Bayesian Markov chain Monte Carlo inversion of geophysical data for the purpose of estimating the hydraulic properties of the unsaturated zone

Ground-penetrating radar (GPR) has the potential to provide valuable information on hydrological properties of the vadose zone because of their strong sensitivity to soil water content. In particular, recent evidence has suggested that the stochastic inversion of crosshole GPR data within a coupled geophysical-hydrological framework may allow for effective estimation of subsurface van-Genuchten-Mualem (VGM) parameters and their corresponding uncertainties. An important and still unresolved issue, however, is how to best integrate GPR data into a stochastic inversion in order to estimate the VGM parameters and their uncertainties, thus improving hydrological predictions. Recognizing the importance of this issue, the aim of the research presented in this thesis was to first introduce a fully Bayesian inversion called Markov-chain-Monte-carlo (MCMC) strategy to perform the stochastic inversion of steady-state GPR data to estimate the VGM parameters and their uncertainties. Within this study, the choice of the prior parameter probability distributions from which potential model configurations are drawn and tested against observed data was also investigated. Analysis of both synthetic and field data collected at the Eggborough (UK) site indicates that the geophysical data alone contain valuable information regarding the VGM parameters. However, significantly better results are obtained when these data are combined with a realistic, informative prior. A subsequent study explore in detail the dynamic infiltration case, specifically to what extent time-lapse ZOP GPR data, collected during a forced infiltration experiment at the Arrenaes field site (Denmark), can help to quantify VGM parameters and their uncertainties using the MCMC inversion strategy. The findings indicate that the stochastic inversion of time-lapse GPR data does indeed allow for a substantial refinement in the inferred posterior VGM parameter distributions. In turn, this significantly improves knowledge of the hydraulic properties, which are required to predict hydraulic behaviour. Finally, another aspect that needed to be addressed involved the comparison of time-lapse GPR data collected under different infiltration conditions (i.e., natural loading and forced infiltration conditions) to estimate the VGM parameters using the MCMC inversion strategy. The results show that for the synthetic example, considering data collected during a forced infiltration test helps to better refine soil hydraulic properties compared to data collected under natural infiltration conditions. When investigating data collected at the Arrenaes field site, further complications arised due to model error and showed the importance of also including a rigorous analysis of the propagation of model error with time and depth when considering time-lapse data. Although the efforts in this thesis were focused on GPR data, the corresponding findings are likely to have general applicability to other types of geophysical data and field environments. Moreover, the obtained results allow to have confidence for future developments in integration of geophysical data with stochastic inversions to improve the characterization of the unsaturated zone but also reveal important issues linked with stochastic inversions, namely model errors, that should definitely be addressed in future research.

Time-lapse and probabilistic inversion strategies for plane-wave electromagnetic data.

