The proposed method relies on the formulation of the model using an implicit model formulation in the form of signed distances to interfaces between rock units (right-hand side of Fig. 1). As proposed by Giraud et al. (2021a), this modelling approach considers rock units to be one or more rock types characterised by the same physical value (e.g. each unit is characterised by a single density value within the modelled area). Each rock unit is modelled by a unique signed-distance scalar field covering the study area. In a study considering nr rock units of known contrasting physical properties, we consider a set of nr signed-distance fields ϕ={ϕk, k=1, …, nr} over nm model cells corresponding to the distance to the boundaries of rock units. These signed distances are calculated using the fast-marching method of Sethian (1996) to maintain the following properties: 1ϕk>0insideunitk,=0attheboundary,<0outsideunitk. In our level set inversions, ϕ is the primary variable considered for inversion. It constitutes the proxy for a direct link that maps the geophysical and geological representations of the subsurface. We map these signed distances to petrophysical properties using 2mϕ1,…,ϕnr=∑i=1nrViHϕi∏j=1,j≠inr1-Hϕj, where V∈Rnr is a vector storing the physical property value assigned to each of the nr geological units (e.g. density contrasts for the different rock units in the case of gravity inversion). Similar to Giraud et al. (2021a), H is the smeared-out Heaviside function, which is calculated following Osher and Fedkiw (2003). This smearing is useful for the calculation of the sensitivity matrix of the calculated data to changes in ϕ from Eq. (2). When adapting it to our problem, H is defined for the kth rock unit in the ith model cell as 3Hϕk,i=0ifϕk,i<-τi,12+ϕk,i2τi+12πsin⁡πϕk,iτiif0≤ϕk,i≤τi,1ifϕk,i>τi, where τ={τi, i=1, …, nm} defines the volume of rock at which the boundary is allowed to vary between two successive iterations of the inversion. To the best of our knowledge, it is common to set all τi values to a constant, equal to 0.5×min(Δx,Δy,Δz) (Li et al., 2017), in a regular mesh of cells with a volume of Δx×Δy×Δz. In our implementation, we extend this to the possibility of using spatially varying boundary thicknesses by allowing the neighbourhood defined by τ to vary in space. This enables the anchoring of the model at observation points where τi= 0, e.g. at surface observations, in boreholes, and along seismic lines. In extreme scenarios, the volume occupied by boundaries with τi≫Δx,Δy,Δz of cells with a volume of Δx×Δy×Δz may cover extensive parts of the study area or, conversely, prevent the model from evolving when τ=0 everywhere.

At each iteration of the inversion, m is updated from the changes in ϕ, δϕ, which are required by the process of optimising the objective function (also called the cost function; see Sect. 2.2) within the domain defined by non-null values of τ.

We formulate the inverse problem in the least squares sense, taking gravity data inversion as an example. By adjusting the words of Giraud et al. (2021a), the choice of a least squares framework is motivated by the flexibility it allows in terms of the number of constraints and the forms of prior information that can be used in the inversion. Note that it corresponds to a maximum a posteriori estimator (Tarantola, 2005).

The objective function to minimise reads as follows: 4Ψϕ,dobs=dobs-dcalc22+λpWpϕ-ϕprior22, where dobs is the observed data, and dcalc is the gravity response of the density contrast model m(ϕ). We use dcalc=dcalc(m)=Smm(ϕ), where Sm is the sensitivity matrix of the gravity data d to changes in densities m(ϕ). The first term of Eq. (4) corresponds to a data misfit term, whereas the second term is a regularisation term that minimises deviations from the prior model. λp is a positive scalar weighting the regularisation term. Wp is an inverse diagonal variance matrix of dimensions (nmnr)×(nmnr), whose values can vary in space according to prior information to favour or discourage specific changes or features in the model. We note that the definition of Wp used here differs from Giraud et al. (2021a), where Wp is constituted of line vectors and has the dimensions nr×(nmnr). This allows for more flexibility when translating prior information into constraints on a cell-by-cell basis. ϕprior is the signed distances of a prior model. For simplicity, we do not include data measurement and modelling errors in the term dobs-dcalc22 of Eq. (4) but instead stop the inversion when the solution reaches a prescribed misfit level. Naturally, variable measurement errors could be integrated by replacing this term by the more general expression Cd-1(dobs-dcalc)22, where Cd would be the data covariance matrix.

