Horizontal loop electromagnetic data inversion through novel metaheuristics and its applications to ore deposit explorations

Geophysical Inverse Theory and Regularization Problems

Inversion of geophysical data relies on knowledge about how to solve the forward problem, that is, computing data from a given set of model parameters. In many applications of inverse problems, the solution to the forward problem is assumed to be known perfectly, without any error. In reality, solving the forward model (forward-modeling process) will almost always be prone to errors, which we referred to as modeling errors. For a specific forward problem, computation of crosshole tomographic first-arrival traveltimes, we evaluated how the modeling error, given several different approximate forward models, can be more than an order of magnitude larger than the measurement uncertainty. We also found that the modeling error is strongly linked to the spatial variability of the assumed velocity field, i.e., the a priori velocity model. We discovered some general tools by which the modeling error can be quantified and cast into a consistent formulation as an additive Gaussian observation error. We tested a method for generating a sample of the modeling error due to using a simple and approximate forward model, as opposed to a more complex and correct forward model. Then, a probabilistic model of the modeling error was inferred in the form of a correlated Gaussian probability distribution. The key to the method was the ability to generate many realizations from a statistical description of the source of the modeling error, which in this case is the a priori model. The methodology was tested for two synthetic ground-penetrating radar crosshole tomographic inverse problems. Ignoring the modeling error can lead to severe artifacts, which erroneously appear to be well resolved in the solution of the inverse problem. Accounting for the modeling error leads to a solution of the inverse problem consistent with the actual model. Further, using an approximate forward modeling may lead to a dramatic decrease in the computational demands for solving inverse problems.

Some applications of statistical modelling to solve inverse problems in geophysics

An implementation of differential evolution algorithm for inversion of geoelectrical data

Electrical resistivity tomography (ERT) is a geophysical method used to imaging the subsurface and widely used in hydrogeological studies due to its sensitivity to electrical resistivity, which is directly related to rock type, porosity, ionic strength of the pore fluids, and surface conductivity of geologic materials. The prediction of subsurface properties from the recorded data at the surface requires solving a challenging geophysical inversion problem. For near-surface characterization studies, this is often accomplished with deterministic electrical resistivity inversion methods. Deterministic geophysical inversion approaches linearize the problem around an initial solution, resulting in a single smooth representation of the subsurface. Deterministic models are unable to capture the natural variability of the subsurface. Moreover, a single solution does not yield enough information for accurate uncertainty assessment. Contrary to deterministic approaches, stochastic inversion methods predict multiple model realizations that fit similarly the recorded geophysical data and allow assessing uncertainties. Lately, deep learning algorithms based on deep generative models have been used to re-parametrize model and data spaces into low-dimensional domains and solve geophysical inverse problems in a more efficient way.We propose an ERT inversion methodology in which a deep convolutional variational autoencoder (VAE) network is trained with a set of electrical resistivity models generated using geostatistical simulation. After training the VAE, the latent space is perturbed and updated iteratively with adaptive stochastic sampling to generate electrical resistivity models by inputting the optimized latent vectors to the decoder part of the VAE. From the set of decoded models, we use a finite volume approximation of Poisson’s equation to compute synthetic apparent resistivity models. The misfit between predicted and observed apparent resistivity data is used to drive the convergence of the iterative procedure and condition the optimization of new models in the subsequent iterations.The proposed methodology is illustrated by applying it to both a two-dimensional synthetic case and to a two-dimensional profile obtained from an ERT survey carried out in an area located in the Southern region of Portugal. In both application examples, the predicted models generate synthetic geophysical data that match the observed one. We show the ability of the model to assess spatial uncertainty and compare the results obtained in the real data set against commercial deterministic ERT inversion methodology.The work presented herein is supported by the PRIMA programme under grant agreement No. 1923, project Innovative and Sustainable Groundwater Management in the Mediterranean (InTheMED). The PRIMA programme is supported by the European Union.

