Markov Chain Monte Carlo Inversion Research Articles

Determination of thermal properties of materials by Monte Carlo inversion of pulsed needle probe data

Needle probe measurements are used in a wide variety of fields for measuring the thermal conductivity of materials. It is most common to extract the thermal conductivity from the temperature rise data using the asymptotic solution to the heat equation. With special reference to, but not limited to Earth materials, this study presents an inversion procedure using pulsed needle probe data to determine both thermal conductivity and thermal diffusivity. The measured temperature response data are interpreted using a finite element forward model and a Markov Chain Monte Carlo inversion algorithm. The thermal properties of the needle probe are part of the forward model and determined by measurements on calibration standards. We examine several factors by synthetic modelling and quantify their effects by Monte Carlo inversions. This include (1) the heat production rate of the probe in order to produce similar temperature increases in materials of different thermal properties, (2) the contact resistance between needle probe and sample, (3) the limitations imposed by sample diameter and the thermal properties of the surrounding medium, and (4) the duration of heating period required to obtain good results. Laboratory test measurements on selected materials (water, glycerol, ceramic standard, clay) demonstrate good agreement with expected values obtained from literature. We demonstrate that the combination of numerical forward modelling and Monte Carlo inversion, applied to the pulsed needle, is a flexible and powerful methodology for accurate laboratory measurements of material thermal properties and with well-defined estimates of uncertainty. With the given equipment and laboratory setup, thermal conductivity is measured to within few percent, while thermal diffusivity generally has a somewhat higher degree of uncertainty, up to about five percent.

Multi-parameter stochastic inversion for first and second moment mass properties of a model-scale ship with topside ice accumulation

This work investigates the indirect monitoring of ice accretion on ship surfaces using a stochastic inversion framework. An accurate assessment of a ship's mass properties during operation is an important concern for ships traveling within the Arctic, where ice accumulation is a concern. Within such contexts, the actual (i.e., current) first and second moment properties of the vessel, including accumulated topside icing, must be considered within the associated equations of motion for a given ship. By leveraging instrumentation such as an existing on-board inertial measurement unit (IMU), in conjunction with existing seakeeping software, the proposed framework recovers a statistical description of two mass properties, simultaneously. The subsequent stochastic inverse problem is solved using the proposed framework in arriving at two mass properties in particular: the vertical center of gravity (a first moment mass property) and the roll gyradius (a second moment mass property).The inversion scheme requires two main inputs: an observed ground truth for the roll period, and an associated signal-to-noise ratio for the roll period measurement. The framework applies a Markov chain Monte Carlo (MCMC) inversion scheme, implemented in Python, that leverages the Standard Ship Motion Program (SMP95) software in order to build a posterior distribution for the desired mass properties. Experimental model results for Research Vessel (R/V) Melville are used to validate the proposed framework. Six different configurations, including one case of no icing and five cases of topside icing, are investigated within the context of this framework in order to invert for their respective roll gyradii and vertical centers of gravity. Icing configurations include both asymmetric and symmetric ice accumulation under moderate to heavy icing conditions. Recommendations concerning strategies for applying the proposed framework are offered.

Parallel multiple-chain DRAM MCMC for large-scale geosteering inversion and uncertainty quantification

Geosteering is the proactive control of a wellbore placement based on the downhole measurements, aiming at maximizing economic production from the well. The recent development of azimuthal resistivity logging-while-drilling (LWD) tools have a much larger depth of detection, thus sets higher demands for geosteering inversion as more unknown parameters need to be taken into account in the earth model to be inverted. For complicated non-linear problems, traditional deterministic inversion methods are more likely to be trapped near local optima. In general, Markov chain Monte Carlo (MCMC) inversion methods are more capable of finding the global optimal solution and providing additional uncertainty analyses by sampling from the target distribution. However, MCMC methods usually have the problem of slow convergence. Even though optimization methods like delayed rejection (DR) and adaptive Metropolis (AM) were proposed to speed up the convergence, using MCMC methods to solve geosteering inversion problems may still incur unacceptable time cost. To reduce the sampling cost of MCMC methods, we use parallel multiple-chain DRAM MCMC methods to solve geosteering inverse problems and estimate the corresponding uncertainty. A clustering method, density-based spatial clustering of applications with noise (DBSCAN), is applied to select the best solution among many results. The simulation results show that running many relatively short MCMC chains for one problem can obtain a similar result as running a long single chain. Besides, avoiding communications between multiple Markov chains during the sampling process yields almost linear scalability.