L'utilisation efficace des systèmes géothermaux, la séquestration du CO2 pour limiter le changement climatique et la prévention de l'intrusion d'eau salée dans les aquifères costaux ne sont que quelques exemples qui démontrent notre besoin en technologies nouvelles pour suivre l'évolution des processus souterrains à partir de la surface. Un défi majeur est d'assurer la caractérisation et l'optimisation des performances de ces technologies à différentes échelles spatiales et temporelles. Les méthodes électromagnétiques (EM) d'ondes planes sont sensibles à la conductivité électrique du sous-sol et, par conséquent, à la conductivité électrique des fluides saturant la roche, à la présence de fractures connectées, à la température et aux matériaux géologiques. Ces méthodes sont régies par des équations valides sur de larges gammes de fréquences, permettant détudier de manières analogues des processus allant de quelques mètres sous la surface jusqu'à plusieurs kilomètres de profondeur. Néanmoins, ces méthodes sont soumises à une perte de résolution avec la profondeur à cause des propriétés diffusives du champ électromagnétique. Pour cette raison, l'estimation des modèles du sous-sol par ces méthodes doit prendre en compte des informations a priori afin de contraindre les modèles autant que possible et de permettre la quantification des incertitudes de ces modèles de façon appropriée. Dans la présente thèse, je développe des approches permettant la caractérisation statique et dynamique du sous-sol à l'aide d'ondes EM planes. Dans une première partie, je présente une approche déterministe permettant de réaliser des inversions répétées dans le temps (time-lapse) de données d'ondes EM planes en deux dimensions. Cette stratégie est basée sur l'incorporation dans l'algorithme d'informations a priori en fonction des changements du modèle de conductivité électrique attendus. Ceci est réalisé en intégrant une régularisation stochastique et des contraintes flexibles par rapport à la gamme des changements attendus en utilisant les multiplicateurs de Lagrange. J'utilise des normes différentes de la norme l2 pour contraindre la structure du modèle et obtenir des transitions abruptes entre les régions du model qui subissent des changements dans le temps et celles qui n'en subissent pas. Aussi, j'incorpore une stratégie afin d'éliminer les erreurs systématiques de données time-lapse. Ce travail a mis en évidence l'amélioration de la caractérisation des changements temporels par rapport aux approches classiques qui réalisent des inversions indépendantes à chaque pas de temps et comparent les modèles. Dans la seconde partie de cette thèse, j'adopte un formalisme bayésien et je teste la possibilité de quantifier les incertitudes sur les paramètres du modèle dans l'inversion d'ondes EM planes. Pour ce faire, je présente une stratégie d'inversion probabiliste basée sur des pixels à deux dimensions pour des inversions de données d'ondes EM planes et de tomographies de résistivité électrique (ERT) séparées et jointes. Je compare les incertitudes des paramètres du modèle en considérant différents types d'information a priori sur la structure du modèle et différentes fonctions de vraisemblance pour décrire les erreurs sur les données. Les résultats indiquent que la régularisation du modèle est nécessaire lorsqu'on a à faire à un large nombre de paramètres car cela permet d'accélérer la convergence des chaînes et d'obtenir des modèles plus réalistes. Cependent, ces contraintes mènent à des incertitudes d'estimations plus faibles, ce qui implique des distributions a posteriori qui ne contiennent pas le vrai modèledans les régions ou` la méthode présente une sensibilité limitée. Cette situation peut être améliorée en combinant des méthodes d'ondes EM planes avec d'autres méthodes complémentaires telles que l'ERT. De plus, je montre que le poids de régularisation des paramètres et l'écart-type des erreurs sur les données peuvent être retrouvés par une inversion probabiliste. Finalement, j'évalue la possibilité de caractériser une distribution tridimensionnelle d'un panache de traceur salin injecté dans le sous-sol en réalisant une inversion probabiliste time-lapse tridimensionnelle d'ondes EM planes. Etant donné que les inversions probabilistes sont très coûteuses en temps de calcul lorsque l'espace des paramètres présente une grande dimension, je propose une stratégie de réduction du modèle ou` les coefficients de décomposition des moments de Legendre du panache de traceur injecté ainsi que sa position sont estimés. Pour ce faire, un modèle de résistivité de base est nécessaire. Il peut être obtenu avant l'expérience time-lapse. Un test synthétique montre que la méthodologie marche bien quand le modèle de résistivité de base est caractérisé correctement. Cette méthodologie est aussi appliquée à un test de trac¸age par injection d'une solution saline et d'acides réalisé dans un système géothermal en Australie, puis comparée à une inversion time-lapse tridimensionnelle réalisée selon une approche déterministe. L'inversion probabiliste permet de mieux contraindre le panache du traceur salin gr^ace à la grande quantité d'informations a priori incluse dans l'algorithme. Néanmoins, les changements de conductivités nécessaires pour expliquer les changements observés dans les données sont plus grands que ce qu'expliquent notre connaissance actuelle des phénomenès physiques. Ce problème peut être lié à la qualité limitée du modèle de résistivité de base utilisé, indiquant ainsi que des efforts plus grands devront être fournis dans le futur pour obtenir des modèles de base de bonne qualité avant de réaliser des expériences dynamiques. Les études décrites dans cette thèse montrent que les méthodes d'ondes EM planes sont très utiles pour caractériser