We introduce a workflow integrating geological modelling uncertainty information to constrain gravity inversions. We test and apply this approach to the Yerrida Basin (Western Australia), where we focus on prospective greenstone belts beneath sedimentary cover. Geological uncertainty information is extracted from the results of a probabilistic geological modelling process using geological field data and their inferred accuracy as inputs. The uncertainty information is utilized to locally adjust the weights of a minimum-structure gradient-based regularization function constraining geophysical inversion. Our results demonstrate that this technique allows geophysical inversion to update the model preferentially in geologically less certain areas. It also indicates that inverted models are consistent with both the probabilistic geological model and geophysical data of the area, reducing interpretation uncertainty. The interpretation of inverted models reveals that the recovered greenstone belts may be shallower and thinner than previously thought.
The integrated interpretation of multiple data types and disciplines in geophysical exploration is a powerful approach to mitigating the limitations inherent to each of the datasets. For instance, gravity data, which have poor horizontal resolution, can be integrated with seismic inversion to mitigate the poor lateral resolution of seismic inversion (Lelièvre et al., 2012). Likewise, geological modelling and geophysical inversions are routinely performed in the same area to obtain a subsurface model consistent with geological and geophysical measurements (Guillen et al., 2008; Lelièvre and Farquharson, 2016; Pears et al., 2017; Williams, 2008). When sufficient prior information is available, petrophysical constraints can be used in inversion (Lelièvre et al., 2012; Paasche and Tronicke, 2007) and integrated with geological modelling to derive local constraints (Giraud et al., 2017). However, in exploration scenarios, this can be impractical as the available petrophysical information may be insufficient to allow us to derive such constraints (Dentith and Mudge, 2014). In such cases, when more than one geophysical dataset is available, practitioners have relied on joint inversion using structural constraints (e.g. Gallardo and Meju, 2003; Haber and Oldenburg, 1997; Zhdanov et al., 2012). Alternatively, when one of the datasets has a spatial resolution that is superior to the others, structural information can be transferred into the gradient regularization constraint for the inversion of the lesser resolving method(s), thus mitigating some of the challenges faced by joint inversion in such cases, in what has been called “guided inversion” (Brown et al., 2012). This strategy has been applied in recent years using the interpretation of predominantly propagative data (e.g. seismics, ground-penetrating radar) to constrain the inversion of diffusive data (e.g. diffusive electromagnetic methods), as reported by Yan et al. (2017) and references reported therein. However, this avenue remains relatively underexplored to date.
In this article, we broaden the applications of guided inversion and explore the integration of non-geophysical information in inversion, such as geological uncertainty, into what we call uncertainty-guided inversion, where we focus on the complementarity of information content between the datasets. We introduce a new technique that integrates local uncertainty information derived from probabilistic geological modelling in the inversion of potential field data, following recommendations of Jessell et al. (2010, 2014, 2018), Lindsay et al. (2013a, 2014) and Wellmann et al. (2014, 2017). In contrast to Giraud et al. (2016, 2017), who derive local petrophysical constraints from petrophysical measurements and geological modelling results, constraints used in uncertainty-guided inversion are based solely on the local conditioning of a gradient regularization function, thereby offering the possibility to integrate probabilistic geological modelling into geophysical inversion in the absence of sufficient petrophysical information. Such conditioning relies on the calculation of local weights derived from prior geological information. In this study, we utilize a probabilistic geological model (PGM) (Pakyuz-Charrier et al., 2018b) consisting of the observation probability of the different lithologies of the area in every model cell. More specifically, we utilize the information entropy (Shannon, 1948; Wellmann and Regenauer-Lieb, 2012), which measures geological uncertainty in probabilistic models. We calculate it in each model cell of the PGM to derive spatially varying weights applied to the gradient regularization function used during inversion.
The integration methodology we develop is similar in philosophy to the work of Brown et al. (2012), Guo et al. (2017) and Wiik et al. (2015), who extract continuous structural information from seismic data to adjust the strength of the regularization term locally in order to promote specific structural features during electromagnetic inversion. However, our work differs from these authors in four main respects. Firstly, the geophysical problem we tackle is different in nature as we constrain potential field data in a hard-rock scenario instead of electromagnetic data in soft-rock studies. Secondly, we use a metric encapsulating geological uncertainty derived from geological measurements, whereas, in contrast, previous studies used other geophysical attributes. Thirdly, we allow inversion to update the model preferably in the most uncertain parts of the geological model, instead of encouraging a certain degree of structural similarity between two geophysical inverse models. Finally, while some of the previous work involves mostly 2-D models, every step of our modelling is performed purely in 3-D.
