Forward EMI response for a given layered earth model is usually calculated by the model, which is generated using the cumulative electrical conductivity distribution over a certain depth range, and is valid under condition of low induction number. The alternative method used to calculate the forward EMI response is to solve the Maxwell equation for the magnetic field measured over a horizontal layered medium . Preliminary analysis indicated that the electromagnetic forward model, which is based on high induction number assumption, returned more reliable apparent electrical conductivity values than the standard sensitivity curves of . Furthermore, increased computational power made it possible to characterize the subsurface by utilizing forward models based on the Maxwell equation . The effective depth of exploration is independent of ECa in a low induction number condition, whereas in high induction number condition an inverse relationship was found between the depth of exploration and ECa . For a combination of a vertical and horizontal dipole source–receiver with an offset ρ over a multilayered earth, the electromagnetic forward model can be written as ECaHCP(x,ρ)=-4ρωμ0Im∫0∞R0J0(ρλ)λ2dλ, ECaVCP(x,ρ)=-4ωμ0Im∫0∞R0J1(ρλ)λdλ.

In these expressions, ECaVCP and ECaHCP represent apparent electrical conductivity – measured in vertical and horizontal coplanar mode, μ0 represents permeability of the free space, λ indicates the radial wave number, J0 and J1 correspond to the zero-order and first-order Bessel functions, x is the depth of layer, ω is the angular frequency, and Im the quadrature component. The reflection factor R0 is obtained recursively, starting from the lowest layer N+1, with RN+1=0: Rn(hn,σn)=Γn-Γn+1Γn+Γn+1+Rn+1exp⁡(-2Γn+1hn+1)1+Γn-Γn+1Γn+Γn+1Rn+1exp⁡(-2Γn+1hn+1),Γn=λ2+ωμ0jσn, where σ0=0, hn is the height, and σn is the electrical conductivity for the nth layer. This is based on the assumption that each layer is uniform with infinite horizontal extent. EMI measurements were carried out under high induction number conditions (ECa> 100 mS m-1) utilizing the full solution of Maxwell's equation to model the forward EMI response.

Bayesian inference is used to express the uncertainties in the system parameters based on a suitable likelihood function and a prior. Given a set of unknown parameters, the so-called posterior distribution of the model parameters, which is the distribution of the parameters conditioned on available observations, is calculated as the product of the prior distribution and the likelihood function . Bayesian inversion has gained a lot of interest in recent years and has been applied in different applications, including climate, ocean and geophysical modeling .

Suppose a set of observations ({yi}i=1n) is available and assume a certain model to predict the data. Let α be the set of unknown parameters in the model; then according to Bayes' rule, p(α|{yi}i=1n)∝p({yi}i=1n|α)p(α), where p(α) is the prior distribution of α that represents the a priori knowledge about α (i.e., before considering the data). The p({yi}i=1n|α) denotes the likelihood function: the probability of predicting the data given α. The p(α|{yi}i=1n) is the posterior probability: the probability of recovering α given the data ({yi}i=1n).

Let us consider the forward model M, for the evaluation of the observations y as a function of the parameters such that y=M(α). Let ϵ be a random variable representing the discrepancy between our model M(α) and the observations, which we refer to as the observational noise: ϵ=y-M(α). Assuming the components of the observational noise to be independent and Gaussian of mean zero and variance σ2, the likelihood function can then be decomposed as p({yi}i=1n|α)=∏i=1n12πσ2exp⁡-(yi-Mi(α))22σ2. Here we consider σ2 as an additional unknown (hyper) parameter and try to estimate its distribution as part of the inference process. The (joint) posterior distribution is then expressed as p(α,σ2|{yi}i=1n)∝∏i=1n12πσ2exp⁡-(yi-Mi(α))22σ2p(α)p(σ2).

The choice of the prior is a key step in the inference process. Here, an informative uniform prior for all five (three conductivities and two thickness) parameters is considered, with αk in the range [αkmaxαkmin]; i.e., p(αk)=1αkmax-αkminforαkmin<αk≤αkmax,0otherwise.

For the noise variance σ2, we consider a Jeffreys prior : p(αk)=1σ2forσ2>0,0otherwise.