et suivre les variations temporelles du sous-sol sur de larges échelles. Les présentes approches améliorent l'évaluation des modèles obtenus, autant en termes d'incorporation d'informations a priori, qu'en termes de quantification d'incertitudes a posteriori. De plus, les stratégies développées peuvent être appliquées à d'autres méthodes géophysiques, et offrent une grande flexibilité pour l'incorporation d'informations additionnelles lorsqu'elles sont disponibles. -- The efficient use of geothermal systems, the sequestration of CO2 to mitigate climate change, and the prevention of seawater intrusion in coastal aquifers are only some examples that demonstrate the need for novel technologies to monitor subsurface processes from the surface. A main challenge is to assure optimal performance of such technologies at different temporal and spatial scales. Plane-wave electromagnetic (EM) methods are sensitive to subsurface electrical conductivity and consequently to fluid conductivity, fracture connectivity, temperature, and rock mineralogy. These methods have governing equations that are the same over a large range of frequencies, thus allowing to study in an analogous manner processes on scales ranging from few meters close to the surface down to several hundreds of kilometers depth. Unfortunately, they suffer from a significant resolution loss with depth due to the diffusive nature of the electromagnetic fields. Therefore, estimations of subsurface models that use these methods should incorporate a priori information to better constrain the models, and provide appropriate measures of model uncertainty. During my thesis, I have developed approaches to improve the static and dynamic characterization of the subsurface with plane-wave EM methods. In the first part of this thesis, I present a two-dimensional deterministic approach to perform time-lapse inversion of plane-wave EM data. The strategy is based on the incorporation of prior information into the inversion algorithm regarding the expected temporal changes in electrical conductivity. This is done by incorporating a flexible stochastic regularization and constraints regarding the expected ranges of the changes by using Lagrange multipliers. I use non-l2 norms to penalize the model update in order to obtain sharp transitions between regions that experience temporal changes and regions that do not. I also incorporate a time-lapse differencing strategy to remove systematic errors in the time-lapse inversion. This work presents improvements in the characterization of temporal changes with respect to the classical approach of performing separate inversions and computing differences between the models. In the second part of this thesis, I adopt a Bayesian framework and use Markov chain Monte Carlo (MCMC) simulations to quantify model parameter uncertainty in plane-wave EM inversion. For this purpose, I present a two-dimensional pixel-based probabilistic inversion strategy for separate and joint inversions of plane-wave EM and electrical resistivity tomography (ERT) data. I compare the uncertainties of the model parameters when considering different types of prior information on the model structure and different likelihood functions to describe the data errors. The results indicate that model regularization is necessary when dealing with a large number of model parameters because it helps to accelerate the convergence of the chains and leads to more realistic models. These constraints also lead to smaller uncertainty estimates, which imply posterior distributions that do not include the true underlying model in regions where the method has limited sensitivity. This situation can be improved by combining planewave EM methods with complimentary geophysical methods such as ERT. In addition, I show that an appropriate regularization weight and the standard deviation of the data errors can be retrieved by the MCMC inversion. Finally, I evaluate the possibility of characterizing the three-dimensional distribution of an injected water plume by performing three-dimensional time-lapse MCMC inversion of planewave EM data. Since MCMC inversion involves a significant computational burden in high parameter dimensions, I propose a model reduction strategy where the coefficients of a Legendre moment decomposition of the injected water plume and its location are estimated. For this purpose, a base resistivity model is needed which is obtained prior to the time-lapse experiment. A synthetic test shows that the methodology works well when the base resistivity model is correctly characterized. The methodology is also applied to an injection experiment performed in a geothermal system in Australia, and compared to a three-dimensional time-lapse inversion performed within a deterministic framework. The MCMC inversion better constrains the water plumes due to the larger amount of prior information that is included in the algorithm. The conductivity changes needed to explain the time-lapse data are much larger than what is physically possible based on present day understandings. This issue may be related to the base resistivity model used, therefore indicating that more efforts should be given to obtain high-quality base models prior to dynamic experiments. The studies described herein give clear evidence that plane-wave EM methods are useful to characterize and monitor the subsurface at a wide range of scales. The presented approaches contribute to an improved appraisal of the obtained models, both in terms of the incorporation of prior information in the algorithms and the posterior uncertainty quantification. In addition, the developed strategies can be applied to other geophysical methods, and offer great flexibility to incorporate additional information when available.

