We propose, test and apply a methodology integrating 1D magnetotelluric (MT) and magnetic data inversion, with a focus on the characterisation of the cover–basement interface. It consists of a cooperative inversion workflow relying on standalone inversion codes. Probabilistic information about the presence of rock units is derived from MT and passed on to magnetic inversion through constraints combining structural constraints with petrophysical prior information. First, we perform the 1D probabilistic inversion of MT data for all sites and recover the respective probabilities of observing the cover–basement interface, which we interpolate to the rest of the study area. We then calculate the probabilities of observing the different rock units and partition the model into domains defined by combinations of rock units with non-zero probabilities. Third, we combine these domains with petrophysical information to apply spatially varying, disjoint interval bound constraints (DIBC) to least-squares magnetic data inversion using the alternating direction method of multipliers (or ADMM). We demonstrate the proof-of-concept using a realistic synthetic model reproducing features from the Mansfield area (Victoria, Australia) using a series of uncertainty indicators. We then apply the workflow to field data from the prospective mining region of Cloncurry (Queensland, Australia). Results indicate that our integration methodology efficiently leverages the complementarity between separate MT and magnetic data modelling approaches and can improve our capability to image the cover–basement interface. In the field application case, our findings also suggest that the proposed workflow may be useful to refine existing geological interpretations and to infer lateral variations within the basement.

Geophysical integration has been gaining traction in recent years, be it when two or more datasets are inverted simultaneously (i.e. joint inversion) or when the inversion of a geophysical dataset is used to constrain another (i.e. cooperative inversion). A number of approaches for joint modelling have been developed with the goal of exploiting the complementarities between different datasets (see, for instance, the reviews of Lelièvre and Farquharson, 2016, and Moorkamp et al., 2016, and references therein). As summarised in the review of Ren and Kalscheuer (2019), “joint inversion of multiple geophysical datasets can significantly reduce uncertainty and improve resolution of the resulting models”. This statement remains valid, be it for the modelling of a single property (e.g. resistivity for joint controlled-source electromagnetic and magnetotelluric (MT) data or density for joint gravity anomaly and gradiometric data) or of multiple properties (e.g. the joint inversion of seismic and gravity data to model P-wave velocity and density). In the second case, joint inversion approaches can be grouped into two main categories based on the hypothesis they rely on. Structural approaches allow for the joint inversion of datasets with differing sensitivities to the properties of the subsurface through the premise that geology requires spatial variations in inverted properties to be co-located. Structural constraints can then be used as a way to link two or more datasets jointly inverted for by encouraging structural similarity between the inverted models (Haber and Oldenburg, 1997; Gallardo and Meju, 2003). Alternatively, petrophysical approaches utilise prior petrophysical information (e.g. from outcrops, boreholes, or the literature) to enforce certain statistics in the recovered model so that it resembles the petrophysical measurements' (Lelièvre et al., 2012; Sun and Li, 2015; Giraud et al., 2017; Astic and Oldenburg, 2019). Whereas structural and petrophysical approaches are well suited to exploit complementarities between datasets in a quantitative manner, running joint inversion might be, in practice, challenging due to, for instance, the risk of increased non-linearity of the inverse problem (see, e.g. the L surface using the cross-gradient constraints in Martin et al., 2021, and approaches adapting coupling during inversion, e.g. Heincke et al., 2017), the necessity to balance the contribution of the different datasets inverted (Bijani et al., 2017) and resolution mismatches (Agostinetti and Bodi, 2018).

In this contribution, we present a new multidisciplinary modelling workflow that relies on sequential, cooperative modelling. It follows the same objectives as the two categories of joint inversion mentioned above in that structural information is passed from one domain to the other and uses petrophysical information to link domains. The development of the sequential inversion scheme we present is motivated by a similar idea as Lines et al. (1988), who states that “the inversion for a particular data set provides the input or initial model estimate for the inversion of a second data set”. A further motivation is to design a workflow capable of integrating the inversion of two or more datasets quantitatively using standalone modelling engines that run independently.

