillen and Menchetti, 1984;Litinsky, 1989;Li and Oldenburg, 1996; Barbosa and Silva 1994;Barbosa, 2004, 2006;Cella and Fedi, 2012;Paoletti et al., 2014;Silva Dias et al., 2009;Wijns and Kowalczyk, 2007). However, there are no 1D algorithms equivalent to those applicable to seismic or electromagnetic methods. ...

We have developed a method to interpret potential fields, which obtains 1D models by inverting vertical soundings of potential field data. The vertical soundings are built through upward continuation of potential field data, measured on either a profile or a surface. The method assumes a forward problem consisting of a volume partitioned in layers, each of them homogeneous and horizontally finite, but with the density changing versus depth. The continuation errors, increasing with the altitude, are automatically handled by determining the coefficients of a third-order polynomial function of the altitude. Due to the finite size of the source volume, we need a priori information about the total horizontal extent of the volume, which is estimated by boundary analysis and optimized by a Markov chain process. For each sounding, a 1D inverse problem is independently solved by a nonnegative least-squares algorithm. Merging of the several inverted models finally yields approximate 2D or 3D models that are, however, shown to generate a good fit to the measured data. The method is applied to synthetic models, producing good results for either perfect or continued data. Even for real data, i.e., the gravity data of a sedimentary basin in Nevada, the results are interesting, and they are consistent with previous interpretation, based on 3D gravity inversion constrained by two gamma-gamma density logs.

... The majority of researchers primarily focus on the prior distribution of the prestack parameters inversion (Buland and Omre, 2003;Wijns and Kowalczyk, 2007;Lelièvre et al., 2009;Karimi et al., 2010;Alemie and Sacchi, 2011). The prior distribution of the inversion parameters is only related to the vertical resolution and sparsity of the inversion results. ...

Comprehensive utilisation of elastic and petrophysical parameters to predict favourable reservoirs can help reduce the possibility of misidentification of resources. Existing simultaneous inversion methods for estimating the elastic and petrophysical parameters are typically based on either the Gassmann equation, with which these parameters are inverted from prestack seismic data through stochastic optimisation methods, or Wyllie's modified equation, with which these parameters are inverted from poststack seismic data using deterministic optimisation methods. The purpose of this work is to develop a strategy for estimating the elastic and petrophysical parameters based on the Gassmann equation using deterministic prestack inversion. We employ the Gassmann equation to construct the relationship between the prestack seismic data and petrophysical parameters. We treat the joint posterior probability of elastic and petrophysical parameters as the objective function under a Bayesian framework. Given the macroscopic geological background and the poor-quality prestack seismic data, seismic facies regularisation constraints were introduced to improve the robustness and accuracy of the inversion. The very fast-simulated annealing method is used to quickly find the optimal solutions for the elastic and petrophysical parameters. Based on a model test and the application of real data demonstrates that the proposed inversion method has high accuracy and strong reliability.

... The scientific literature is extraordinarily rich in algorithms concerning 2D and 3D potential field data inversion (e.g., Last and Kubik, 1983;Guillen and Menchetti, 1984;Litinsky, 1989;Li and Oldenburg, 1996; Barbosa and Silva 1994;Barbosa, 2004, 2006;Cella and Fedi, 2012;Paoletti et al., 2014;Silva Dias et al., 2009;Wijns and Kowalczyk, 2007). However, there are no 1D algorithms equivalent to those applicable to seismic or electromagnetic methods. ...

We present a new 1D inversion algorithm for potential field data, which consists in the inversion of vertical soundings of data at different altitudes. The inversion of multiple 1D vertical gravity/magnetic soundings is made over the same source-volume, which is a finite volume of layers of different densities, for interpreting anomalies along a profile or over a measurement surface. The inversion algorithm was tested on synthetic data and on a real case, relative to gravity data referred to a sedimentary basin in Nevada, USA.

... Clapp et al. (2004) use seismic reflector dip estimates based on migrated data to construct a nonstationary anisotropic regularization operator to be used in the objective function. Others have built on the work of Clapp et al. (2004) to allow for increased input from the interpreter to help ensure a good geologic inversion solution (Barbosa and Silva, 2006;Wijns and Kowalczyk, 2007). Guitton et al. (2012) present constrained full-waveform inversion using dip information obtained from migrated seismic sections. ...

