Xin Zhang

SUMMARY: To address the challenge of assessing the reliability of 3-D magnetotelluric (MT) inversion results, we have developed a variational inference (VI) inversion framework (VI-MT) based on the Stein Variational Gradient Descent (SVGD) method. Through parallel particle optimization, we efficiently approximate the model posterior distribution, overcoming the computational limitation of traditional Markov Chain Monte Carlo methods. Synthetic tests show that the VI-MT inversion can effectively recover the synthetic model while reducing the tailing effect, and it can approximate the multimodal posterior distribution, providing quantitative uncertainty estimates. Furthermore, the VI-MT inversion is applied to the field MT data collected at the Weishan volcano in northeast China, with the posterior mean model consistent with the deterministic inversion model. A clear low resistivity body of ~5 Ω m with small uncertainties is imaged at depths of ~2–6 km beneath the volcanic crater, suggesting the existence of a shallow magma chamber. Our study shows that the VI-MT inversion based on the SVGD method can efficiently solve the 3-D MT Bayesian inversion, providing reliable model uncertainties.

SUMMARY: Seismic surface wave tomography uses surface wave information to obtain velocity structures in the subsurface. Due to data noise and nonlinearity of the problem, surface wave tomography often has non-unique solutions. It is therefore required to quantify uncertainty of the results in order to better interpret the resulting images. Bayesian inference is the most widely used method for this purpose. However, the commonly used Monte Carlo methods require huge computational cost and remains intractable in high-dimensional problems. Variational inference uses optimization to solve Bayesian inverse problems, and therefore can be more efficient in the case of large data sets and high-dimensional parameter spaces. Variational inference has been widely applied to 2-D phase velocity map inversion. In this study, we extend the method to 3-D surface wave tomography by directly inverting for 3-D spatial seismic velocity structures from frequency-dependent traveltime measurements. Specifically, we apply three variational methods, mean-field automatic differential variational inference (mean-field ADVI), physically structured variational inference (PSVI) and stochastic Stein varational gradient descent (sSVGD) to surface wave tomographic problems using both synthetic data and real data in the Southwest China. The results show that all methods can provide accurate velocity mean estimates, while sSVGD produces more reasonable uncertainty estimates than mean-field ADVI and PSVI because of Gaussian assumption used in these methods. In the real data case, the variational methods provide more detailed velocity structures than those obtained using traditional linearized methods, along with reliable uncertainty estimates. We therefore conclude that variational surface wave tomography can be applied fruitfully to many realistic problems.

To address the challenge of assessing the reliability of three-dimensional (3D) magnetotelluric (MT) inversion results, we have developed a variational inference (VI) inversion framework (VI-MT) based on the Stein Variational Gradient Descent (SVGD) method. Through parallel particle optimization, we efficiently approximate the model posterior distribution, overcoming the computational limitation of traditional Markov Chain Monte Carlo (MCMC) methods. Synthetic test shows the VI-MT inversion results align with deterministic solutions while better recovering resistivity amplitudes. In comparison, the VI-MT inversion can effectively identify large model uncertainties in the boundary region by the associated multimodal posterior characteristics. Furthermore, the VI-MT inversion is applied to the field MT data collected at the Weishan volcano in northeast China, with the posterior mean model consistent with the deterministic inversion model. Depth-dependent model uncertainties indicate strong data constraints in the upper crust. A clear low resistivity body of ~5 Ω·m with small uncertainties is imaged at depths of ~2-6 km beneath the volcanic crater, suggesting the existence of a shallow magma chamber. Our study shows that the VI-MT inversion based on the SVGD method can efficiently solve the 3D MT Bayesian inversion, providing reliable model uncertainties.

