Trans-dimensional Bayesian inversion of airborne electromagnetic data for 2D conductivity profiles

Geophysical Inverse Theory and Regularization Problems

As an efficient geophysical exploration method, the time-domain airborne electromagnetic (AEM) data often show sign reversal in late-time channels due to induced polarization (IP) effect. The traditional imaging and inversion methods without considering the IP effect cannot recover the true electrical structure of the earth, so it is necessary to develop 3D EM forward modeling and inversion techniques with IP effect. In this paper, we propose a 3D forward modeling method for time-domain AEM with IP effect based on unstructured finite-element (FE) method. To describe the IP effect of a medium, we introduce the Cole-Cole model and transform it into fractional derivative form using frequency-time conversion. Then, we discretize it using the difference discretization format of Caputo fractional derivative. Finally, we use the vector FE method based on unstructured tetrahedral mesh and unconditionally stable second-order backward Euler’s scheme to discretize Maxwell’s equations in space and time. In this way, we can solve the 3D forward modeling problem for AEM with IP effect in time domain. We verify the accuracy of our algorithm by comparing it with the 1D semi-analytical solution for a half-space model, and then calculate EM responses for typical abnormal models and analyze the characteristics of IP effect.

We present two approaches to solving the airborne electromagnetic (AEM) problems given the induced polarization (IP) effect: with calculating the field in a polarizable medium directly in the time domain (TD) and with calculating the EM+IP field in the frequency domain (FD) followed by a transition to the time domain. The first approach is based on calculating the field at each time step given the sources that depend on the currents excited in the medium at previous time steps. This approach allows us to use any IP decay functions. The frequency domain approach is based on the Fourier series expansion of a non-stationary source and the Cole-Cole model. In order to reduce the computational cost, we use the Hermite spline interpolation. Both approaches allow modeling EM + IP processes in complex media with topography and the curved boundaries of layers containing 3D heterogeneities. 3D modeling is performed on non-conforming hexahedral meshes generated fully automatically. The analysis of computational efficiency and verification of the developed approaches are presented in comparison with the results from other researchers. Moreover, we compare the results obtained in the time and frequency domains. The results of 3D modeling the IP effects, which are obtained for some geoelectrical models typical for AEM exploration problems, are demonstrated.

Following a successful bathymetric mapping demonstration in a previous study, the potential of airborne EM for seafloor characterisation has been investigated. The sediment thickness inferred from 1D inversion of helicopter-borne time-domain electromagnetic (TEM) data has been compared with estimates based on marine seismic studies. Generally, the two estimates of sediment thickness, and hence depth to resistive bedrock, were in reasonable agreement when the seawater was ~20 m deep and the sediment was less than ~40 m thick. Inversion of noisy synthetic data showed that recovered models closely resemble the true models, even when the starting model is dissimilar to the true model, in keeping with the uniqueness theorem for EM soundings. The standard deviations associated with shallow seawater depths inferred from noisy synthetic data are about ± 5% of depth, comparable with the errors of approximately ± 1 m arising during inversion of real data. The corresponding uncertainty in depth-to-bedrock estimates, based on synthetic data inversion, is of order of ± 10%. The mean inverted depths of both seawater and sediment inferred from noisy synthetic data are accurate to ~1 m, illustrating the improvement in accuracy resulting from stacking. It is concluded that a carefully calibrated airborne TEM system has potential for surveying sediment thickness and bedrock topography, and for characterising seafloor resistivity in shallow coastal waters.

An increase in demand for commodities coupled with a decrease in world-class ore deposit discoveries in the last three decades is the driving force for exploring for a new generation of world-class ore deposits at depth. Exploration through cover is becoming one of the critical challenges for the mineral exploration industry. This paper explores methods of integrating geophysical, geological and landscape data so as to reduce uncertainty in landscape evolution models interpreted from inversion of electromagnetic (EM) data. Inversion of EM data is, in general, non-unique: many different models will be able to fit the EM data equally well, resulting in large uncertainty. However, EM model uncertainty can be significantly constrained when geological and landscape context are taken into account. This study aims to characterise the regolith (weathered and transported cover), and understand how its conductivity varies with the landscape and with the regolith architecture. This is assisted by logging information from 104 boreholes penetrating through the regolith and into the basement rocks. The study area is associated with the DeGrussa Cu-Au deposit located in the Capricorn Orogen of Western Australia, a regolith-dominated terrain where the regolith varies in thickness between <5 and ~150m. We first select EM decay curves, extracted from an airborne EM survey, whose locations are close to those of the boreholes. We then use a layered-earth (1D) forward model, and invert those data for the electrical resistivities of each of the lithologies identified on the geological borehole logs. Layer boundary depths are fixed to the borehole depths. We show how the non-uniqueness associated with EM inversion can be reduced by the inclusion of decay curves from geology with different layer thickness ratios. Lithological models derived from the integration of electromagnetic, geological and landscape data show less uncertainty and are therefore more reliable for mineral exploration targeting.