We solve Eq. (4) iteratively and calculate the update of the signed distances δϕk that reduces Ψ(ϕk,dobs) at each iteration k. To solve for δϕk, we build the system of equations given in Appendix A, which requires the sensitivity matrix Sϕ,k of dcalc(m(ϕk)) with respect to changes in ϕk, using the chain rule: 5Sϕ,k=∂dcalcmϕk∂ϕk=∂d∂mϕk∂mϕk∂ϕk=Sm∂mϕk∂ϕk, where ∂m(ϕk)∂ϕk is obtained analytically from Eqs. (2) and (3) (see Giraud et al., 2021a, for details).

It can be shown that the objective function Ψ(ϕ,dobs) is equal to the log-posterior probability density distribution, as formulated in the Bayesian framework (Tarantola, 2005; see their Chapters 1 and 3 for more details). Here, the problem is therefore cast as a maximum a posteriori estimation. At the kth iteration, we calculate ϕk+1, such that ϕk+1=ϕk+δϕk and the updated model m(ϕk+1) is calculated consistently with Eq. (2) by selecting the rock unit with the largest signed-distance value at each model cell i=1, …, nm. 6mik+1=Vswheres=argmaxsϕs=1,…,nrk+1i We remind the reader that the vector Vs∈Rnr contains the physical property values assigned to the different geological units (see Eq. 2). In Eq. (6), the argmax function leads to the selection of the density contrast corresponding to the highest value of ϕ, which, intuitively, corresponds to the “innermost” rock unit. Following the same rationale as Zheglova et al. (2013), we then calculate the signed distances corresponding to the updated boundaries of mk+1 to maintain the signed-distance properties of ϕk+1, as introduced in Sect. 2.1. We note that at any given iteration, the search space is restricted to the vicinity of boundaries between rock units, as defined by the boundary's neighbourhood controlled by τ, which determines the current inversion's domain. This localisation dramatically reduces the volume of rock and the number of model cells considered for modification between two successive iterations to satisfy geophysical data fit requirements.

Using a prior structural geological model, the corresponding signed distances ϕprior to boundaries can be calculated. We remind the reader that in Eq. (4), the prior model constraints on signed distances are given as λpWp(ϕ-ϕprior)22.

This allows the inversion to explore the part of the model space remaining within the neighbourhood of ϕprior. The size of this neighbourhood globally depends on λp, which controls the relative importance assigned to the prior model term during inversion compared to the geophysical data misfit term. It is also locally determined by Wp, which tunes the importance of the prior model term for each model cell. Similar to other least squares inversions, λp can be set manually by trial and error, for example, by starting with a high value that will be reduced until model changes occur. Alternatively, the L-curve (Hansen and Johnston, 2001; Hansen and O'Leary, 1993) or general cross-validation principle (Farquharson and Oldenburg, 2004) may be used. In instances where a geological prior model is used, it is generally obtained before geophysical inversion and remains constant throughout inversion. In this contribution, our objective is to use, as a prior model, the result of a geological modelling process anchored only to the geological data and principles to better explore the admissible geological and geophysical parameter space.