In complex geology, the presence of highly dipping structures can complicate impedance inversion. We have developed a structurally constrained inversion in which a computationally well-behaved objective function is minimized subject to structural constraints. This approach allows the objective function to incorporate structural orientation in the form of dips into our inversion algorithm. Our method involves a multitrace impedance inversion and a rotation of an orthogonal system of derivative operators. Local dips used to constrain the derivative operators were estimated from migrated seismic data. In addition to imposing structural constraints on the inversion model, this algorithm allows for the inclusion of a priori knowledge from boreholes. We investigated this algorithm on a complex synthetic 2D model as well as a seismic field data set. We compared the result obtained with this approach with the results from single trace-based inversion and laterally constrained inversion. The inversion carried out using dip information produces a model that has higher resolution that is more geologically realistic compared with other methods. © 2016 Society of Exploration Geophysicists and American Association of Petroleum Geologists.

... Most of these aforementioned references incorporate the a priori information to the original set up of the inversion. Wijns & Kowalczyk (2007) and Barbosa & Silva (2006) develop user interactive inversion approaches, which take user input to direct the inversion towards a geologically reasonable solution. A good summary of incorporating a priori information-especially from geological data-into seismic observations can be obtained in Lelièvre (2009). ...

Conventional traveltime seismic tomography methods with Tikhonov regularization (L2 norm) typically produce smooth models, but these models may be inappropriate when subsurface structure contains discontinuous features, such as faults or fractures, indicating that tomographic models should contain sharp boundaries. For this reason, we develop a double-difference (DD) traveltime tomography method that uses a modified total-variation regularization scheme incorporated with a priori information on interfaces to preserve sharp property contrasts and obtain accurate inversion results. In order to solve the inversion problem, we employ an alternating minimization method to decouple the original DD tomography problem into two separate subproblems: a conventional DD tomography with Tikhonov regularization and a L2 total-variation inversion. We use the LSQR linear solver to solve the Tikhonov inversion and the split-Bregman iterative method to solve the total-variation inversion. Through our numerical examples, we show that our new DD tomography method yields more accurate results than the conventional DD tomography method at almost the same computational cost.

... The authors also presented an adaptive learning procedure for incorporating prior knowledge (Silva & Barbosa 2006). Wijns & Kowalczyk (2007) proposed a semi-automatic procedure that allows the interpreter to set a geologically reasonable solution. With the aim of incorporating depth information and regularizing the solution, Barnes & Barraud (2012) developed an inversion algorithm that solves for the geometric interface between geological bodies. ...

We present a potential-ﬁeld-constrained inversion procedure based on apriori information
derived exclusively from the analysis of the gravity and magnetic data (self-constrained in-
version). The procedure is designed to be applied to underdetermined problems and involves
scenarios where the source distribution can be assumed to be of simple character. To set up
effective constraints, we ﬁrst estimate through the analysis of the gravity or magnetic ﬁeld
some or all of the following source parameters: the source depth-to-the-top, the structural
index, the horizontal position of the source body edges and their dip. The second step is in-
corporating the information related to these constraints in the objective function as depth and
spatial weighting functions. We show, through 2-D and 3-D synthetic and real data examples,
that potential ﬁeld-based constraints, for example, structural index, source boundaries and
others, are usually enough to obtain substantial improvement in the density and magnetization
models.

... The authors also presented an adaptive learning procedure for incorporating prior knowledge (Silva & Barbosa 2006). Wijns & Kowalczyk (2007) proposed a semi-automatic procedure that allows the interpreter to set a geologically reasonable solution. With the aim of incorporating depth information and regularizing the solution, Barnes & Barraud (2012) developed an inversion algorithm that solves for the geometric interface between geological bodies. ...

We present a potential-field constrained inversion procedure based on a priori information derived exclusively from the analysis of the gravity and magnetic data (self-constrained inversion). The procedure is designed to be applied to underdetermined problems and involves scenarios where the source distribution can be assumed to be of simple character. To set up effective constraints we first estimate through analysis of the gravity or magnetic field some or all of the following source parameters: the source depth-to-the-top, the structural index, the horizontal position of the source body edges and their dip. The second step is incorporating the information related to these constraints in the objective function as depth and spatial weighting functions. We show, through 2D and 3D synthetic and real data examples, that potential field-based constraints, e.g., structural index, source boundaries and others, are usually enough to obtain substantial improvement in the density and magnetization models.

... Another novel approach is that of Wijns & Kowalczyk (2007) who, similarly to the approach of Barbosa & Silva (2006), allow for input from the interpreter to help ensure a geologically reasonable solution. Several inversions are performed with random values for several control parameters. ...

In this paper, we investigate options for incorporating structural orientation information into under-determined inversions in a deterministic framework (i.e. minimization of an objective function). The first approach involves a rotation of an orthogonal system of smoothness operators, for which there are some important practical details in the implementation that avoid asymmetric inversion results. The second approach relies on addition of linear constraints into the optimization problem, which is solved using a logarithmic barrier method. A 2-D synthetic example is provided involving a synclinal magnetic structure and we invert two sets of real survey data in 3-D (one gravity data, the other magnetic data). Using those examples, we demonstrate how different types of orientation information can be incorporated into inversions. Incorporating orientation information can yield bodies that have expected aspect ratios and axis orientations. Physical property increase or decrease in particular directions can also be obtained.

