Maximilian Weigand’s research while affiliated with University of Bonn and other places

In this work, we introduce the probabilistic inversion of tomographic complex resistivity (CR) measurements using the Hamiltonian Monte Carlo (HMC) method. The posterior model distribution on which our approach operates accounts for the underlying complex-valued nature of the CR imaging problem accurately by including the individual errors of the measured impedance magnitude and phase, allowing for the application of independent regularization on the inferred subsurface conductivity magnitude and phase, and incorporating the effects of cross-sensitivities. As the tomographic CR inverse problem is non-linear, of high dimension and features strong correlations between model parameters, efficiently sampling from the posterior model distribution is challenging. To meet this challenge we use HMC, a Markov-chain Monte Carlo method that incorporates gradient information to achieve efficient model updates. To maximize the benefit of a given number of forward calculations, we use the No-U-Turn sampler (NUTS) as a variant of HMC. We demonstrate the probabilistic inversion approach on a synthetic CR tomography measurement. The NUTS succeeds in creating a sample of the posterior model distribution that provides us with the ability to analyze correlations between model parameters and to calculate statistical estimators of interest, such as the mean model and the covariance matrix. Our results provide a strong basis for the characterization of the posterior model distribution and uncertainty quantification in the context of the tomographic CR inverse problem.

Background: Root systems are key contributors to plant health, resilience, and, ultimately, yield of agricultural crops. To optimize plant performance, phenotyping trials are conducted to breed plants with diverse root traits. However, traditional analysis methods are often labour-intensive and invasive to the root system, therefore limiting high-throughput phenotyping. Spectral electrical impedance tomography (sEIT) could help as a non-invasive and cost-efficient alternative to optical root analysis, potentially providing 2D or 3D spatio-temporal information on root development and activity. Although impedance measurements have been shown to be sensitive to root biomass, nutrient status, and diurnal activity, only few attempts have been made to employ tomographic algorithms to recover spatially resolved information on root systems. In this study, we aim to establish relationships between tomographic electrical polarization signatures and root traits of different fine root systems (maize, pinto bean, black bean, and soy bean) under hydroponic conditions. Results: Our results show that, with the use of an optimized data acquisition scheme, sEIT is capable of providing spatially resolved information on root biomass and root surface area for all investigated root systems. We found strong correlations between the total polarization strength and the root biomass ( R 2 = 0.82 ) and root surface area ( R 2 = 0.8 ). Our findings suggest that the captured polarization signature is dominated by cell-scale polarization processes. Additionally, we demonstrate that the resolution characteristics of the measurement scheme can have a significant impact on the tomographic reconstruction of root traits. Conclusion: Our findings showcase that sEIT is a promising tool for the tomographic reconstruction of root traits in high-throughput root phenotyping trials and should be evaluated as a substitute for traditional, often time-consuming, root characterization methods.

We introduce a novel probabilistic framework for the solution of non-linear geophysical inverse problems in complex variables. By using complex probability distributions, this approach can simultaneously account for individual errors of real and imaginary data parts, independently regularize real and imaginary parts of the complex model, and still take into account cross-sensitivities resulting from a complex forward calculation. The inverse problem is solved by means of optimization. An application of the framework to complex resistivity (CR) imaging demonstrates its advantages over the established inversion approach for CR measurements. We show that CR data, with real and imaginary parts being subject to different errors, can be fitted adequately, accounting for the individual errors and applying independent regularization to the real and imaginary part of the subsurface conductivity. The probabilistic framework itself serves as a basis for the future application of global sampling approaches, such as Markov chain Monte Carlo methods.