Bayesian Markov-chain-Monte-Carlo inversion of time-lapse cross hole ground-penetrating radar data to characterize the vadose zone at the Arrenaes field site, Denmark

The ground-penetrating radar (GPR) geophysical method has the potential to provide valuable information on the hydraulic properties of the vadose zone because of its strong sensitivity to soil water content. In particular, recent evidence has suggested that the stochastic inversion of crosshole GPR traveltime data can allow for a significant reduction in uncertainty regarding subsurface van Genuchten-Mualem (VGM) parameters. Much of the previous work on the stochastic estimation of VGM parameters from crosshole GPR data has considered the case of steady-state infiltration conditions, which represent only a small fraction of practically relevant scenarios. We explored in detail the dynamic infiltration case, specifically examining to what extent time-lapse crosshole GPR traveltimes, measured during a forced infiltration experiment at the Arreneas field site in Denmark, could help to quantify VGM parameters and their uncertainties in a layered medium, as well as the corresponding soil hydraulic properties. We used a Bayesian Markov-chain-Monte-Carlo inversion approach. We first explored the advantages and limitations of this approach with regard to a realistic synthetic example before applying it to field measurements. In our analysis, we also considered different degrees of prior information. Our findings indicate that the stochastic inversion of the time-lapse GPR data does indeed allow for a substantial refinement in the inferred posterior VGM parameter distributions compared with the corresponding priors, which in turn significantly improves knowledge of soil hydraulic properties. Overall, the results obtained clearly demonstrate the value of the information contained in time-lapse GPR data for characterizing vadose zone dynamics.

Two-dimensional probabilistic inversion of plane-wave electromagnetic data: Methodology, model constraints and joint inversion with electrical resistivity data

Probabilistic inversion methods based on Markov chain Monte Carlo (MCMC) simulation are well suited to quantify parameter and model uncertainty of nonlinear inverse problems. Yet, application of such methods to CPU-intensive forward models can be a daunting task, particularly if the parameter space is high dimensional. Here, we present a 2-D pixel-based MCMC inversion of plane-wave electromagnetic (EM) data. Using synthetic data, we investigate how model parameter uncertainty depends on model structure constraints using different norms of the likelihood function and the model constraints, and study the added benefits of joint inversion of EM and electrical resistivity tomography (ERT) data. Our results demonstrate that model structure constraints are necessary to stabilize the MCMC inversion results of a highly discretized model. These constraints decrease model parameter uncertainty and facilitate model interpretation. A drawback is that these constraints may lead to posterior distributions that do not fully include the true underlying model, because some of its features exhibit a low sensitivity to the EM data, and hence are difficult to resolve. This problem can be partly mitigated if the plane-wave EM data is augmented with ERT observations. The hierarchical Bayesian inverse formulation introduced and used herein is able to successfully recover the probabilistic properties of the measurement data errors and a model regularization weight. Application of the proposed inversion methodology to field data from an aquifer demonstrates that the posterior mean model realization is very similar to that derived from a deterministic inversion with similar model constraints.

Evaluation of the reconstruction limits of a frequency-independent crosshole georadar waveform inversion scheme in the presence of dispersion

Waveform tomographic imaging of crosshole georadar data is a powerful method to investigate the shallow subsurface because of its ability to provide images of pertinent petrophysical parameters with extremely high spatial resolution. All current crosshole georadar waveform inversion strategies are based on the assumption of frequency-independent electromagnetic constitutive parameters. However, in reality, these parameters are known to be frequency-dependent and complex and thus recorded georadar data may show significant dispersive behavior. In this paper, we evaluate synthetically the reconstruction limits of a recently published crosshole georadar waveform inversion scheme in the presence of varying degrees of dielectric dispersion. Our results indicate that, when combined with a source wavelet estimation procedure that provides a means of partially accounting for the frequency-dependent effects through an "effective" wavelet, the inversion algorithm performs remarkably well in weakly to moderately dispersive environments and has the ability to provide adequate tomographic reconstructions.