In this paper, we introduce the methodology and field application as follows. In the methodology section, we first introduce the inversion and integration scheme, and provide essential background information about probabilistic geological modelling. We then provide the essential background about information entropy before detailing its usage in inversion. In the ensuing section, we investigate the applicability of the proposed technique using a realistic synthetic case study. Following this, we present a field application case focused on the Yerrida Basin (Western Australia), starting with the introduction of the geological context and modelling procedure. We then analyse the influence of local regularization conditioning on inverted models and demonstrate how it improves the clarity and improves the reliability of the interpretation of the buried greenstone belts.
The inversion procedure we propose integrates spatially varying prior information to weight the regularization function locally (e.g. in each cell). It is implemented in an expanded version of the least-square inversion platform TOMOFAST-X (Martin et al., 2013, 2018), which offers the possibility to condition the regularization function (Tikhonov and Arsenin, 1977) of Li and Oldenburg (1996) locally using geological uncertainty. This is achieved by incorporating prior information into a structure-based regularization function in a fashion similar to Brown et al. (2012), Wiik et al. (2015) and Yan et al. (2017) by locally adjusting the related weight.
Solving the inverse problem regularized in this fashion consists of
finding a model
The structural regularization term in Eq. (1) enforces structural
constraints during inversion. It is weighted locally by matrix
We utilize the integrated sensitivity technique of Portniaguine and Zhdanov (2002) to balance the decreasing sensitivity of gravity data with depth. We chose this technique because it offers the advantage of providing “equal sensitivity of the observed data to the cells located at different depths and at different horizontal positions” (Vatankhah and Renaut, 2017).
Probabilistic geological modelling is performed using the Monte Carlo
uncertainty estimator (MCUE) method of (Pakyuz-Charrier et al.,
2018a, b), which extends previous works from Jessell et al. (2010), Lindsay et
al. (2012) and Wellmann et al. (2010). It is a 3-D
uncertainty propagation method for implicit geological modelling that uses
geological rules and geological orientation measurements (foliation and
interface of the geological units sampled at surface level or in boreholes)
as inputs. The sampling algorithm perturbs orientation data used to derive a
reference model by sampling probability distributions describing the
uncertainty of orientation data to produce a series of unique altered
geological models. Realizations that do not respect a series of geological
rules are considered implausible and are rejected. Coupled to the 3-D
geological modelling engine of Geomodeller© (Calcagno et al., 2008), it produces a
set of plausible geological models honouring the geological input
measurements that represent the geological model space (Lindsay et al., 2013b). The
observation probabilities constituting the probabilistic geological model (PGM)
are obtained, in each model cell, by calculating the relative
observation frequencies of the different lithologies from the set of
geological models. For the
Information entropy was introduced for geological modelling by
Wellmann and Regenauer-Lieb (2012) and is gaining popularity as a measure of uncertainty in
probabilistic geological modelling (de la Varga et al., 2018; de la Varga and
Wellmann, 2016; Lindsay et al., 2013, 2014; Pakyuz-Charrier et al., 2018b; Schweizer et al., 2017;
Thiele et al., 2016; Wellmann et al., 2017; Yamamoto et al., 2014). Quoting
Schweizer et al. (2017), information entropy
“quantifies the amount of missing information and hence, the uncertainty at
a discrete location”. For the
This section introduces the proof of concept of the proposed method through an idealized case study illustrating the potential of the proposed inverse modelling scheme to ameliorate inversion results and to reduce interpretation uncertainty. We use the same 3-D density contrast model as Giraud et al. (2017), which is obtained by populating the structural framework of Pakyuz-Charrier et al. (2018b). We simulate a series of PGMs sought to represent expected values as well as possible extreme scenarios. The short presentation of the model below and the analysis of results provides essential information about the synthetic survey and shows the proof of concept of the methodology used in the paper.
The 3-D unperturbed reference geological model was generated from contact
(interface points) and surface orientation (foliations) field measurements
collected in the Mansfield area (Victoria, Australia). It presents a
Carboniferous mudstone–sandstone basin oriented 170
The true density contrast model (Fig. 1a) was
obtained by assigning density contrasts consistently with the structural
setting of the reference geological model, assuming a flat topography.