In this paper, the workflow is applied to the sequential inversion of MT followed by magnetic data, taking into account the importance of robustly constraining the thickness of the regolith in hard-rock imaging and mineral exploration. This is motivated by the relative paucity of works considering cooperative workflows to integrate MT and magnetic data together with the recent surge in interest for the characterisation of the depth to basement interface in mineral exploration, despite these two geophysical methods being part of the geoscientists' toolkit for depth to basement imaging. Historically, MT has often been integrated with other electromagnetic methods or with seismic data (e.g. Gustafson et al., 2019; Peng et al., 2019), and with gravity to a lower extent (see review of the topic of Moorkamp, 2017). It is, however, seldom modelled jointly with magnetic data unless a third dataset is considered (e.g. Oliver-Ocaño et al., 2019; Zhang et al., 2020; Gallardo et al., 2012; Le Pape et al., 2017). We surmise that this is because (i) the interest for integrating MT with other disciplines arose primarily in oil and gas and geothermal studies and relied on structural similarity constraints for reservoir or (sub)salt imaging, (ii) the difference in terms of spatial coverage between the two methods elsewhere, (iii) the differences in terms of sensitivity to exploration targets and (iv) the difficulty to robustly correlate electrical conductivity and magnetic susceptibility. Bearing these considerations in mind, we developed a workflow incorporating MT and magnetic inversion with petrophysical information and geological prior knowledge.

MT–magnetic integration workflow summary showing the role of the different techniques.

In the workflow we develop, we exploit the differences in sensitivity between MT and magnetic data. On the one hand, the MT method, used in a 1D probabilistic workflow as presented here, is well suited to recover vertical resistivity variations and interfaces, especially in a sedimentary basin environment (Seillé and Visser, 2020). MT data are, however, poorly sensitive to resistors, particularly when they are overlaid by conductors (e.g. Chave et al., 2012), which makes it difficult to differentiate between highly resistive features, such as intra-basement resistive intrusions. On the other hand, magnetic data inversion is more sensitive to lateral magnetic susceptibility changes and to the presence of vertical or tilted structures or anomalies. Bearing this in mind, we first derive structural information across the studied area in the form of probability distributions of the interfaces between geological units, extracted from the interpolation of probabilistic 1D MT data inversion. From there, the probability of occurrence of geological units can be estimated in 2D or 3D. These probabilities are used to divide the area into domains where only specific units can be observed (e.g. basement, sedimentary cover, or both). Such domains are then passed to magnetic data inversion, where they are combined with prior petrophysical information to derive spatially varying, disjoint interval bound constraints that can consider multiple intervals in every model cell. Such constraints are enforced using the alternating direction method of multipliers for 2D or 3D inversion (ADMM; see Ogarko et al., 2021a, for application to gravity data using geological prior information and Giraud et al., 2021c, for MT-constrained gravity inversion). Finally, uncertainty analysis of the recovered magnetic susceptibility model is performed, and rock unit differentiation allows for controlling the compatibility of magnetic inversion results with the MT data. The workflow is summarised in Fig. 1, as applied to 1D MT inversions. In this paper, we apply this workflow to 2D magnetic data inversion, but it is applicable in 3D.

The remainder of this paper is organised as follows. We first introduce the methodology and summarise the MT and magnetic standalone modelling procedures we rely on. We then introduce the proof of concept in detail using a realistic synthetic case study based on a geological model of the Mansfield area (Victoria, Australia), which we use to explore the different possibilities for integrating MT-derived information and petrophysics offered by our workflow. Following this, we present a field application using data from Cloncurry (Queensland, Australia) where we tune our approach to the specificity of the area. Finally, this work is placed in the broader context of geoscientific modelling, and perspectives for future work are exposed in the discussion section.

The MT method is a natural source electromagnetic method. Simultaneous measurements of the fluctuations of the magnetic and electric fields are
recorded at the Earth's surface under the assumption of a plane wave source. The relationship between the input magnetic field

Resistivity models derived from MT data are found by forward modelling and inversion of the impedance tensor

Within the context of Bayesian inversion, the solution of the inverse problem consists of a posterior probability distribution, calculated from an
ensemble of models fitting the data within uncertainty. The posterior probability distribution

The prior distribution

The term inside the exponential is the data misfit, which is the distance between observed data

data processing errors, which we model introducing a matrix

errors introduced by the violation of the 1D assumption when using 1D models, which we model introducing

Both sources of uncertainty are included in the calculation of

Following this, we define

We use a 1D MT trans-dimensional Markov chain Monte Carlo algorithm (Seillé and Visser, 2020). Trans-dimensional Bayesian inversions have gained traction for applications to geophysical inversion (Malinverno, 2002; Sambridge et al., 2006; Bodin et al., 2009 Xiang et al., 2018) in recent years as an efficient means to sample the model space. This algorithm solves for the resistivity distribution at depth and the number of layers in the model. Having the number of layers treated as an unknown is convenient because it does not require the formulation of assumptions about the inversion regularisation or the model parameterisation. Thus, the mesh discretisation and the natural parsimony of the trans-dimensional algorithms favour models that fit the data with fewer model parameters, thereby penalising complex models (Malinverno, 2002).

