Deterministic Inversion Research Articles

Generating 3D lithology probability volumes using poststack inversion, probabilistic neural networks, and Bayesian classification — A case study from the mixed carbonate and siliciclastic deposits of the Cisco Group of the Eastern Shelf of the Permian Basin, north-central Texas

The deposition and mixing of carbonates and siliciclastics in the Cisco Group of the Eastern Shelf of the Permian Basin are complicated by the temporal overlap between icehouse eustatic sea-level oscillations and fluctuations in sediment influx due to the rejuvenation of the Ouachita fold belt. Previous investigators have used well-log correlation as the primary tool in their interpretations of the area’s reciprocal depositional model, but well-log correlation alone cannot explain the full range of spatial lithology variations in the system. To better understand the lithology variation in the area, we use an integrated technique that combines wireline log information from 17 wells with 625 km2 3D seismic data through poststack seismic inversion, probabilistic neural networks (PNNs), and Bayesian classification. We use deterministic matrix inversion to derive lithology classes from well logs. Crossplot analyses reveal that the acoustic impedance and neutron porosity log pair can be used to differentiate lithologies. We perform model-based poststack inversion to generate a P-impedance volume and use PNNs to generate a neutron porosity volume. We combine these volumes through supervised Bayesian classification to generate lithology probability volumes for each lithology and a most probable lithology volume throughout the seismic data. The lithology volumes highlight the dominant lithologies (carbonate, shale, sand, and mixed) that allowed the interpretation of major carbonate platforms, sand-to-shale ratio variations, carbonate buildups between wells, and channel fill lithologies. Our semiautomated lithology detection workflow applies to regional studies and is also valid for reservoir-scale studies to determine variations in lithologies.

Determination of the seismic structure of the Earth based on joint analysis of gravimetric and seismometric data – a case study

The evolution of the Earth’s surface is driven by external and internal forces, the latter of which can only be studied indirectly. Knowledge about the structure of the Earth’s interior is very important for modeling and predicting the processes occurring at the surface. This study presents a new concept of joint analysis of the gravimetric and seismometric recordings of earthquakes for determining the seismic structure of the Earth down to the depth of 1250 km. The proposed method allows the use of gravimetric data without the known full transfer function of the instrument. Group velocity dispersion curves of the fundamental mode of Rayleigh waves up to the period of 550 s are measured based on the joint analysis of the recordings of superconducting gravimeter and broadband seismometers operating at the same location in five testing sites in Europe, allowing for the exploration of a broader response for incoming seismic waves. Averaged dispersion curves for earthquakes around the world for each site are inverted by the weighted linear inversion and Monte Carlo methods to estimate the distribution of shear-wave seismic velocity in the Earth’s mantle. A comparison of the deterministic and probabilistic inversion methods can excellently demonstrate surface waves’ ability to determine the Earth’s mantle structure. The inversion results are compared with the global ak135 seismic model (Kennett et al. 1995) to verify the proposed method.

Three-dimensional cooperative inversion of airborne magnetic and gravity gradient data using deep-learning techniques

Using multiple geophysical methods has become a prevailing approach in numerous geophysical applications to investigate subsurface structures and parameters. These multimethod-based exploration strategies have the potential to greatly diminish uncertainties and ambiguities encountered during geophysical data analysis and interpretation. One of the applications is the cooperative inversion of airborne magnetic and gravity gradient data for the interpretation of data obtained in mineral, oil and gas, and geothermal explorations. In this paper, a unified cooperative inversion framework is designed by combining the standard separate inversions with a deep neural network (DNN), which serves as the link between different types of data. A well-trained DNN takes the separately inverted susceptibility and density models as the inputs and provides improved models that will be used as the initial models of deterministic inversions. A two-round iteration strategy is adopted to guarantee the reasonability of the recovered models and overall efficiency of the inversion. In addition, this deep-learning (DL)-based framework demonstrates excellent generalization abilities when tested on models that are entirely distinct from the training data sets. The framework can easily incorporate multiphysics without necessitating any structural changes to the network. Synthetic experiments validate that our DL-based method outperforms conventional separate inversions and cross-gradient-based joint inversion in view of the accuracy of the recovered models and inversion efficiency. Successful application to field data further verifies the effectiveness of our DL-based method.

