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Spatially 3-dimensional seismic full waveform inversion (3D FWI) is a highly nonlinear and computationally demanding inverse problem that constructs 3D subsurface seismic velocity structures using seismic waveform data. To characterise non-uniqueness in the solutions we demonstrate Bayesian 3D FWI using an efficient method called physically structured variational inference applied to 3D acoustic Bayesian FWI. The results provide reasonable posterior uncertainty estimates, at a computational cost that is only an order of magnitude greater than that of standard, deterministic FWI. Furthermore, we deploy variational prior replacement to calculate Bayesian solutions corresponding to different classes of prior information at low additional cost, and analyse those prior hypotheses by constructing Bayesian L-curves. This reveals the sensitivity of the inversion process to different prior assumptions. Thus we show that fully probabilistic 3D FWI can be performed at a cost that may be practical in small FWI problems, and can be used to test different prior hypotheses.

Many scientific investigations require that the values of a set of model parameters are estimated using recorded data. In Bayesian inference, information from both observed data and prior knowledge is combined to update model parameters probabilistically by calculating the posterior probability distribution function. Prior information is often described by a prior probability distribution. Situations arise in which we wish to change prior information during the course of a scientific project. However, estimating the solution to any single Bayesian inference problem is often computationally costly, as it typically requires many model samples to be drawn, and the data set that would have been recorded if each sample was true must be simulated. Recalculating the Bayesian inference solution every time prior information changes can therefore be extremely expensive. We develop a mathematical formulation that allows the prior information that is embedded within a solution, to be changed using variational methods, without recalculating the original Bayesian inference. In this method, existing prior information is removed from a previously obtained posterior distribution and is replaced by new prior information. We therefore call the methodology variational prior replacement (VPR). We demonstrate VPR using a 2D seismic full waveform inversion example, in which VPR provides similar posterior solutions to those obtained by solving independent inference problems using different prior distributions. The former can be completed within minutes on a laptop computer, whereas the latter requires days of computations using high-performance computing resources. We demonstrate the value of the method by comparing the posterior solutions obtained using three different types of prior information: uniform, smoothing and geological prior distributions.

Communication and position sensing are among the most important capabilities for swarm robots to interact with their peers and perform tasks collaboratively. However, the hardware required to facilitate communication and position sensing is often too complicated, expensive, and bulky to be carried on swarm robots. Here we present Maneuverable Piccolissimo 3 (MP3), a minimalist, single motor drone capable of executing inter-robot communication via infrared light and triangulation-based sensing of relative bearing, distance, and elevation using message arrival time. Thanks to its novel design, MP3 can communicate with peers and localize itself using simple components, keeping its size and mass small and making it inherently safe for human interaction. We present the hardware and software design of MP3 and demonstrate its capability to localize itself, fly stably, and maneuver in the environment using peer-to-peer communication and sensing.

Despite their growing popularity, swarms of robots remain limited by the operating time of each individual. We present algorithms which allow a human to sculpt a swarm of robots into a shape that persists in space perpetually, independent of onboard energy constraints such as batteries. Robots generate a path through a shape such that robots cycle in and out of the shape. Robots inside the shape react to human initiated changes and adapt the path through the shape accordingly. Robots outside the shape recharge and return to the shape so that the shape can persist indefinitely. The presented algorithms communicate shape changes throughout the swarm using message passing and robot motion. These algorithms enable the swarm to persist through any arbitrary changes to the shape. We describe these algorithms in detail and present their performance in simulation and on a swarm of mobile robots. The result is a swarm behavior more suitable for extended duration, dynamic shape-based tasks in applications such as agriculture and emergency response., Comment: 20 pages, 17 figures

Geoscientists use observed data to estimate properties of the Earth's interior. This often requires non-linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic framework, in which uncertainty is represented by a so-called posterior probability distribution. Recently, variational inference has emerged as an efficient method to estimate Bayesian solutions. By seeking the closest approximation to the posterior distribution within any chosen family of distributions, variational inference yields a fully probabilistic solution. It is important to define expressive variational families so that the posterior distribution can be represented accurately. We introduce boosting variational inference (BVI) as a computationally efficient means to construct a flexible approximating family comprising all possible finite mixtures of simpler component distributions. We use Gaussian mixture components due to their fully parametric nature and the ease with which they can be optimised. We apply BVI to seismic travel time tomography and full waveform inversion, comparing its performance with other methods of solution. The results demonstrate that BVI achieves reasonable efficiency and accuracy while enabling the construction of a fully analytic expression for the posterior distribution. Samples that represent major components of uncertainty in the solution can be obtained analytically from each mixture component. We demonstrate that these samples can be used to solve an interrogation problem: to assess the size of a subsurface target structure. To the best of our knowledge, this is the first method in geophysics that provides both analytic and reasonably accurate probabilistic solutions to fully non-linear, high-dimensional Bayesian full waveform inversion problems.