The output of the probabilistic inversion consists of an ensemble of models describing the posterior probability distribution. Each model of the
ensemble fits the input data, the logarithm of the determinant of the impedance tensor

In the context of depth to basement imaging, this allows us to derive three domains characterised by

In this section, we summarise the method used to enforce disjoint interval bound constraints during magnetic data inversion. We largely follow Ogarko
et al. (2021a), which we extend to locally weighted bound constraints and magnetic data inversion. The geophysical inverse problem is formulated in
the least-squares sense (see chap. 3 in Tarantola, 2005). The cost function we minimise during inversion is given as

To balance the decreasing sensitivity of magnetic field data with the depth, we utilise the integrated sensitivity technique of Portniaguine and Zhdanov (2002), which we use as a preconditioner multiplying the constraints terms in the system of equations representing Eq. (5) (see Giraud et al., 2021b for details).

We solve Eq. (5) while constraining the inversion using the disjoint interval bound constraints of Ogarko et al. (2021a) (which we further refer to as
DIBC). The problem can be expressed in its generic form as

The set of intervals

Inversion results are assessed using indicators calculated from the difference between reference and recovered models. We calculate three
complementary global indicators and one local indicator with the aim to characterise the similarity between causative bodies and retrieved models in
terms of both the petrophysical properties and the corresponding rock units. These indicators are listed below in the order they are introduced in
this subsection:

root-mean-square model misfit, which measures the discrepancy between the inverted and true models in terms of the values of physical properties that have been inverted for;

the membership value to the different intervals used as constraints, which is a local metric indicative of the geological interpretation ambiguity from which two global metrics are calculated (average model entropy and Jaccard distance);

average model entropy, which is a statistical indicator that we use to estimate geological interpretation uncertainty;

Jaccard distance, which measures the dissimilarity between sets and is used here to evaluate the difference between the recovered and true rock unit models.

In the synthetic study we present, we evaluate the capability of inversions to recover the causative magnetic susceptibility model using the commonly
used root mean square (rms) of the misfit between the true and inverted models (rms model misfit,

In the context geophysical inverse modelling, membership analyses provide a quantitative estimation of interpretation uncertainty to interpretation of
recovered petrophysical properties. We calculate the membership values to rock units based on the distance between the recovered magnetic
susceptibility and interval bounds, on the premise that magnetic susceptibility intervals for the rock types or group of rock types do not
overlap. We distinguish between three cases:

When the recovered magnetic susceptibility falls within an interval as defined in Eqs. (7) and (8), its membership to the corresponding unit is set to 1 and all others are set to 0.

When the recovered value falls in between two intervals, the membership value is calculated for the two corresponding units, with all others being
set to 0. In such cases, the membership value is calculated from the relative distance to the intervals' respective upper and lower bound. Assuming
that for the

When

Using the membership values

In addition to calculating

Calculating the index of the corresponding rock unit in each model cell, we obtain a rock unit model

Using

Stratigraphic column showing geological topological relationships and average physical properties. Lithologies are indexed from 1 through 6 by order of genesis.

The synthetic case study that we use to test our workflow is built using a structural geological framework initially introduced in Pakyuz-Charrier et al. (2018). It presents geological features that reproduce field geological measurements from the Mansfield area (Victoria, Australia). The choice of resistivity and magnetic susceptibility values to populate the structural model was made to test the limits of this sequential, cooperative workflow and to show its potential to alleviate some of the limitations inherent to potential field and MT inversions. To this end, we have selected a part of the synthetic model where MT data is affected by 2D and 3D effects to challenge the workflow we propose. The objective of this exercise is to assess the workflow's efficacy to recover the sediment–basement interface. To this end, we rely on the magnetic inversion's sensitivity to magnetic susceptibility contrast to model the interface between highly susceptible units (basement) and rocks presenting little to no magnetic susceptibility (sedimentary units). The magnetic susceptibility model we use consists of 2D structures.

The structural geological model was derived from foliations and contact points using the Geomodeller^{®} software
(Calcagno et al., 2008; Guillen et al., 2008; Lajaunie et al., 1997). It is constituted of a sedimentary syncline abutting a faulted contact with a
folded basement. The model's complexity was increased with the addition of a fault and an ultramafic intrusion. Details about the original
3D geological model are provided in Pakyuz-Charrier et al. (2018b). Here, we increase the maximum depth of the model to 3150

Simulated total magnetic field anomaly.

