Integration of geological uncertainty into geophysical inversion by means of local gradient regularization

We introduce a workflow integrating geological uncertainty information in order to constrain gravity inversions. We test and apply this approach to data from 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 uncertainty as input. It is utilized to locally adjust the weights of a 15 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 finally reveals that the recovered greenstone belts may be shallower and thinner than previously thought. 20


Introduction
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 has poor horizontal resolution, can be integrated with seismic inversion to mitigate the poor lateral resolution of seismic inversion . Likewise, geological modelling and geophysical inversions are routinely performed in the same area to obtain an subsurface model 25 consistent with geological and geophysical measurements Pears et al., 2017;Williams, 2008). When sufficient prior information is available, petrophysical constraints can be derived for inversion Paasche and Tronicke, 2007), and integrated with geological modelling to derive local constraints . However, in exploration scenarios, this might be impractical as the available petrophysical information may be insufficient to allow us to derive such constraints . In such cases, when more than one 30 geophysical dataset is available, practitioners may rely on joint inversion using structural constraints (e.g., 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 into what  termed guided inversion. This strategy 5 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), see (Yan et al., 2017) and references reported therein. However, this avenue remains relatively unexplored 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 10 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 Lindsay et al., 2013;Wellmann et al., , 2017. In contrast to  who derives 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 15 gradient regularization function, thereby offering the possibility to integrate probabilistic geological modelling into geophysical inversion in the absence of sufficient petrophysical information. This conditioning relies on the calculation of local weights derived from prior geological information. In this study, we utilize a probabilistic geological model (PGM) consisting of the observation probability of the different lithologies of the area in every model cell. More specifically, we utilize the information entropy , which 20 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 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 25 in four main respects. Firstly, the geophysical problem we tackle is different in nature as we constrain potential field data in 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 use 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 previous work 30 involve mostly 2D models, every step of our modelling is performed purely in 3D. the inversion and integration scheme and algorithm, 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 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 5 conditioning on inverted models and demonstrate how it allows clearer and more reliable interpretation of the buried greenstone belts than when it is not utilized.

Inversion methodology
The inversion procedure we propose integrates spatially varying prior information to weight the regularization function locally 10 (e.g., in each cell). It is implemented in an expanded version of the least-square inversion platform Tomofast-x (Martin et al., 2013(Martin et al., , 2018, which offers the possibility to condition the regularization function  of  locally using geological uncertainty. This is achieved by incorporating prior information into a structurebased regularization function in a fashion similar to Wiik et al., 2015;Yan et al., 2017) by locally adjusting the related weight. 15 Solving the inversion problem regularized in this fashion consists of finding a model that minimizes the objective function given below: Data term Model term Structural regularization term where relates to the geophysical measurements to be inverted, is the forward modelling operator; relates to the model being searched for, and is the prior model; , and are diagonal weighting matrices corresponding to data noise, 20 model weighting and gradient regularization, respectively. The model term is a ridge regression constraint term .
The structural regularization term in Eq. (1) enforces structural constraints during inversion. It is weighted locally by matrix , which can be derived from prior information (see Subsect. 2.3 for details). The positive free parameter controls the overall weight of the regularization term; ∇ is the spatial gradient operator. Note that ‖∇ ‖ , estimates the amount of structure 25 in inverted physical property model . Note that parts of the model where = 0 are excluded from the calculation of the structural regularization and can change freely to accommodate geophysical data.

Probabilistic geological modelling
Probabilistic geological modelling is performed using the Monte-Carlo Uncertainty Estimator (MCUE) method of , which is an uncertainty propagation method for 3D implicit geological modelling using geological rules and geological orientation measurements (foliation and interface) as inputs. The sampling algorithm perturbs orientation data by sampling probability distributions describing the uncertainty of orientation data to produce a series of unique altered models. 5 Realizations that do not respect a series of geological rules are considered implausible and are rejected. Coupled to the 3D geological modelling engine of Geomodeller© , it produces a set of plausible geological models representing 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 th model cell of a PGM containing lithologies, vector = 10 [ =1 , … , = ] contains the observation probabilities of each lithology. As we show in the next subsection, the resulting PGM can be used to estimate uncertainty levels and as a source of prior information.

Utilisation of information entropy for local conditioning
Information entropy has recently been introduced in geological modelling by ) and is gaining popularity as a measure of uncertainty in probabilistic geological modelling (de la Varga et al., 2018; de la Varga and 15 Lindsay et al., 2013;Wellmann et al., 2017;Yamamoto et al., 2014). Quoting , information entropy "quantifies the amount of missing information and hence, the uncertainty at a discrete location". For the th model-cell, it is given as ): (2) 20 Instead of using directly, we calculate utilising its normalized complementary, which reflects the degree of certainty across the model. Let us express as follows, for the th model cell: The consequence of Eq.
(2) and 3 is that is minimum at interfaces and in areas poorly constrained by geological information, and equal to unity in areas where the geology is well resolved. Consequently, the conditioning process attaches small weights to the structural term of Eq. (1) in uncertain cells, while consistently high values will be applied to the most 25 geologically certain cells. As a result, it enables the inversion algorithm to favour nearly constant changes in the inverted model in contiguous certain groups of cells (e.g., where → 1) while relaxing the regularisation constraints in the most uncertain parts (e.g., where → 0).
For proof-of-concept validation, we simulated an idealized case study to assess the capability of inversion using as per Eq.
(3) to improve inversion results compared to the non-conditioned case (e.g., with equal to the identity matrix). We tested the proposed methodology using synthetic geophysical data calculated from the structural geological model of the Mansfield area of , which we populated in the same fashion as . The analysis 5 of inverted models demonstrates the potential of the proposed inverse modelling scheme to ameliorate inversion results and to reduce interpretation uncertainty (see details in Appendix A). Importantly, in this synthetic case, local conditioning allows geophysical inversion to significantly improve the imaging of geologically uncertain areas. From the success of that theoretical proof-of-concept study, we are confident that our integration method can be tested here using real world data for field validation. 10