Buried Impact Features on Mercury as Revealed by Gravity Data

AbstractGravity data provide information about the internal mass distribution of celestial bodies. The latest gravity field model of Mercury, HgM007, has a higher spatial resolution than previously published gravity models, allowing smaller crustal structures to be resolved. In this study, a global free‐air gravity anomaly grid of Mercury was derived from the HgM007 gravity model using the spherical expansion method, and a Bouguer anomaly map was calculated based on the free‐air gravity and topography data. We used this gravity model to search for quasi‐circular Bouguer anomalies (QCBAs) larger than ∼109 km in diameter within the Northern Smooth Plains (NSP) of Mercury, where the resolution of the gravity data is the highest. Assuming that these QCBAs were correlated with mantle uplifts caused by ancient buried impact structures, we applied the prism rectangular shape model and the Markov chain Monte Carlo inversion method to determine the subsurface structure of these QCBAs. Four ancient buried basins, not clearly visible in the surface topography, were detected. We estimated the crustal thickness ratios and original rim‐to‐rim diameters for these four impact basins, based on empirical cratering scaling laws, finding that the thickness of the infill lava within the NSP might be 5 times larger than previously believed and the ancient terrain beneath the NSP might be older than previously thought.

Re‐parameterisations of the Cole–Cole model for improved spectral inversion of induced polarization data

ABSTRACTThe induced polarization phenomenon, both in time domain and frequency domain, is often parameterised using the empirical Cole–Cole model. To improve the resolution of model parameters and to decrease the parameter correlations in the inversion process of induced polarization data, we suggest here three re‐parameterisations of the Cole–Cole model, namely the maximum phase angle Cole–Cole model, the maximum imaginary conductivity Cole–Cole model, and the minimum imaginary resistivity Cole–Cole model. The maximum phase angle Cole–Cole model uses the maximum phase φmax and the inverse of the phase peak frequency, τφ, instead of the intrinsic charge‐ability m0 and the time constant adopted in the classic Cole–Cole model. The maximum imaginary conductivity Cole–Cole model uses the maximum imaginary conductivity instead of m0 and the time constant τσ of the Cole–Cole model in its conductivity form. The minimum imaginary resistivity Cole–Cole model uses the minimum imaginary resistivity instead of m0 and the time constant τρ of the Cole–Cole model in its resistivity form.The effects of the three re‐parameterisations have been tested on synthetic time‐domain and frequency‐domain data using a Markov chain Monte Carlo inversion method, which allows for easy quantification of parameter uncertainty, and on field data using 2D gradient‐based inversion. In comparison with the classic Cole–Cole model, it was found that for all the three re‐parameterisations, the model parameters are less correlated with each other and, consequently, better resolved for both time‐domain and frequency‐domain data. The increase in model resolution is particularly significant for models that are poorly resolved using the classic Cole–Cole parameterisation, for instance, for low values of the frequency exponent or with low signal‐to‐noise ratio. In general, this leads to a significantly deeper depth of investigation for the , , and parameters, when compared with the classic m0 parameter, which is shown with a field example. We believe that the use of re‐parameterisations for inverting field data will contribute to narrow the gap between induced polarization theory, laboratory findings, and field applications.

Estimating P- and S-wave inverse quality factors from observed seismic data using an attenuative elastic impedance

P- and S-wave inverse quality factors quantify seismic wave attenuation, which is related to several key reservoir parameters (porosity, saturation, and viscosity). Estimating the inverse quality factors from observed seismic data provides additional and useful information during gas-bearing reservoir prediction. First, we have developed an approximate reflection coefficient and attenuative elastic impedance (QEI) in terms of the inverse quality factors, and then we established an approach to estimate elastic properties (P- and S-wave impedances, and density) and attenuation (P- and S-wave inverse quality factors) from seismic data at different incidence angles and frequencies. The approach is implemented as a two-step inversion: a model-based and damped least-squares inversion for QEI, and a Bayesian Markov chain Monte Carlo inversion for the inverse quality factors. Synthetic data tests confirm that P- and S-wave impedances and inverse quality factors are reasonably estimated in the case of moderate data error or noise. Applying the established approach to a real data set is suggestive of the robustness of the approach, and furthermore that physically meaningful inverse quality factors can be estimated from seismic data acquired over a gas-bearing reservoir.