We assign magnetic susceptibility in the model considering non-magnetic sedimentary rocks in the basin units (lithologies 3, 5 and 6 in Table 1) and
literature values (see Lampinen et al., 2016) to dolerite (lithology 4), diorite (lithology 2) and ultramafic rocks (lithology 1). We assign
electrical resistivities assuming relatively conductive sedimentary rocks and resistive basement and intrusive formations. Resistivities in
sedimentary rocks might vary by orders of magnitude and mainly depend on porosity, which is linked to the degree of compaction and the type of
lithology and the salinity of pore fluid (Evans et al., 2012). The three sedimentary layers are assigned different resistivity values of 30, 10
and 50

The core 2D model is discretised into

Airborne magnetic data are simulated for a fixed-wing aircraft flying at an altitude of 100

We add normally distributed noise with an amplitude equal to 4 % of the maximum amplitude of the data. We simulate noise contamination by adding
noise sampled randomly from by a normal distribution characterised by a standard deviation of 3.8

The synthetic MT data is computed using the complete 3D resistivity model derived from the 3D geological model. The 3D resistivity model and the MT
responses can be found online (Giraud and Seillé 2022). The core of the electrical conductivity model used the same discretisation as the magnetic
susceptibility model (cells of dimension 127

In the following subsections, we present the results of the modelling of synthetic MT data along a 2D section (see Fig. 2c) of the 3D resistivity volume, following the workflow proposed in Sect. 2. Along this section, 16 MT sites are used as mentioned above. We start with the modelling of MT data to derive constraints and prior information for the inversion of magnetic data.

Example of posterior

We perform the 1D MT inversions of the 16 MT sounding independently using the 1D trans-dimensional Bayesian inversion described in
Sect. 2.1. Synthetic data for three MT sites are shown in Fig. 4a. The phase tensor skewness

Probability of interfaces between sedimentary cover

All the inversions ran using 60 Markov chains with 10

The model posterior distribution for three MT sites is shown in Fig. 4b. The interface probability within the posterior ensemble of 1D models is
described by a change point histogram. From the posterior ensembles of models and interfaces, a cover–basement interface probability
distribution

Figure 4 shows the interface probability and the cover–basement interface probability distribution

Starting from

The interpolated probabilities

domain 1 (sediments only): [

domain 2 (non-sediment units only): [0.024 0.055] SI;

domain 3 (sediments and non-sediment units): [

Inversion results for the different scenarios tested. Cases

The prior model for magnetic inversion is obtained using the MT-derived rock unit probabilities (Fig. 5b, c) and the magnetic susceptibility of the
rock units given in Table 1. We proceed in the same spirit as Giraud et al., (2017), who calculate the mathematical expectation from probabilistic
geological modelling to obtain a starting model for least-squares inversion. Here, we propose a different approach and combine information from the
intervals with MT probabilities as follows. For the

We remind that

Scenarios tested for the utilisation of MT-derived information in magnetic data inversion. “High confidence” refers to the case where constraints are applied only to models cells with MT-derived rock unit probabilities equal to 1.

In this section, we study the influence of MT-derived prior information onto magnetic inversion and estimate the related reduction of interpretation
uncertainty. In what follows, we consider that the prediction from MT can be considered with “high confidence” when the probability of one of the
units is predicted with a probability of 1. We perform inversions for six cases, consisting of the following scenarios.

Membership values for the non-magnetic lithologies. Cases

The different scenarios tested in the synthetic example are summarised in Table 2. The corresponding inversion results are shown in Fig. 6. We note that magnetic susceptibility models shown in this section are equivalent from the magnetic data inversion point of view as they present a similar data misfit, which we assume to be acceptable when it is of the same magnitude as the estimated noise in the data.

Metrics calculated for the assessment of inverted models for cases (a)–(e).

We complement the calculation of

A visual comparison of the membership values in Fig. 7e and e with the MT-derived domains (Fig. 5d) indicates good consistency with MT domains (1) and (2) (single rock units inferred). It also shows that the proposed workflow has the capability to improve the recovery of the sedimentary cover thickness significantly when compared to cases that do not use MT-derived DIBC across the entire model (Fig. 7a–c).

Two main observations can be made from the results shown in Figs. 6, 7 and Table 3. First, the use of DIBC at all locations of the model reduces
interpretation ambiguity (lower

Interpreted solid geological map of the area (Dhnaram and Greenwood, 2013). The small dots are the MT sites of the Cloncurry MT survey. The red line is the profile used in this study, and the red dots are the MT sites associated to this profile. The Constantine domain to the west and the Soldiers Cap domain to the east are separated by the Mount Margaret fault. The dashed red line delineates the area we focus on.

We propose an application example illustrating potential utilisations of the proposed sequential inversion workflow in the Cloncurry district (Queensland, Australia; see Fig. 8). Using observations made in the synthetic case, our aim here is to integrate MT with magnetic data inversion using the case relying on MT-derived DIBC with a homogenous starting model and smoothing constraints.