Some applications of statistical modelling to solve inverse problems in geophysics

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.

We describe the main properties of the EQUATOR airborne geophysical system and how the system can be employed to solve some engineering and environmental problems. To date, the EQUATOR system has been used for groundwater search in 2014 and 2015 in Siberia. The name “EQUATOR” originates from the intention to EQUATE, or to balance time and frequency domain approaches in airborne electromagnetics, which is why the system was originally designed. To the best of our knowledge, EQUATOR is the only system to collect electromagnetic data both in time and in frequency domains. We further explain the difference between the two methods on some numerical and field examples. Also we describe some advantages of their coupling. In particular, by virtue of our approach, a numerically more robust solution of the one-dimensional electromagnetic inverse problem may be achieved. The recorded full-time measurements are further converted to in-phase, quadrature and off-time responses, as opposed to standard frequency or time domain solutions, which have been widely adapted within the industry. When inverting the data, our main focus is within the top 100 m of the Earth, which are the most relevant for engineering and environmental studies. We emphasize that a combined use of frequency and time domain data gives more detailed information about geological structures in the near-surface. As an example, we present some case studies obtained over a platform (Siberia) and a crystalline shield (Western Africa) setting. We further suggest using these results in such near-surface studies as sediment thickness estimation, soil state analysis, groundwater search, and geological structural analysis.

Neural networks (NNs) are successfully used to solve inverse and other problems in geophysics. The aim of this work, which is a continuation of a series of works by a group of authors, is to improve the efficiency of the NN method for solving nonlinear inverse 3D problems of geoelectrics, based on the construction of the author’s convolutional neural network. The network includes a number of additional special transformations (data compression, suppression of the influence of an unknown background environment, etc.) preceding the training of a classical MLP neural network and adapted to the inverse problem that is being solved. This allows us to formally, excluding the human factor, solve inverse problems of geoelectrics of large dimensions without specifying a first approximation based on data measured in areas whose dimensions exceed the dimensions of the network training area. The inversion speed is a few tens of seconds and does not depend on the physical dimensionality (2D or 3D) of the data. The solution to the inverse problem found using a trained neural network can, if necessary, be refined using a random search method. Numerical results of solving 3D geoelectric problems on model and field data are presented, confirming the stated development parameters.

SUMMARY Probabilistic inversion methods, typically based on Markov chain Monte Carlo, exist that allow exploring the full uncertainty of geophysical inverse problems. The use of such methods is though limited by significant computational demands, and non-trivial analysis of the obtained set of dependent models. Here, a novel approach, for sampling the posterior distribution is suggested based on using pre-calculated lookup tables with the extended rejection sampler. The method is (1) fast, (2) generates independent realizations of the posterior, and (3) does not get stuck in local minima. It can be applied to any inverse problem (and sample an approximate posterior distribution) but is most promising applied to problems with informed prior information and/or localized inverse problems. The method is tested on the inversion of airborne electromagnetic data and shows an increase in the computational efficiency of many orders of magnitude as compared to using the extended Metropolis algorithm.

Hood 10 is a small polymetallic (Cu - Zn) VHMS mineral deposit located in the Archean aged Slave Craton in arctic Canada, and is 100% owned by MMG Ltd. The deposit is hosted in the Napaktulik volcanic belt, a bimodal greenstone belt of approximately 2.67 Ga age. The deposit was initially discovered in the early 1970’s by Texas Gulf and covered with airborne and ground frequency domain EM (FDEM) and magnetics. The property has changed hands several times over the intervening years, with drill programs in the mid 1970’s, early 1980’s and early 1990’s resulting in a non-JORC compliant resource of 1.2 million tonnes of 4.4% zinc and 4.1% copper, and defining the ore body along a 250 m strike length and to a depth of 220 m. The property was acquired by MMG Ltd in 2009. In 2012, MMG Ltd conducted significant geophysical exploration on the property for the first time, acquiring surface and airborne time domain EM (TDEM) coverage of the deposit, as well as TDEM on 11 of 12 holes drilled into Hood 10 in 2012. New geophysical modelling of the TDEM results, drilling results, recent surface geological mapping and structural studies have significantly increased the understanding of the deposit. The current non-JORC compliant resource stands at 2.4 million tonnes of 3.5% zinc and 4.5% copper.