In implicit modelling, geological structures (e.g. faults, foliations, intrusions, and stratigraphic horizons) are represented by iso values of one or several 3D scalar fields (see Wellmann and Caumon, 2018, for a review). For example, fault surfaces are generated as iso values of signed-distance functions (possibly restricted to a given region of space), and strata are generated as iso values of a relative geological time function. For each geological surface or series of surfaces, the 3D scalar field is obtained by least squares interpolation between the spatial measurement points. In this paper, we use LoopStructural, which is an open-source Python library for implicit 3D geological modelling (Grose et al., 2021). In LoopStructural, geological features are modelled backwards in time, starting with the most recent. Faults are modelled by first modelling the fault surface and fault displacement vector by building a structural frame consisting of three signed-distance fields representing the fault geometry and kinematics. The fault can then be applied to the faulted features by restoring the observations of the faulted surface prior to interpolating the faulted surface. This means that the kinematics of the fault are directly incorporated into the surface description.

LoopStructural uses a discrete implicit modelling approach, where the implicit function is approximated using a piecewise combination of basis functions on a predefined support, such as a linear tetrahedron on a tetrahedral mesh or a trilinear basis function on a Cartesian grid. Discrete implicit modelling forms an under-constrained system of equations because geological observations are sparse, and there are usually more degrees of freedom than geological constraints (location or orientation of geological features). To ensure the stability of the solution, a continuous regularisation term is added. Usual choices for regularisation constraints are some type of discrete smoothness constraint (Frank et al., 2007; Irakarama et al., 2021) or the minimisation of a continuous energy (Irakarama et al., 2022; Renaudeau et al., 2019). In this study, we use the finite difference regularisation, as implemented in LoopStructural, which minimises the second derivative of the scalar fields in all directions, using a finite difference scheme on a Cartesian grid (following Irakarama et al., 2021).

Geological observations such as the location of contacts, form lines, fault locations, and structural measurements can constrain the value and/or the gradient of the implicit function (Frank et al., 2007). Geological observations (further denoted as dgeol) are incorporated by finding the mesh element which contains the observation point and adding the linear constraint for the relevant degrees of freedom (nodes of the element). Orientation data can be used to constrain the gradient of the implicit function g as follows: 7∇g(x)=n, where n is the normal vector of the geological surface at the location x. As Eq. (7) constrains both the direction and magnitude of the gradient of the scalar field, an alternative formulation, used to impose only the orientation, is to find two tangent vectors, e.g. the strike vector vstrike and dip vector vdip, and to set the following: 8∇g(x)⋅vstrike=0,∇g(x)⋅vdip=0. Geological contacts or the location of geological features are integrated into the implicit modelling by setting the value of the implicit function as follows: 9g(x)=val, where val is the value of the implicit function given at the location x. The value should represent the distance to a reference horizon (for example, zero when the observation is located directly on the surface, such as a fault surface, being modelled) or the cumulative stratigraphic thickness to some reference horizon for different conforming stratigraphic interfaces. The implicit function is determined by solving the resulting regularised, overdetermined problem using least squares minimisation. For this reason, LoopStructural uses a conjugate gradient algorithm to iteratively find the solution of the system of equations.

In the next few subsections, we introduce how to recover input data for implicit geological modelling, as mentioned above, from models obtained through geophysical inversion. More specifically, we detail how to

extract the location of contacts between units from geophysical regions to constrain stratigraphic contacts; and

The overlap coefficient (OC; Szymkiewicz, 2017) is a similarity measure related to the Jaccard index (Jaccard, 1901) that measures the overlap between two sets. When applied to geological modelling, it is a measure of the dissimilarity between discrete representations of the subsurface. To compare the inverted rock type model minv and the reference model mref, OC can be written as 14OCminv,mref=cardmref⋂minvmincardmref,cardminv=1nm∑i=1nm1miref=miinv, where ⋂ denotes the intersection of values of sets; card is the cardinality operator, which returns the size of a given set; card(mref⋂minv) is number of model cells for which the rock types from mref and minv are the same, while min(card(mref),card(minv)) returns the size of the smallest set (here the number of model cells); and 1 is the indicator function, such that 1miref=miinv=1 if miref and miinv are equal, and 1miref=miinv = 0 otherwise. In this paper, mref and minv have the same discretisation. Therefore, OC represents the relative volume of rock assigned with the correct rock unit. Full dissimilarity is characterised by a value of OC equal to zero, and perfect similarity is characterised by a value of one.