The most commonly used computational strategy to numerically solve a multidimensional parameters Bayesian inference problem is the Markov chain Monte Carlo (MCMC) method. We have applied an adaptive Metropolis MCMC algorithm to sample the posterior distribution, as described in details in and

Two sets of experimental setups were considered to test the MCMC approach and to evaluate the estimated model parameters and associated uncertainties using synthetic data for CMD Mini-Explorer configurations. Figure a and b show a three-layer earth model setups of low and high conductivity for non-saline soil and saline soil salinity, respectively. In both setups, thicknesses for the three-layer earth model were conceptualized by a plow horizon (0.25 m thick), with an intermediate subsoil layer (0.50 m thick) and underlying consolidated layer up to 1.5 m depth. The plowing horizon generally has less soil moisture as compared to the deeper horizon because of evaporation and infiltration processes. The scenario of non-saline soil therefore used a plowing horizon with low electrical conductivity of 15 mS m-1 as compared to the intermediate and consolidated soil layers (Fig. a). In the saline soil scenario, salt accumulates on the surface of soil due to evaporation of water. As a result, the electrical conductivity of plowing horizon is considered higher 1800 mS m-1 as compared to the deeper layers (Fig. b). In the agricultural field, the increase in the soil salinity is generally due to the use of poor water quality or the excessive use of fertilizers. The forward response of both scenarios was calculated in HCP and VCP via Eqs. () and (), respectively, for EMI configuration setups using the characteristics of CMD Mini-Explorer of three receiver coils respectively placed at 0.32, 0.71 and 1.18 m distance from the receiver.

In both scenarios, six configurations, three for each HCP and VCP with different spacings, were taken as an output for forward models. Let α=(σ1,σ2,σ3,h1,h2)T be a vector of model parameters. σ1, σ2, and σ3 are layer conductivities, and h1 and h2 thicknesses. Bayesian inference was used to estimate these five parameters that minimize the errors between observed and modeled HCP and VCP. An adaptive MCMC method was used to sample the posterior distributions and consequently update α distributions according to the observed data. All the results presented below are based on 104 MCMC samples. Parameter range for h1 and h2 was fixed between 0.05 and 0.6 m in each scenario. In the non-saline scenario, parameter range for σ1, σ2, and σ3 was considered between 5 and 100 mS m-1 and the saline soil scenario range was fixed between 5 and 3000 mS m-1. A uniform prior distribution function was considered in both scenarios.

Field measurements were also carried out in a farm, where Acacia trees were irrigated with saline groundwater. The farm is located at a distance of 6 km from the Red Sea coast at Al-Qadeimah, Makkah province, Saudi Arabia. EMI measurements were collected at an interval of 2 m over a 40 m long transect, along which three Acacia trees were irrigated using drip irrigation. At each location, EMI measurements using CMD Mini-Explorer system gives six different values of apparent electrical conductivity (using two coil orientations and three offsets); each responds to different depth ranges. Ten pits were dug along the same transect and in each pit the bulk electrical conductivity σb profile was measured at 15 locations within a depth range of 0.05–1.5 m via 5TE capacitance sensors (Decagon Devices, Pullman, USA). EMI and 5TE measurements were performed 8 h after the drip irrigation system was stopped, so that the soil moisture is not concentrated below the drippers and to give enough time to reduce the soil moisture impact due to evaporation, root water uptake, and infiltration .

Figure a and b depict the observed, estimated (modeled) and range of ECa as they result from the chain of MCMC simulation for six configurations of the synthetic case with saline and non-saline soil. The x axis represents VCP and HCP with three coil spacing (ρ32, ρ71, ρ118). In a non-saline scenario, the layer electrical conductivity increases with depth (Fig. a), and this is reflected in the observed and modeled ECa in the VCP and HCP with increasing trend for larger spacing (Fig. a). The ECa value for the VCP and HCP with maximum spacing of 1.8 m between transmitter and receiver corresponds to deeper horizon; in the case of saline soil scenario the layer conductivity decreases (Fig. b) and as a result ECa values in VCP and HCP configuration exhibit a decreasing trend (Fig. b). The electromagnetic forward model is sensitive to high electrical conductive soil, so the modeled ECa values for the saline soil scenario match well with the observed as compared to the non-saline scenario. The mismatch between the observed and modeled ECa values for non-saline soil is due to the weak sensitivity of the forward electromagnetic model to the low electrical conductivity.

