The nonuniqueness problem in geophysical inversion, especially potential-field inversion, is widely recognized. It is argued that uncertainty analysis of a recovered model should be as important as finding an optimal model. However, quantifying uncertainty still remains challenging, especially for 3D inversions in both deterministic and Bayesian frameworks. Our objective is to develop an efficient method to empirically quantify the uncertainty of the physical property models recovered from 3D potential-field inversion. We work in a deterministic framework in which an objective function consisting of a data misfit term and a regularization term is minimized. We perform inversions using a mixed Lp-norm formulation in which various combinations of Lp (0≤p≤2) norms can be implemented on different components of the regularization term. Specifically, we randomly sample the p-norm values in multiple times, and generate a large and diverse sequence of physical property models that all reproduce the observed geophysical data equally well. This suite of models offers practical insights into the uncertainty of the recovered model features. We quantify the uncertainty through calculation of standard deviations and interquartile range, as well as visualizations in box plots and histograms. The numerical results for a realistic synthetic density model created based on a ring-shaped igneous intrusive body quantitatively illustrate uncertainty reduction due to different amounts of prior information imposed on inversions. We also apply the method to a field data set over the Decorah area in northeastern Iowa. We adopt an acceptance-rejection strategy to generate 31 equivalent models based on which the uncertainties of the inverted models as well as the volume and mass estimates are quantified.

Joint inversion has drawn considerable attention due to the availability of multiple geophysical data sets, ever-increasing computational resources, the development of advanced algorithms, and its ability to reduce inversion uncertainty. A key issue of joint inversion is to develop effective strategies to link different geophysical data in a unified mathematical framework, in which the information obtained from different models can complement each other. We have developed a deep learning-enhanced joint inversion framework to simultaneously reconstruct different physical models by fusing different types of geophysical data. Traditionally, structure similarity constraints are pursued by joint inversion algorithms using manually crafted formulations (e.g., cross gradient). The constraint is constructed by a deep neural network (DNN) during the learning process. The framework is designed to combine the DNN and a traditional independent inversion workflow and improve the joint inversion result iteratively. The network can be easily extended to incorporate multiphysics without structural changes. Numerical experiments on the joint inversion of 2D DC resistivity data and seismic traveltime are used to validate our method. In addition, this learning-based framework demonstrates excellent generalization abilities when tested on data sets using different geologic structures. It also can handle different sensing configurations and nonconforming discretization.

Mineral exploration under a thick sedimentary cover naturally relies on geophysical methods. We have used high-resolution airborne magnetic and gravity gradient data over northeast Iowa to characterize the geology of the concealed Precambrian rocks and evaluate the prospectivity of mineral deposits. Previous researchers have interpreted the magnetic and gravity gradient data in the form of a 2D geologic map of the Precambrian basement rocks, which provides important geophysical constraints on the geologic history and mineral potentials over the Decorah area located in the northeast of Iowa. However, their interpretations are based on 2D data maps and are limited to the two horizontal dimensions. To fully tap into the rich information contained in the high-resolution airborne geophysical data, and to further our understanding of the undercover geology, we have performed separate and joint inversions of magnetic and gravity gradient data to obtain 3D density contrast models and 3D susceptibility models, based on which we carried out geology differentiation. Based on separately inverted physical property values, we have identified 10 geologic units and their spatial distributions in 3D which are all summarized in a 3D quasi-geology model. The extension of 2D geologic interpretation to 3D allows for the discovery of four previously unidentified geologic units, a more detailed classification of the Yavapai country rock, and the identification of the highly anomalous core of the mafic intrusions. Joint inversion allows for the classification of a few geologic units further into several subclasses. We have demonstrated the added value of the construction of a 3D quasi-geology model based on 3D separate and joint inversions.

The physical property models obtained from geophysical inversions can be converted to a 3D quasi-geology model via a process called geology differentiation. Recent works indicate that the geology differentiation can help maximize the value of the information contained in geophysical data. However, it remains largely unexplored as to how to quantify the uncertainties of a 3D quasi-geology model. We approach this problem by using a recently developed mixed Lp norm regularization and a priori physical property measurements. We use mixed Lp norm joint inversion to construct a large sequence of physical property models based on the Gzz component of the airborne gravity gradient and magnetic measurements. The available physical property measurements are used to determine which physical property models to accept. We then construct a sequence of 3D quasi-geology models by performing the geology differentiation for all of the accepted models, which allows us to compute the probabilities of our geology differentiation results. We apply our approach to a set of field data collected over the Decorah area located in northeast Iowa. We successfully quantify the uncertainties of the spatial extents for the identified geologic units and compute probabilities of geologic units at any location in our study area. The proposed workflow has broad implications for 3D geologic model building based on multiple geophysical and/or rock sample measurements.

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.

SUMMARY: Accurate delineation of salt body shapes is critical for hydrocarbon exploration. Various imaging methods based on seismic data have been developed. Due to the density contrast between salt and sedimentary rocks, gravity data have also been used as a de-risking tool to constrain the salt body shapes. However, quantifying uncertainties of the salt body shapes recovered from gravity data remains underexplored. Our goal is to understand and quantify how different constraints affect uncertainties of the salt body shapes reconstructed from gravity data. We adopt a trans-dimensional Markov chain Monte Carlo (MCMC) approach to explore the uncertainties. To address the computational challenges with MCMC sampling, we resort to two methods: sparse geometry parametrization and randomized parallel tempering. The first uses a set of simple geometries (e.g. ellipses) to approximate the complex shapes of salt bodies, greatly reducing the number of parameters to be sampled and making the MCMC approach computationally feasible. The second serves to further improve the acceptance ratio and computational efficiency. To quantify the uncertainties of the recovered salt body shapes, we design several scenarios to simulate different constraints on the top boundary of salt bodies from seismic imaging. We develop a new method to impose structural constraints on the top boundaries of salt bodies. This new method combines a set of fixed ellipses with randomly sampled ellipses through a concave hull. The results from different scenarios are compared to understand how uncertainties are reduced when stronger constraints are imposed. In addition, to make our uncertainty quantification results more relevant for practitioners, we propose to compute the salt probability models which show the spatial distribution of probabilities of salt materials at each cell. Finally, we investigate the effect of an uncertain salt density on the salt body reconstruction and the case of depth-varying densities in the sedimentary background. We apply our methods to the modified 2-D SEG-EAGE and Sigsbee salt models and quantify the uncertainties of the recovered salt body shapes in different scenarios. Our results highlight the importance of properly interpreting the uncertainty estimates in light of prior information and information content in the data.