Electrical impedance tomography (EIT) is a promising method to image the frequency-dependent complex electrical conductivity distribution of the subsurface in the mHz to kHz frequency range. In contrast to the well-developed electrical resistivity tomography (ERT) method, the inversion approach for EIT data is less established. Different inversion strategies have been proposed, but the implications of the differences between these methods have not been investigated yet. In this study, we aim to compare four different inversion strategies for EIT measurements. The first strategy (CVI) formulates the inverse problem in the complex number domain and is mathematically the most elegant method. The second strategy (RVI) is the established real-valued inversion method, which decouples the inversion of the real and imaginary parts and completely ignores the complex nature. The third strategy (ALT) is very similar to the RVI strategy in case of small phase angles, but it considers the complex coupling in the forward operator and alternately updates the real and imaginary parts of the model in the case of large phase angles. The fourth and final strategy (CVI+) was newly formulated in this study. It fully considers the complex nature of EIT measurements but separates the treatment of the real and imaginary part in terms of the data weighting and regularization. The different inversion strategies were tested with two synthetic models. The first model has a small phase contrast and the second model has a large phase contrast. In the case of a small phase contrast, the CVI strategy was able to resolve the distribution of electrical conductivity amplitude, but the inversion result for the phase angle was less reliable. The other three strategies presented similar results and the models were well resolved within the expected data misfit. In the case of a model with large phase contrast, only the newly formulated CVI + strategy was able to produce reliable results. It was found that the extremely large phase angle can have a significant influence on the modelled amplitude of data. The cross-sensitivity (i.e. the imaginary part of the sensitivity) that describes the influence on the real part of data due to a change in the imaginary part of model, or that on the imaginary part of data due to a change in the real part of model, provided unique information during the inversion. It was concluded that the CVI + strategy is theoretically the most comprehensive and correct approach for EIT inversion, but that in the case of small phase angles the RVI strategy has the practical advantage that no complex calculations are required, which substantially reduces the required computational effort.

Spectral electrical impedance tomography (sEIT) is increasingly used to characterise the structure of subsurface systems using measurements in the megahertz to kilohertz range. Additionally, hydrogeophysical and biogeophysical processes are characterised and monitored using sEIT. The method combines multiple, spatially distributed, spectroscopic measurements with tomographic inversion algorithms to obtain images of the complex electrical resistivity distribution in the subsurface at various frequencies. Spectral polarisation measurements provide additional information about the systems under investigation and can be used to reduce ambiguities that occur if only the in-phase resistivity values are analysed. However, spectral impedance measurements are very sensitive to details of the measurement setup as well as to external noise and error components. Despite promising technical progress in improving measurement quality as well as progress in the characterisation and understanding of static polarisation signatures of the subsurface, long-term (i.e. multi-month to multi-year) monitoring attempts with fixed setups are still rare. Yet, measurement targets often show inherent non-stationarity that would require monitoring for a proper system characterisation. With the aim of improving operating foundations for similar endeavours, we here report on the design and field deployment of a permanently installed monitoring system for sEIT data. The specific aim of this monitoring installation is the characterisation of crop root evolution over a full growing season, requiring multiple measurements per day over multiple months to capture relevant system dynamics. In this contribution, we discuss the general layout and design of the monitoring setup, including the data acquisition system, additional on-site equipment, required corrections to improve data quality for high frequencies, data management and remote-processing facilities used to analyse the measured data. The choice and installation of electrodes, cables and measurement configurations are discussed and quality parameters are used for the continuous assessment of system functioning and data quality. Exemplary analysis results of the first season of operation highlight the importance of continuous quality control. It is also found that proper cable elevation decreased capacitive leakage currents and in combination with the correction of inductive effects led to consistent tomographic results up to 1 kHz measurement frequency. Overall, the successful operation of an sEIT monitoring system over multiple months with multiple daily tomographic measurements was achieved.

Spectral electrical impedance tomography (sEIT) is increasingly used to characterize the structure of subsurface systems. Additionally, petrophysical and biogeophysical processes are characterized and monitored using sEIT. The method combines multiple, spatially distributed, spectroscopic measurements with tomographic inversion algorithms to obtain images of the complex electrical resistivity distribution in the subsurface at various frequencies. Spectral data, as well as polarization measurements provide additional information about the systems under investigation, and can be used to reduce ambiguities that occur if only the in-phase resistivity values are analysed. However, spectral impedance measurements are very sensitive to details of the measurement setup, as well as external noise and error components. Despite promising technical progress in improving measurement quality, as well as progress in the static characterisation and understanding of electrical polarisation signatures of the subsurface, long-term monitoring attempts are still rare. Yet, measurement targets often show inherent non-stationarity that would require such approaches for a proper system characterisation. With the aim of improving operating foundations for similar endeavours, we here report on the design and field deployment of a permanently installed monitoring system for sEIT data. The specific aim of this monitoring installation is the characterisation of crop root evolution over a full growing season, requiring multiple measurements per day over multiple months to capture relevant system dynamics. In this contribution, we discuss the general layout and design of the monitoring system, including the core measurement system, additional on-site equipment, required corrections to improve data quality for high frequencies, data management, and remote processing facilities used to analyse the generated data. The choice and installation of electrodes, cables, and measurement configurations are discussed, as well as quality parameters used for the continuous assessment of system functioning and data quality. Exemplary analysis results of the first season of operation highlight the importance of continuous quality control. It is also found that proper cable elevation decreased capacitive leakage currents and in combination with the correction of inductive effects lead to consistent tomographic results up to 1 kHz measurement frequency. Overall, the successful operation of an sEIT monitoring system over multiple months with multiple daily tomographic measurements was achieved.