Geophysical investigations which commenced thousands of years ago in China from observations of the Earth shaking caused by large earthquakes (Lee et al., 2003) have gone a long way in their development from an initial, intuitive stage to a modern science employing the newest technological and theoretical achievements. In spite of this enormous development, geophysical research still faces the same basic limitation. The only available information about the Earth comes from measurement at its surface or from space. Only very limited information can be acquired by direct measurements. It is not surprising, therefore, that geophysicists have contributed significantly to the development of the inverse theory—the theory of inference about sought parameters from indirect measurements.

Inversion algorithms numerically evaluate the mis- match between model and data to guide the search for minima in parameter spaces. In an alternative approach, the numerical evaluation of data misfit can be replaced by subjectively judging the solution's quality. This widens the class of problems that can be treated within the framework of formal inverse theory—in particular, var- ious geophysical/geological/geodynamic applications in which structural similarity between model and data de- termines the quality of the fit. In this situation, prior knowledge, experience, and even personal intuition are crucial. This approach also provides a simple way to in- clude such expertise in more traditional numeric appli- cations, e.g., to treat ambiguous problems and disregard geologically unfeasible solutions from the inverse search.

The use of genetic algorithms in geophysical inverse problems is a relatively recent development and of- fers many advantages in dealing with the nonlinearity inherent in such applications. However, in their appli- cation to specific problems, as with all algorithms, prob- lems of implementation arise. After extensive numerical tests, we implemented a genetic algorithm to efficiently invert several sets of synthetic seismic refraction data. In particular, we aimed at overcoming one of the main problems in the application of genetic algorithms to geophysical problems: i.e., high dimensionality. The ad- dition of a pseudo-subspace method to the genetic algo- rithm, whereby the complexity and dimensionality of a problem is progressively increased during the inversion, improves the convergence of the process. The method allows the region of the solution space containing the global minimum to be quickly found. The use of local optimization methods at the last stage of the search fur- ther improves the quality of the inversion. The genetic algorithm has been tested on a field data set to determine the structure and base of the weathered layer (regolith) overlaying a basement of granite and greenstones in an Archaean terrain of Western Australia.

In this paper, the term "depth of investigation" refers generically to the depth below which surface data are insensitive to the value of the physical property of the earth. Estimates of this depth for de resistivity and induced polarization (IP) surveys are essential when interpreting models obtained from any inversion because structure beneath that depth should not be interpreted geologically. We advocate carrying out a limited exploration of model space to generate a few models that have minimum structure and that differ substantially from the final model used for interpretation, Visual assessment of these models often provides answers about existence of deeper structures. Differences between the models can be quantified into a depth of investigation (DOI) index that can be displayed with the model used for interpretation. An explicit algorithm for evaluating the DOI is presented. The DOI curves are somewhat dependent upon the parameters used to generate the different models, but the results are robust enough to provide the user with a first-order estimate of a depth region below which the earth structure is no longer constrained by the data. This prevents overinterpretation of the inversion results. The DOI analysis reaffirms the generally accepted conclusions that different electrode array geometries have different depths of penetration. However, the differences between the inverted models for different electrode arrays are far less than differences in the pseudosection images. Field data from the Century deposit are inverted and presented with their DOI index.

Inverse modelling of geological processes, in the absence of established numerical criteria to act as inversion targets, requires an approach that uses human interaction to assess forward model results. The method of interactive evolutionary computation provides for the inclusion of qualitative geological expertise within a rigorous mathematical inversion scheme, by simply asking an expert user to evaluate a sequence of forward geological models. The traditional numerical misfit is replaced by a human appraisal of misfit. We use this interactive technique to successfully invert a geodynamic model for a conceptual pattern of fault spacing during crustal extension.

We survey the research on interactive evolutionary computation
(IEC). The IEC is an EC that optimizes systems based on subjective human
evaluation. The definition and features of the IEC are first described
and then followed by an overview of the IEC research. The overview
primarily consists of application research and interface research. In
this survey the IEC application fields include graphic arts and
animation, 3D computer graphics lighting, music, editorial design,
industrial design, facial image generation, speed processing and
synthesis, hearing aid fitting, virtual reality, media database
retrieval, data mining, image processing, control and robotics, food
industry, geophysics, education, entertainment, social system, and so
on. The interface research to reduce human fatigue is also included.
Finally, we discuss the IEC from the point of the future research
direction of computational intelligence. This paper features a survey of
about 250 IEC research papers