Bayesian inference has become an important tool to solve inverse problems and to quantify uncertainties in their solutions. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently by using optimization. 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.

Time-lapse seismic full-waveform inversion (FWI) provides estimates of dynamic changes in the 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. Markov chain Monte Carlo (McMC) methods have been used for this purpose, but due to their high computational cost, those studies often require an 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. We test two Bayesian strategies: separate Bayesian inversions for each seismic survey, and a single join inversion for baseline and repeat surveys, and compare the methods with the standard linearised 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, but that the two Bayesian methods generate more accurate results when the acquisition geometry changes between surveys. Furthermore the joint inversion provides the most accurate velocity change and uncertainty estimates in all cases. We therefore conclude that Bayesian time-lapse inversion, especially adopting a joint inversion strategy, may be useful to image and monitor the subsurface changes, in particular where uncertainty in the results might lead to significantly different decisions.

This paper introduces variational design methods that are novel to Geophysics, and discusses their benefits and limitations in the context of geophysical applications and more established design methods. Variational methods rely on functional approximations to probability distributions and model-data relationships. They can be used to design experiments that best resolve either all model parameters, or the answer to specific questions about the system to be interrogated. The methods are tested in three schematic geophysical applications: (i) estimating a source location given arrival times at sensor locations, and (ii) estimating the contrast in seismic wavefield velocity across a stratal interface given measurements of the amplitudes of seismic wavefield reflections from that interface, and (iii) designing a survey to best constrain CO2 saturation in a subsurface storage scenario. Variational methods allow the value of an experiment to be calculated and optimised simultaneously, which results in substantial savings in computational cost. In the context of designing a survey to best constrain CO2 in a subsurface storage scenario, we show that optimal designs may change substantially depending on the questions of interest. Overall, this work indicates that optimal design methods should be used more widely in Geophysics, as they are in other scientifically advanced fields., Comment: Submitted to Geophysical Journal International in June 2023

Wavefield travel time tomography is used for a variety of purposes in acoustics, geophysics and non-destructive testing. Since the problem is non-linear, assessing uncertainty in the results requires many forward evaluations. It is therefore important that the forward evaluation of travel times and ray paths is efficient, which is challenging in anisotropic media. Given a computed travel time field, ray tracing can be performed to obtain the fastest ray path from any point in the medium to the source of the travel time field. These rays can then be used to speed up gradient based inversion methods. We present a forward modeller for calculating travel time fields by localised estimation of wavefronts, and a novel approach to ray tracing through travel time fields. These methods have been tested in a complex anisotropic weld and give travel times comparable to those obtained using finite element modelling while being computationally cheaper.

Geological process models simulate a range of dynamic processes to evolve a base topography into a final 2D cross-section or 3D geological scenario. In principle, process parameters may be updated to better align with observed geophysical or geological data; however, many realisations of process models that embody different conceptual models may provide similar consistency with observed data, and finding all such realisations may be infeasible due to the computational demands of the task. Alternatively, geophysical probabilistic tomographic methods may be used to estimate the family of models of a target subsurface structure that are consistent both with information obtained from previous experiments and with new data (the Bayesian posterior distribution). However, this family seldom embodies geologically reasonable images. We show that the posterior distribution of tomographic images can be enhanced by injecting geological prior information into Bayesian inference, and we can do this near-instantaneously by trained MDNs. We invoke two geological concepts as geological prior information: a braided river system, and a set of marine parasequences, each parameterised by a GAN. Data from a target structure can then be used to infer the image parameter values using either geological concept using MDNs. Our MDN solutions closely resemble those found using expensive McMC methods, and while the use of incorrect geological conceptual models provides less accurate results the mean structures still approximate the target. We then show that a classification network can infer the correct geological conceptual model. Thus, imposing even incorrect geological prior information may improve geophysical tomographic images compared to those obtained without geological prior information, and in principle geological conceptual models can be inferred directly from geophysical travel time data.