3
Field validation: Yerrida Basin case study

Geological context and geophysical survey setup
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. 1a), and extends approximately 150km N-S and 180 km E-W (Fig. 1b). The studied area is delimited in the northwest by the Goodin Fault, which represents a faulted contact between the Yerrida Basin and the Bryah-Padbury 15 Basin. The structures of interest in this work are: Archean greenstone belts extending north-northwest that are unconformably overlain by Paleoproterozoic sedimentary rocks the form the Yerrida Basin. Features A and B ( Fig. 1a and Fig. 1b) indicate the interpreted position of buried Wiluna Greenstone Belt. Where the Wiluna Greenstone Belt is exposed, it is host to base and precious metal mineralisation (Williams, 2009). With a relatively high positive density contrast (expected to be between 190 and 270 kg.m -³) to the Yilgarn Craton granite-gneiss host, mafic greenstone belts A, B, and C are suitable targets for gravity 20 inversion. Interpretations from multiple studies in the region, e.g, (Johnson et al., 2013;Pirajno et al., 1998; were compiled while additional geological measurements acquired in 2015, 2016 and 2017 complemented legacy data (Occhipinti et al., 2017;Olierook et al., 2018). This allowed the revision of existing models and improved our understanding of the area. This, in turn, also highlights the challenges presented by the characterization of greenstone belts A, B and C, and that further geophysical analysis through constrained inversion is a useful 25 pathway for reducing exploration risk.
Post-processing includes spherical-cap and terrain corrections and the removal of the regional trend to obtain the complete Bouguer anomaly, which we forward model following (Boulanger and Chouteau, 2001). As most data points were acquired 1 to 4 km apart, the dataset was resampled to match the geological model discretization, making up a total of 4882 measurement 30 points. The model is discretized into 100 × 100 × 42 cells of dimensions 2.335 km × 1.875 km × 1.0475 km, down to a depth  to precondition the data term in Eq. (1) in order to balance the decreasing sensitivity of gravity field data with depth.

Geological modelling
Geological data consists of in-situ structural measurements (interfaces and foliations) and interpretation of aeromagnetic, 10 airborne electromagnetic, Landsat 8 and ASTER data. Legacy data from the Geological Survey of Western Australia  and CSIRO  were used, to which about 600 data points and petrophysical measurements from recent geological field campaigns were added. Although the available petrophysical measurements are not used to derive petrophysical constraints because of the insufficient sampling of several of the modelled lithologies, they are a useful source of information to populate geological models and for interpretation. 15 These datasets were used jointly to build a reference geological model for MCUE simulations, after which lithologies with similar density contrasts were merged and subsequently treated as a single rock type. Uncertainty related to structural measurements was subsequently used as inputs to the MCUE perturbations ) of the reference model to generate a collection of 500 accepted models. Information extracted from the PGM displayed in Fig. 2. Figure 2a shows the lithologies with the highest probability for each cell of the PGM. The associated values used in inversion are shown in Fig. 2b. The starting model for inversion, which we also use as prior model , is equal to the mean model of the 500 plausible models generated by MCUE, is shown in Fig. 2c. 5

Inversion results and analysis
Our analysis aims at determining the influence of the local conditioning of structural constraints on inversion through comparison with the non-conditioned case, all other things remaining constant.

Comparative analysis strategy
Prior to examination of the inverted models, we analyse geophysical data misfit after inversion for a fixed number of major 5 iterations (100) of the least-square geophysical inverse solver superior to that needed for convergence of the inversion algorithm (~10 in this case). 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. 3a) and conditioned regularization function (Fig. 3b) using (Fig. 2b). In addition to departures from the starting model, variations between the two cases are studied by visual comparison of Fig. 3a and Fig. 3b, through qualitative ( Fig. 3c) and quantitative 10 comparative analysis . Our interpretation of inversion results is complemented by metrics quantifying the differences between models. We give particular attention to model cells where the probability of mafic greenstone is superior to zero. For these cells, we classify lithologies by identifying cells with a density contrast corresponding to mafic greenstone.

Results
Data root-mean-square (RMS) error decreases during inversion from 12.46 mGal to reach 1.59 mGal and 1.53 mGal for the 15 non-conditioned and conditioned cases, respectively. The corresponding data misfit maps show a linear correlation coefficient of 0.999 (see details in Appendix A3 and Fig. A3). 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 nonuniqueness Gao et al., 2012;20 Juhojuntti and Kamm, 2015;Molodtsov et al., 2013;Moorkamp et al., 2013;Li, 2016, 2017 certain group). This suggests that local regularization conditioning allows inversion to update the model preferentially in geologically uncertain areas. In turn, differences with the starting model in more geologically certain areas are reduced compared to the non-conditioned case. This effect of conditioning is corroborated by Fig. 3c where the longest distances to the dashed line, which represents equal model update for the two studied cases, occur in geologically uncertain areas. This also translates in higher difference between model updates of the two compare cases in Fig. 3d for lower values of . In addition, 30 we observe that local conditioning produces stronger density contrasts in Fig. 3b in some of the areas where the conditioning values are higher in Fig. 2b. Furthermore, structures in the inverted model are easier to identify when local conditioning is used. It is confirmed by global roughness measures ‖∇ ‖ 2 equal to 3.4 ( 3 ⁄ )/ and 4 ( 3 ⁄ )/ for the nonconditioned and conditioned cases, respectively. More specifically, as shown by Fig. 3e, this difference arise in parts of the model associated with lower , which characterize uncertain areas, including interfaces between lithologies. 5 The recovered greenstone belts are shown in Fig. 3a and Fig. 3b. In Fig. 3b, the extension of recovered mafic greenstone belts is significantly different than when geological uncertainty is not accounted for (Fig. 3a). In particular, belt A is significantly larger in Fig. 3b than in Fig. 3a (2.4×10 2 km 3 vs 4.6×10 2 km 3 ). Similarly, the extent of belt C is increased overall (volume of 5.3 * 10 2 km³ vs 14×10 2 km 3 ), while its different portions reconnect; the northern half is also significantly shallower and broader than in Fig. 2a and Fig. 3a. It appears that belt A remains thinner and shallower (Fig. 3b) than suggested by the preferred 10 lithology volume (Fig. 2a). While similar geometries for belt B are recovered in Fig. 3a and Fig. 3b, they both differ from Fig.   2a as only the eastern part is preserved. Note that it is larger in Fig. 3b, with a volume 40% higher than in Fig. 3a. As discussed in the next subsection, these differences have a signification impact on the interpretation of inversions results and are important to understand the influence of local conditioning on inversion.