Density contrasts of 0 and 100 kg m
True density contrast model and
MCUE perturbations of the reference geological model were first performed
using standard measurement uncertainty values recommended by metrological
studies, as reported by Allmendinger et al. (2017) and Novakova and Pavlis (2017). We generated a series of 300 models that
were subsequently combined into a PGM. The resulting volume representing the
To assess the impact of local conditioning of the regularization function
onto inversion, we compare inversions using a non-conditioned (Fig. 2a) and a locally
conditioned (Fig. 2b) regularization function, respectively. Please note that
when simulating the absence of prior petrophysical information, a homogenous prior
model set to 0 kg m
Besides qualitative visual comparison of the models, we interpret inversion results (Fig. 2) through the commonly used model and data root-mean-square errors (RMSEs), which correspond to the model and data terms calculated with weights and covariances set to unity. We evaluate the geometrical similarity between inverted and true model through the Bravais–Pearson correlation (also often called “linear correlation coefficient”) between their gradients (Table 1).
Indicators for comparison of inversion results in terms of model, data and structure.
Comparison of inversion results.
Comparison of the true model (Fig. 1a) with inversion results in Fig. 2a and b shows that, while the structures in
the shallowest part of the model are well retrieved in both cases, it appears
that they are considerably better recovered through usage of conditioned
regularization overall (Fig. 2b). The guiding effect of
Please also note that we do not show the recovered geophysical measurements because visual differences between recovered and inverted measurements are minimal.
From these observations, we conclude that the application of the local
conditioning scheme can fulfil the objectives of data integration in
inversion as it allows inversion to recover models that are closer to the
causative bodies and easier to interpret, while honouring geophysical data.
Nevertheless, it remains important to test the methodology in cases where
the uncertainty indicator
In this subsection, we investigate the effect of inaccurate geological
models and the propagation of the related uncertainty in inversion. For this
purpose, we calculate a second PGM from MCUE perturbations in which we split
the ultramafic basement into two independent units, without changing the
density contrast values (Fig. 3a). This results
in the existence of a fictitious geological unit that is invisible to
gravity data and presents no density contrast but which increases geological
complexity and uncertainty (Fig. 3b) (we further
refer to it as a “ghost” geological unit). Notably, it increases geological
uncertainty and smears interfaces that are well-constrained as per Fig. 1. It also
decreases
Comparison of the inverted models obtained without (Fig. 4a) and with the ghost unit
(Fig. 4b) reveals that they exhibit broadly similar features except in the most
geologically complex parts of the model as per Fig. 1b, where differences are minor. This
indicates that while geophysical inversion updates the prior density
contrast model preferably in geologically uncertain regions, low values
of
To complete this series of tests, we generated a third PGM showing
exaggerated geological uncertainty. To this end, we used a half-aperture
95 % confidence interval of
The features visible in Fig. 5a reflect the high geological input
measurement uncertainty. Geologically uncertain areas cover large portions
of the volume and only the simplest geological structures (e.g. the basin)
seem to be well constrained by geology. Areas of the model previously well
constrained (Fig. 1b) present varying degrees of
uncertainty. This illustrates that, as can be expected, increasing
geological input uncertainty translates in relaxing the guiding effect of
local conditioning using
Comparison of inverted model using
The analysis and comparison of the results shown in this section demonstrates the potential of the proposed inverse modelling scheme to ameliorate inversion results and to reduce interpretation uncertainty. It also illustrates the capability of our methodology to deal with high or biased conditioning uncertainty estimates. In this synthetic case, local conditioning allows geophysical inversion to significantly improve the imaging of geologically uncertain areas and to exploit complementarities between geological modelling and geophysical inversion. From the success of this proof-of-concept study, we are confident that our integration method can be tested here using real world data for field validation.
The Yerrida Basin is located in the southern part of the Capricorn Orogen,
at the northern margin of the Yilgarn Craton in Western Australia
(Fig. 6a), and extends approximately 150 km N–S
and 180 km E–W (Fig. 6b). The studied area is
bounded in the north-west by the Goodin Fault, which represents a faulted
contact between the Yerrida Basin and the Bryah–Padbury Basin. The
structures of interest in this work are the Archean greenstone belts
extending north-northwest that are unconformably overlain by
Paleoproterozoic sedimentary rocks the form the Yerrida Basin. Features A
and B (Fig. 6a and b) indicate the interpreted position of buried Wiluna Greenstone Belt.
Where the Wiluna Greenstone Belt is exposed, it is host to base and precious
metal mineralization (Williams, 2009). With a
relatively high positive density contrast (expected to be between 190 and
270 kg m
Geophysical data consist of ground surveys obtained from Geoscience
Australia (
Geological information consists of in situ structural measurements (interfaces and foliations) and interpretation of aeromagnetic, airborne electromagnetic, Landsat 8 and ASTER hyperspectral data. Legacy data from the Geological Survey of Western Australia (Pirajno and Adamides, 2000) and CSIRO (Ley-Cooper et al., 2017) were used, to which about 600 surface geological and petrophysical measurements from recent geological field campaigns were added. Although the available petrophysical measurements were not used to derive petrophysical constraints because of the insufficient sampling of several of the modelled lithologies, they were a useful source of information to populate geological models and for interpretation. Remote-sensing data were used to test interpretations.
