In this section, we describe the tools that are required to perform a joint inversion of direct current (DC) resistivity and small-loop electromagnetic (EM) data. In particular, these include (1) the two forward models used for describing the relation between electrical conductivity (EC) and DC resistivity and EM data and (2) the inversion method used, in this case the Kalman ensemble generator (KEG).

#### 2.1. Frequency-Domain Electromagnetics

Several different measurement configurations for EM data exist, and all of them are sensitive to the EC of the subsurface. The EM method that is discussed in this work is the small loop-loop frequency-domain electromagnetic (FDEM) induction method [

16]. Applying small-loop FDEM techniques, a primary electromagnetic field is generated by an alternating current of a single low frequency, in our case 9 kHz, in a transmitter coil that is approximated by a magnetic dipole [

17]. The resulting EM field propagation can be described by a diffusion equation. This field induces eddy currents in conductive material that is affected by the primary field. These eddy currents generate a secondary field. The secondary field, together with the primary field, is recorded at one or multiple receiver coils placed at a defined distance from the transmitter coil. Besides the transmitter-receiver offset, the measurement sensitivity depends largely on the respective coil geometries (see

Figure 1b). The strength of the secondary field can be related to the subsurface EC, the measurement quantity of interest. The secondary field is usually expressed as in-phase (IP) and quadrature-phase (QP) components in relation to the primary field [

16], below expressed in parts-per-million [ppm].

In this work, we simulate FDEM data using the code that was provided by Hanssens et al. (2019) [

18], applying the physical equations, as given by Ward and Hohmann (1988) [

17]. Their description, using a one-dimensional full solution of Maxwell’s equations, denotes a non-linear relation between the measurement and electrical conductivity of the subsurface volume affected by the induction phenomena.

It is well-known that small loop-loop FDEM field data are prone to severe systematic errors [

19]. Thus, for the field data application described at the end of our text, we modify the FDEM forward model by adding a simple offset term to the QP and IP responses [

20]. These offsets denote the possible systematic errors that are included in the IP and QP responses.

#### 2.3. Bayesian Inference and the Kalman Ensemble Generator

In Bayesian inversion approaches, all of the available information is translated into a probability distribution. Bayesian updates modify prior information by relevant measurement data producing posterior probability distributions for model parameters.

The prior information is given as a probability density function (PDF)

$\rho \left(\mathbf{m}\right)$ for the model parameters

$\mathbf{m}\in {\mathbb{R}}^{{n}_{par}}$. Inserting

$\mathbf{m}$ into the relevant geophysical forward model

g, and combining with the observations

$\mathbf{d}\in {\mathbb{R}}^{{n}_{obs}}$, we derive the likelihood function

$\rho \left(\mathbf{d}\right|\mathbf{m})$, the probability for a set of observations given the parameters. Here,

${n}_{par}$ and

${n}_{obs}$ denote the number of parameters and number of observations, respectively. Bayes’ theorem defines the posterior probability density function in terms of the PDFs discussed above:

where random variables left of a vertical bar are conditional to a set of values for the random variables on the right side of the bar and

$\kappa $ denotes a normalization factor.

In joint inversions,

$\mathbf{d}$ consists of data of two different types of geophysical measurements. Assuming that the measurement errors between the two data sets are uncorrelated, Blatter et al. [

23] point out that the likelihood in a joint Bayesian framework can simply be given as the product of the individual likelihoods:

with a joint data vector

for VES data

${\mathbf{d}}_{ves}$ and FDEM data

${\mathbf{d}}_{fdem}$.

Dealing with high-dimensional non-linear problems, no analytical solution is available to solve Equation (

1). Instead, the posterior needs to be numerically sampled. Most commonly, this is done using Markov chain-Monte Carlo (MCMC) methods applying a Metropolis- or Gibbs-sampler. Using an infinite number of samples, this approach converges to the exact solution of the Bayesian update problem [

24]. However, in practice, the computational resources are limited and the applicability of MCMC methods is therefore restricted. Computational restrictions are particularly pressing when expensive forward models are necessary and large data sets are used, which is usually the case in geophysical applications. Often, an MCMC sampler is no longer feasible for computing solutions to such problems as no convergence of the chain can be achieved within the number of affordable forward model runs.

Here, we will apply a specific Monte Carlo implementation of the Bayesian update problem: the ensemble Kalman filter (EnKf, [

15]). The EnKf was introduced as an efficient alternative to the original Kalman filter [

25] for solving large-dimensional and non-linear problems in data assimilation [

26]. In the EnKf context, the set of Monte Carlo samples is called an ensemble. The ensemble formulation makes the EnKF more efficient than the Kalman filter for two reasons: (1) in contrast to the the Kalman filter, the covariance equation can be replace by sample covariance and (2) Jacobian computations are unnecessary avoiding many calls of the forward model.