On the Grain Size Sensitivity of Olivine Rheology

AbstractUsing the Markov chain Monte Carlo (MCMC) inversion technique of Mullet et al. (2015), we reassess the validity of the conventionally accepted values of the grain size exponent of diffusion creep in olivine aggregates. A systematic and comprehensive analysis of individual experimental runs taken from three widely cited studies reveals that these data do not tightly constrain the grain size exponent or any other flow law parameter for diffusion and dislocation creep. Our analysis indicates that large data uncertainties can cause inversion results to deviate significantly from true values because of the covariance between the grain size and stress exponents, and that even resolving a grain size exponent of 2 from 3 is difficult. The versatility of our MCMC inversion technique can, however, be exploited to improve this situation by identifying optimal conditions for future experimental studies. Because the uncertainties of the grain size and stress exponents are highly correlated, for example, increasing the range of grain size variation can help better constrain both exponents simultaneously.

Training‐Image Based Geostatistical Inversion Using a Spatial Generative Adversarial Neural Network

AbstractProbabilistic inversion within a multiple‐point statistics framework is often computationally prohibitive for high‐dimensional problems. To partly address this, we introduce and evaluate a new training‐image based inversion approach for complex geologic media. Our approach relies on a deep neural network of the generative adversarial network (GAN) type. After training using a training image (TI), our proposed spatial GAN (SGAN) can quickly generate 2‐D and 3‐D unconditional realizations. A key characteristic of our SGAN is that it defines a (very) low‐dimensional parameterization, thereby allowing for efficient probabilistic inversion using state‐of‐the‐art Markov chain Monte Carlo (MCMC) methods. In addition, available direct conditioning data can be incorporated within the inversion. Several 2‐D and 3‐D categorical TIs are first used to analyze the performance of our SGAN for unconditional geostatistical simulation. Training our deep network can take several hours. After training, realizations containing a few millions of pixels/voxels can be produced in a matter of seconds. This makes it especially useful for simulating many thousands of realizations (e.g., for MCMC inversion) as the relative cost of the training per realization diminishes with the considered number of realizations. Synthetic inversion case studies involving 2‐D steady state flow and 3‐D transient hydraulic tomography with and without direct conditioning data are used to illustrate the effectiveness of our proposed SGAN‐based inversion. For the 2‐D case, the inversion rapidly explores the posterior model distribution. For the 3‐D case, the inversion recovers model realizations that fit the data close to the target level and visually resemble the true model well.

Estimation of modified fluid factor and dry fracture weaknesses using azimuthal elastic impedance

We consider the problem of fluid identification and fracture detection in unconventional reservoir (tight gas sand and shale gas) characterization. We begin with a simplification of the stiffness parameters and the derivation of a linearized reflection coefficient and azimuthal elastic impedance (EI). The accuracy of the simplification is confirmed in application to gas-bearing fractured rocks with low porosity and small fracture density. We have developed a modified fluid factor that is more sensitive to fluid type and less influenced by porosity. A two-step inversion workflow is evaluated based on the derived linearized reflection coefficient and azimuthal EI, including (1) a damped least-squares inversion for azimuthal EI, constrained by an initial model, and (2) a Bayesian Markov chain Monte Carlo inversion for the modified fluid factor and dry fracture weaknesses. Stability and accuracy are examined with synthetic data, from which we conclude that the modified fluid factor and dry fracture weaknesses can be stably determined in the presence of moderate data error/noise. The stability of our approach is further confirmed on a fractured tight gas sand field data set, within which we observe that geologically reasonable parameters (Lamé constants, the modified fluid factor, and dry fracture weaknesses) are determined. We conclude that our inversion workflow and its underlying assumptions form realistic predictions/discriminations of reservoir fracture and fluid parameters.