Interpretation
Note that, in contrast to the differences between inversion results highlighted above for belts A and C, differences between the inverted models in the north-eastern part of the model and the different interpretations of belt B ( Fig. 3a and Fig. 3b) are small.
This shows that locally conditioned regularization does not enforce changes in the inverted model everywhere geological uncertainty is high as uncertainty is only a reflection of potential errors. Rather, this indicates that in such cases, the guiding 5 effect of such regularization will be exerted provided that it does not prevent the data term in ( , ) as per Eq. (1) from decreasing. This also confirms that geophysical data is the main driver of the model updates in geologically uncertain areas.
Instead of smooth departures from the starting model to match geophysical data regardless of geological considerations, local regularization constraints allow inversion to account for the probabilistic geological modelling of the area and for geological uncertainty. It can therefore provide results that conform better to known geology. 10 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 the geoscientific data in the area.
Geophysical inversion using geological uncertainty information (Fig. 2b) confirms the presence of high density anomalies that 15 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, B and C to obtain a model honouring geological uncertainty information. In particular, mafic greenstone A and B may be smaller than the extent suggested by the PGM, and mafic greenstone C shallower than anticipated. Inversion results interpretation also reveal that greenstone B might be extended further to the east than indicated by the preferred lithology volume (Fig. 2a) and that greenstone C may be thinner in its central part. 20

Conclusions
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. Contrarily to previous work, 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 25 the physical property model preferably in geologically uncertain parts of the studied area during what we called uncertaintyguided inversion. This therefore partly mitigates inversion's non-uniqueness through the addition of constraints encouraging inversion to produce models that account for geological uncertainty across the entire inverted volume. We have demonstrated that it can be used collaboratively with geological modelling efficiently through field application in the Yerrida Basin.
Inversion results show that our integration methodology has the capability to refine the recovered physical property model and 30 interpretations in portions of the model where geological uncertainty is high. Another advantage of the proposed technique pertains to its time and cost-effectiveness as our workflow utilizes the PGM resulting from geological modelling and requires the same parameterization as non-conditioned inversion.
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 5 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 the geological knowledge of the southern Capricorn Orogen as they indicate 10 reduced (compared to the preferred lithology volume) mafic greenstone volumes under the Yerrida Basin on one hand, and decreased cover thickness on the other hand, thereby opening the door to updates in the geological interpretation of the history of the Yerrida Basin and potential new undercover exploration prospects.
Future research include the utilization of local petrophysical constraints of  in the uncertainty-guided inversion scheme we presented, as well as its application to weight the cross-gradient term of  in 15 joint inversion schemes. 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  or through cross-gradient joint inversion. This Appendix 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 3D density contrast model as , which is obtained using the structural framework and PGM of . The short presentation of the model below and the analysis of results provides essential 30 information and support about the proof-of-concept of the methodology used in the paper.

A1 Synthetic survey setup
The 3D unperturbed 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 N170, abutting a faulted contact with a folded ultramafic basement to the South-West. Model complexity was artificially increased through the addition of a North-South fault and of a mafic intrusion. 5 The reference density contrast model (Fig. A4a) was obtained assigning density contrasts consistently with the structural setting of the unperturbed model, assuming a flat topography. Density contrasts of 0 and 100 kg.m -³ were assigned to the upper and lower basin lithotypes, respectively. Mafic rocks were assigned a density contrast of 200 kg.m -³ while the density contrast of the ultramafic basement was set to 300 kg.m -³.
MCUE perturbations of the reference model were performed using standard measurement uncertainty values recommended 10 by metrological studies as reported by Novakova and Pavlis, 2017)

A2 Comparison of inversion results
To assess the impact of local conditioning of the regularization function, we compare inversions using non-conditioned ( Fig.   A5a) and locally conditioned (Fig. A5b) regularization function, respectively. Please note that, simulating the absence of prior petrophysical information, a homogenous starting model set to 0 kg.m-³ was used in both cases. Besides qualitative visual comparison of the models, we interpret inversion results through the commonly used model and data root-mean-square error (RMSE). We evaluate the geometrical similarity between insverted and true model through the Bravais-10 Pearson correlation (also often called 'linear correlation coefficient') between their gradients (Table A1).
Comparison of the reference model ( Fig. A4a) with inversion results in Fig. A5a and Fig. A5b shows that while the structures in 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. A5b). The guiding effect of is visible in Fig. A5b where the main structures at depth follow the genesral features of the conditioning volume (Fig. A4b). Moreover, in order to minimize the 15 conditioned regularization constraint simultaneously to data misfit, inversion was driven to accommodate inverted model values ( Fig. A5b) such that they are closer to the causative model ( Fig. A4a) than without conditioning (Fig. A5a). This leads to reduced model RMSE on one hand, and data RMSE on the other hand (Table A1). This reduction in data RMSE can also be explained by the relaxation of the constraints in several portions of the model, thus increasing the degree of liberty to accommodate the model towards lower data misfit. Importantly, the Bravais-Pearson correlation between the inverted and reference model gradients is nearly three times higher when information from information entropy is used, which indicates that local conditioning of the regularization function also allows for significantly better retrieval of the causative bodies' (e.g., 5 true model) structural features.

