Non destructive investigation of soil properties is crucial when trying to identify inhomogeneities in the ground or the presence of conductive substances. This kind of survey can be addressed with the aid of electromagnetic induction measurements taken with a ground conductivity meter. In this paper, starting from electromagnetic data collected by this device, we reconstruct the electrical conductivity of the soil with respect to depth, with the aid of a regularized damped Gauss–Newton method. We propose an inversion method based on the low-rank approximation of the Jacobian of the function to be inverted, for which we develop exact analytical formulae. The algorithm chooses a relaxation parameter in order to ensure the positivity of the solution and implements various methods for the automatic estimation of the regularization parameter. This leads to a fast and reliable algorithm, which is tested on numerical experiments both on synthetic data sets and on field data. The results show that the algorithm produces reasonable solutions in the case of synthetic data sets, even in the presence of a noise level consistent with real applications, and yields results that are compatible with those obtained by electrical resistivity tomography in the case of field data.

Frequency-domain electromagnetic instruments allow the collection of data in different configurations, that is, varying the intercoil spacing, the frequency, and the height above the ground. Their handy size makes these tools very practical for near-surface characterization in many fields of applications, for example, precision agriculture, pollution assessments, and shallow geological investigations. To this end, the inversion of either the real (in-phase) or the imaginary (quadrature) component of the signal has already been studied. Furthermore, in many situations, a regularization scheme retrieving smooth solutions is blindly applied, without taking into account the prior available knowledge. The present work discusses an algorithm for the inversion of the complex signal in its entirety, as well as a regularization method that promotes the sparsity of the reconstructed electrical conductivity distribution. This regularization strategy incorporates a minimum gradient support stabilizer into a truncated generalized singular value decomposition scheme. The results of the implementation of this sparsity-enhancing regularization at each step of a damped Gauss-Newton inversion algorithm (based on a nonlinear forward model) are compared with the solutions obtained via a standard smooth stabilizer. An approach for estimating the depth of investigation, that is, the maximum depth that can be investigated by a chosen instrument configuration in a particular experimental setting, is also discussed. The effectiveness and limitations of the whole inversion algorithm are demonstrated on synthetic and real data sets.

Electromagnetic induction surveys are among the most popular techniques for non-destructive investigation of soil properties in order to detect the presence of either ground inhomogeneities or of particular substances. In this paper we develop a regularized algorithm for the inversion of a nonlinear mathematical model well established in applied geophysics, starting from noisy electromagnetic data collected by varying both the height of the measuring device with respect to the ground level and its operating frequency. Assuming the conductivity to be known in advance, we focus on the determination of the magnetic permeability of the soil with respect to depth, and give the analytical expression of the Jacobian matrix of the forward model, which is indispensable for the application of the inversion algorithm. Finally, numerical experiments on synthetic data sets illustrate the effectiveness of the method.

Abstract. This paper deals with the issue of monitoring the spatial distribution of bulk electrical conductivity, σ b , in the soil root zone by using electromagnetic induction (EMI) sensors under different water and salinity conditions. To deduce the actual distribution of depth-specific σ b from EMI apparent electrical conductivity (EC a ) measurements, we inverted the data by using a regularized 1-D inversion procedure designed to manage nonlinear multiple EMI-depth responses. The inversion technique is based on the coupling of the damped Gauss-Newton method with truncated generalized singular value decomposition (TGSVD). The illposedness of the EMI data inversion is addressed by using a sharp stabilizer term in the objective function. This specific stabilizer promotes the reconstruction of blocky targets, thereby contributing to enhance the spatial resolution of the EMI results in the presence of sharp boundaries (otherwise smeared out after the application of more standard Occamlike regularization strategies searching for smooth solutions). Time-domain reflectometry (TDR) data are used as groundtruth data for calibration of the inversion results. An experimental field was divided into four transects 30 m long and 2.8 m wide, cultivated with green bean, and irrigated with water at two different salinity levels and using two different irrigation volumes. Clearly, this induces different salinity and water contents within the soil profiles. For each transect, 26 regularly spaced monitoring soundings (1 m apart) were selected for the collection of (i) Geonics EM-38 and (ii) Tektronix reflectometer data. Despite the original discrepancies in the EMI and TDR data, we found a significant correlation of the means and standard deviations of the two data series; in particular, after a low-pass spatial filtering of the TDR data. Based on these findings, this paper introduces a novel methodology to calibrate EMI-based electrical conductivities via TDR direct measurements. This calibration strategy consists of a linear mapping of the original inversion results into a new conductivity spatial distribution with the coefficients of the transformation uniquely based on the statistics of the two original measurement datasets (EMI and TDR conductivities).

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