The success of horizontal loop electromagnetic method in determining ore body model parameters is intricately tied to the inversion process. However, geophysical inverse problems often grapple with ill‐posed and non‐unique mathematical difficulties. To handle this situation, we presented two new metaheuristics: the Barnacles Mating Optimizer and the Hunger Games Search algorithm. This pioneering approach is the first in the literature to employ both optimizers simultaneously to invert electromagnetic anomalies. To test the core functionality of these inversion tools, we simplified the horizontal loop electromagnetic problem by using a single frequency. Before carrying out the data inversions, a synthetic anomaly caused by a simple multi‐source model was used to perform some modal analyses. These analyses revealed the composite modality character of the inversion process and highlighted the necessity of an algorithm that effectively balances global exploration (thoroughly searching the model space while avoiding local minima) and local exploitation, which focuses on approximating the global optimum closely. In addition, the findings indicated the importance of conducting post‐inversion analyses to evaluate potential uncertainties in the estimations. Initial experiments with nine challenging benchmark functions revealed that Hunger Games Search outperforms in tackling optimization problems. The performances of both algorithms were then examined using synthetic horizontal loop electromagnetic anomalies and two field data sets from Australia. To achieve a balanced comparison, we fine‐tuned the control parameters of both algorithms, optimizing their performance under comparable conditions. These assessments provided valuable insights into the effectiveness of each algorithm when properly configured. Uncertainty appraisal analyses mostly confirmed the consistency and mathematical consistency of the obtained solutions. In the real data inversion cases, the outputs of Hunger Games Search showed strong agreement with the drilling data. This study illustrates the stronger performance of the Hunger Games Search algorithm relative to Barnacles Mating Optimizer in the presented scenarios and demonstrates its effectiveness in estimating source parameters using horizontal loop electromagnetic anomalies.

PreviousNext You have accessSymposium on the Application of Geophysics to Engineering and Environmental Problems 2011Practical 3D Inversion of AEM Data for Environmental Applications in Complex Regolith SettingsAuthors: Glenn WilsonTimothy MundayAndrew FitzpatrickLeif CoxMichael ZhdanovGlenn WilsonSearch for more papers by this author, Timothy MundaySearch for more papers by this author, Andrew FitzpatrickSearch for more papers by this author, Leif CoxSearch for more papers by this author, and Michael ZhdanovSearch for more papers by this authorhttps://doi.org/10.4133/1.3614252 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Abstract This paper presents the application of an innovative approach to 3D inversion of airborne electromagnetic (AEM) data which renders the routine inversion of entire surveys realistic from the computational perspective. The method uses a moving footprint, and is based on the 3D integral equation method for computing data and sensitivities, and the re‐weighted regularized conjugate gradient method for minimizing the objective functional. Specifically, we examine results from several well constrained AEM surveys that target the geometry, distribution and water quality associated with palaeochannel aquifers which are developed in contrasting, variably complex regolith settings. The TEMPEST and SkyTEM surveys targeted anastomozing coarse sands and gravels beds in a sandy clay background. The surveys also targeted the abruptly changing and varying basement (due to changes in lithology and structure) which defines the base of the alluvial aquifer and/or the weathered regolith. Results from the 3D inversions are compared against drill hole geological and other geophysical data. The integrated interpretations are able to define the location and groundwater salinity within the palaeochannels. As a result of the surveys and drilling, local farm water and irrigation systems are being developed.Permalink: https://doi.org/10.4133/1.3614252FiguresReferencesRelatedDetailsCited By3D inversion of airborne electromagnetic dataGEOPHYSICS, Vol. 77, No. 4 Symposium on the Application of Geophysics to Engineering and Environmental Problems 2011ISSN (online):1554-8015Copyright: 2011 Pages: 642 publication data© 2011 Copyright © 2011 The Environmental and Engineering Geophysical SocietyPublisher:Environmental & Engineering Geophysical Society HistoryPublished: 06 Jul 2011 CITATION INFORMATION Glenn Wilson, Timothy Munday, Andrew Fitzpatrick, Leif Cox, and Michael Zhdanov, (2011), "Practical 3D Inversion of AEM Data for Environmental Applications in Complex Regolith Settings," Symposium on the Application of Geophysics to Engineering and Environmental Problems Proceedings : 101-101. https://doi.org/10.4133/1.3614252 Plain-Language Summary PDF DownloadLoading ...