Introduction
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 has poor horizontal resolution, can be integrated with seismic inversion to mitigate the poor lateral resolution of seismic inversion . Likewise, 25 geological modelling and geophysical inversions are routinely performed in the same area to obtain a subsurface model consistent with geological and geophysical measurements Pears et al., 2017;Williams, 2008). When sufficient prior information is available, petrophysical constraints can be used in inversion Paasche and Tronicke, 2007), and integrated with geological modelling to derive local constraints . However, in exploration scenarios, this can be impractical as the available petrophysical information 30 geophysical dataset is available, practitioners have relied on joint inversion using structural constraints (e.g., 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 into 5 what has been called guided inversion . 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 10 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 . In contrast to   specifically, we utilize the information entropy , 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 , , and Wiik et al. (2015), who extract continuous structural information from seismic data to adjust the strength of the regularization 25 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 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 use other geophysical attributes. Thirdly, we allow inversion to update the model preferably in the most uncertain parts of the geological model, 30 instead of encouraging a certain degree of structural similarity between two geophysical inverse models. Finally, while some of the previous work involve mostly 2D models, every step of our modelling is performed purely in 3D. 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 5 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.

Inversion methodology 10
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(Martin et al., , 2018, which offers the possibility to condition the regularization function  of  locally using geological uncertainty. This is achieved by incorporating prior information into a structurebased regularization function in a fashion similar to Wiik et al., 2015;Yan et al., 2017) by locally adjusting 15 the related weight.
Solving the inversion problem regularized in this fashion consists of finding a model that minimizes the objective function given below:

Data term Model term Structural regularization term
where relates to the geophysical measurements to be inverted, is the forward modelling operator; relates to the model 20 being searched for, and is the prior model; , and are diagonal weighting matrices corresponding to data noise, model weighting and gradient regularization, respectively. The model term is a ridge regression constraint term .
The structural regularization term in Eq. (1) enforces structural constraints during inversion. It is weighted locally by matrix , which can be derived from prior information (see Subsect. 2.3 for details). The positive free parameters and 25 control the overall weight of model and structural regularization terms; ∇ is the spatial gradient operator. Note that ‖∇ ‖ , estimates the amount of structure in inverted physical property model . Also note that parts of the model where = 0 are excluded from the calculation of the structural regularization and can change freely to accommodate geophysical data.
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 5
Probabilistic geological modelling is performed using the Monte-Carlo Uncertainty Estimator (MCUE) method of (Pakyuz-Charrier et al., 2018b, 2018a, which extends previous works from . It is a 3D 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 borehole) as inputs. The sampling algorithm perturbs orientation data used to derive a reference model by sampling probability distributions describing 10 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 3D geological modelling engine of Geomodeller© , 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 15 the different lithologies from the set of geological models. For the th model cell of a PGM containing lithologies, vector = [ =1 , … , = ] contains the observation probabilities of each lithology. As we show in the next subsection, the resulting PGM can be used to estimate uncertainty levels and as a source of prior information.

Utilisation of information entropy for local conditioning
Information entropy was introduced for geological modelling by  and is gaining 20 popularity as a measure of uncertainty in probabilistic geological modelling de la Varga and Wellmann, 2016;Lindsay et al., 2013;Wellmann et al., 2017;Yamamoto et al., 2014). Quoting , information entropy "quantifies the amount of missing information and hence, the uncertainty at a discrete location". For the th model-cell, it is given as : 25 (2) Fundamentally, geological uncertainty contained in encapsulates information about possible locations of interfaces between units and areas where the geological data is insufficiently informative. Instead of using directly, we calculate utilising its normalized complementary, which reflects the degree of certainty across the model. Let us express as follows, for the th model cell: The consequence of Eq. (2) and 3 is that is minimum at interfaces and in areas poorly constrained by geological information, and equal to unity in areas where the geology is well resolved. Consequently, the conditioning process attaches small weights to the structural term of Eq. (1) in uncertain cells, while consistently high values will be applied to the most 5 geologically certain cells. As a result, it enables the inversion algorithm to favour nearly constant changes in the inverted model in contiguous certain groups of cells (e.g., where → 1) while relaxing the regularisation constraints in the most uncertain parts (e.g., where → 0).

Proof of concept: synthetic case study
This section introduces the proof of concept of the proposed method through an idealized case study illustrating the potential 10 of the proposed inverse modelling scheme to ameliorate inversion results and to reduce interpretation uncertainty. We use the same 3D density contrast model as , which is obtained by populating the structural framework of (Pakyuz-Charrier et al., 2018b). We simulate a series of PGM sought to represent expected values as well as possible extreme scenarii.
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. 15

Survey setup
The 3D 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 mudstonesandstone basin oriented N170, abutting a faulted contact with a folded ultramafic basement to the South-West. Model complexity was artificially increased through the addition of a North-South fault and of a mafic intrusion. 20 The true density contrast model (Figure 1a) was obtained 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 -³ were assigned to the upper and lower basin lithotypes, respectively. Mafic rocks were assigned a density contrast of 200 kg.m -³ while the density contrast of the ultramafic basement was set to 300 kg.m -³.
MCUE perturbations of the reference geological model were first performed using standard measurement uncertainty values 25 recommended by metrological studies as reported by Novakova and Pavlis, 2017). We generated a series of 300 models that were subsequently combined into a PGM. The resulting volume representing the values calculated from this PGM in each cell of the model as per Eq.