Bayesian inversion of a CRN depth profile to infer Quaternary erosion of the northwestern Campine Plateau (NE Belgium)

Abstract. The rate at which low-lying sandy areas in temperate regions, such as the Campine Plateau (NE Belgium), have been eroding during the Quaternary is a matter of debate. Current knowledge on the average pace of landscape evolution in the Campine area is largely based on geological inferences and modern analogies. We performed a Bayesian inversion of an in situ-produced 10Be concentration depth profile to infer the average long-term erosion rate together with two other parameters: the surface exposure age and the inherited 10Be concentration. Compared to the latest advances in probabilistic inversion of cosmogenic radionuclide (CRN) data, our approach has the following two innovative components: it (1) uses Markov chain Monte Carlo (MCMC) sampling and (2) accounts (under certain assumptions) for the contribution of model errors to posterior uncertainty. To investigate to what extent our approach differs from the state of the art in practice, a comparison against the Bayesian inversion method implemented in the CRONUScalc program is made. Both approaches identify similar maximum a posteriori (MAP) parameter values, but posterior parameter and predictive uncertainty derived using the method taken in CRONUScalc is moderately underestimated. A simple way for producing more consistent uncertainty estimates with the CRONUScalc-like method in the presence of model errors is therefore suggested. Our inferred erosion rate of 39 ± 8. 9 mm kyr−1 (1σ) is relatively large in comparison with landforms that erode under comparable (paleo-)climates elsewhere in the world. We evaluate this value in the light of the erodibility of the substrate and sudden base level lowering during the Middle Pleistocene. A denser sampling scheme of a two-nuclide concentration depth profile would allow for better inferred erosion rate resolution, and including more uncertain parameters in the MCMC inversion.

Bayesian Markov chain Monte Carlo inversion for anisotropy of PP- and PS-wave in weakly anisotropic and heterogeneous media

A single set of vertically aligned cracks embedded in a purely isotropic background may be considered as a long-wavelength effective transversely isotropy (HTI) medium with a horizontal symmetry axis. The crack-induced HTI anisotropy can be characterized by the weakly anisotropic parameters introduced by Thomsen. The seismic scattering theory can be utilized for the inversion for the anisotropic parameters in weakly anisotropic and heterogeneous HTI media. Based on the seismic scattering theory, we first derived the linearized PP- and PS-wave reflection coefficients in terms of P- and S-wave impedances, density as well as three anisotropic parameters in HTI media. Then, we proposed a novel Bayesian Markov chain Monte Carlo inversion method of PP- and PS-wave for six elastic and anisotropic parameters directly. Tests on synthetic azimuthal seismic data contaminated by random errors demonstrated that this method appears more accurate, anti-noise and stable owing to the usage of the constrained PS-wave compared with the standards inversion scheme taking only the PP-wave into account.

An introduction of Markov chain Monte Carlo method to geochemical inverse problems: Reading melting parameters from REE abundances in abyssal peridotites

Markov chain Monte Carlo (MCMC) simulation is a powerful statistical method in solving inverse problems that arise from a wide range of applications. In Earth sciences applications of MCMC simulations are primarily in the field of geophysics. The purpose of this study is to introduce MCMC methods to geochemical inverse problems related to trace element fractionation during mantle melting. MCMC methods have several advantages over least squares methods in deciphering melting processes from trace element abundances in basalts and mantle rocks. Here we use an MCMC method to invert for extent of melting, fraction of melt present during melting, and extent of chemical disequilibrium between the melt and residual solid from REE abundances in clinopyroxene in abyssal peridotites from Mid-Atlantic Ridge, Central Indian Ridge, Southwest Indian Ridge, Lena Trough, and American-Antarctic Ridge. We consider two melting models: one with exact analytical solution and the other without. We solve the latter numerically in a chain of melting models according to the Metropolis–Hastings algorithm. The probability distribution of inverted melting parameters depends on assumptions of the physical model, knowledge of mantle source composition, and constraints from the REE data. Results from MCMC inversion are consistent with and provide more reliable uncertainty estimates than results based on nonlinear least squares inversion. We show that chemical disequilibrium is likely to play an important role in fractionating LREE in residual peridotites during partial melting beneath mid-ocean ridge spreading centers. MCMC simulation is well suited for more complicated but physically more realistic melting problems that do not have analytical solutions.