Physics-Informed Deep Learning Inversion with Application to Noisy Magnetotelluric Measurements

Despite demonstrating exceptional inversion production for synthetic data, the application of deep learning (DL) inversion methods to invert realistic magnetotelluric (MT) measurements, which are inevitably contaminated by noise in acquisition, poses a significant challenge. Hence, to facilitate DL inversion for realistic MT measurements, this work explores developing a noise-robust MT DL inversion method by generating targeted noisy training datasets and constructing a physics-informed neural network. Different from most previous works that only considered the noise of one fixed distribution and level, we propose three noise injection strategies and compare their combinations to mitigate the adverse effect of measurement noise on MT DL inversion results: (1) add synthetic relative noise obeying Gaussian distribution; (2) propose a multiwindow Savitzky–Golay (MWSG) filtering scheme to extract potential and possible noise from the target field data and then introduce them into training data; (3) create an augmented training dataset based on the former two strategies. Moreover, we employ the powerful Swin Transformer as the backbone network to construct a U-shaped DL model (SwinTUNet), based on which a physics-informed SwinTUNet (PISwinTUNet) is implemented to further enhance its generalization ability. In synthetic examples, the proposed noise injection strategies demonstrate impressive inversion effects, regardless of whether they are contaminated by familiar or unfamiliar noise. In a field example, the combination of three strategies drives PISwinTUNet to produce considerably faithful reconstructions for subsurface resistivity structures and outperform the classical deterministic Occam inversions. The experimental results show that the proposed noise-robust DL inversion method based on the noise injection strategies and physics-informed DL architecture holds great promise in processing MT field data.

Probabilistic seismic inversion based on physics-guided deep mixture density network

Deterministic inversion based on deep learning has been widely utilized in model parameters estimation. Constrained by logging data, seismic data, wavelet and modeling operator, deterministic inversion based on deep learning can establish nonlinear relationships between seismic data and model parameters. However, seismic data lacks low-frequency and contains noise, which increases the non-uniqueness of the solutions. The conventional inversion method based on deep learning can only establish the deterministic relationship between seismic data and parameters, and cannot quantify the uncertainty of inversion. In order to quickly quantify the uncertainty, a physics-guided deep mixture density network (PG-DMDN) is established by combining the mixture density network (MDN) with the deep neural network (DNN). Compared with Bayesian neural network (BNN) and network dropout, PG-DMDN has lower computing cost and shorter training time. A low-frequency model is introduced in the training process of the network to help the network learn the nonlinear relationship between narrowband seismic data and low-frequency impedance. In addition, the block constraints are added to the PG-DMDN framework to improve the horizontal continuity of the inversion results. To illustrate the benefits of proposed method, the PG-DMDN is compared with existing semi-supervised inversion method. Four synthetic data examples of Marmousi II model are utilized to quantify the influence of forward modeling part, low-frequency model, noise and the pseudo-wells number on inversion results, and prove the feasibility and stability of the proposed method. In addition, the robustness and generality of the proposed method are verified by the field seismic data.

Reversing Unknown Qubit-Unitary Operation, Deterministically and Exactly.

We report a deterministic and exact protocol to reverse any unknown qubit-unitary operation, which simulates the time inversion of a closed qubit system. To avoid known no-go results on universal deterministic exact unitary inversion, we consider the most general class of protocols transforming unknown unitary operations within the quantum circuit model, where the input unitary operation is called multiple times in sequence and fixed quantum circuits are inserted between the calls. In the proposed protocol, the input qubit-unitary operation is called 4 times to achieve the inverse operation, and the output state in an auxiliary system can be reused as a catalyst state in another run of the unitary inversion. We also present the simplification of the semidefinite programming for searching the optimal deterministic unitary inversion protocol for an arbitrary dimension presented by M. T. Quintino and D. Ebler [Quantum 6, 679 (2022)2521-327X10.22331/q-2022-03-31-679]. We show a method to reduce the large search space representing all possible protocols, which provides a useful tool for analyzing higher-order quantum transformations for unitary operations.