Locally constrained inversion: validation
To assess the impact of local conditioning of the regularization function onto inversion, we compare inversions using non-5 conditioned ( Figure 2a) and locally conditioned ( Figure 2b) regularization function, respectively. Please note that, simulating the absence of prior petrophysical information, a homogenous prior model set to 0 kg.m-³ was used in both cases. Besides qualitative visual comparison of the models, we interpret inversion results (Figure 2) through the commonly used model and data root-mean-square error (RMSE), 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 5 correlation (also often called 'linear correlation coefficient') between their gradients (Table 1).
Comparison of the true model ( Figure 1a) with inversion results in Figure 2a and Figure 2b shows that while the structures in 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 (Figure 2b). The guiding effect of is visible in Figure 2b where the main structures at depth follow the general features of the conditioning volume ( Figure 1b). Moreover, in order to minimize the 10 conditioned regularization constraint simultaneously to data misfit, inversion was driven to accommodate inverted model values (Figure 2b) such that they are closer to the causative model (Figure 1a) than without conditioning (Figure 2a). This leads to reduced model RMSE on the one hand, and data RMSE on the other (Table 1). This reduction in data RMSE can also be explained by the relaxation of the constraints in several portions of the model, thus increasing the degree of liberty to accommodate the model towards lower data misfit. More importantly, the Bravais-Pearson correlation between the inverted 15 and true model gradients is much higher when information from information entropy is used. This indicates that local 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 10 indicator is biased and/or shows high geological uncertainty levels away from faults and contacts. A thorough analysis lying beyond the scope of this paper, the remainder of this section presents an elementary sensitivity analysis using a series of two volumes representing distinct extreme scenarios.

Inversion constrained by biased geological uncertainty model
In this subsection, we investigate the effect of inaccurate geological models and the propagation of the related uncertainty in 15 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 (Figure 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 ( Figure 3b) (we further refer to it as 'ghost' geological unit). Notably, it increases geological uncertainty and smears interfaces that are well-constrained as per Figure 1. It also decreases in large parts of the model where → 1 20 previously, thereby favouring model changes in these areas during inversion and encouraging it to place larger density contrast or interfaces in these areas.

Inversion constrained using exaggerated geological uncertainty
To complete this series of tests, we generated a third PGM showing exaggerated geological uncertainty. To this end, we used 5 a half aperture 95% confidence interval of ~50 degrees for orientation data measurements in our MCUE simulations. This is far higher than for the rest of the MCUE simulations used in this paper. All other simulations (Subsect. 3.2 and 3.3) use a value of ~11 degrees, which is representative of realistic measurement uncertainty as proposed by recent metrological studies Novakova and Pavlis, 2017). Figure 5 below shows the resulting volume ( Figure 5a) and the inverted model obtained using it for local conditioning of the regularization constraint (Figure 5b). 10 , which results in geophysical inversion being less strongly guided by geological information. As can be seen in Fig   4b, the inverted model obtained in this case shows structures that present weaker contrast around interfaces than when 10 geological uncertainty is lower (Figure 2b). Importantly, however, most structures are well preserved and the overall model values for the different lithologies remain closer to the true model than for the non-conditioned case (Figure 2a). This indicates that even in high geological uncertainty scenarios, interpretation outcome may be largely more reliable when local regularisation is used.

15
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 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. it is host to base and precious metal mineralisation (Williams, 2009). With a relatively high positive density contrast (expected to be between 190 and 270 kg.m -³) to the Yilgarn Craton granite-gneiss host, mafic greenstone belts A, B, and C are suitable targets for gravity inversion. Interpretations from multiple studies in the region, e.g, (Johnson et al., 2013;Pirajno et al., 1998; were compiled while additional geological measurements acquired 15 in 2015, 2016 and 2017 complemented legacy data (Occhipinti et al., 2017;Olierook et al., 2018). This allowed the revision of existing models and improved our understanding of the area. This, in turn, also highlights the challenges presented by the characterization of greenstone belts A, B and C, and that further geophysical analysis through constrained inversion is a useful pathway for reducing exploration risk.
Geophysical data consists of ground surveys obtained from Geoscience Australia (http://www.ga.gov.au/data-pubs). Post-20 processing includes spherical-cap and terrain corrections and the removal of the regional trend to obtain the complete Bouguer anomaly. As most data points were acquired 1 to 4 km apart, the dataset was resampled to match the geological model discretization, making up a total of 4882 measurement points. The model is discretized into 100 × 100 × 42 cells of dimensions 2.335 km × 1.875 km × 1.0475 km, down to a depth of 44 km, making up a total of 420000 cells.

Geological modelling 5
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  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 10 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

Inversion results and analysis
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.

Comparative analysis strategy
Prior to examination of the inverted models, we analyse geophysical data misfit after inversion. This enables us to ensure that 5 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 ( Figure 8a)  interpretation of inversion results is complemented by metrics quantifying the differences between models. We give particular 10 attention to model cells where the probability of mafic greenstone is larger than zero. For these cells, we classify lithologies by identifying cells with a density contrast corresponding to mafic greenstone.

Results
Data root-mean-square (RMS) error decreases during inversion from 12.46 mGal to reach 1.59 mGal and 1.53 mGal for the non-conditioned and conditioned cases, respectively. The corresponding data misfit maps show a linear correlation coefficient 15 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 Gao et al., 2012;Molodtsov et al., 2013;Moorkamp et al., 2013;Sun 20 andLi, 2016, 2017). 1 (most certain group). This suggests that local regularization conditioning allows inversion to update the model preferentially in geologically uncertain areas. In turn, differences with the prior model in more geologically certain areas are reduced compared to the non-conditioned case. This effect of conditioning is corroborated by Figure 8c where the longest distances to the dashed line, which represents equal model update for the two studied cases, occur in geologically uncertain areas. This also translates in higher difference between model updates of the two cases in Figure 7d for lower values of . In addition, we 30 observe that local conditioning produces stronger density contrasts in Figure 8b in some of the areas where the conditioning values are higher in Figure 8b. Furthermore, structures in the inverted model are easier to identify when local conditioning is used. It is confirmed by global roughness measures ‖∇ ‖ 2 equal to 3.4 ( 3 ⁄ )/ and 4 ( 3 ⁄ )/ for the nonconditioned and conditioned cases, respectively. More specifically, as shown by Figure 7e, this difference arise in parts of the model associated with lower , which characterize uncertain areas, including interfaces between lithologies. 5 The recovered greenstone belts are shown in Figure 8a and Figure 8b. In Figure 8b, the extension of recovered mafic greenstone belts is significantly different than when geological uncertainty is not accounted for (Figure 8a). In particular, belt A is significantly larger in Figure 8b than in Figure 8a (2.4×10 2 km 3 vs 4.6×10 2 km 3 ). Similarly, the extent of belt C is increased overall (volume of 5.3×10 2 km 3 vs 14×10 2 km 3 ), while its different portions reconnect; the northern half is also significantly shallower and broader than in Figure 7a and Figure 8a. It appears that belt A remains thinner and shallower (Figure 8b Figure 8b, with a volume 40% higher than in Figure 8a. As discussed in the next subsection, these differences have a signification impact on the interpretation of inversions results and are important to understand the influence of local conditioning on inversion.