Comparison of time-lapse GPR data collected under natural and forced infiltration conditions to estimate vadose zone hydraulic parameters

Time-lapse crosshole ground-penetrating radar (GPR) data, collected while infiltration occurs, can provide valuable information regarding the hydraulic properties of the unsaturated zone. In particular, the stochastic inversion of such data provides estimates of parameter uncertainties, which are necessary for hydrological prediction and decision making. Here, we investigate the effect of different infiltration conditions on the stochastic inversion of time-lapse, zero-offset-profile, GPR data. Inversions are performed using a Bayesian Markov-chain-Monte-Carlo methodology. Our results clearly indicate that considering data collected during a forced infiltration test helps to better refine soil hydraulic properties compared to data collected under natural infiltration conditions

Functional mapping of the human visual cortex with intravoxel incoherent motion MRI.

Functional imaging with intravoxel incoherent motion (IVIM) magnetic resonance imaging (MRI) is demonstrated. Images were acquired at 3 Tesla using a standard Stejskal-Tanner diffusion-weighted echo-planar imaging sequence with multiple b-values. Cerebro-spinal fluid signal, which is highly incoherent, was suppressed with an inversion recovery preparation pulse. IVIM microvascular perfusion parameters were calculated according to a two-compartment (vascular and non-vascular) diffusion model. The results obtained in 8 healthy human volunteers during visual stimulation are presented. The IVIM blood flow related parameter fD* increased 170% during stimulation in the visual cortex, and 70% in the underlying white matter.

Inversion prepared coronary MR angiography: direct visualization of coronary blood flow.

PURPOSE: Visualization of coronary blood flow by means of a slice-selective inversion pre-pulse in concert with bright-blood coronary MRA. MATERIALS AND METHODS: Coronary magnetic resonance angiography (MRA) of the right coronary artery (RCA) was performed in eight healthy adult subjects on a 1.5 Tesla MR system (Gyroscan ACS-NT, Philips Medical Systems, Best, NL) using a free-breathing navigator-gated and cardiac-triggered 3D steady-state free-precession (SSFP) sequence with radial k-space sampling. Imaging was performed with and without a slice-selective inversion pre-pulse, which was positioned along the main axis of the coronary artery but perpendicular to the imaging volume. Objective image quality parameters such as SNR, CNR, maximal visible vessel length, and vessel border definition were analyzed. RESULTS: In contrast to conventional bright-blood 3D coronary MRA, the selective inversion pre-pulse provided a direct measure of coronary blood flow. In addition, CNR between the RCA and right ventricular blood pool was increased and the vessels had a tendency towards better delineation. Blood SNR and CNR between right coronary blood and epicardial fat were comparable in both sequences. CONCLUSION: The combination of a free-breathing navigator-gated and cardiac-triggered 3D SSFP sequence with a slice-selective inversion pre-pulse allows for direct and directional visualization of coronary blood flow with the additional benefit of improved contrast between coronary and right ventricular blood pool.

Total hemoglobin mass--a new parameter to detect blood doping?

PURPOSE: All kinds of blood manipulations aim to increase the total hemoglobin mass (tHb-mass). To establish tHb-mass as an effective screening parameter for detecting blood doping, the knowledge of its normal variation over time is necessary. The aim of the present study, therefore, was to determine the intraindividual variance of tHb-mass in elite athletes during a training year emphasizing off, training, and race seasons at sea level. METHODS: tHb-mass and hemoglobin concentration ([Hb]) were determined in 24 endurance athletes five times during a year and were compared with a control group (n = 6). An analysis of covariance was used to test the effects of training phases, age, gender, competition level, body mass, and training volume. Three error models, based on 1) a total percentage error of measurement, 2) the combination of a typical percentage error (TE) of analytical origin with an absolute SD of biological origin, and 3) between-subject and within-subject variance components as obtained by an analysis of variance, were tested. RESULTS: In addition to the expected influence of performance status, the main results were that the effects of training volume (P = 0.20) and training phases (P = 0.81) on tHb-mass were not significant. We found that within-subject variations mainly have an analytical origin (TE approximately 1.4%) and a very small SD (7.5 g) of biological origin. CONCLUSION: tHb-mass shows very low individual oscillations during a training year (<6%), and these oscillations are below the expected changes in tHb-mass due to Herythropoetin (EPO) application or blood infusion (approximately 10%). The high stability of tHb-mass over a period of 1 year suggests that it should be included in an athlete's biological passport and analyzed by recently developed probabilistic inference techniques that define subject-based reference ranges.