Seismic full-waveform inversion (FWI) provides high resolution images of the subsurface by exploiting information in the recorded seismic waveforms. This is achieved by solving a highly nonnlinear and nonunique inverse problem. Bayesian inference is therefore used to quantify uncertainties in the solution. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently using optimization. The method has been applied to 2D FWI problems to produce full Bayesian posterior distributions. However, due to higher dimensionality and more expensive computational cost, the performance of the method in 3D FWI problems remains unknown. We apply three variational inference methods to 3D FWI and analyse their performance. Specifically we apply automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD), to a 3D FWI problem, and compare their results and computational cost. The results show that ADVI is the most computationally efficient method but systematically underestimates the uncertainty. The method can therefore be used to provide relatively rapid but approximate insights into the subsurface together with a lower bound estimate of the uncertainty. SVGD demands the highest computational cost, and still produces biased results. In contrast, by including a randomized term in the SVGD dynamics, sSVGD becomes a Markov chain Monte Carlo method and provides the most accurate results at intermediate computational cost. We thus conclude that 3D variational full-waveform inversion is practically applicable, at least in small problems, and can be used to image the Earth's interior and to provide reasonable uncertainty estimates on those images.

Expert knowledge is required to interpret data across a range of fields. Experts bridge gaps that often exists in our knowledge about relationships between data and the parameters of interest. This is especially true in geoscientific applications, where knowledge of the Earth is derived from interpretations of observable features and relies on predominantly unproven but widely accepted theories. Thus, experts facilitate solutions to otherwise unsolvable problems. However, experts are inherently subjective, and susceptible to cognitive biases and adverse external effects. This work examines this problem within geoscience. Three compelling examples are provided of the prevalence of cognitive biases from previous work. The problem is then formally defined, and a set of design principles which ensure that any solution is sufficiently flexible to be readily applied to the range of geoscientific problems. No solutions exist that reliably capture and reduce cognitive bias in experts. However, formal expert elicitation methods can be used to assess expert variation, and a variety of approaches exist that may help to illuminate uncertainties, avoid misunderstandings, and reduce herding behaviours or single-expert over-dominance. This work combines existing and future approaches to reduce expert suboptimality through a flexible modular design where each module provides a specific function. The design centres around action modules that force a stop-and-perform step into interpretation tasks. A starter-pack of modules is provided as an example of the conceptual design. This simple bias-reduction system may readily be applied in organisations and during everyday interpretations through to tasks for major commercial ventures.

In a variety of scientific applications we wish to characterize a physical system using measurements or observations. This often requires us to solve an inverse problem, which usually has non-unique solutions so uncertainty must be quantified in order to define the family of all possible solutions. Bayesian inference provides a powerful theoretical framework which defines the set of solutions to inverse problems, and variational inference is a method to solve Bayesian inference problems using optimization while still producing fully probabilistic solutions. This chapter provides an introduction to variational inference, and reviews its applications to a range of geophysical problems, including petrophysical inversion, travel time tomography and full-waveform inversion. We demonstrate that variational inference is an efficient and scalable method which can be deployed in many practical scenarios.

Seismic surface wave dispersion inversion is used widely to study the subsurface structure of the Earth. The dispersion property is usually measured by using frequency-phase velocity (f-c) analysis and by picking phase velocities from the obtained f-c spectrum. However, because of potential contamination the f-c spectrum often has multimodalities at each frequency for each mode. These introduce uncertainty and errors in the picked phase velocities, and consequently the obtained shear velocity structure is biased. To overcome this issue, in this study we introduce a new method which directly uses the spectrum as data. We achieve this by solving the inverse problem in a Bayesian framework and define a new likelihood function, the energy likelihood function, which uses the spectrum energy to define data fit. We apply the new method to a land dataset recorded by a dense receiver array, and compare the results to those obtained using the traditional method. The results show that the new method produces more accurate results since they better match independent data from refraction tomography. This real-data application also shows that it can be applied efficiently since it removes the need to pick phase velocities, and with relatively little adjustment to current practice since it uses standard f-c panels to define the likelihood. We therefore recommend using the energy likelihood function rather than explicitly picking phase velocities in surface wave dispersion inversion.