Bayesian estimation of parameters and uncertainty of a 3-D dipping fault using gravity anomalies

SUMMARY Many significant geophysical anomalies are generated by fault systems, and effectively estimating fault parameters is a key step to understand the characteristics of these field sources. In general, the 2-D dipping fault is frequently utilized to simplify the representation of fault geometry. However, this approach could lead to larger errors when comparing realistic scenarios with simplifications. In this study, we use a parametrized 3-D dipping model to represent fault systems and compute gravity anomalies using an analytic method in the wavenumber domain. To estimate the marginal distribution of the 3-D dipping fault parameters, we utilize the Bayesian theory that integrates the likelihood function and prior information. Subsequently, we use the Markov chain Monte Carlo sampling method to assess the marginal distribution of the parameters. Compared with other deterministic inversion approaches, our method provides an effective means of quantifying parameter uncertainty. We first use synthetic data to validate the effectiveness of the proposed method. The results demonstrate the efficiency of our method in revealing 3-D dipping fault parameters and evaluating their uncertainty. We subsequently apply our method to the Pudu River fault, located in Yunnan Province, China, to demonstrate its practical applicability. Two alternative realistic fault models referred the prior seismic profile have been proposed. The optimal parameters and uncertainty with the specific model have been estimated. The results show that the fault characteristics retrieved by our method deviate slightly from existing knowledge. Specifically, the estimated depth of the bottom surface is greater than that reported in previous studies. This could potentially offer us new perspectives on the geological evolution and structural attributes of this fault.

A high resolution inversion method for fluid factor with dynamic dry-rock VP/VS ratio squared

As an important indicator parameter of fluid identification, fluid factor has always been a concern for scholars. However, when predicting Russell fluid factor or effective pore-fluid bulk modulus, it is necessary to introduce a new rock skeleton parameter which is the dry-rock VP/VS ratio squared (DVRS). In the process of fluid factor calculation or inversion, the existing methods take this parameter as a static constant, which has been estimated in advance, and then apply it to the fluid factor calculation and inversion. The fluid identification analysis based on a portion of the Marmousi 2 model and numerical forward modeling test show that, taking the DVRS as a static constant will limit the identification ability of fluid factor and reduce the inversion accuracy. To solve the above problems, we proposed a new method to regard the DVRS as a dynamic variable varying with depth and lithology for the first time, then apply it to fluid factor calculation and inversion. Firstly, the exact Zoeppritz equations are rewritten into a new form containing the fluid factor and DVRS of upper and lower layers. Next, the new equations are applied to the four parameters simultaneous inversion based on the generalized nonlinear inversion (GNI) method. The testing results on a portion of the Marmousi 2 model and field data show that dynamic DVRS can significantly improve the fluid factor identification ability, effectively suppress illusion. Both synthetic and filed data tests also demonstrate that the GNI method based on Bayesian deterministic inversion (BDI) theory can successfully solve the above four parameter simultaneous inversion problem, and taking the dynamic DVRS as a target inversion parameter can effectively improve the inversion accuracy of fluid factor. All these results completely verified the feasibility and effectiveness of the proposed method.

Inversion géostatistique par itérations de la tomographie de résistivité électrique

AbstractElectrical resistivity tomography (ERT) is a geophysical method used to create an image of the subsurface due to its sensitivity to porosity, water saturation, and pore fluid salinity. This geophysical method has been widely applied in the investigation of mineral and groundwater resources, as well as in archaeological, environmental, and engineering studies. The prediction of subsurface properties, such as electrical conductivity, from measured ERT data requires solving a challenging geophysical inversion problem. This work proposes an iterative geostatistical resistivity inversion method using stochastic sequential simulation and co-simulation as stochastic model perturbation and update techniques. Electrical resistivity models are generated conditioned to a target histogram, often retrieved from available resistivity borehole data, and assuming a spatial continuity pattern described by a variogram model. From the electrical resistivity models, a finite-volume approximation of Poisson’s equation is used to compute synthetic ERT data. The misfit between predicted and observed data drives the convergence of an iterative procedure and conditions the co-simulation of new models in the subsequent iterations. This methodology is applied to a two-dimensional synthetic case, and a set of two-dimensional profiles obtained from an ERT survey carried out in southern Portugal. In both application examples, the final models predict ERT data that match the observed ones while reproducing borehole data and imposed variogram models. The results obtained in both data sets are compared against a commercial deterministic ERT inversion methodology, showing the ability of the proposed method to model small-scale variability and assess spatial uncertainty.