We use existing results of the depth to basement derived using MT within a probabilistic workflow (Seillé et al., 2021) in an area of the Cloncurry district. These results are used to constrain the magnetic inversion.

The depth to basement interface probability used to constrain the magnetic inversion was derived as part of a previous study using a similar workflow
as presented in Sects. 2.1 and 3.3, details about the survey can be found in Seillé et al.(2021) and are summarised in what follows. The study
consisted of modelling the full Cloncurry MT dataset using 1D probabilistic inversions. For each MT site, the cover–basement interface probability
distribution

Data preparation.

In this study, we focus on a 2D profile (L26; see location on map in Fig. 8a) and invert the corresponding magnetic data extracted from the anomaly map shown in Fig. 9a and b. The choice of an east–west-oriented profile is motivated by the north–south orientation of the main structures in the area and by the geological features that the known geology and the geophysical measurements suggest. The profile is nearly perpendicular to these structures, making it suitable for use within a 2D inversion scheme. It crosses the north–south-oriented Mount Margaret fault, which is thought to belong to the northern part of the regional Cloncurry fault structure, a major crustal boundary that runs north–south over the Mount Isa province (Austin and Blenkinsop, 2008; Blenkinsop, 2008). This boundary separates two major Paleoproterozoic sedimentary sequences (Austin and Blenkinsop, 2008). The geological modelling performed by Dhnaram and Greenwood (2013) also indicates that the Mount Margaret fault separates two distinct domains, the Constantine domain to the west and the Soldier Caps domain to the east. In our study area, the Constantine domain is covered by non-magnetic cover constituted of Mesozoic and Cenozoic sediments, lying on what is believed to be the Mount Fort Constantine volcanics, which is in some places intruded by the Williams Supersuite pluton. On the eastern side, the Soldier Caps domain is also covered by Mesozoic and Cenozoic sediments, and the basement is interpreted to be a succession of volcanic and metamorphic rocks (Dhnaram and Greenwood, 2013).

The depth to basement probabilistic surface derived by Seillé et al. (2021) along the E–W profile (see Fig. 9c) presents shallow basement depths
in the western part of the profile (top basement at a depth of approximately 100

In this work, we assume a non-magnetic sedimentary cover and a magnetic basement. In addition, we assume little to no remanent magnetisation and little to no self-demagnetisation. Important remanence and self-demagnetisation can be observed in the vicinity of magnetite-rich iron oxide, copper, and gold ore deposits (e.g. Anderson and Logan, 1992; Austin et al., 2013), but we consider there to be no indication of such features along L26. Further to this, we make this assumption for the sake of simplicity as the main object of this paper is the introduction of a new sequential inversion workflow and to show that it is applicable to field data.

Under these premises, the features the magnetic data presents can be exploited to improve the image of the cover–basement interface when integrated
with prior information about the thickness of cover. In this context, magnetic data inversion constrained by MT performs multiple roles:

constraining the depth and extent of the magnetic anomalies and refine their geometry;

analysing the compatibility between the constraints derived from MT and the magnetic data and resolving some small-scale structures not defined by the MT constraints;

reducing the interpretation uncertainty of the cover–basement interface;

proposing new scenarios in relation to the composition of the basement (in terms of its magnetic susceptibility) and structure (through its lateral variations).

The depth of the cover–basement interface probability shown in Fig. 9c is used to derive the domains required by the spatially varying bound constraints used in magnetic inversion.

MT-derived domains for cases with (i) sedimentary units only, (ii) sedimentary and non-sedimentary units and (iii) non-sedimentary units only. The magnetic susceptibilities for the different domains are also indicated.

Scenarios tested for the utilisation of MT-derived information in the field case and corresponding

We use the gridded reduced-to-pole (RTP) magnetic data from the Geological Survey of Queensland shown in Fig. 9
(

We convert the interface probability shown in Fig. 9c into basement and sedimentary rock probabilities using the method described in Sects. 2.1 and 3.4.1. We assume that the sedimentary basin domain overlies the basement domain and derive the corresponding domains for the DIBC using the domain procedure described above. The resulting domains are shown in Fig. 10.

In what follows, we assume that sedimentary rocks have a low magnetic susceptibility comprised within the range [

domain 1 (sediments only): [

domain 2 (basement and sediments): [

domain 3 (basement only): [0.015, 0.09] SI.

Note that the lowest magnetic susceptibility values are negative (

Similar to the synthetic model used in Sect. 3, padding cells were added in both horizontal directions. The resulting model covers a surface area
defined by a rectangle of 157

To examine the impact of different type of constraints, we first perform inversions using minimum prior information and successively increase the
amount of prior information from unconstrained inversions by using MT-derived intervals for multiple-bound constraints. In the scenarios investigated
here, we perform inversion using global smoothness constraints (

constrained by global smoothness constraints,

constrained by global smoothness constraints with lower- and upper-bound constraints,

constrained by global smoothness constraints with global multiple-bound constraints,

constrained by global smoothness constraints with local DIBC defined from MT probabilities.