Time series of temperature in Fram Strait determined from the 2008–2009 DAMOCLES acoustic tomography measurements and an ocean model

AbstractA pilot acoustic tomography program in Fram Strait during 2008–2009 measured a year‐long record of acoustic travel times along a 130 km range acoustic path crossing the West Spitsbergen Current. Individual ray arrivals were not observed. Rather, the arrival patterns consisted of a single, stable, broad arrival pulse of about 100 ms duration. Travel time variations of ±0.15 s recorded the vigorous mesoscale environment of the region and the seasonal cycle. To estimate ocean temperature from the tomography data an inverse scheme employed a high‐resolution ocean model for Fram Strait as the reference ocean. The information from the tomographic measurements is primarily average temperature. Estimated temperatures, averaged over 0–1000 m depth and over range, had a mean of 1.11°C and variations of ±0.33°C; the uncertainty of the tomography estimates was about 60m°C. Agreement with an alternate inverse approach based on EOFs and a Markov Chain Monte Carlo inversion scheme relying on a matched‐peak approach was excellent, indicating a robust estimate for ocean temperature. The inverse estimates for average temperature agreed with the equivalent estimates from hydrographic sections obtained along the acoustic path at the start and end of the program. Among other deficiencies, the ocean model greatly underestimated the intensity of the mesoscale fluctuations and exhibited a warm bias of about 0.38°C in section‐averaged temperature. Tomographic measurements in Fram Strait offer unique large‐scale temperature constraints for ocean models through data assimilation. It is anticipated that these constraints will lead to more accurate estimates of the circulation and transports in Fram Strait.

Markov Chain Monte Carlo inversion of temperature and salinity structure of an internal solitary wave packet from marine seismic data

AbstractMarine seismic reflection technique is used to observe the strong ocean dynamic process of nonlinear internal solitary waves (ISWs or solitons) in the near‐surface water. Analysis of ISWs is problematical because of their transient nature and limitations of classical physical oceanography methods. This work explores a Markov Chain Monte Carlo (MCMC) approach to recover the temperature and salinity of ISW field using the seismic reflectivity data and in situ hydrographic data. The MCMC approach is designed to directly sample the posterior probability distributions of temperature and salinity which are the solutions of the system under investigation. The principle improvement is the capability of incorporating uncertainties in observations and prior models which then provide quantified uncertainties in the output model parameters. We tested the MCMC approach on two acoustic reflectivity data sets one synthesized from a CTD cast and the other derived from multichannel seismic reflections. This method finds the solutions faithfully within the significantly narrowed confidence intervals from the provided priors. Combined with a low frequency initial model interpreted from seismic horizons of ISWs, the MCMC method is used to compute the finescale temperature, salinity, acoustic velocity, and density of ISW field. The statistically derived results are equivalent to the conventional linearized inversion method. However, the former provides us the quantified uncertainties of the temperature and salinity along the whole section whilst the latter does not. These results are the first time ISWs have been mapped with sufficient detail for further analysis of their dynamic properties.

Image synthesis with graph cuts: a fast model proposal mechanism in probabilistic inversion

Geophysical inversion should ideally produce geologically realistic subsurface models that explain the available data. Multiple-point statistics is a geostatistical approach to construct subsurface models that are consistent with site-specific data, but also display the same type of patterns as those found in a training image. The training image can be seen as a conceptual model of the subsurface and is used as a non-parametric model of spatial variability. Inversion based on multiple-point statistics is challenging due to high nonlinearity and time-consuming geostatistical resimulation steps that are needed to create new model proposals. We propose an entirely new model proposal mechanism for geophysical inversion that is inspired by texture synthesis in computer vision. Instead of resimulating pixels based on higher-order patterns in the training image, we identify a suitable patch of the training image that replace a corresponding patch in the current model without breaking the patterns found in the training image, that is, remaining consistent with the given prior. We consider three cross-hole ground-penetrating radar examples in which the new model proposal mechanism is employed within an extended Metropolis Markov chain Monte Carlo (MCMC) inversion. The model proposal step is about 40 times faster than state-of-the-art multiple-point statistics resimulation techniques, the number of necessary MCMC steps is lower and the quality of the final model realizations is of similar quality. The model proposal mechanism is presently limited to 2-D fields, but the method is general and can be applied to a wide range of subsurface settings and geophysical data types.