Figure a shows the true parameter values (red line) with the estimated parameters using MCMC (blue dashed line) for the non-saline soil scenario. The MCMC samples were used to obtain the marginalized posterior distributions based on kernel density estimation (KDE) . The 95 % confidence interval of the KDE for each parameter is shown by the shaded gray background (Fig. a). The resulting marginalized posterior probability density functions (PDFs) of the three conductivities and two thicknesses are shown in Fig. b–f. The estimated parameters (Fig. b–f) show a single peak, corresponding to the best parameter values. The electrical conductivities of the three model layers (σ1, σ2, and σ3) are reasonably well estimated as compared to the layer thicknesses. Different uniform prior distributions were also tested for the layer thicknesses, but the MCMC solution converged close to the prior instead of the true layer thicknesses. The topography of the objective function was too flat in this case to allow consequent changes in the direction of layer thicknesses. This suggests that the electromagnetic model is not sensitive to the layer thicknesses for the low-conductive soil layer.

Figure  illustrates the true and estimated depth profile of electrical conductivity for saline scenario, and the KDE of the marginalized posterior distributions for the three layer conductivities (σ1, σ2, and σ3) and the two layer thicknesses (h1 and h2). The shaded gray background shows the 95 % of the KDE for each parameter (Fig. a). The vertical electrical conductivity profile is well recovered by MCMC. The electrical conductivity of the top two layers are well estimated as compared to the consolidated layer with low electrical conductivity. Furthermore, for the six tested configurations of CMD Mini-Explorer, the HCP and VCP configuration with spacing 1.18 m are mostly sensitive to the consolidated layer while the remaining four configurations are more sensitive to the upper horizon. A large range of the parameter space was explored by MCMC (Fig. b–e), illustrating the sensitivity of the electromagnetic model to the considered parameters.

Measurements were carried out in a farm, where acacia trees were irrigated with saline groundwater. The farm is located at a distance of 6 km from the Red Sea coast at Al-Qadeimah, Makkah province, Saudi Arabia. EMI measurements were collected at an intervals over a 40 m long transect, along which three acacia trees were irrigated using drip irrigation. At each location, EMI measurements using the CMD Mini-Explorer system provides six different values of apparent electrical conductivity (using two coil orientations and three offsets); each responds to different depth ranges. Ten pits were dug along the same transect and in each pit the vertical σb profile was measured at 15 locations within a depth range of 0.05–1.5 m via 5TE capacitance sensors (Decagon Devices, Pullman, USA). 5TE and EMI measurements were carried out on the same day 8 h after the drip irrigation system was stopped, so that the soil moisture concentration below the drippers is avoided, and enough time is given for the reduction of soil moisture impact due to root water uptake, evaporation and infiltration .

Figure  shows the soil electrical conductivity measured in ten pits along a transect and the modeled soil electrical conductivity as estimated by the MCMC using the multi-configuration EM induction measurements. Pit locations along the transect are indicated by black triangle and cubic interpolation of 150 5TE sensor measurements were used to construct the two-dimensional profile of measured soil electrical conductivity σ (Fig. a). The groundwater used to irrigate the acacia trees has an electrical conductivity of 4200 mS m-1. The three patterns of high electrical conductivity is due to the infiltration front and soil salinity near the three acacia trees. In total, 21 multi-configuration EMI measurements were performed along a transect and calibrated with in situ measurements collected using capacitance sensors . The three-layer earth model was considered for Bayesian inference of the five parameters (σ1,σ2,σ3,h1,h2) and their uncertainty based on the 15 000 MCMC samples. For all MCMC simulations, the parameter search space was set relatively large, with the range of low and high values of electrical conductivity of soil; 0<σ1<3000 mS m-1, 0<σ2<3000 mS m-1, 0<σ3<3000 mS m-1, 0.05<h1<0.6 m, and 0.05<h1<0.6 m. In the depth section of soil electrical conductivity resulting from the EMI MCMC simulations, the effect of infiltration patterns and the soil salinity due to the drip irrigation near the three acacia trees is clear (Fig. b). The estimated soil electrical conductivity values by MCMC are in a good agreement with the sensor measurements performed in pits (Fig. a).