These datasets were used jointly to build a reference geological model reconciling the available geological information in Geomodeller.
Lithologies with similar density contrasts were merged and subsequently
treated as a single rock type in MCUE simulations. Uncertainty related to
structural measurements was subsequently used as inputs to the MCUE
perturbations (Pakyuz-Charrier et al., 2018b) of the reference model to generate a collection of
500 accepted models. Information extracted from the PGM is displayed in Fig. 7. Figure 7a shows
the lithologies with the highest probability for each cell of the PGM. The
associated
The aim of our analysis is to assess the influence of the local conditioning of structural constraints on inversion through comparison with the non-conditioned case, all other things remaining constant.
Geological context and geophysical data.
Prior to examination of the inverted models, we analyse geophysical data
misfit after inversion. This enables us to ensure that the inversion results
we compare produce, in our case, similar gravity anomalies. Our study of
inverted models focuses on results obtained through usage of non-conditioned
(Fig. 8a) and conditioned regularization functions (Fig. 8b) using
Geological modelling results.
Data root-mean-square (RMS) error decreases during inversion from 12.46 mGal to reach 1.59 and 1.53 mGal for the non-conditioned and conditioned cases, respectively. The corresponding data misfit maps show a linear correlation coefficient of 0.999 (see details in Appendix A). This similarity illustrates that, as in many other studies, most changes related to holistic data integration in geophysical inversion occur primarily in model space, hence reducing the effect of non-uniqueness (Abtahi et al., 2016; Abubakar et al., 2012; Brown et al., 2012; Demirel and Candansayar, 2017; Gallardo et al., 2012; Gallardo and Meju, 2004, 2007, 2011; Gao et al., 2012; Giraud et al., 2017; Guo et al., 2017; Heincke et al., 2017; Jardani et al., 2013; Juhojuntti and Kamm, 2015; Kalscheuer et al., 2015; Molodtsov et al., 2013; Moorkamp et al., 2013; Rittgers et al., 2016; Sun and Li, 2016, 2017).
Comparison of inversion results.
Qualitatively, comparison of Fig. 8a and b reveals that departures from the prior
model (Fig. 7c) are more significant in the most
geologically uncertain areas. Quantitatively, the RMS model update for cells
characterized by
The recovered greenstone belts are shown in Fig. 8a and b. In
Fig. 8b, the extension of recovered mafic
greenstone belts is significantly different than when geological uncertainty
is not accounted for (Fig. 8a). In particular, belt A is significantly larger in Fig. 8b than in
Fig. 8a (
Note that, in contrast to the differences between inversion results
highlighted above for belts A and C, there are only small differences
between the inverted models in the north-eastern part of the model and the
different interpretations of belt B (Fig. 7a and b). This shows that locally conditioned
regularization does not enforce changes in the inverted model in all places
where geological uncertainty is high, as uncertainty is only a reflection of
potential errors. Rather, this indicates that in such cases, the guiding
effect of such regularization will be exerted on the condition that it does
not prevent the data term in
In consequence, by confronting a probabilistic geological model encapsulating all MCUE realizations with geophysical measurements in an inversion scheme favouring model updates in the most geologically uncertain areas, inversion complements probabilistic geological modelling in that it guides and refines the interpretation of other geoscientific data in the area.
Geophysical inversion using geological uncertainty information (Fig. 7b) confirms the presence of high-density anomalies that we interpret to be the mafic components of the greenstone as suggested by MCUE in several portions of the model. It also adjusts the outline and geometry of belts A–C to obtain a model honouring geological uncertainty information. In particular, mafic greenstone belts A and B may be smaller than the extent suggested by the PGM, and mafic greenstone C shallower than anticipated. The interpretation of inversion results also reveals that greenstone B might extend further to the east than indicated by the preferred lithology volume (Fig. 7a) and that greenstone C may be thinner in its central part.