Interpretation
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 ( Figure   7a and Figure 7b). This shows that locally conditioned regularization does not enforce changes in the inverted model everywhere geological uncertainty is high as uncertainty is only a reflection of potential errors. Rather, this indicates that in 5 such cases, the guiding effect of such regularization will be exerted on the condition that it does not prevent the data term in ( , ) as per Eq. (1) from decreasing. This also confirms that geophysical data is the main driver of the model updates in geologically uncertain areas. Instead of smooth departures from the prior model to match geophysical data regardless of geological considerations, local regularization constraints allow inversion to account for the probabilistic geological modelling of the area and for geological uncertainty. It can therefore provide results that conform better to known geology. Geophysical inversion using geological uncertainty information (Figure 7b) confirms the presence of high density anomalies 15 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, B and 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 reveal that greenstone B might extend further to the east than indicated by the preferred lithology volume (Figure 7a) and that greenstone C may be thinner in its central part. 20

Concluding remarks
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 25 model preferably in geologically uncertain parts of the studied area during what we called uncertainty-guided inversion. This therefore partly mitigates the non-uniqueness of the inversion through the addition of constraints encouraging inversion to produce models that account for geological uncertainty across the entire inverted volume. We have demonstrated that it can be used collaboratively with geological modelling efficiently through field application in the Yerrida Basin. Inversion results show that our integration methodology has the capability to refine the recovered physical property model and interpretations 30 in portions of the model where geological uncertainty is high. Another advantage of the proposed technique is that it is time the same parameterization as non-conditioned inversion.
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 5 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 10 (compared to the preferred lithology volume) mafic greenstone volumes under the Yerrida Basin on 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 technique or single-discipline workflows. However, despite its robustness to misplaced interface (e.g., bias) or to high 15 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. 20 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  in the uncertainty-guided inversion scheme we presented, as well as the utilisation of geological uncertainty to weight the cross-gradient term of  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 25  or through cross-gradient joint inversion.

Appendix A: Data misfit maps from inversion in the Yerrida Basin
Figure A 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. Ad and Fig. Ah, respectively). It is complemented by the corresponding plots of 5 starting (Fig. Aa and Fig. Ae), observed ( Fig. Ab and Fig. Af), and calculated data ( Fig. Ac and Fig. Ah). Note that Fig Giraud are the main developers. Mark Jessell has been involved in the validation of the methodology at the initial development of gravity data from the Yerrida Basin. Evren Pakyuz-Charrier assisted Mark Lindsay with the utilisation of MCUE. All coauthors contributed to the final version of this article. Mark Lindsay and Vitaliy Ogarko were the most actively involved in the revision process of the drafts leading to this paper.

Introduction
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 has poor horizontal resolution, can be integrated with seismic inversion to mitigate the poor lateral resolution of seismic inversion . Likewise, 25 geological modelling and geophysical inversions are routinely performed in the same area to obtain a subsurface model consistent with geological and geophysical measurements Pears et al., 2017;Williams, 2008). When sufficient prior information is available, petrophysical constraints can be used in inversion Paasche and Tronicke, 2007), and integrated with geological modelling to derive local constraints . However, in exploration scenarios, this can be impractical as the available petrophysical information 30 geophysical dataset is available, practitioners have relied on joint inversion using structural constraints (e.g., 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 into 5 what has been called guided inversion . 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 10 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 . In contrast to   specifically, we utilize the information entropy , 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 , , and Wiik et al. (2015), who extract continuous structural information from seismic data to adjust the strength of the regularization 25 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 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 use other geophysical attributes. Thirdly, we allow inversion to update the model preferably in the most uncertain parts of the geological model, 30 instead of encouraging a certain degree of structural similarity between two geophysical inverse models. Finally, while some of the previous work involve mostly 2D models, every step of our modelling is performed purely in 3D. 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 5 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.

Inversion methodology 10
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(Martin et al., , 2018, which offers the possibility to condition the regularization function  of  locally using geological uncertainty. This is achieved by incorporating prior information into a structurebased regularization function in a fashion similar to Wiik et al., 2015;Yan et al., 2017) by locally adjusting 15 the related weight.
Solving the inversion problem regularized in this fashion consists of finding a model that minimizes the objective function given below:

Data term Model term Structural regularization term
where relates to the geophysical measurements to be inverted, is the forward modelling operator; relates to the model 20 being searched for, and is the prior model; , and are diagonal weighting matrices corresponding to data noise, model weighting and gradient regularization, respectively. The model term is a ridge regression constraint term .
The structural regularization term in Eq. (1) enforces structural constraints during inversion. It is weighted locally by matrix , which can be derived from prior information (see Subsect. 2.3 for details). The positive free parameters and 25 control the overall weight of model and structural regularization terms; ∇ is the spatial gradient operator. Note that ‖∇ ‖ , estimates the amount of structure in inverted physical property model . Also note that parts of the model where = 0 are excluded from the calculation of the structural regularization and can change freely to accommodate geophysical data.
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 5
Probabilistic geological modelling is performed using the Monte-Carlo Uncertainty Estimator (MCUE) method of (Pakyuz-Charrier et al., 2018b, 2018a, which extends previous works from . It is a 3D 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 borehole) as inputs. The sampling algorithm perturbs orientation data used to derive a reference model by sampling probability distributions describing 10 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 3D geological modelling engine of Geomodeller© , it produces a set of plausible geological models honouring the geological input measurements that represent the geological model space (Lindsay et al., 2013b).