Geothermal systems provide a low-carbon, renewable source of heat, whose performance depends on the presence of permeable, fluid-filled rock at depth. Tomographic images of compressional and shear-wave velocities, Vp and Vs, and their ratio, Vp/Vs, are typically used to constrain the lithology, porosity, fluid content and extent of fracturing in such systems: contrasts in seismic velocity delineate lithological boundaries, identify zones of fracture damage or fluid saturation, and thereby indicate areas of elevated permeability.Passive seismic acquisition is attractive for geothermal exploration, because it is minimally invasive and can exploit microseismicity recorded by dense nodal seismological arrays. Combining data recorded from microseismic events with Bayesian joint inversion of seismic velocity and source location – here implemented with stochastic Stein Variational Gradient Descent (sSVGD) and double difference tomography – yields relocated earthquake events and three-dimensional estimates of Vp, Vs, and Vp/Vs together with their respective uncertainty. sSVGD approximates the statistical description of all possible models that fit the data, referred to as the posterior distribution, using an ensemble of particles or samples. These are initialized from a prior distribution, which encodes the prior information about the domain, and driven toward the posterior by iterative transforms that minimise the Kullback-Leibler divergence between the particle density and the posterior.We apply this workflow to the Eden Project geothermal site (Cornwall, UK), using microseismic events recorded by an array of 450 STRYDE nodes deployed around the injection site. The objective is to recover mean models of Vp and Vs, and Vp/Vs with their corresponding uncertainty from passive sources alone, enabling probabilistic assessment of the subsurface structure and potential future well-placement targets.Owing to the nodal geometry and the spatial distribution of microseismic sources, ray-path coverage is highly heterogeneous across the survey volume. Consequently, the posterior uncertainty is large over much of the domain and decreases substantially where ray coverage is dense – mostly around the geothermal well. Within this region, we observe velocity anomalies consistent with fractured and fluid-saturated rock, while regions distant from the well remain poorly constrained. By providing a clearer understanding of uncertainties inherent to tomographic inversions, the probabilistic imaging framework enables more robust and reliable analysis of the results, which is crucial in geothermal exploration.

Seismic tomography is a methodology to image subsurface properties of the Earth. In order to better interpret the resulting images, it is important to assess uncertainty in the results. Mixture density networks (MDNs) provide an efficient way to estimate Bayesian posterior probability density functions (pdfs) that describe the uncertainty of tomographic images. However, the method can only be applied in cases where the number of data is fixed, and consequently a large number of practical applications that have variable data sizes cannot be solved. To resolve this issue, we introduce graph neural networks (GNNs) to solve seismic tomographic problems. Graphs are data structure which provides flexible representation of complex, variable systems. GNNs are neural networks that manipulates graph data, and can be combined with MDNs (called graph MDNs) to provide efficient estimates of posterior pdfs for graph data. In this study we apply graph MDNs to seismic tomography by representing travel time data with a graph. We demonstrate the method using both synthetic and real data, and compare the results with those obtained using Markov chain Monte Carlo (McMC). The results show that graph MDNs can provide comparable posterior pdfs to those obtained using McMC at significantly lower cost. We thus conclude that graph MDNs can be used in a range of practical applications that require many similar seismic tomographic problems with different number of data to be solved.

A wide range of academic and practical applications require that we interrogate the Earth’s subsurface for answers to scientific questions. A common approach is to image the subsurface properties using data recorded at or above the Earth’s surface, and to interpret those images to address questions of interest. Seismic tomograph is one such method which has been used widely to generate those images. In order to obtain robust and well-justified answers, it is important to assess uncertainties in property estimates.To solve seismic tomographic problems efficiently, mixture density networks (MDNs) have been used to estimate Bayesian posterior probability density functions (pdfs) which describe the uncertainty of tomographic images. However, the method can only be applied in cases where the number of data is fixed, and consequently cannot be used in a large number of practical applications that have variable sizes of data. To resolve this issue, we introduce graph neural networks (GNNs) to solve seismic tomographic problems. Graphs are data structures that provide flexible representation of complex, variable systems. GNNs are neural networks that manipulate graphs. GNNs can be combined with MDNs (called graph MDNs) to provide estimates of posterior pdfs for graph data. In this study we use graph MDNs to solve seismic tomographic problems by representing seismic travel time data using a graph. We demonstrate the method using both synthetic and real data, and compare the results with those obtained using Monte Carlo sampling methods. The results show that graph MDNs can provide comparable solutions to those obtained using Monte Carlo methods for problems with variable number of data. After training, graph MDNs estimate posterior pdfs in seconds on a typical desktop computer. Hence the method can be used to provide rapid solutions for similar problems with variable sizes of data. We therefore conclude that graph MDNs can be an important tool to solve many practical tomographic problems.