Figure a and b show the measured, estimated (modeled) and range of ECa as they result from the MCMC chain for the six multi-configurations of CMD Mini-Explorer for saline and non-saline soil. Three coil spacings for each VCP and HCP are represented on the x axis. EMI measurement is shown for non-saline and saline soil at locations 4 and 9 of the pit (Fig. a), respectively. The soil was completely dry for non-saline soil as no irrigation was applied, whereas in the case of saline soil the moisture in the soil varied between 0.005 and 0.19 at the time of EMI and sensor measurements. In non-saline soil, the measured six ECa values range between 5 and 60 mS m-1 and the modeled ECa between 23 and 38 mS m-1 (Fig. a). The range of ECa estimated from the last 10 000 MCMC samples is in the range of 0–75 mS m-1. As observed in the synthetic non-saline soil scenario, the electromagnetic forward model was not sensitive to the low electrical conductive soil. Similarly, the fit between the measured and modeled ECa is not in good agreement with the real measurements (Fig. a). Furthermore, the misfit may be due to the large search parameter space in the MCMC simulations. In the case of saline soil, the electrical conductivity of the top 50 cm soil is high due to the saline infiltration and soil salinity. This effect can be seen in the decreasing trend of the measured ECa for the VCP and HCP measurements with larger coil spacing (Fig. b). The measured and modeled ECa are in good agrement and this is due to the high sensitivity of the electromagnetic forward model to high electrical conductive soil.

Figure  plots the vertical profile of electrical conductivity for non-saline soil as measured by capacitance sensors (red line), the value of the MCMC estimated parameters (blue dashed line), and the KDE of the marginalized posterior distributions for the three layer conductivities and the two layer thicknesses. The CMD Mini-Explorer measurements at pit 4 for non-saline soil were used for the analysis. In Fig. a the measured vertical profile of soil electrical conductivity falls within the shaded area in the top 95 % KDE distribution limit 0–0.7 m depth and below this depth the modeled soil electrical conductivity is overestimated. The mismatch between the measured and modeled ECa for the maximum coil separation Hρ118 and Vρ118 is behind the overestimation of the soil electrical conductivity. The marginalized posterior PDFs of the three conductivities and two thicknesses are shown in Fig. b–f. The PDFs of the parameters (Fig. b–f) exhibit a single peak, corresponding to the best parameters. The peak of the σ3 is flat between 30 and 38 mS m-1 and it seems that the topography of the objective function does not change within this range of conductivity in each iteration of the MCMC simulation.

Figure  plots the vertical profile of electrical conductivity for the saline soil measured by capacitance sensors (red line), the value of the MCMC estimated parameters (blue dashed line), and the KDE of the marginalized posterior distributions for the three layer conductivities and the two layer thicknesses. CMD Mini-Explorer measurements at pit 9 for saline soil was used for the analysis. The shaded area in Fig. a indicates the 95 % KDE distribution limits. The whole measured vertical profile of soil electrical conductivity falls within the shaded area, suggesting that the electrical conductivity is well estimated. The marginalized posterior PDFs of the three conductivities and two thicknesses, as shown in Fig. b–f, exhibit a single peak for all parameters except the layer thickness h2 which is flat, suggesting that the data were not informative to refine our prior knowledge about h2. The posterior PDFs of the first two conductivities (σ1 and σ2) and layer thickness h1 exhibit a clear Gaussian shape with an obvious maximum a posteriori (MAP) estimate. For the conductivity parameter σ3, we notice a posterior with a well-defined peak, but no standard PDF shape.

Figure  shows the spatial distribution of the soil salinity as estimated from EMI measurement using MCMC. Soil salinity ECe is related to bulk electrical conductivity σb via a linear relationship (ECe=13.74σb+0.001) established by for the same site. Infiltration front and high soil salinity range between 0.01 and 0.5 m at three locations where acacia trees are irrigated with brackish water. The results show that the Bayesian inversion of multi-configuration EMI measurements successfully estimates the soil salinity caused by the brackish water infiltration. In the field, acacia tree roots concentrated in the top 70 cm of soil and the low soil salinity below 30 cm shows that acacia are capable of extracting salt solutions and reducing subsoil salinity.