Utilisation of information entropy for local conditioning
Information entropy was introduced for geological modelling by  and is gaining 20 popularity as a measure of uncertainty in probabilistic geological modelling de la Varga and Wellmann, 2016;Lindsay et al., 2013;Wellmann et al., 2017;Yamamoto et al., 2014). Quoting , information entropy "quantifies the amount of missing information and hence, the uncertainty at a discrete location". For the th model-cell, it is given as : 25 (2) Fundamentally, geological uncertainty contained in encapsulates information about possible locations of interfaces between units and areas where the geological data is insufficiently informative. Instead of using directly, we calculate utilising its normalized complementary, which reflects the degree of certainty across the model. Let us express as follows, for the th model cell: The consequence of Eq. (2) and 3 is that is minimum at interfaces and in areas poorly constrained by geological information, and equal to unity in areas where the geology is well resolved. Consequently, the conditioning process attaches small weights to the structural term of Eq. (1) in uncertain cells, while consistently high values will be applied to the most 5 geologically certain cells. As a result, it enables the inversion algorithm to favour nearly constant changes in the inverted model in contiguous certain groups of cells (e.g., where → 1) while relaxing the regularisation constraints in the most uncertain parts (e.g., where → 0).

Proof of concept: synthetic case study
This section introduces the proof of concept of the proposed method through an idealized case study illustrating the potential 10 of the proposed inverse modelling scheme to ameliorate inversion results and to reduce interpretation uncertainty. We use the same 3D density contrast model as , which is obtained by populating the structural framework of (Pakyuz-Charrier et al., 2018b). We simulate a series of PGM sought to represent expected values as well as possible extreme scenarii.
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. 15

Survey setup
The 3D 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 mudstonesandstone basin oriented N170, abutting a faulted contact with a folded ultramafic basement to the South-West. Model complexity was artificially increased through the addition of a North-South fault and of a mafic intrusion. 20 The true density contrast model (Figure 1a) was obtained 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 -³ were assigned to the upper and lower basin lithotypes, respectively. Mafic rocks were assigned a density contrast of 200 kg.m -³ while the density contrast of the ultramafic basement was set to 300 kg.m -³.
MCUE perturbations of the reference geological model were first performed using standard measurement uncertainty values 25 recommended by metrological studies as reported by Novakova and Pavlis, 2017). We generated a series of 300 models that were subsequently combined into a PGM. The resulting volume representing the values calculated from this PGM in each cell of the model as per Eq.

Locally constrained inversion: validation
To assess the impact of local conditioning of the regularization function onto inversion, we compare inversions using non-5 conditioned ( Figure 2a) and locally conditioned ( Figure 2b) regularization function, respectively. Please note that, simulating the absence of prior petrophysical information, a homogenous prior model set to 0 kg.m-³ was used in both cases. Besides qualitative visual comparison of the models, we interpret inversion results (Figure 2) through the commonly used model and data root-mean-square error (RMSE), 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 5 correlation (also often called 'linear correlation coefficient') between their gradients (Table 1).
Comparison of the true model ( Figure 1a) with inversion results in Figure 2a and Figure 2b shows that while the structures in 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 (Figure 2b). The guiding effect of is visible in Figure 2b where the main structures at depth follow the general features of the conditioning volume ( Figure 1b). Moreover, in order to minimize the 10 conditioned regularization constraint simultaneously to data misfit, inversion was driven to accommodate inverted model values (Figure 2b) such that they are closer to the causative model ( Figure 1a) than without conditioning (Figure 2a). This leads to reduced model RMSE on the one hand, and data RMSE on the other (Table 1). This reduction in data RMSE can also be explained by the relaxation of the constraints in several portions of the model, thus increasing the degree of liberty to accommodate the model towards lower data misfit. More importantly, the Bravais-Pearson correlation between the inverted 15 and true model gradients is much higher when information from information entropy is used. This indicates that local 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 10 indicator is biased and/or shows high geological uncertainty levels away from faults and contacts. A thorough analysis lying beyond the scope of this paper, the remainder of this section presents an elementary sensitivity analysis using a series of two volumes representing distinct extreme scenarios.

Inversion constrained by biased geological uncertainty model
In this subsection, we investigate the effect of inaccurate geological models and the propagation of the related uncertainty in 15 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 (Figure 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 ( Figure 3b) (we further refer to it as 'ghost' geological unit). Notably, it increases geological uncertainty and smears interfaces that are well-constrained as per Figure 1. It also decreases in large parts of the model where → 1 20 previously, thereby favouring model changes in these areas during inversion and encouraging it to place larger density contrast or interfaces in these areas.

Inversion constrained using exaggerated geological uncertainty
To complete this series of tests, we generated a third PGM showing exaggerated geological uncertainty. To this end, we used 5 a half aperture 95% confidence interval of ~50 degrees for orientation data measurements in our MCUE simulations. This is far higher than for the rest of the MCUE simulations used in this paper. All other simulations (Subsect. 3.2 and 3.3) use a value of ~11 degrees, which is representative of realistic measurement uncertainty as proposed by recent metrological studies Novakova and Pavlis, 2017). Figure 5 below shows the resulting volume ( Figure 5a) and the inverted model obtained using it for local conditioning of the regularization constraint ( Figure 5b). 10 , which results in geophysical inversion being less strongly guided by geological information. As can be seen in Fig   4b, the inverted model obtained in this case shows structures that present weaker contrast around interfaces than when 10 geological uncertainty is lower (Figure 2b). Importantly, however, most structures are well preserved and the overall model values for the different lithologies remain closer to the true model than for the non-conditioned case (Figure 2a). This indicates that even in high geological uncertainty scenarios, interpretation outcome may be largely more reliable when local regularisation is used.