Application of audio-frequency magnetotelluric data to cover characterisation – validation against borehole petrophysics in the East Tennant region, Northern Australia

The characterisation of the thickness and geology of cover sequences significantly improves targeting for mineral exploration in buried terrains. Audio-frequency Magnetotelluric (AMT) data is applicable to characterise cover sequences, where their conductivity (inverse resistivity) can be differentiated. We present a regional study from the under-cover East Tennant region in the Northern Territory (Australia) where we have applied deterministic and probabilistic inversion methods to derive 2D and 1D resistivity models. We integrated these models with information of co-located basement penetrating boreholes (lithological and geophysical logs) to ground-truth and validate the models and to improve geophysical interpretations. In the East Tennant region, borehole lithology and wireline logging demonstrate that the modelled AMT response is largely controlled by the mineralogy of the cover and basement rocks. The bulk conductivity is due primarily to bulk mineralogy and the success of using the AMT models to predict cover thickness is shown to be dependent on whether the bulk mineralogy of cover and basement rocks are sufficiently different to provide a detectable conductivity contrast. Our investigation of a range of geological scenarios that differ in thickness, complexity and geology of the cover and basement rocks suggests that in areas where there is sufficient difference in bulk mineralogy and where the stratigraphy is relatively simple, AMT models predict the cover thickness with high certainty. In more complex scenarios interpretation of AMT models may be more ambiguous and requires integration with other data (e.g. drilling, wireline logging, potential field modelling). Overall, we conclude that the application of the method has been validated and the results compare favourably with borehole stratigraphy logs once geological (i.e. bulk mineralogical) complexity is understood. This demonstrates that the method is capable of identifying major litho-stratigraphic units with resistivity contrasts. Our results have assisted with the planning of regional drilling programs and have helped to reduce the uncertainty and risk associated with intersecting targeted stratigraphic units in covered terrains.

An information theoretic Bayesian uncertainty analysis of AEM systems over Menindee Lake, Australia

SUMMARY Long-range, active-source airborne electromagnetic (AEM) systems for near-surface conductivity imaging fall into two categories: helicopter (rotary-wing) borne or fixed-wing aircraft borne. A multitude of factors such as flying height, transmitter loop area and current, source waveforms, aerodynamic stability and data stacking times contribute to the geological resolvability of the subsurface. A comprehensive comparison of the relative merits of each system considering all such factors is difficult, but test flights over well-constrained subsurface geology with downhole induction logs are extremely useful for resolution studies. However, given the non-linear nature of the electromagnetic inverse problem, handling transmitter–receiver geometries in fixed-wing aircraft is especially challenging. As a consequence of this non-linearity, inspecting the closeness of downhole conductivities to deterministic inversion results is not sufficient for studying resolvability. A more comprehensive picture is provided by examining the variation in probability mass of the depth-wise Bayesian posterior conductivity distributions for each kind of AEM system within an information theoretic framework. For this purpose, probabilistic inversions of data must be carried out. Each acquiring system should fly over the same geology, survey noise levels must be measured and the same prior probabilities on conductivity must be used. With both synthetic models as well as real data from over the Menindee calibration range in New South Wales, Australia, we shed new light on the matter of AEM inverse model uncertainty. We do this using two information theoretic attributes derived from different Kullback–Leibler divergences—Bayesian information gain, and a strictly proper scoring rule, to assess posterior probabilities estimated by a novel Bayesian inversion scheme. The inversion marginalizes fixed-wing geometry attributes as generic nuisance parameters during Markov chain sampling. This is the first time-domain AEM study we know of, that compares nuisance marginalized subsurface posterior conductivities from a fixed-wing system, with rotary-wing derived posterior conductivities. We also compare field results with induction log data where available. Finally, we estimate the information gain in each case via a covariate shift adaptation technique that has not been used before in geophysical work. Our findings have useful implications in AEM system selection, as well as in the design of better deterministic AEM inversion algorithms.

Bayesian ensemble machine learning-assisted deterministic and stochastic groundwater DNAPL source inversion with a homotopy-based progressive search mechanism