The constraints used in each case are summarised in Table 4.

Inversion results. Panels

Similarly to the synthetic case, we determine the value of

The inversions reached a satisfactory data fit, with the exception of the constrained inversion 4 (see the data fit in Fig. 11e). In that case a
significant underfit of the magnetic data is observed within certain areas, which points to an incompatibility between the magnetic data and the
constraints applied. Four areas in the central part of the model are slightly underfit, as shown by double arrows between approximately 458

From Fig. 11, we identify five main areas where hypotheses made for the utilisation of MT-derived domains need to be adjusted. In each case, the
domain allowing sedimentary units may be deeper than expected or the basement may be less susceptible. We test the plausibility of such alternative
scenarios by adapting the MT-derived domains by adjusting the domains. We increased the depth of the non-sedimentary (i.e. basement) units in the
eastern part of the model and between the areas delimited by dashed lines in Fig. 11d. From a geological point of view, this corresponds to adjusting
our working hypothesis to a case where rocks previously identified as basement only may be less susceptible than expected. The domains we use after
adjustment are shown in Fig. 12a, and inversion results are shown in Fig. 12b and c, respectively. Figure 12d proposes an automated interpretation with
membership values

Beyond the possibility to review hypotheses made at earlier stages of the workflow, we get insights into the structure and magnetic susceptibility of the basement. While electrical conductivity and magnetic susceptibility may be sensitive to changes in rock type, there are scenarios where they exhibit differing sensitivity to texture and grain properties, respectively. For instance, metamorphism and alteration might affect electrical conductivity and magnetic susceptibility differently (Clark, 2014; Dentith et al., 2020). Under these circumstances, our results can provide indications about plausible geological processes given sufficient prior geological information about the deformation history.

From a multi-physics modelling point of view, the results presented in the previous section show a general agreement between the MT-derived
constraints and the magnetic data. However, the results also show incompatibilities in a few parts of the model. We identified two major areas where
incompatibility occurs:

a smaller inconsistent area in the western part of the survey,

a large inconsistent area east of the Mount Margaret fault.

We interpret these incongruities as being mainly due to the different sensitivities of the two geophysical methods to different geological features and to the petrophysical variability of the basement in the area.

The greater depth extent of some of the lower magnetic susceptibility zones required by the magnetic data in the western part of the survey suggests that the depth to magnetic source is greater than suggested by the constraints. Adjustments to the constraints allowed a better data fit. A low magnetic response between 460 and 470 km east (Fig. 9) is assumed to be the consequence of low magnetic susceptibility contrasts and is interpreted to be granitic intrusions of the Williams Supersuite (Dhnaram and Greenwood, 2013). The presence of such intrusions offers a plausible explanation for the discrepancies between the magnetic and MT modelling. On the one hand, MT data modelling might not able to distinguish between an electrically resistive basement and an electrically resistive intrusion, while magnetic data modelling could not distinguish between the non-magnetic cover and a non-magnetic intrusion. On the other hand, magnetic data inversion can differentiate the low-susceptibility intrusion from the higher-susceptibility volcanic rocks, and the MT data are sensitive to the basal cover interface above both the volcanic rock and the intrusion. The constrained inversion permits detection of the lateral extent of the intrusion while estimating cover thickness. While detailed modelling of higher-resolution data would be required to refine the geometry of these intrusive bodies, our modelling suggests that the intrusion could be modelled as several smaller intrusions.

East of the Mount Margaret fault, the incompatibility between the original MT-derived constraints and the magnetic data points to regional-scale
structures. Drill hole observations indicating basement do not exceed 350

We have presented a workflow for sequential joint modelling of geophysical data and applied it to synthetic and field measurements. In this study we used constraints in the form of interface probabilities derived from a probabilistic workflow driven by MT data, but it is general in nature and is not limited to a particular geological or geophysical modelling method to generate the inputs. This has allowed us to report the utilisation of the ADMM algorithm to constrain magnetic data inversion using disjoint interval bound constraints for the first time.