In our analysis of synthetic cases, we assess the ability of the inversion to recover the reference density contrast model using the root mean square error ERRm as a measure of the difference between the reference and inverted models. It is calculated as 15ERRmmref,minv=1nm∑inmmiref-miinv2. It corresponds to the standard deviation of the misfit between the retrieved and reference models. It is routinely used to evaluate the capacity of inversion algorithms to recover the reference petrophysical model in synthetic studies.

We assess whether the estimated model adequately reflects the measured geophysical data and monitor the inversion's stability using the root mean square error ERRd as a measure of the geophysical data misfit. It corresponds to a normalisation of the data misfit term in Eq. (4). We calculate it as 16ERRdminv=1nd∑inddiobs-dicalc2. It is one of the metrics most commonly used to evaluate the capability of the inversion to reproduce field measurements.

Similar to the posterior analysis of Giraud et al. (2019), we analyse rock unit models recovered from inversion using adjacency matrices. Adjacency defines which rock bodies are in contact (Egenhofer, 1989) and is one of the simplest ways to assess a geological model from a quantitative point of view. For details, we refer the reader to Pellerin et al. (2015) and Thiele et al. (2016), who show its usefulness in the context of geological modelling. In this work, we simply use the number of grid faces located at the boundary between units with the indices i and j, respectively, as a coefficient for Ai,j of the corresponding nr×nr adjacency matrix A. From a more abstract standpoint, this representation amounts to a consideration of the geological model as a non-oriented graph (Godsil and Royle, 2001), where the nodes correspond to the rock units and the edges correspond to adjacency relationships. It can be calculated globally for a general overview (i.e. one adjacency matrix calculated for the full model) or locally for more detailed analysis (i.e. adjacency matrices calculated only at certain locations).

To quantify the difference between rock unit boundary locations in the reference and recovered models from an implicit modelling point of view, we propose a metric using signed distances to these interfaces. It is calculated as 17ERRϕϕref,ϕinv=1nr1nm∑inr∑jnmϕijref-ϕijinv2. Like the density contrast model misfit measures ERRm introduced above, the signed-distance misfit ERRϕ measures the discrepancy between two models. Here, it offers quantitative insight into the distance between the interfaces of two structural models with the same discretisation.

In this section, we investigate a more challenging geological setting and explore the possibility of using automatic geological modelling to define the prior model term of the inversion's objective function. We also test the possibility of combining it with the application of geological correction to the model update. Additionally, we examine the possibility of using geological correction a posteriori to ensure geological realism (to “geologify”) in an existing model presenting features that conflict with the geological principles and/or data.

In this section, we investigate the possibility of increasing the geological realism of a pre-existing model provided a priori from, e.g., an already performed inversion or classification of the inversion results. To simulate this, we first start from the model inverted with the level set method without geological correction, as obtained in Sect. 5.1 (Fig. 14b), and run an inversion with the geological correction applied. The inverted model obtained in this fashion is shown in Fig. 14a. An animation showing the evolution of the inverted model together with geological inconsistences is shown in the Supplement provided by Giraud and Caumon (2023). The application of geological correction manages to remove a number of unrealistic features present in the starting model. However, the effect of the geological correction seems visually smaller than in the application of geological correction from the onset of inversion (Fig. 14c). Based on this premise, if large unrealistic features appear, then we recommend using the geological correction from the beginning of inversion instead of as an ad hoc process. We note that in the transition between Fig. 14a and b by the application of geological correction, all models are geophysically equivalent or nearly equivalent in that they present similar geophysical data misfit values. This suggests that, for this dataset, a continuum of models exists that fit the geophysical data to a similar level, while presenting different degrees of geological realism. In what follows, we investigate this possibility further, using the synthetic dataset presented in Sect. 5.2.