The Department of Defense (DoD) has significantly increased its investment in the design, evaluation, and deployment of Artificial Intelligence and Machine Learning (AI/ML) capabilities to address national security needs. While there are numerous AI/ML successes in the academic and commercial sectors, many of these systems have also been shown to be brittle and nonrobust. In a complex and ever-changing national security environment, it is vital that the DoD establish a sound and methodical process to evaluate the performance and robustness of AI/ML models before these new capabilities are deployed to the field. This paper reviews the AI/ML development process, highlights common best practices for AI/ML model evaluation, and makes recommendations to DoD evaluators to ensure the deployment of robust AI/ML capabilities for national security needs.

The goal of a scientific investigation is to find answers to specific questions. In geosciences this is typically achieved by solving an inference or inverse problem and interpreting the solution. However, the answer obtained is often biased because the solution to an inverse problem is nonunique and human interpretation is a biased process. Interrogation theory provides a systematic way to find optimal answers by considering their full uncertainty estimates, and by designing an objective function that defines desirable qualities in the answer. In this study we demonstrate interrogation theory by quantifying the size of a particular subsurface structure. The results show that interrogation theory provides an accurate estimate of the true answer, which cannot be obtained by direct, subjective interpretation of the solution mean and standard deviation. This demonstrates the value of interrogation theory. It also shows that fully nonlinear uncertainty assessments may be critical in order to address real-world scientific problems, which goes some way towards justifying their computational expense.

Rendering objects invisible to impinging acoustic waves (cloaking) and creating acoustic illusions (holography) has been attempted using active and passive approaches. While passive methods are applicable only to narrow frequency bands, active approaches attempt to respond dynamically, interfering with broadband incident or scattered wavefields by emitting secondary waves. Without prior knowledge of the primary wavefield, the signals for the secondary sources need to be estimated and adapted in real-time. This has thus far impeded active cloaking and holography for broadband wavefields. We present experimental results of active acoustic cloaking and holography without prior knowledge of the wavefield so that objects remain invisible and illusions intact even for broadband moving sources. This opens novel research directions and facilitates practical applications including architectural acoustics, and stealth.

It is a well-established principle that cross-correlating seismic observations at different receiver locations can yield estimates of band-limited inter-receiver Green's functions. This principle, known as seismic interferometry, is a powerful technique that can transform noise into signals which allow us to remotely image and interrogate subsurface Earth structures. In practice it is often necessary and even desirable to rely on noise already present in the environment. Theory that underpins many applications of ambient noise interferometry makes an assumption that the noise sources are uncorrelated in space and time. However, many real-world noise sources such as trains, highway traffic and ocean waves are inherently correlated both in space and time, in direct contradiction to the current theoretical foundations. Applying standard interferometric techniques to recordings from correlated energy sources makes the Green's function liable to estimation errors that so far have not been fully accounted for theoretically nor in practice. We show that these errors are significant for common noise sources, always perturbing and sometimes obscuring the phase one wishes to retrieve. Our analysis explains why stacking may reduce the phase errors, but also shows that in commonly-encountered circumstances stacking will not remediate the problem. This analytical insight allowed us to develop a novel workflow that significantly mitigates effects arising from the use of correlated noise sources. Our methodology can be used in conjunction with already existing approaches, and improves results from both correlated and uncorrelated ambient noise. Hence, we expect it to be widely applicable in real life ambient noise studies.

Estimating the spatially varying microstructures of heterogeneous and locally anisotropic media non-destructively is necessary for the accurate detection of flaws and reliable monitoring of manufacturing processes. Conventional algorithms used for solving this inverse problem come with significant computational cost, particularly in the case of high dimensional non-linear tomographic problems. In this paper, we propose a framework which uses deep neural networks (DNNs) with full aperture, pitch-catch and pulse-echo transducer configurations to reconstruct material maps of crystallographic orientation. We also present the first ever application of generative adversarial networks (GANs) to achieve super resolution of ultrasonic tomographic images, providing a factor-four increase in image resolution and up to a 50% increase in structural similarity. The importance of including appropriate prior knowledge in the GAN training dataset to increase inversion accuracy is highlighted; known information about the material's structure should be present in the training data. We show that after a computationally expensive training process, the DNNs and GANs can be used in less that one second (0.9 seconds on a standard desktop computer) to provide a high resolution map of the material's grain orientations.