Joint probabilistic inference of multi-Gaussian conductivity fields and their associated variograms from indirect hydrological data

PUBLICATIONS Water Resources Research RESEARCH ARTICLE 10.1002/2014WR016395 Key Points: Joint Bayesian inference of Gaussian conductivity fields and their variograms A dimensionality reduction that systematically honors the underlying variogram Distributed multiprocessor implementation is straightforward Correspondence to: E. Laloy, elaloy@sckcen.be Citation: Laloy, E., N. Linde, D. Jacques, and J. A. Vrugt (2015), Probabilistic inference of multi-Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction, Water Resour. Res., 51, 4224–4243, doi:10.1002/2014WR016395. Received 10 SEP 2014 Accepted 14 MAY 2015 Accepted article online 19 MAY 2015 Published online 12 JUN 2015 Probabilistic inference of multi-Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction Eric Laloy 1,2 , Niklas Linde 3 , Diederik Jacques 1 , and Jasper A. Vrugt 2,4 Institute for Environment, Health and Safety, Belgian Nuclear Research Centre, Mol, Belgium, 2 Department of Civil and Environmental Engineering, University of California, Irvine, California, USA, 3 Applied and Environmental Geophysics Group, Institute of Earth Sciences, University of Lausanne, Lausanne, Switzerland, 4 Department of Earth Systems Science, University of California, Irvine, California, USA Abstract We present a Bayesian inversion method for the joint inference of high-dimensional multi- Gaussian hydraulic conductivity fields and associated geostatistical parameters from indirect hydrological data. We combine Gaussian process generation via circulant embedding to decouple the variogram from grid cell specific values, with dimensionality reduction by interpolation to enable Markov chain Monte Carlo (MCMC) simulation. Using the Mat ern variogram model, this formulation allows inferring the conductivity values simultaneously with the field smoothness (also called Mat ern shape parameter) and other geostatisti- cal parameters such as the mean, sill, integral scales and anisotropy direction(s) and ratio(s). The proposed dimensionality reduction method systematically honors the underlying variogram and is demonstrated to achieve better performance than the Karhunen-Loe`ve expansion. We illustrate our inversion approach using synthetic (error corrupted) data from a tracer experiment in a fairly heterogeneous 10,000-dimensional 2-D conductivity field. A 40-times reduction of the size of the parameter space did not prevent the posterior simulations to appropriately fit the measurement data and the posterior parameter distributions to include the true geostatistical parameter values. Overall, the posterior field realizations covered a wide range of geostatistical models, questioning the common practice of assuming a fixed variogram prior to inference of the hydraulic conductivity values. Our method is shown to be more efficient than sequential Gibbs sampling (SGS) for the considered case study, particularly when implemented on a distributed computing cluster. It is also found to outperform the method of anchored distributions (MAD) for the same computational budget. 1. Introduction High-parameter dimensionality poses considerable challenges for the inversion of groundwater flow and transport data [e.g., Kitanidis, 1995; Hendricks-Franssen et al., 2009; Laloy et al., 2013; Zhou et al., 2014, and references therein]. What is more, conceptual and structural inadequacies of the subsurface model and measurement errors of the model input (boundary conditions) and output (calibration) data introduce uncertainty in the estimated parameters and model simulations. Another important source of uncertainty originates from sparse data coverages that rarely contain sufficient information to uniquely characterize the subsurface at a spatial resolution deemed necessary for accurate modeling. This results in an ill-posed inverse problem with many different sets of model parameter values that fit the data acceptably well. Inver- sion methods should consider this inherent uncertainty and provide an ensemble of model realizations that accurately span the range of possible models that honor the available calibration data and prior information. C 2015. American Geophysical Union. V All Rights Reserved. LALOY ET AL. Hydraulic conductivity (K) fields are typically assumed to be stationary and log-normally distributed with a spatial structure determined by a two-point geostatistical model or variogram [e.g., Rubin, 2003]. Unfortu- nately, a lack of (sufficient) point K measurements (if any) makes it difficult to estimate directly the geostatis- tical parameters (mean, sill, variogram model, integral scales and anisotropy factors) from variographic analysis [Ortiz and Deutsch, 2002; Nowak et al., 2010]. Simultaneous (inverse) inference of conductivity values and associated geostatistical parameters is therefore attractive yet computationally challenging. Indeed, only a few studies can be found in the literature that have attempted simultaneous estimation using global and probabilistic search methods. For example, Jafarpour and Tarrahi [2011] used the Ensemble Kalman JOINT FIELD/VARIOGRAM MCMC INVERSION