15
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 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. it is host to base and precious metal mineralisation (Williams, 2009). With a relatively high positive density contrast (expected to be between 190 and 270 kg.m -³) to the Yilgarn Craton granite-gneiss host, mafic greenstone belts A, B, and C are suitable targets for gravity inversion. Interpretations from multiple studies in the region, e.g, (Johnson et al., 2013;Pirajno et al., 1998; were compiled while additional geological measurements acquired 15 in 2015, 2016 and 2017 complemented legacy data (Occhipinti et al., 2017;Olierook et al., 2018). This allowed the revision of existing models and improved our understanding of the area. This, in turn, also highlights the challenges presented by the characterization of greenstone belts A, B and C, and that further geophysical analysis through constrained inversion is a useful pathway for reducing exploration risk.
Geophysical data consists of ground surveys obtained from Geoscience Australia (http://www.ga.gov.au/data-pubs). Post-20 processing includes spherical-cap and terrain corrections and the removal of the regional trend to obtain the complete Bouguer anomaly. As most data points were acquired 1 to 4 km apart, the dataset was resampled to match the geological model discretization, making up a total of 4882 measurement points. The model is discretized into 100 × 100 × 42 cells of dimensions 2.335 km × 1.875 km × 1.0475 km, down to a depth of 44 km, making up a total of 420000 cells.

Geological modelling 5
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  and CSIRO  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 10 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

Inversion results and analysis
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.

Comparative analysis strategy
Prior to examination of the inverted models, we analyse geophysical data misfit after inversion. This enables us to ensure that 5 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 ( Figure 8a) and conditioned regularization function (Figure 8b) using ( Figure 7b). In addition to departures from the prior model, variations between the two cases are studied by visual comparison of Figure 8a and Figure 7b, through qualitative ( Figure 7c) and quantitative comparative analysis (Figure 7d-e). Our interpretation of inversion results is complemented by metrics quantifying the differences between models. We give particular 10 attention to model cells where the probability of mafic greenstone is larger than zero. For these cells, we classify lithologies by identifying cells with a density contrast corresponding to mafic greenstone.

Results
Data root-mean-square (RMS) error decreases during inversion from 12.46 mGal to reach 1.59 mGal and 1.53 mGal for the non-conditioned and conditioned cases, respectively. The corresponding data misfit maps show a linear correlation coefficient 15 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 Gao et al., 2012;Molodtsov et al., 2013;Moorkamp et al., 2013;Sun 20 andLi, 2016, 2017). 1 (most certain group). This suggests that local regularization conditioning allows inversion to update the model preferentially in geologically uncertain areas. In turn, differences with the prior model in more geologically certain areas are reduced compared to the non-conditioned case. This effect of conditioning is corroborated by Figure 8c where the longest distances to the dashed line, which represents equal model update for the two studied cases, occur in geologically uncertain areas. This also translates in higher difference between model updates of the two cases in Figure 7d for lower values of . In addition, we 30 observe that local conditioning produces stronger density contrasts in Figure 8b in some of the areas where the conditioning values are higher in Figure 8b. Furthermore, structures in the inverted model are easier to identify when local conditioning is used. It is confirmed by global roughness measures ‖∇ ‖ 2 equal to 3.4 ( 3 ⁄ )/ and 4 ( 3 ⁄ )/ for the nonconditioned and conditioned cases, respectively. More specifically, as shown by Figure 7e, this difference arise in parts of the model associated with lower , which characterize uncertain areas, including interfaces between lithologies. 5 The recovered greenstone belts are shown in Figure 8a and Figure 8b. In Figure 8b, the extension of recovered mafic greenstone belts is significantly different than when geological uncertainty is not accounted for (Figure 8a). In particular, belt A is significantly larger in Figure 8b than in Figure 8a (2.4×10 2 km 3 vs 4.6×10 2 km 3 ). Similarly, the extent of belt C is increased overall (volume of 5.3×10 2 km 3 vs 14×10 2 km 3 ), while its different portions reconnect; the northern half is also significantly shallower and broader than in Figure 7a and Figure 8a. It appears that belt A remains thinner and shallower (Figure 8b Figure 7a as only the eastern part is preserved. Note that it is larger in Figure 8b, with a volume 40% higher than in Figure 8a. As discussed in the next subsection, these differences have a signification impact on the interpretation of inversions results and are important to understand the influence of local conditioning on inversion.

Interpretation
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 ( Figure   7a and Figure 7b). This shows that locally conditioned regularization does not enforce changes in the inverted model everywhere geological uncertainty is high as uncertainty is only a reflection of potential errors. Rather, this indicates that in 5 such cases, the guiding effect of such regularization will be exerted on the condition that it does not prevent the data term in ( , ) as per Eq. (1) from decreasing. This also confirms that geophysical data is the main driver of the model updates in geologically uncertain areas. Instead of smooth departures from the prior model to match geophysical data regardless of geological considerations, local regularization constraints allow inversion to account for the probabilistic geological modelling of the area and for geological uncertainty. It can therefore provide results that conform better to known geology. 10 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 (Figure 7b) confirms the presence of high density anomalies 15 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, B and 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 reveal that greenstone B might extend further to the east than indicated by the preferred lithology volume (Figure 7a) and that greenstone C may be thinner in its central part. 20

Concluding remarks
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 25 model preferably in geologically uncertain parts of the studied area during what we called uncertainty-guided inversion. This therefore partly mitigates the non-uniqueness of the inversion through the addition of constraints encouraging inversion to produce models that account for geological uncertainty across the entire inverted volume. We have demonstrated that it can be used collaboratively with geological modelling efficiently through field application in the Yerrida Basin. Inversion results show that our integration methodology has the capability to refine the recovered physical property model and interpretations 30 in portions of the model where geological uncertainty is high. Another advantage of the proposed technique is that it is time the same parameterization as non-conditioned inversion.
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 5 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 10 (compared to the preferred lithology volume) mafic greenstone volumes under the Yerrida Basin on 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 technique or single-discipline workflows. However, despite its robustness to misplaced interface (e.g., bias) or to high 15 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. 20 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  in the uncertainty-guided inversion scheme we presented, as well as the utilisation of geological uncertainty to weight the cross-gradient term of  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 25  or through cross-gradient joint inversion.