Seismic full-waveform inversion (FWI) uses full seismic records to estimate subsurface velocity structure. This requires a highly nonlinear and nonunique inverse problem to be solved, and Bayesian methods have been used to quantify uncertainties in the solution. Variational Bayesian inference uses optimization to provide solutions efficiently. The method has been applied to solve a transmission FWI problem using data generated by known earthquake-like sources, with strong prior information imposed on the velocity. Unfortunately such prior information about velocity structure and earthquake sources is never available in practice. We present the first application of the method in a seismic reflection setting, and with realistically weak prior information. We thus demonstrate that the method can produce high resolution images and reliable uncertainties given practically reasonable prior information., Comment: 7 pages, 3 figures

Surface wave tomography uses measured dispersion properties of surface waves to infer the spatial distribution of subsurface properties such as shear-wave velocities. These properties can be estimated vertically below any geographical location at which surface wave dispersion data are available. As the inversion is significantly non-linear, Monte Carlo methods are often used to invert dispersion curves for shear-wave velocity profiles with depth to give a probabilistic solution. Such methods provide uncertainty information but are computationally expensive. Neural network based inversion provides a more efficient way to obtain probabilistic solutions when those solutions are required beneath many geographical locations. Unlike Monte Carlo methods, once a network has been trained it can be applied rapidly to perform any number of inversions. We train a class of neural networks called mixture density networks, to invert dispersion curves for shear-wave velocity models and their non-linearised uncertainty. Mixture density networks are able to produce fully probabilistic solutions in the form of weighted sums of multivariate analytic kernels such as Gaussians, and we show that including data uncertainties in the mixture density network gives more reliable mean velocity estimates when data contains significant noise. The networks were applied to data from the Grane field in the Norwegian North sea to produce shear-wave velocity maps at several depth levels. Post-training we obtained probabilistic velocity profiles with depth beneath 26,772 locations to produce a 3D velocity model in 21 seconds on a standard desktop computer. This method is therefore ideally suited for rapid, repeated 3D subsurface imaging and monitoring., Comment: Submitted to Geophysical Journal International

Seismic tomography is a methodology to image the interior of solid or fluid media, and is often used to map properties in the subsurface of the Earth. In order to better interpret the resulting images it is important to assess imaging uncertainties. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose, but they are generally computationally intractable for large datasets and high-dimensional parameter spaces. To extend uncertainty analysis to larger systems we use variational inference methods to conduct seismic tomography. In contrast to Monte Carlo sampling, variational methods solve the Bayesian inference problem as an optimization problem, yet still provide probabilistic results. In this study, we applied two variational methods, automatic differential variational inference (ADVI) and Stein variational gradient descent (SVGD), to 2D seismic tomography problems using both synthetic and real data and we compare the results to those from two different Monte Carlo sampling methods. The results show that variational inference methods can produce accurate approximations to the results of Monte Carlo sampling methods at significantly lower computational cost, provided that gradients of parameters with respect to data can be calculated efficiently. We expect that the methods can be applied fruitfully to many other types of geophysical inverse problems., Comment: 26 pages, 14 figures

Travel time tomography for the velocity structure of a medium is a highly non-linear and non-unique inverse problem. Monte Carlo methods are becoming increasingly common choices to provide probabilistic solutions to tomographic problems but those methods are computationally expensive. Neural networks can often be used to solve highly non-linear problems at a much lower computational cost when multiple inversions are needed from similar data types. We present the first method to perform fully non-linear, rapid and probabilistic Bayesian inversion of travel time data for 2D velocity maps using a mixture density network. We compare multiple methods to estimate probability density functions that represent the tomographic solution, using different sets of prior information and different training methodologies. We demonstrate the importance of prior information in such high dimensional inverse problems due to the curse of dimensionality: unrealistically informative prior probability distributions may result in better estimates of the mean velocity structure, however the uncertainties represented in the posterior probability density functions then contain less information than is obtained when using a less informative prior. This is illustrated by the emergence of uncertainty loops in posterior standard deviation maps when inverting travel time data using a less informative prior, which are not observed when using networks trained on prior information that includes (unrealistic) a priori smoothness constraints in the velocity models. We show that after an expensive program of training the networks, repeated high-dimensional, probabilistic tomography is possible on timescales of the order of a second on a standard desktop computer., Comment: To be published in Neural Computing and Applications