This workflow presents several advantages. It is computationally inexpensive via the use of standalone inversions. The inversion of the MT dataset used to derive the constraints is performed only once. Following this, a series of constrained magnetic inversions is run to test different geophysical and petrophysical hypotheses. It shows the example of a fast and flexible approach to test different structural and petrophysical assumptions while modelling data sensitive to different physical parameters. It allows us to focus the modelling efforts on survey-specific features (anomalies, geological structures) when appropriate petrophysical information is available. However, as with generalisable methods, strengths become limitations under certain circumstances. For instance, in the case of MT and magnetic data inversions as proposed in this work, the electrical resistivity and magnetic susceptibility for the rock types of interest is dependent on a range of factors and processes (such as porosity, permeability, rock alteration) such that their correlation may be case dependent (see Dentith et al., 2020; Dentith and Mudge, 2014). While we may surmise that it remains reasonable to assume the existence of such correlation in hard rock scenarios, it may not always hold in basin environments. For example, one can easily think of a basin exploration case where electrical resistivity increases rapidly with increasing hydrocarbon concentration in reservoirs, while changes in magnetic susceptibility might make the use of magnetic data inversion redundant. In such cases, property pairings other than magnetic susceptibility and resistivity could be considered for variables such as electrical resistivity and seismic attributes (see examples in Le et al., 2016, and Tveit et al., 2020, who use seismic inversion to extract prior information for CSEM inversion). Further to this, the utilisation of magnetic data inversion for the deeper part of the crust is limited to depths shallower than the Curie point (typically from approximately 10 km to a few tens of kilometres under continents). For deeper imaging of the crust, the workflow we propose may be suited to the utilisation of gravity data with MT.

An assumption that is worth examining is whether the study area is adequately represented by two geological domains. In the cases we investigated, these domains are defined by the probability of observing only two rock classes (basement and non-basement). While this assumption reduces the risk of misinterpretation as no hypotheses are made to distinguish between different sedimentary units or rocks of different nature in the basement, it also then limits the interpretations that can be made from the results presented. We expect that if the rock units present discriminative features, i.e. distinctive magnetic susceptibility and resistivities (or other properties depending on the geophysical techniques considered), several rock types can be considered in the modelling. Such discriminative aspects of the petrophysics need to be ascertained while defining the number of distinctive domains that may be present in the study area. Ideally, robust petrophysical data are available given the strong constraint that these domains may impart on inversion. However, in the absence of petrophysical data or the number and character of geological domains, literature values or broad intervals can be used to define constraints. In these cases, the magnitude of the data misfit can inform us as to whether a proposed number of domains or magnetic susceptibility ranges are plausible, driving data acquisition or refinement of the conceptual geological model. Methods that exploit this approach remain to be investigated further in future case studies.

The application case is performed in 2D to illustrate the workflow. Extending the presented work to large-scale problems in 3D is straightforward as the inversion methods employed in this study are designed for 3D modelling. The 1D MT modelling and interpolation schemes present excellent scalability. The Tomofast-x engine (Giraud et al., 2021b; Ogarko et al., 2021b) is implemented using 3D grids. It presents good scalability and it offers the possibility to reduce the size of the computation domain to save memory when calculating the sensitivity matrix in the same fashion as Èuma et al. (2012) and Èuma and Zhdanov (2014) for large-scale potential field data modelling. Ongoing developments on Tomofast-x comprise the application of wavelet compression operators to accelerate the inversion in the same way as Li and Oldenburg (2003) and Martin et al. (2013) while maintaining a sufficiently low modelling error and developing joint inversions using the DIBC.

Another straightforward extension of the workflow is the use of gravity data simultaneously with (or instead of) magnetic data since it is already implemented in Tomofast-x (Giraud et al., 2021b). Giraud et al. (2021c) presented a synthetic MT-constrained gravity inversion using a similar workflow to the one presented here. This would be of particular interest in the Cloncurry region (Queensland, Australia), where, for instance, Moorkamp (2021) recently investigated the joint inversion of gravity and MT data, and where our workflow could be applied using the MT modelling results of Seillé et al. (2021).

From a geophysical point of view, magnetic inversion is affected by the non-uniqueness of the solution to the inverse potential field problem despite prior information and constraints being used. The workflow could be improved by using a series of models representative of the geological archetypes that can be derived from the ensembles of 1D MT models. Geological archetypes are distinctly different structural configurations (or topologies) that plausibly exist for a given location with available data (Pakyuz-Charrier et al., 2019, Wellmann and Caumon, 2018). Identification of the archetypes could be achieved from the ensemble of geological model realisations in the same spirit as Pakyuz-Charrier et al. (2019), who use a Monte Carlo approach to generate a range of topologies that are then examined for distinct clusters representing the archetypes.