We present statistical and interval techniques for evaluating the uncertainties associated with geophysical tomographic inversion problems, including estimation of data errors, model errors, and total solution uncertainties. These techniques are applied to the inversion of traveltime data collected in a cross well seismic experiment. The inversion method uses the conjugate gradient technique, incorporating expert knowledge of data and model uncertainty to stabilize the solution. The technique produced smaller uncertainty than previous tomographic inversion of the data.

<div>We present a framework to solve geophysical inverse problems using the Hamiltonian Monte Carlo (HMC) method, with a focus on Bayesian tomography. Recent work in the geophysical community has shown the potential for gradient-based Monte Carlo sampling for a wide range of inverse problems across several fields.</div><div> </div><div>Many high-dimensional (non-linear) problems in geophysics have readily accessible gradient information which is unused in classical probabilistic inversions. Using HMC is a way to help improve traditional Monte Carlo sampling while increasing the scalability of inference problems, allowing access to uncertainty quantification for problems with many free parameters (>10'000). The result of HMC sampling is a collection of models representing the posterior probability density function, from which not only "best" models can be inferred, but also uncertainties and potentially different plausible scenarios, all compatible with the observed data. However, the amount of tuning parameters required by HMC, as well as the complexity of existing statistical modeling software, has limited the geophysical community in widely adopting a specific tool for performing efficient large-scale Bayesian inference.</div><div> </div><div>This work attempts to make a step towards filling that gap by providing an HMC sampler tailored for geophysical inverse problems (by e.g. supplying relevant priors and visualizations) combined with a set of different forward models, ranging from elastic and acoustic wave propagation to magnetic anomaly modeling, traveltimes, etc.. The framework is coded in the didactic but performant languages Julia and Python, with the possibility for the user to combine their own forward models, which are linked to the sampler routines by proper interfaces. In this way, we hope to illustrate the usefulness and potential of HMC in Bayesian inference. Tutorials featuring an array of physical experiments are written with the aim of both showcasing Bayesian inference and successful HMC usage. It additionally includes examples on how to speed up HMC e.g. with automated tuning techniques and GPU computations.</div>

We introduce Recursive Ant Colony Optimization (RACO), a new technique for the estimation of function parameters from field data obtained from various geophysical surveys. RACO is a modified form of the ant colony method (ACO) and can be used to determine the best solution for various geophysical problems. As RACO is an extension of ACO, it simulates the social behaviour of ants, optimizing their path from the nest to the food source. RACO applies ACO recursively, introducing an additional term ‘depth’ that decides the extent of the recursion. The results of each depth contribute towards the construction of models for the following depth and the range of values for each parameter is condensed around the actual solution. The algorithm is tested on a two parameter mathematical function and to further test its efficiency and stability on real world problems, we applied it on a few geophysical inversion problems. We show its application on synthetic data sets for the self‐potential anomaly (noise free and 10% noise) caused in the case of an inclined sheet like a body buried inside the earth, synthetic 1D vertical electrical sounding (VES) geo‐electric data sets for three‐layer and five‐layer earth models, a real 1D VES data set obtained from the Tangasol region, India and P‐wave logs for two sites located in the Krishna‐Godavari (KG) Basin, India. The results generated by RACO inversions are found to be in good agreement with previous studies conducted using these data sets.