Zohdy's method for the inversion of dc-resistivity data has been adapted to the inversion of airborne electromagnetic (AEM) data. AEM responses are first transformed into apparent-conductivity depth profiles, followed by an iterative adjustment of layer thicknesses and interval conductivities. The start model, including the number of layers, is determined from the data. This approach optimizes model flexibility without the need for parameter regularization. Results from Zohdy's inversion applied to TEMPEST, GEOTEM, and [Formula: see text] data acquired in a range of conductivity scenarios including the Bull Creek prospect in Queensland, Australia; the Boteti area, Botswana; and the Reid-Mahaffy test site in Ontario, Canada, show well-delineated target zones. A comparison with Occam's inversion shows good agreement between the conductivity-depth models recovered by the two methods, with Zohdy's inversion being 25 to 80 times faster.

Inverse problems play a crucial role in geophysics because one of the main tasks in this field is to probe the Earth's interior both for economic reasons, such as oil prospecting, and for the pursuit of academic knowledge about our planet. A variety of different physical fields are used for this: elastic waves form the basis of seismic prospecting, electromagnetic or magnetic fields are used to make inferences about the electrical conductivity in the Earth and the magnetic properties of the core, the gravity field constrains the mass distribution within the Earth, and many other examples can be given where inverse problem theory is crucial for inferring the properties of the Earth from measurements at its surface. Examples of the theory and practice of geophysical inverse problems are given by Iyer and Hirahara (1993), Lines and Levin (1988) and Parker (1994). This special section is aimed at presenting the inverse problems community with the theoretical barriers that geophysicists encounter, and providing some unorthodox examples of geophysical inverse problems. Every image obtained from an inverse problem with an incomplete data set gives an incomplete impression of the object under consideration. Trampert shows how limitations in global tomography influence the interpretation and implications of global maps of the seismic velocity in the Earth. In addition he shows how difficult it is to quantify the limitation of the images that are obtained. As shown by Snieder, nonlinearity aggravates this problem. He makes the point that currently there are no satisfactory theoretical tools for the error and resolution analysis of truly nonlinear inverse problems. Mosegaard attacks this problem, not by seeking a single solution to inverse problems, but by constructing many models and making inferences on the likelihood of these models using Bayesian statistics. That such an approach is needed for many practical problems is shown by Sambridge who shows that in practical problems the misfit function (or probability density function) can be extremely complex. In addition he develops tools to characterize this complexity. That inverse problem theory can be applied to problems other than making images of the Earth's interior is shown by Peltier, who makes inferences about the viscosity in the Earth from the measurement of sea-level variations, and by Gallagher, who reconstructs the thermal history of oil reservoirs from measurements at the surface. A major challenge in geophysics is to use different datasets to make inferences about the Earth's interior. Barghazi and Sansó present theoretical tools that can be used for the joint inversion of seismological and gravity data. It will be clear from this special section that inverse problems is a rich and diverse field of research within geophysics. Despite the success of many of its applications (western industrial society would not exist without effective methods for seismic exploration), geophysicists run into theoretical barriers. Within the geophysics community, solutions are usually sought that rely heavily on numerical techniques. It could be that this is the wisest strategy. However, it is also likely that an increased interaction with theoreticians who are active in the field of inverse problems could be beneficial both for the geophysics community and for those active in inverse problem theory. The goal of this special section is to help to close the gap between theory and practice within the inverse problems community. References Iyer H M and Hirahara K (ed) 1993 Seismic Tomography; Theory and Practice (London: Chapman and Hall) Lines L R and Levin F L (ed) 1988 Inversion of Geophysical Data SEG reprint series, 9 Parker R L 1994 Geophysical Inverse Theory (Princeton, NJ: Princeton University Press)