Probabilistic 3-D time-lapse inversion of magnetotelluric data: application to an enhanced geothermal system

SUMMARY Surface-based monitoring of mass transfer caused by injections and extractions in deep boreholes is crucial to maximize oil, gas and geothermal production. Inductive electromagnetic methods, such as magnetotellurics, are appealing for these applications due to their large penetration depths and sensitivity to changes in fluid conductivity and fracture connectivity. In this work, we propose a 3-D Markov chain Monte Carlo inversion of time-lapse magnetotelluric data to image mass transfer following a saline fluid injection. The inversion estimates the posterior probability density function of the resulting plume, and thereby quantifies model uncertainty. To decrease computation times, we base the parametrization on a reduced Legendre moment decomposition of the plume. A synthetic test shows that our methodology is effective when the electrical resistivity structure prior to the injection is well known. The centre of mass and spread of the plume are well retrieved. We then apply our inversion strategy to an injection experiment in an enhanced geothermal system at Paralana, South Australia, and compare it to a 3-D deterministic time-lapse inversion. The latter retrieves resistivity changes that are more shallow than the actual injection interval, whereas the probabilistic inversion retrieves plumes that are located at the correct depths and oriented in a preferential north–south direction. To explain the time-lapse data, the inversion requires unrealistically large resistivity changes with respect to the base model. We suggest that this is partly explained by unaccounted subsurface heterogeneities in the base model from which time-lapse changes are inferred.

Linearized two-hydrophone localization of a pulsed acoustic source in the presence of refraction: Theory and simulations

This paper develops an efficient three-dimensional localization method for transient acoustic sources, with uncertainty estimation, based on time differences between direct and surface-reflected arrivals at two hydrophones. The localization method accounts for refraction caused by a depth-dependent sound-speed profile using a ray-theoretic approach for calculating eigenray travel times and partial derivatives. Further, the method provides localization error estimates accounting for uncertainties of the arrival times and hydrophone locations, as well as for depth-dependent uncertainties in the sound-speed profile. In the first of two steps, source depth and range to each hydrophone are estimated using an iterative, linearized Gauss-Markov inversion scheme. In the second step, the estimated source ranges are combined with the hydrophone locations to obtain the source location in the horizontal. Localization performance is analyzed in a simulation study, and the linearized localization estimates and uncertainties are validated by comparison with a fully nonlinear (but numerically intensive) Markov-chain Monte Carlo inversion.

Thin-sheet electromagnetic modeling of magnetovariational data for a regional-scale study

Naturally existing electromagnetic (EM) fields recorded at the surface of the Earth can be used to infer the electrical conductivity distribution of the subsurface. In the magnetovariational sounding (MVS) technique, the transient variations of orthogonal components of the Earth’s magnetic field are measured. In the frequency domain, the magnetic transfer function relates the vertical component to the horizontal components of the magnetic field. This paper describes the thin-sheet modeling of MVS data on a regional scale. The integrated conductivity variations in the thin-sheet model were estimated by applying the Markov chain Monte Carlo (MCMC) inversion algorithm. The application of this method to MVS data from the Finland part of the Fennoscandian Shield has illustrated the utility of both the thin-sheet approximation and MCMC inversion modeling. The conductivity anomalies obtained from this study confirmed the regional-scale geology of the area.

Markov chain Monte Carlo inversion for the rheology of olivine single crystals

AbstractWe present major modifications to the Markov chain Monte Carlo inversion method of Korenaga and Karato (2008), which was developed to analyze rock deformation data and determine a corresponding flow law and its uncertainty. The uncertainties of state variables, e.g., temperature, pressure, and stress, are now taken into account by data randomization, to avoid parameter bias that could be introduced by the original implementation of the cost function. Also, it is now possible to handle a flow law composed of both parallel and sequential deformation mechanisms, by using conjugate gradient search to determine scaling constants. We test the new inversion algorithm extensively using synthetic data as well as the high‐quality experimental data of Bai et al. (1991) on the deformation of olivine single crystals. Our reanalysis of this experimental data set reveals that a commonly adopted value for the stress exponent (∼3.5) is considerably less certain than previously reported, and we offer a detailed account for the validity of our new estimates. The significance of fully reporting parameter uncertainties including covariance is also discussed with a worked example on flow law prediction under geological conditions.