The starting model we consider for an improvement of the geological realism (Fig. 15a) shows very convoluted geometry for the unconformity and significant thickness variations in the pre-erosional sequence, while presenting a geophysical data misfit value ERRd close to the objective data misfit value of 0.5 mGal (dashed line in Fig. 13a). In real-world studies, this could correspond to the results of the rock type classification obtained a posteriori from geophysical inversion only or a legacy inversion. We will further refer to this case scenario as case (5). As can be seen in Fig. 15a, the starting model corresponding to case (5) is in strong disagreement with the reference model (Fig. 8) from a structural geological point of view as shown, for instance, by the high starting OC value (Fig. 16b). In particular, the unconformity is poorly recovered, and the expected planar contact is significantly distorted. In contrast, the layered stratigraphy is well resolved.

We use this model as a starting model for the inversion that applies only geological correction (Sect. 3.3.1; Eq. 4; flow (1) in Fig. 2). We use this case to evaluate the capability of our method to restore geological consistency between inversion results and geological observation while maintaining a geophysical data fit within the prescribed levels. As in the previous synthetic example, we focus the analysis on the unconformity, since it is the main feature targeted by the inversion in this example.

A visual inspection of inverted models shown in Fig. 16b indicates that the application of geological correction effectively drives the optimisation process towards models that are in better agreement with the unconformity observed in the geological map. However, the thickness variations in the pre-erosional sequence are only partly resolved. Nonetheless, the resulting model is in a region of the model space that is considerably closer to geologically plausible scenarios than the starting model. This is illustrated by the metrics used to monitor the inversion, which show an overall increase in the OC and a decrease in the model misfit ERRm (Fig. 16b and c, respectively). This implies that using a geological correction term may reduce the risk of the geophysical inversion converging to a geologically unrealistic local minima of the cost function.

In terms of geophysical data misfit, the data error ERRd presents a plateau that decreases only after iteration 40 (Fig. 16a), indicating the gradual deformation of models with similar ERRd values. This shows that the inversion navigates a region of the model space comprising models that are equivalent in terms of geophysical data misfit but which are gradually more consistent with the available geological data. This implies that the proposed approach may be effectively used to navigate the space of geophysically equivalent models. We note that the exploration of geophysically equivalent models can be performed using null-space shuttles to modify an already existing model, while maintaining a nearly constant geophysical data misfit (Deal and Nolet, 1996; Muñoz and Rath, 2006; Fichtner and Zunino, 2019). In our case, it could be achieved by the gradual deformation of the starting model to accommodate geological information. Based on these premises, the use of geological correction, as performed in this section, may open up avenues to

derive models not proposed by implicit geological modelling or geophysical inversion alone but through combining them instead; and

In the last decade, the sampling of models from regions of the geological solution space to fit geological field observations has gained traction. The models proposed by these methods are usually derived from the interpolation of geological observations (Wellmann and Caumon, 2018). The inversion algorithm we proposed here can be used in conjunction with this strategy to modify such models using geophysical inversion. As geophysical inversion may be sensitive to features that geological modelling has little to no sensitivity to, geophysical inversion can be used to explore different regions of the solution space. This can be achieved with limited development efforts by using models that are representative of families sampled from the geological model space using, for example, topological analysis (Pakyuz-Charrier et al., 2019) or a similarity distance (Suzuki et al., 2008), to select starting models for inversion. Following the same idea, it may be possible to use deep learning for 3D geological structure inversion results (Jessell et al., 2022; Guo et al., 2021) as starting points to run series of inversions using the method presented here.