Geophysical inverse problems are inherently ill-posed due to sparse, noisy, and indirect observations, making Uncertainty Quantification (UQ) a fundamental requirement for reliable subsurface characterization. Bayesian inversion provides a comprehensive probabilistic framework for inferring subsurface parameters by coherently combining prior knowledge with observational data through the likelihood function. However, the practical deployment of Bayesian methods in large-scale geophysical settings is often hampered by the prohibitive computational cost of repeated forward model evaluations. In this context, uncertainty is often not solely driven by observational noise; a substantial and sometimes dominant contribution arises from model error, resulting from simplified physical descriptions, numerical discretization, and uncertain boundary conditions. When these sources of uncertainty are neglected or inadequately represented, Bayesian inversions may yield biased posterior estimates and unrealistically narrow uncertainty bounds. These limitations are particularly acute in deep Earth applications, where complex rheologies, poorly constrained geometries, and computationally intensive forward models coexist.A key challenge is the accurate delineation of the Lithosphere–Asthenosphere Boundary (LAB), which plays a central role in controlling mantle dynamics, lithospheric deformation, and deep geothermal processes. Despite the necessity of relying on Bayesian approaches to estimate the LAB and its associated uncertainties, the high computational cost of repeated evaluations of the forward solver makes this unfeasible within realistic time frames [1]. To address these limitations, this work investigates Reduced-Order Modeling (ROM) techniques to enable efficient Bayesian inversion of LAB geometry in geodynamical Stokes flow models. ROMs construct low-dimensional surrogates of high-fidelity solvers, allowing rapid forward simulations while preserving the dominant physical behavior of mantle flow. By integrating ROMs with Bayesian inference, the proposed framework enables effective and reliable UQ for LAB characterization.Keywords: Geophysical inverse problems; Bayesian inversion; Uncertainty Quantification; Reduced-Order Modeling; Lithosphere–Asthenosphere BoundaryAcknowledgement This research was conducted within the EarthSafe Doctoral Network and has received funding from the European Union’s Horizon Europe research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 101120556.

One of the most important stages in addressing geophysical problems is the inversion procedure of the observed data, which involves incorporating several interdependent model parameters that describe the geophysical properties of the Earth’s subsurface. In this article, we propose a new modification to the inversion technique by applying global optimization method based on systematic search. Global optimization methods are particularly useful because geophysical inverse problems are often nonlinear making traditional local optimization methods less effective. A systematic search using the permutation method for inverting geophysical data refers to a comprehensive and methodical approach that explores different parameter combinations or configurations. The guided permutation method ensures that all possible configurations of model parameters are tested to find the best match between the geophysical model and the observed data. While this method is highly reliable in finding global solutions, its computational cost can be prohibitive for large-scale problems. This issue can be mitigated by conducting an analysis of geophysical parameter sensitivity, which helps determine which model parameters can be reliably estimated, and which are less influential. Furthermore, this analysis enhances the understanding of model uncertainties, guides the inversion process, reduces computational costs, and improves the reliability of the final geophysical model by employing this new inversion technique with the specific method used that is systematic search, permutation, sensitivity analysis. The computer program codes based on systematic search and guided permutation method demonstrates the robustness and accuracy of geophysical inversions. The inversion results from this program, applied to multimode Love wave dispersion curves, show a significant improvement in the quality of the match between the observed and modelled dispersion curves. This improvement is quantitatively confirmed by a decrease in root mean square (RMS) error from 13.435 m/s to 2.887×10-5 m/s, representing a significant reduction in misfit. Based on the obtained results, the program implementing this new inversion technique, incorporating systematic search, guided permutation, and sensitivity analysis, indicates the robustness and accuracy of geophysical inversions. Finally, we ensure that this inversion technique can be applied to solve other geophysical data problems.

Nowadays, the utilization of quantum computing is on the rise, with its applications extending across numerous fields. This new computing paradigm is likely to inject fresh blood into the development of multiple disciplines. For a long time, solving the ill‐conditioned problem in geophysics has been an incredibly challenging topic. Especially when the condition number of linear systems is too large, the solution method of linear equation system based on matrix inversion tends to fail. In this paper, an iterative quantum annealing algorithm is proposed and successfully applied to the solution process of staggered‐grid finite‐difference operators based on dispersion preservation method. Then, the dispersion analysis and forward simulation of the obtained operators are carried out, which verifies that the algorithm yields results close to the classical optimization method. Our experiments verify that the algorithm can effectively deal with the ill‐conditioning of linear systems, and the algorithm is essentially a global optimization algorithm, which is suitable for solving ill‐conditioned problems and ill‐posed problems in geophysics. With the flourishing development of quantum annealer devices, it is expected that this method may be applied in solving large‐scale geophysical inverse problem in the future.