A hyperheuristic homotopy algorithm (HH-HA) and a homotopy-based swarm intelligence algorithm for parallel stochastic search are proposed to improve the search ergodicity and stability of deterministic and stochastic inversion approaches, respectively, for the identification of dense nonaqueous phase liquid (DNAPL) source characteristics and contaminant transport parameters in groundwater. Meanwhile, a Bayesian ensemble machine learning (BEML) method for numerical simulation surrogate modeling is proposed to enhance the learning and generalization capability, while significantly reducing the computational cost of repetitive CPU-demanding likelihood evaluations in inversion iterations. Two cases are presented for the verification and application of the proposed inverse modeling methods. The results show that the BEML surrogate model is powerful in approximating the numerical model outputs, reducing the mean relative errors of the contaminant concentration and pressure head predictions to 1.18% and 1.29%, respectively. The homotopy-based progressive search mechanism is highly effective in achieving more accurate identification values for deterministic inversion and a more reliable posterior distribution for stochastic inversion. The HH-HA shows stable performance, approaching the global optimum in wide areas and reducing the mean relative error of identification from 11.95% to 4.11%. The homotopy-based parallel stochastic search lines adequately cover the search space tracking the change of homotopy parameter, and the mean relative error of point estimation is reduced to 2.03%. Though the deterministic inversion approach can provide intuitive identification values, the reliability of the inversion outputs cannot be quantified. In contrast, the proposed stochastic inversion system is of great significance for providing decision makers with comprehensive reference information.

An Assessment of the Laminar Hypersonic Double-Cone Experiments in the LENS-XX Tunnel

This is an investigation on two experimental datasets of laminar hypersonic flows, over a double-cone geometry, acquired in Calspan—University at Buffalo Research Center’s Large Energy National Shock (LENS)-XX expansion tunnel. These datasets have yet to be modeled accurately. A previous paper suggested that this could partly be due to mis-specified inlet conditions. The authors of this paper solved a Bayesian inverse problem to infer the inlet conditions of the LENS-XX test section and found that in one case they lay outside the uncertainty bounds specified in the experimental dataset. However, the inference was performed using approximate surrogate models. In this paper, the experimental datasets are revisited and inversions for the tunnel test-section inlet conditions are performed with a Navier–Stokes simulator. The inversion is deterministic and can provide uncertainty bounds on the inlet conditions under a Gaussian assumption. It was found that deterministic inversion yields inlet conditions that do not agree with what was stated in the experiments. An a posteriori method is also presented to check the validity of the Gaussian assumption for the posterior distribution. This paper contributes to ongoing work on the assessment of datasets from challenging experiments conducted in extreme environments, where the experimental apparatus is pushed to the margins of its design and performance envelopes.

Integrating Deep Learning and Deterministic Inversion for Enhancing Fault Detection in Electrical Resistivity Surveys

Clays in fault zones have low electrical resistivity, making electrical resistivity tomography (ERT) effective for fault investigations. However, traditional ERT inversion methods struggle to find a unique solution and produce unstable results owing to the ill-posed nature of the problem. To address this, a workflow integrating deep-learning (DL) technology with traditional ERT inversion is proposed. First, a deep-learning model named DL-ERT inversion that maps apparent resistivity data to subsurface resistivity models is developed. To create target-oriented training data, we use approximately 150 field borehole data acquired from various survey areas in South Korea. The DL-ERT inversion algorithm is based on a U-Net structure and includes an additional network called the borehole mixer to incorporate borehole information when available. The DL-ERT inversion model is trained in three stages: base model training, borehole mixer training, and fine-tuning. Results showed that the fine-tuning model provided the highest prediction accuracy for all test datasets. Next, the prediction of the trained model is used as the initial model for the deterministic inversion method to predict the final subsurface model. The efficiency and accuracy of the proposed workflow are demonstrated in fault detection using a field data example compared with traditional deterministic inversion.

Machine-learning inversion via adaptive learning and statistical sampling: Application to airborne micro-TEM for seismic sand corrections

We develop a novel physics-adaptive machine-learning (ML) inversion scheme showing optimal generalization capabilities for field data applications. We apply the physics-driven deep-learning inversion to a massive helicopter-borne transient electromagnetic (TEM) field data set. The objective is the accurate modeling of the near surface for enhancing the exploration of low-relief structures in a sand covered desertic area. Enhanced generalization of a neural network (NN) or other ML techniques is obtained from automatic physics-based adaptive training using images of real-world data. Data reduction schemes, based on statistical sampling techniques, enable the use of small and fully informative data sets for accelerated training. Adaptive learning is implemented via an iterative physics-driven data augmentation strategy. A deterministic inversion is regularized by a penalty term built from the difference between the inverted models and the models predicted by ML. The resulting inverted models and calculated responses are used for augmenting the ML training base. The automated procedure converges rapidly, producing a trained network model, which captures the general background data statistics as well as the characteristics of field-specific geophysical data as embedded in the augmented samples. The high-resolution ML inversions with acceptable levels of data misfit are obtained for large volumes of the field geophysical data. We test several different learning models, such as artificial NNs, convolutional NN (U-Net), and the Gaussian process, which provide stable and comparable results with network-specific characteristics. The developed scheme is successfully applied to micro-TEM data providing accurate near-surface seismic corrections for the exploration of low-relief structures.