SUMMARY: Time-lapse seismic full-waveform inversion (FWI) provides estimates of dynamic changes in the Earth’s subsurface by performing multiple seismic surveys at different times. Since FWI problems are highly non-linear and non-unique, it is important to quantify uncertainties in such estimates to allow robust decision making based on the results. Markov chain Monte Carlo (McMC) methods have been used for this purpose, but due to their high computational cost, those studies often require a pre-existing accurate baseline model and estimates of the locations of potential velocity changes, and neglect uncertainty in the baseline velocity model. Such detailed and accurate prior information is not always available in practice. In this study we use an efficient optimization method called stochastic Stein variational gradient descent (sSVGD) to solve time-lapse FWI problems without assuming such prior knowledge, and to estimate uncertainty both in the baseline velocity model and the velocity change over time. We test two Bayesian strategies: separate Bayesian inversions for each seismic survey, and a single joint inversion for baseline and repeat surveys, and compare the methods with standard linearized double difference inversion. The results demonstrate that all three methods can produce accurate velocity change estimates in the case of having fixed (exactly repeatable) acquisition geometries. However, the two Bayesian methods generate significantly more accurate results when acquisition geometries changes between surveys. Furthermore, joint inversion provides the most accurate velocity change and uncertainty estimates in all cases tested. We therefore conclude that Bayesian time-lapse inversion using a joint inversion strategy may be useful to image and monitor subsurface changes, in particular where variations in the results would lead to different consequent decisions.

Bayesian inference has become an important methodology to solve inverse problems and to quantify uncertainties in their solutions. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently by using optimisation. In this study we present a Python Variational Inversion Package (VIP), to solve inverse problems using variational inference methods. The package includes automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD), and provides implementations of 2D travel time tomography and 2D full waveform inversion including test examples and solutions. Users can solve their own problems by supplying an appropriate forward function and a gradient calculation code. In addition, the package provides a scalable implementation which can be deployed easily on a desktop machine or using modern high performance computational facilities. The examples demonstrate that VIP is an efficient, scalable, extensible and user-friendly package, and can be used to solve a wide range of low or high dimensional inverse problems in practice.

SUMMARY: Ambient noise seismic tomography has proven to be an effective tool for subsurface imaging, particularly in volcanic regions such as the Reykjanes Peninsula (RP), SW Iceland, where ambient seismic noise is ideal with isotropic illumination. The primary purpose of this study is to obtain a reliable shear wave velocity model of the RP, to get a better understanding of the subsurface structure of the RP and how it relates to other geoscientific results. This is the first tomographic model of the RP which is based on both on- and off-shore seismic stations. We use the ambient seismic noise data and apply a novel algorithm called one-step 3-D transdimensional tomography. The main geological structures in the study area (i.e. covered by seismic stations) are the four NE–SW trending volcanic systems, orientated highly oblique to the plate spreading on the RP. These are from west to east; Reykjanes, Eldvörp-Svartsengi, Fagradalsfjall and Krýsuvík, of which all except Fagradalsfjall host a known high-temperature geothermal field. Using surface waves retrieved from ambient noise recordings, we recovered a 3-D model of shear wave velocity. We observe low-velocity anomalies below these known high-temperature fields. The observed low-velocity anomalies below Reykjanes and Eldvörp-Svartsengi are significant but relatively small. The low-velocity anomaly observed below Krýsuvík is both larger and stronger, oriented near-perpendicular to the volcanic system, and coinciding well with a previously found low-resistivity anomaly. A low-velocity anomaly in the depth range of 5–8 km extends horizontally along the whole RP, but below the high-temperature fields, the onset of the velocity decrease is shallower, at around 3 km depth. This is in good agreement with the brittle–ductile transition zone on the RP. In considerably greater detail, our results confirm previous tomographic models obtained in the area. This study demonstrates the potential of the entirely data-driven, one-step 3-D transdimensional ambient noise tomography as a routine tomography tool and a complementary seismological tool for geothermal exploration, providing an enhanced understanding of the upper crustal structure of the RP.