We have introduced a new integration scheme for the inversion of gravity
data that utilizes a measure of geological uncertainty to calculate
locally conditioned gradient regularization constraints. This approach
enables the integration of probabilistic geological modeling in geophysical
inversion in the absence of petrophysical information sufficient to the
calculation of petrophysical constraints. It uses geophysical measurements
to optimize the inverse problem by updating the physical property model
preferably in geologically uncertain parts of the studied area during what
we called
In the Yerrida Basin study area, application of the proposed methodology provided the effective delineation of the greenstone belts by quantitatively integrating geological measurements and geophysical data. Our findings suggest that some of the greenstone belts covered by the basin might be shallower than previously anticipated and occupy smaller volumes. This is particularly pronounced in the north-east (belt C), where the resulting model is in agreement with the shallowest cases allowed by the PGM. Likewise, in the south (belt A), only the shallowest part of the mafic greenstone body can be resolved, while the south-eastern (belt B) greenstone belt appears to be limited in extension to the eastern part of the volume where it is the preferred lithology in the PGM. In such cases, this can also indicate that these greenstone bodies might be too thin to be imaged by gravity data. These results have implications for our knowledge of the southern Capricorn Orogen as they indicate reduced (compared to the preferred lithology volume) mafic greenstone volumes under the Yerrida Basin on the one hand, and decreased cover thickness on the other hand, thereby opening the door to updates in the geological interpretation of geometry of the Yerrida Basin and potential new undercover exploration prospects.
The quantitative integration technique we presented reduces uncertainty and ambiguity compared to qualitative interpretation techniques or single-discipline workflows. However, despite its robustness to misplaced interfaces (e.g. bias) or to high geological uncertainty (e.g. sparse or very uncertain geological input measurements) as shown in the synthetic case, interpreters need to bear in mind the specificities of the geophysical data inverted for (resolution of specific geometries, depth of investigation) and the shortcomings of geological modelling workflows. As for all geological modelling, MCUE is oblivious to geological units or faults that are not sampled by field geological measurements, which can lead to biases in final models due to, for instance, inclusions not be accounted for.
Current research comprises the development of sensitivity and resolution analyses in an effort to mitigate the risk of the model being affected by uncertainty sources not accounted for. Future research will include the utilization of local petrophysical constraints of Giraud et al. (2017) in the uncertainty-guided inversion scheme we presented, as well as the utilization of geological uncertainty to weight the cross-gradient term of Gallardo and Meju (2003) locally. With this last respect, uncertainty-guided inversion can be assisted in the most uncertain parts of the model by guided inversion (in the sense of Brown et al., 2012) or through cross-gradient joint inversion.
True property models, inversion results and recovered
models relating to the Yerrida Basin shown in this article are made available
online at
Figure A1 below relates to the analysis of data misfit in Sect. 4 of the article through the plot of the data misfit maps for the non-conditioned and conditioned cases (Fig. A1d and h, respectively). It is complemented by the corresponding plots of starting (Fig. A1a and e), observed (Fig. A1b and f) and calculated data (Fig. A1c and h). Note that Fig. A1c and g show few visual differences, and that Fig. A1d and h exhibit similar features while showing limited coherent signal.
Comparison of input and output geophysical data. Panels
JG performed the integrated inverse modelling of geophysical data for both the Mansfield synthetic study and the Yerrida Basin. He performed posterior analysis and interpretation of results and he is the main contributor to the writing of this article. ML acquired part of the geological field measurements from the Yerrida Basin and performed the geological modelling of the area. He participated in the writing of the geological setting subsection and he produced the geological map shown in Fig. 6a. VO and JG worked together on the implementation and testing of the proposed methodology in TOMOFAST-X, of which VO, RM and JG are the main developers. MJ has been involved in the validation of the methodology at the initial development stage and supervised the progress of the presented work. RM provided support at the initial stage of the inversion of gravity data from the Yerrida Basin. EP-C assisted ML with the utilization of MCUE. All co-authors contributed to the final version of this article. ML and VO were the most actively involved in the revision process of the drafts leading to this paper.
The authors declare that they have no conflict of interest.
Appreciation is expressed to the CALMIP supercomputing centre (Toulouse, France), for their support through Roland Martin's computing projects no. P1138_2017 and no. P1138_2018 and for the computing time provided on the EOS machine. Jeremie Giraud is a recipient of the International Postgraduate Research Scholarship from the Australian Federal Government and he received a grant from the Australian Society of Exploration Geophysics Research Foundation. The authors also thank Peter Lelievre for interesting discussions and constructive feedback relating to the utilization of gradient-related constraints. Mark D. Lindsay and Mark W. Jessell thank the Geological Survey of Western Australia (Royalties for Regions Exploration Incentive Scheme) and the Minerals Research Institute of Western Australia for their support. Mark W. Jessell is supported by a Western Australian Fellowship. The authors acknowledge use of the Zenodo research date repository to share the data necessary to reproduce the presented work. They finally thank Colin Farquharson and an anonymous reviewer for their review of the paper. Edited by: Michal Malinowski Reviewed by: Colin Farquharson and one anonymous referee