Instead of the Jacobian, the full forward response

g is computed for all ensemble realizations:

where

${n}_{ens}$ is the size of the ensemble,

${\u03f5}_{err}$ is the measurement error, and

${m}_{i}$ and

${d}_{i}$ are realisations of the Gaussian prior PDF and Gaussian data PDF, respectively. The ensemble is generated by drawing multivariate normal random numbers from the multivariate Gaussian prior using the Matlab function ‘mvnrnd’ [

27].

The EnKf update equation is given by [

15]:

for

$i\in \{1,\dots ,{n}_{ens}\}$. The covariances

${\mathbf{C}}_{mg}^{e}\in {\mathbb{R}}^{{n}_{par}\times {n}_{obs}}$, and

${\mathbf{C}}_{gg}^{e}\in {\mathbb{R}}^{{n}_{obs}\times {n}_{obs}}$ are derived from the prior ensemble given by matrix

$\mathbf{A}\in {\mathbb{R}}^{{n}_{par}\times {n}_{ens}}$ and the forward response ensemble given by matrix

$\mathbf{G}\in {\mathbb{R}}^{{n}_{obs}\times {n}_{ens}}$, where

${n}_{obs}$ is the number of observations and

${n}_{par}$ is the number of model parameters. Correcting the matrices

$\mathbf{A}$ and

$\mathbf{G}$ for their respective mean values, those are named

${\mathbf{A}}^{\prime}$ and

${\mathbf{G}}^{\prime}$, and can be used for covariance computation, as follows:

where

${\mathbf{A}}^{\prime}$ is derived from the matrix of ensemble means

where

${\mathbf{1}}_{{n}_{ens}}{\mathbb{R}}^{({n}_{ens}\times {n}_{ens})}$ corresponds to the matrix with each element equal to

$1/{n}_{ens}$ [

15]. The mean-corrected matrix is computed according to

Matrix ${\mathbf{G}}^{\prime}$ is computed analogously to matrix ${\mathbf{A}}^{\prime}$.

In the context of stationary inversion theory, the ensemble Kalman filter is called Kalman ensemble generator (KEG, [

14]). In contrast to the MCMC samplers, the KEG does not use a Markov chain to explore the model parameter space. Instead, the KEG requires that the Gaussian assumption is applicable to all PDFs [

28]. Likewise, the posterior PDF derived from the KEG update is considered a Gaussian; therefore, it can be understood as a Gaussian approximation to the true (non-Gaussian) posterior PDF.

In contrast, MCMC approaches converge to the exact posterior solution for near-exhaustive sampling. However, in MCMC computations, consecutive samples in the chain are correlated, causing lower efficiency than KEG for the following reasons. First, the early Monte Carlo samples might be biased and must be skipped; this corresponds to the so-called burn-in phase [

8]. Second, due to the requirement of independent random samples in statistical analysis, only every

n-th sample can be used to characterize the posterior PDF. Third, the acceptance rate of samples (using the mostly used Metropolis sampler) is between 30 and 50% [

1]. Finally, the chain computations cannot be parallelized. In contrast, the KEG is built on independent random sampling without the need of rejection sampling and, consequently, each sample can (1) be used for the characterization of the posterior PDF and (2) be run fully in parallel.

The KEG was only recently adopted for the inversion of geophysical data. Bobe et al. [

12] applied it to the inversion of frequency-domain electromagnetic data to the probabilistic estimation of electrical conductivity and magnetic susceptibility. As for other probabilistic methods, a joint framework is readily available while using the KEG. In such frameworks, no weighing of different data types is necessary as long as significant errors in the parameter estimation are limited to Gaussian noise. Instead, the covariances available from Monte Carlo sampling determining the relative weight of each measurement. This holds for all types of measurement, irrelevant of the fact of whether the data are EM or DC resistivity data.

The likelihood is generated by passing on all prior realizations in

$\mathbf{A}$ to both forward models

${g}_{ves}$ and

${g}_{fdem}$. Thus, for the same EC realizations of the subsurface, signals for both measurement techniques are computed. Those responses are summarized in a forward response ensemble matrix

An efficient implementation of the KEG is given by [

14]:

This matrix equation summarizes the

${n}_{ens}$ computations described in Equation (

5), where matrix

$\mathbf{D}\in {\mathbb{R}}^{{n}_{obs}\times {n}_{ens}}$ is the ensemble data matrix, with uncorrelated Gaussian noise, defining a Gaussian PDF that ultimately defines the overall diagonal data covariance matrix

${\mathbf{C}}_{D}$. The mean corrected version of matrix

$\mathbf{D}$ is named

$\mathbf{E}$.

Matrix

${\mathbf{A}}^{Update}$ is characterized by its mean value, being considered as the best fit to the measurement data, and a standard deviation (STD) as corresponding uncertainty around the best fit. For a more detailed description of this inversion approach applied in a geophysical context see [

12].

Restricting the posterior estimate to its Gaussian approximation has computational advantages over MCMC processing. Gaussian PDFs can be sampled directly and the samples are truly random, such that (1) no model rejection or burn-in phase is necessary [

24] and (2) forward computations can be performed in parallel.