From a methodological point of view, it could be argued that simultaneous joint geophysical inversion combining structural and petrophysical constraints might outperform the workflow we propose here. However, this would make the modelling process more demanding when combined with limitations based on cases where determining the causative relationships between petrophysics supporting joint approaches poses a challenge. The workflow we propose here presents a few advantages over a joint inversion scheme, in the sense that it does not require both datasets to be inverted simultaneously under a defined set of petrophysical and/or structural constraints. As the time required to run a joint inversion is limited by the running time of the more computationally expensive technique, it can limit the range of tests to be performed. In this study, we could rapidly run many 2D constrained magnetic inversions, even if the 1D probabilistic inversions of the MT data (and posterior fusion) required significantly longer running times compared to the 2D constrained magnetic inversion. This point would be particularly relevant in the case of large 3D datasets. This approach may represent a step in the modelling workflow that is useful to explore, understand and refine structural and petrophysical relationships between different physical parameters before undertaking more demanding joint inversions.

In the field application case presented here, the probabilistic depth to basement is derived assuming lateral continuity of the depth to basement estimates at a large scale, not accounting for small-scale lateral variations. Thus, uncertainty in the depth to the basement may be underestimated at some locations, in particular in between MT sites as shallow depths. In such cases, the existence of incompatibilities between MT-derived constraints and the magnetic data might require reconsidering the spatial continuity assumptions taken during the calculation of the probabilistic depth to basement surface. Extensions of this work may be devised to alleviate some of the limitations of the workflow. For instance, magnetic susceptibility from the inversion of magnetic data could be mapped back to a resistivity model to calculate forward MT data for validation (dashed line in Fig. 1) or to constrain the next cycle of MT inversions in the case the workflow is extended to cooperative joint inversion. It would also be straightforward to use to a level set inversion that can consider an arbitrary number of geological units (e.g. Giraud et al., 2021a) using MT modelling as a source of prior information and constraints. We have used hard bounds using the ADMM algorithm, which can easily be complemented or replaced by the use of multi-modal petrophysical distributions as available in Tomofast-x (e.g. mixture models as in Giraud et al., 2017, 2019b) as an alternative.

We have introduced, synthetically tested, and applied to field data a cooperative inversion scheme for the integration of MT and magnetic inversions. We have shown that despite its simplicity, the workflow we propose efficiently leverages the complementarities between the two methods and has the capability to improve our understanding of the cover–basement interface and of the basement itself. We have tested our workflow on a synthetic study that illustrates the flexibility of the method and the different possibilities our workflow offers, as well as its limitations. In the field application case (Cloncurry area, Queensland), we have shown how the quantitative integration of MT and magnetic data may bring insightful results on geological structural and petrophysical aspects, opening up new avenues for interpretations of the geology of the area and prompting future works.

The modified version of the structural model of Pakyuz-Charrier (2018) used here is given in Giraud and Seillé (2022). It also contains the synthetic MT and magnetic data used. The field data can be obtained from the Geological Survey of Queensland.

The version of the Tomofast-x inversion code used here was made publicly available by Ogarko et al. (2021b); the latest version is freely available at

JG and HS designed the study with input from the rest of the authors. JG and HS and generated the magnetic and MT synthetic data, respectively. JG performed the magnetic data inversions. JG redacted the manuscript with input from HS, who is the main contributor to MT-related aspects, and comments from the rest of the authors. HS performed the synthetic MT data modelling with support from GV. HS and GV derived the MT depth to basement results used in the case study. MDL and MWJ provided geological support and knowledge. MDL participated in both the geological and geophysical interpretation of the results and edited the relevant sections of the manuscript. MDL provided detailed comments during its redaction. GV participated in the design of the methodology and the interpretation of results. JG and HS prepared the datasets for public release.

VO and JG worked together on the development and testing of functionalities of Tomofast-x that are introduced and applied in the presented work. All authors provided feedback on the different versions of the manuscript and during the progress of the study.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Jérémie Giraud, Mark D. Lindsay, and Mark W. Jessell were supported, in part, by Loop – 25 Enabling Stochastic 3D Geological Modelling (LP170100985) and the Mineral Exploration Cooperative Research Centre, whose activities are funded by the Australian Government's Cooperative Research Centre Program. This is MinEx CRC Document 2021/36. Mark D. Lindsay was supported by ARC DECRA DE190100431. Hoël Seillé and Gerhard Visser were supported by the CSIRO Deep Earth Imaging Future Science Platform. We acknowledge the developers of the ModEM code for making it available. Jérémie Giraud has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 101032994.

This research has been supported by the Department of Industry, Science, Energy and Resources of the Australian Government (grant no. GA22270). Australia Research Council (grant DE190100431), and European Commission (grant no. 101032994).

This paper was edited by Charlotte Krawczyk and reviewed by Kristina Tietze and Max Moorkamp.

^{®}in Machine Learning, 3, 1–122,