In some cases, geological modelling and geophysical inversion are difficult to reconcile. This may indicate that the modelling of geology is not sufficiently well informed by the available geological measurements and/or that hypotheses about the area need to be revisited. Under these circumstances, the geological model space might not contain a sufficiently good representation of the subsurface to fit the geophysical data. It could be the case, for instance, when structures are invisible to the available geological observations but can be sensed by geophysical data. In such cases, geophysical inversion, as performed here could be used as a tool to adjust these models and to infer the presence of unseen geological features.

As mentioned above, our inversion algorithm enables the estimation of the magnitude of adjustments required to reconcile geological models with geophysical measurements not only by comparing the forward response of the given models but also, and more importantly, by adjusting the model's geometry to fit the geophysical data. For scenario testing, the parameter controlling the amplitude of perturbation of interfaces between two successive iterations, τ, can be set accordingly with uncertainty information. For instance, it can be set arbitrarily small in locations of low uncertainty such as in the vicinity of boreholes, outcrops, or high-resolution seismic images. Following the same idea, τ could also be set using the uncertainty about domain boundaries derived using the implicit approach of Fouedjio et al. (2021). Further considerations of uncertainty may be required to better evaluate and understand the inversion results. For instance, the approach of Wei and Sun (2022), who generate a series of inverted models by varying their deterministic inversions' hyperparameters, could be a source of inspiration for uncertainty estimation. Likewise, the scalar field perturbation of Henrion et al. (2010), Clausolles et al. (2023), or Yang et al. (2019) could be transposed to the modelling approach we propose here.

We note that while experienced interpreters may be able to interpret geologically unrealistic inversion results “correctly”, it is nonetheless safer to ensure that geological principles are not violated by inversion. It removes an important source of uncertainty, and it reduces the impact of human bias and ambiguous interpretation, while ensuring that results can be robustly passed on to the next stage of modelling. More fundamentally, adding constraints essentially reduces the non-uniqueness of the inverse problem by making the search space smaller. In practice, however, more studies should be performed to assess whether the geological constraints change the “landscape” (or rugosity) or the objective function and to assess whether it always helps optimisation methods to avoid convergence to local minima.

Finally, the approach presented here belongs to a family of inversion approaches that could be referred to as “geometry driven”. That is, the main driver for fitting the geophysical data is the geometry of the subsurface model. One of our working assumptions is that the geological data used to derive structural geological models are complemented by geophysical data, with the possibility of altering the shape of geological models. Another geometry-driven approach consists of sampling points controlling the interpolated geological models within uncertainty and integrating their forward geophysical response to the calculation of a posterior distribution (Güdük et al., 2021; Liang et al., 2023). This method is very elegant, as it uses a unique geological level set parameterisation, but it does not explicitly address the integration of spatial geological data as provided by, e.g., drill holes. In contrast, our method can honour (up to discretisation errors) spatial data. The mapping of the geophysical signed distances ϕ onto their geological counterparts through fgeol makes it challenging to derive sensitivities directly like in Liang et al. (2023), but it probably makes it possible to explore a larger model space by possibly departing from geologically acceptable solutions during the non-linear iterations. Although further tests and comparisons between the two approaches are probably needed, this feature could be useful for preventing the optimisation from converging to local minima and could possibly be used in the future for stochastic inversions.

A potential limitation of the proposed approach that is inherent to the use of geological constraints pertains to the projection of an implicit rock model realisation onto the discrete mesh used to model geophysical data. Quoting Scalzo et al. (2022), “Naively exporting voxelised geology … can easily produce a poor approximation to the true geophysical [response]”. While it can be alleviated with the appropriate material averaging approach, aliasing can affect the geophysical response of the geological model projected onto the aforementioned mesh. This consideration is absent from our investigations, but it may be important to address in real-world case studies, where stakes are higher than in the work presented here.