VALIDATION OF EXTENDED ELASTIC IMPEDANCE BASED ON MINIMUM ENERGY ANGLE. A CASE STUDY APPLICATION FOR OPTIMIZED PETROLEUM RESERVOIR CHARACTERIZATION

This work intends to showcase a validation of the applicability of Extended Elastic Impedance (EEI) inversion method in reservoir characterization and modeling. In order to achieve that, deterministic seismic inversion and extended elastic impedance (EEI) analysis were applied to obtain quantitative estimates of reservoir properties over the Pu field of the West African Congo basin. Optimum EEI angles corresponding to the reservoir properties were then analyzed using well logs data, together with a lithology indicator. Pre-stack seismic data were simultaneously inverted into density, acoustic and gradient impedances cubes, through model based inversion algorithm. The last two broadband inverted volumes were then projected to corresponding Chi angles proportionate to petrophysical indicators, thus resulting to two broadband EEI volumes. At well locations, the EEI versus petrophysical parameters linear trends were then applied to convert EEI volumes into porosity and shale volumes based on specified lithology. In order to generate reservoir facies distribution, minimum angle was applied based on background EEI, thus allowing for mapping of reservoir facies. In order to validate the EEI approach, a Geo-statistical model was further developed for the same field. Hence, from porosity, shale content and background EEI cubes, a comparison was made between the properties generated from the EEI and that generated based on geo-statistical method, which shows that EEI is a robust way of reservoir characterization that pinpointing favorable reservoir potentials which shall guide future drilling locations.

A Reconstructed Method of Acoustic Logging Data and Its Application in Seismic Lithological Inversion for Uranium Reservoir

As a sedimentary mineral, most sandstone type uranium deposits are formed in petroliferous basins. Therefore, we can fully tap the residual economic value of historical logging and 3D seismic data measured for oil and gas to search for sandstone type uranium deposits. However, a large amount of acoustic logging data are missing in the target stratum of the uranium reservoir in that it is not the main stratum of oil and gas. A reconstructed method of acoustic logging data based on clustering analysis and with the low-frequency compensation of deterministic inversion is proposed to solve this problem. Secondly, we can use these logging data with seismic data to obtain the 3D inversion data volume representing the sand body of the uranium reservoir based on seismic lithological inversion. Then, we can also delimit the 3D spatial range of sandstone type uranium deposits in petroliferous basins based on the calibration of uranium anomaly and sub-body detection. Finally, a 3D field data example is given to test and analyze the effectiveness of the above research schemes.

Efficient Bayesian inversion of borehole geophysical measurements with a gradient‐based Markov chain Monte Carlo method

AbstractWell logs are geophysical measurements of rock properties acquired continuously along a borehole. Because of the physics underlying their operation, borehole logging instruments perform local spatial averages of rock properties in the vicinity of the borehole, yielding well logs that are often affected by tool design, layer boundaries, borehole environmental conditions and mud‐filtrate invasion. Such environmental effects are ubiquitous and can lead to errors in petrophysical–property estimations from well logs if not accounted for in the interpretation. Separate well‐log inversion mitigates environmental effects by matching well logs with their numerical simulations. The latter simulations rely on specific assumptions about the relative geometry of the borehole and the rocks penetrated by the well. For vertical wells penetrating horizontal layers, it is common to assume piecewise‐constant layer properties that we collectively refer to as the earth model; some of these properties have a direct relationship to well logs (e.g. resistivity, gamma ray, density and acoustic slowness). Well‐log inversion is traditionally approached with deterministic methods such as the Levenberg–Marquardt algorithm to minimize the differences between well logs and their numerical simulations, often without accounting for the relationship between measurement noise and model uncertainty. Bayesian inversion, on the other hand, yields the posterior probability distribution of estimated properties and intrinsically quantifies their uncertainty. However, Bayesian Well‐log inversion usually involves the implementation of Markov chain Monte Carlo sampling, which requires a prohibitive number of forward simulations and is therefore not suitable for rapid petrophysical and/or elastic and mechanical evaluations of rocks. We introduce an efficient Bayesian Well‐log inversion method using a gradient‐based Markov chain Monte Carlo method. Gradient‐based Markov chain Monte Carlo is a group of relatively new Markov chain Monte Carlo algorithms that combine gradient updates with Hessian‐based sampling. gradient‐based Markov chain Monte Carlo draws samples efficiently from the posterior probability distribution using the gradient information to guide samples towards high‐probability regions and the Hessian to approximate the posterior probability distribution locally. We verify the gradient‐based Markov chain Monte Carlo inversion method with the inversion of synthetic well logs, including gamma ray, resistivity, density, neutron porosity, photoelectric factor and compressional/shear‐wave slowness. Results show that gradient‐based Markov chain Monte Carlo decreases the computational cost by more than 90% compared to conventional Markov chain Monte Carlo sampling. Next, gradient‐based Markov chain Monte Carlo inversion is applied to a field example from the Central North Sea, where conventional interpretation methods yield shale concentration, sandstone porosity and water saturation with errors up to 19.2%, 14.9% and 48.8%, respectively. Gradient‐based Markov chain Monte Carlo inversion products satisfactorily reproduce the available measurements with only a modest increase in computational cost compared to deterministic inversion approaches.