Estimation of the lateral correlation structure of subsurface water content from surface-based ground-penetrating radar reflection images

Over the past decade, significant interest has been expressed in relating the spatial statistics of surface-based reflection ground-penetrating radar (GPR) data to those of the imaged subsurface volume. A primary motivation for this work is that changes in the radar wave velocity, which largely control the character of the observed data, are expected to be related to corresponding changes in subsurface water content. Although previous work has indeed indicated that the spatial statistics of GPR images are linked to those of the water content distribution of the probed region, a viable method for quantitatively analyzing the GPR data and solving the corresponding inverse problem has not yet been presented. Here we address this issue by first deriving a relationship between the 2-D autocorrelation of a water content distribution and that of the corresponding GPR reflection image. We then show how a Bayesian inversion strategy based on Markov chain Monte Carlo sampling can be used to estimate the posterior distribution of subsurface correlation model parameters that are consistent with the GPR data. Our results indicate that if the underlying assumptions are valid and we possess adequate prior knowledge regarding the water content distribution, in particular its vertical variability, this methodology allows not only for the reliable recovery of lateral correlation model parameters but also for estimates of parameter uncertainties. In the case where prior knowledge regarding the vertical variability of water content is not available, the results show that the methodology still reliably recovers the aspect ratio of the heterogeneity.

Trabecular bone score (TBS) the new parameter of 2D texture analysis for the evaluation of 3D bone micro architecture status

X-ray is a technology that is used for numerous applications in the medical field. The process of X-ray projection gives a 2-dimension (2D) grey-level texture from a 3- dimension (3D) object. Until now no clear demonstration or correlation has positioned the 2D texture analysis as a valid indirect evaluation of the 3D microarchitecture. TBS is a new texture parameter based on the measure of the experimental variogram. TBS evaluates the variation between 2D image grey-levels. The aim of this study was to evaluate existing correlations between 3D bone microarchitecture parameters - evaluated from μCT reconstructions - and the TBS value, calculated on 2D projected images. 30 dried human cadaveric vertebrae were acquired on a micro-scanner (eXplorer Locus, GE) at isotropic resolution of 93 μm. 3D vertebral body models were used. The following 3D microarchitecture parameters were used: Bone volume fraction (BV/TV), Trabecular thickness (TbTh), trabecular space (TbSp), trabecular number (TbN) and connectivity density (ConnD). 3D/2D projections has been done by taking into account the Beer-Lambert Law at X-ray energy of 50, 100, 150 KeV. TBS was assessed on 2D projected images. Correlations between TBS and the 3D microarchitecture parameters were evaluated using a linear regression analysis. Paired T-test is used to assess the X-ray energy effects on TBS. Multiple linear regressions (backward) were used to evaluate relationships between TBS and 3D microarchitecture parameters using a bootstrap process. BV/TV of the sample ranged from 18.5 to 37.6% with an average value at 28.8%. Correlations' analysis showedthat TBSwere strongly correlatedwith ConnD(0.856≤r≤0.862; p<0.001),with TbN (0.805≤r≤0.810; p<0.001) and negatively with TbSp (−0.714≤r≤−0.726; p<0.001), regardless X-ray energy. Results show that lower TBS values are related to "degraded" microarchitecture, with low ConnD, low TbN and a high TbSp. The opposite is also true. X-ray energy has no effect onTBS neither on the correlations betweenTBS and the 3Dmicroarchitecture parameters. In this study, we demonstrated that TBS was significantly correlated with 3D microarchitecture parameters ConnD and TbN, and negatively with TbSp, no matter what X-ray energy has been used. This article is part of a Special Issue entitled ECTS 2011. Disclosure of interest: None declared.