A clear limitation of this work, which is intrinsic to the level set inversion, is the discreteness of the physical properties that are inverted for, which probably makes this kind of inversion one of the most parsimonious approaches. It presupposes accurate knowledge of the density of rock units and/or that they can be described by constant densities, neglecting geological and petrophysical phenomena leading to, for instance, lateral facies variations, compaction, and alteration. It is therefore possible for several rock types from the same stratigraphy to fall under the same “unit” in the modelling approach that we follow. Polynomials describing properties could be used instead of constant values. This limitation precludes the application of the method in certain scenarios. This may also call for integration with more advanced statistical rock physics and geostatistical modelling, using, for instance, the approach of Phelps (2016) to generate densities using a geostatistical approach. Additionally, porting the continuous value inversion component of the shape optimisation workflow of Dahlke et al. (2020) to 3D potential field inversion may help with generalising the potential use of level set inversion. Alternatively, the mapping from signed distances to a fixed-density value used here could be replaced by a mapping to an interval that defines the bound constraints enforced during inversion using the same approach as, e.g., Ogarko et al. (2021).

Finally, as mentioned in several places in the paper, further considerations on uncertainty are also required to better understand inversion results and find a set of admissible models, both geologically and geophysically.

We formulate and solve the inverse problem in the least squares sense, using Tikhonov regularisation to stabilise the inversion and to infuse geological information. The regularisation functional we employ addresses only the difference between the proposed model and a given model. It is straightforward to consider other regularisation terms such as a gradient or Laplacian minimisation to encode geological information. Another avenue could be to use the same parameterisation for the geophysical and geological modelling and to solve for all information simultaneously.

In addition, the flexibility offered by the least squares framework allows for the design of constraint terms specific to the use of signed distances to enforce physical principles. For instance, one can think of maintaining the sum of the updates of signed distances to zero in each model cell such that, for the ith model cell, ∑k=1Nδϕik=0. This would preserve the signed-distance properties of ϕ at each iteration, thereby eliminating the need to reinitialise the signed-distance fields at each iteration using the fast-marching method.

The available geological information for constraining a geophysical inversion can take many forms, ranging from sparse information about rock properties to dense borehole data and seismic interpretations (Grana et al., 2012). We have explored the possibility of constraining a geophysical inverse model using surface geological data and seismic sections. Borehole information can without a doubt be considered in the same manner, and there is no restriction on the use of other sources of information, such as the modelling of other geophysical techniques (e.g. depth-to-basement from electromagnetic methods and reflectors from passive seismics).

An obvious and straightforward extension of this work is to extend it to magnetic data, the inversion of which shares many features with gravity inversion.

Finally, extensions of our method may allow for null-space exploration and the mitigation of some of the limitations identified in the previous subsection. This paper investigates the importance of geological information in a level set inversion. Previous work focusing on level set inversion and following an approach similar to ours has investigated the importance of accurate knowledge on the geometry and the number of rock units a priori (Giraud et al., 2021a). Giraud et al. (2021a) and Rashidifard et al. (2021) suggest that inversion is somewhat robust to errors in the starting model geometry and in the petrophysics of the rock units. Nonetheless, relatively small deviations between cases (2) and (5) illustrate that the proposed methodology is not sufficient to address the ill-posedness of the potential field inverse problem. Moreover, the results from Giraud et al. (2021a) suggest that level set inversion “presents limitations when an important geologic unit is missing from the initial model”. To alleviate this, ways to generate the “birth” of new geological units in order for inversion to consider geological bodies previously not accounted for due to the lack of information may be devised. One possibility could be to use the sensitivity of geophysical data to changes in the physical property. This may be useful, for instance, for modelling intrusions invisible to surface geology. Further to this, when starting from a discrete model resulting from level set inversion, it is possible to explore regions of the model space with intra-rock unit variations (lateral facies variations, compaction, etc.) simply through using the null-space shuttles, as proposed by Deal and Nolet (1996) and Fichtner and Zunino (2019). We believe that these two possibilities are two sides of the same coin that constitute a promising area of research for future work on uncertainty quantification.