Trans-dimensional Bayesian joint inversion of magnetotelluric and geomagnetic depth sounding responses to constrain mantle electrical discontinuities

SUMMARYJoint inversion of magnetotelluric (MT) and geomagnetic depth sounding (GDS) responses can constrain the crustal and mantle conductivity structures. Previous studies typically use either deterministic inversion algorithms that provide limited information on model uncertainties or using stochastic inversion algorithms with a predetermined number of layers that is generally not known a priori. Here, we present a new open-source Bayesian framework for the joint inversion of MT and GDS responses to probe 1D layered Earth’s conductivity structures. Within this framework, model uncertainties can be accurately estimated by generating numerous models that fit the observed data. A trans-dimensional Markov Chain Monte Carlo (MCMC) method is employed to self-parametrize the model parameters, where the number of layers is treated as an inversion parameter that is determined automatically by the data. This adaptability can overcome the under or over-parametrization problem and may be able to automatically detect the conductivity discontinuities in the Earth’s interior. To accelerate the computations, a large number of Markov chains with different initial states can be run simultaneously using the MPI parallel technique. Synthetic data sets are used to validate the feasibility of our method and illustrate how separate and joint inversions, as well as various priors affect the posterior model distributions. The trans-dimensional MCMC algorithm is then applied to jointly invert the MT and GDS responses estimated at the Tucson geomagnetic observatory, North America. Our results not only contain model uncertainty estimates but also indicate two distinct conductivity discontinuities at around 85 and 440 km, which are likely related to the lithosphere-asthenosphere boundary and the upper interface of the mantle transition zone, respectively.

A 3-D Full Convolution Electromagnetic Reconstruction Neural Network (3-D FCERNN) for Fast Super-Resolution Electromagnetic Inversion of Human Brain.

Three-dimensional (3-D) super-resolution microwave imaging of human brain is a typical electromagnetic (EM) inverse scattering problem with high contrast. It is a challenge for the traditional schemes based on deterministic or stochastic inversion methods to obtain high contrast and high resolution, and they require huge computational time. In this work, a dual-module 3-D EM inversion scheme based on deep neural network is proposed. The proposed scheme can solve the inverse scattering problems with high contrast and super-resolution in real time and reduce a huge computational cost. In the EM inversion module, a 3-D full convolution EM reconstruction neural network (3-D FCERNN) is proposed to nonlinearly map the measured scattered field to a preliminary image of 3-D electrical parameter distribution of the human brain. The proposed 3-D FCERNN is completely composed of convolution layers, which can greatly save training cost and improve model generalization compared with fully connected networks. Then, the image enhancement module employs a U-Net to further improve the imaging quality from the results of 3-D FCERNN. In addition, a dataset generation strategy based on the human brain features is proposed, which can solve the difficulty of human brain dataset collection and high training cost. The proposed scheme has been confirmed to be effective and accurate in reconstructing the distribution of 3-D super-resolution electrical parameters distribution of human brain through noise-free and noisy examples, while the traditional EM inversion method is difficult to converge in the case of high contrast and strong scatterers. Compared with our previous work, the training of FCERNN is faster and can significantly decrease computational resources.