This paper presents the application of a novel trans-dimensional sampling approach to a time domain airborne electromagnetic (AEM) inverse problem to solve for plausible conductivities of the subsurface. Geophysical inverse field problems, such as time domain AEM, are well known to have a large degree of non-uniqueness. Common least-squares optimisation approaches fail to take this into account and provide a single solution with linearised estimates of uncertainty that can result in overly optimistic appraisal of the conductivity of the subsurface. In this new non-linear approach, the spatial complexity of a 2D profile is controlled directly by the data. By examining an ensemble of proposed conductivity profiles it accommodates non-uniqueness and provides more robust estimates of uncertainties.

In the paper a combined inversion algorithm solving the nonlinear geophysical well-logging inverse problem is presented. We apply a successive combination of a float-encoded Genetic Algorithm as a global optimization method and the well-known linearized Marquardt algorithm forming a fast inversion procedure. The technique is able to decrease the CPU run time at least one order of magnitude compared to the Genetic Algorithm and gives the parameter estimation errors having a few linearized optimization steps at the end of the iteration process. We use depth-dependent tool response equations to invert all the data of a greater depth-interval jointly in order to determine petrophysical parameters of homogeneous or inhomogeneous layers in one inversion procedure. The so-called interval inversion method gives more accurate and reliable estimation for the petrophysical model parameters than the conventional point by point inversion methods. It also enables us to determine the layer-thicknesses that can not be extracted from the data set by means of conventional inversion techniques. We test the combined interval inversion method on synthetic data, and employ it to the interpretation of well logs measured in a Hungarian hydrocarbon exploratory borehole.

Summary We have developed a novel probabilistic approach to the joint inversion of multi-modal geophysical data based on the Gramian constraint. The multi-modal geophysical survey is the most effective technique for geophysical exploration because different physical data reflect distinct physical properties of the different components of the geological system. The joint inversion of multi-modal data can produce enhanced subsurface images of the physical property distributions, which enhances our ability to explore natural resources. One effective method of joint inversion is based on the Gramian constraint. This technique enforces the relationships between different model parameters during the inversion process. We demonstrate that the Gramian can be interpreted as a determinant of the covariance matrix between different physical models representing subsurface geology in the framework of the probabilistic approach to inverse theory. This interpretation enables us to use all the power of the modern probability theory and statistics in developing new methods for the joint inversion of multi-modal geophysical data. We apply the developed joint inversion methodology to inversion of gravity gradiometry and magnetic data in the Nordkapp Basin, Barents Sea to image salt diapirs.

Recently, Love wave applications on geological and geophysical problems have been frequently reported by numerous authors. Unlike Rayleigh wave, Love wave modelling has a simpler algorithm since depends only on three earth properties: SH-wave velocity, density, and thickness of layers. The fewer the parameters more stable the inversion result. There are several tasks, however, that have not been reported related to the modelling and inversion of Love wave data. The tasks are to give answer to several following questions: (a) can Love wave propagate in an inversely dispersive medium of the earth? (b) what is the method recommended to use for explaining appropriate Love wave dispersion curve for the inversely dispersive medium? (c) is that possible to have ‘effective’ Love wave dispersion curve for the inversely dispersive medium and to utilize it in the inversion process? Numerical simulations show that modal matrix method cannot prepare appropriate Love wave dispersion for such a medium. The resulting Love wave dispersion curve still shows high phase velocities at low frequencies, which is not as desired. Full SH-waveform method, in the other hand, successful simulates appropriate Love wave dispersion for such a medium. By this method, an effective Love wave dispersion curve is also introduced. Inversion of the effective dispersion curve can excellently estimate the shear-wave velocity.