Electrical impedance spectroscopy has long been considered a promising technique for noninvasive, in‐situ root investigation because of its sensitivity to anatomy and physiology. However, the complexity of the root system and its coupling with stem and soil have hindered the signal interpretation and methodological upscaling to field applications. This study addresses these key issues by introducing three‐channel acquisitions and their interpretation through Cole–Cole fitting. This solution could successfully decouple the impedance response of stem, roots, and soil, as well as provide convenient parametrization and comparison of their impedance signals. The methodological solution was tested on 80 wheat (Triticum aestivum L.) and 10 pecan [Carya illinoensis (Wangenh.) K. Koch] plants, the first extensive and field investigation. The investigation provided evidence of (a) proximal current leakage in herbaceous root systems, extending recent laboratory results and previous indirect field studies. (b) Major role of the plant stem, which has been a substantial concern raised in numerous studies. (c) Minor contribution from the soil, addressing the doubts on the comparability of results obtained in different soil conditions. All together, these evidences lead to indirect correlations between impedance signals and root traits. The explored solution is expected to support the adoption of the impedance spectroscopy, in line with the diffusion of multichannel impedance meters and growing interest in root physiology and phenotyping.

Geoelectrical methods are increasingly being used for non-invasive characterization and monitoring of permafrost sites, since the electrical properties are sensitive to the phase change of liquid to frozen water. Here, electrical resistivity tomography (ERT) is most commonly applied, using resistivity as a proxy for various quantities, such as temperature or ice content. However, it is still challenging to distinguish between air and ice in the pore space of the rock based on resistivity alone due to their similarly low electrical conductivity. Meanwhile, geoelectrical methods that utilize electrical polarization effects to characterize permafrost are also being explored. For example, the usage of the spectral induced polarization (SIP) method, in which the complex, frequency-dependent impedance is measured, can reduce ambiguities in the subsurface conduction properties, considering the SIP signature of ice. These measurements seem to be suitable for the quantification of ice content (and thus the differentiation of ice and air), and for the improved thermal characterization of alpine permafrost sites. However, to improve the interpretation of SIP measurements, it is necessary to understand in more detail the electrical conduction and polarization properties as a function of temperature, ice content, texture, and mineralogy under frozen and partially frozen conditions. In the study presented here, electrical impedance was measured continuously using SIP in the frequency range of 10 mHz to 45 kHz on various water-saturated solid rock and loose sediment samples during controlled freeze-thaw cycles (+20°C to -40°C). These measurements were performed on rock samples from different alpine permafrost sites with different mineralogical compositions and textures. For all samples, the resistance (impedance magnitude) shows a similar temperature dependence, with increasing resistance for decreasing temperature. Also, hysteresis between freezing and thawing behavior is observed for all measurements. During freezing, a jump within the temperature-dependent resistance is observed, suggesting a lowering of the freezing point to a critical temperature where an abrupt transition from liquid water to ice occurs. During thawing, on the other hand, there is a continuous decrease in the measured resistance, suggesting a continuous thawing of the sample. The spectra of impedance phase, which is a measure for the polarization, exhibit the same qualitative, well-known temperature-dependent relaxation behaviour of ice at higher frequencies (1 kHz - 45 kHz), with variations in shape and strength for different rock texture and mineralogy. At lower frequencies (1 Hz - 1 kHz), a polarization with a weak frequency dependence is observed in the unfrozen state of the samples. We interpret this response as membrane polarization, which likewise depends on the texture as well as on the mineralogy of the respective sample. This polarization response partially vanishes during freezing. Overall, the investigated SIP spectra do not only show a dependence on texture and mineralogy, but mainly a dependence on the presence of ice in the sample as well as temperature. This indicates the possibility of a thermal characterization, as well as a determination of the ice content, of permafrost rocks using SIP.