arXiv subjects

Andrew Curtis

Publications and source records attributed to Andrew Curtis.

At least 19 recordsLinked to original sources

arXiv2026

CellNet -- Localizing Cells using Sparse and Noisy Point Annotations

Counting living cells is an important step in many biological research workflows. Our collaborators at the Wellcome Sanger Institute study vital genes in humans via large scale saturation genome editing screening, which requires repeatedly counting cells a great number of times. Computer Vision based automation is crucial for high throughput and resource efficiency. In this work, we develop a regression-based deep learning computer vision algorithm to detect and count cells in phase-contrast microscopy images. To reduce annotation effort, which in practice often becomes a bottleneck, we focus on counting cells only using sparse point annotations, which are fast and easy to acquire. By comparison to state-of-the-art 0-shot methods, we show that regression-based counting is a promising alternative in low data regimes. Through developing methods to automatically count living cells in microscopy images, we contribute to valuable research on the human genome. The code is available at https://github.com/beijn/cellnet.

cs.CV↗

arXiv2026

Response to: "A note on conditional densities, Bayes' rule, and recent criticisms of Bayesian inference" by Yan et al., 2026

In a recent preprint (Mosegaard and Curtis, 2024, arXiv:2411.13570v2) we analyzed the consequences of ignoring the well-known inconsistency of classical conditional probability densities. We explained how this inconsistency, together with acausality in hierarchical methods, invalidate a variety of commonly applied Bayesian methods when applied to problems in the physical world. Yan et al., 2026, (arXiv:2603.27038v1) published a note, in which they claim, contrary to our preprint, that there are no inconsistencies if one uses the method of conditional expectations to derive probabilities. Furthermore, they believe that there are mathematical errors in our exposition and in our use of the Bayesian framework. This note is a response to the claims made by Yan et al. Yan et al. do not discriminate between physical and statistical consistency. Their note addresses statistical consistency of a solution under a change of variables; this is already known to be resolved by using the theory of conditional expectations. By contrast, our preprint concerns the physical consistency of any solution under a change of mathematics used to derive that solution. It demonstrates that widely used methods to compute Bayesian posterior solutions are physically inconsistent under a change of variables. Their note does not, therefore, address the tenet of our preprint. We show herein that the theory of conditional expectations does not resolve physical inconsistency, and that Yan et al. make mathematical errors. We conclude that their claims are unfounded, and in some cases we show that their critique is meaningless. The conclusions of our preprint therefore stand.

stat.ME↗

arXiv2026

Designing Solutions to Geophysical Inverse Problems by Changing Variables

Geoscientists often solve inverse problems to estimate values of parameters of interest given relevant data sets. Bayesian inference solves these problems by combining probability distributions that describe uncertainties in both observations and unknown parameters, and we require that the solution provides unbiased uncertainty estimates in order to inform risk-based decisions. It has been known for over a century that employing different, but equivalent parametrisations of the same information can yield conditional probabilities that are mathematically inconsistent, a property referred to as the BK-inconsistency. Recently this inconsistency was shown to invalidate the solutions to physical problems found using several well-established methods of Bayesian inference. In this study, we explore the extent to which this inconsistency affects solutions to common geophysical problems. We demonstrate that changes in parametrisations result in inconsistent conditional probability densities, even though they represent exactly the same information. We show that this can affect Bayesian posterior solutions dramatically across various geoscientific problems using real and synthetic data. Given that deterministic inversion is often equivalent to finding the maximum a posteriori solution to specific Bayesian problems (the mathematical equations to be solved are identical), the BK-inconsistency also results in inconsistent solutions to deterministic inverse problems. Indeed, we show that solutions can potentially be designed, simply by changing the parametrisation. This study highlights that a careful rethinking of Bayesian inference and deterministic inversion may be required in physical problems: the effects that we demonstrate are likely to affect past and present inverse problem solutions in a variety of different fields of application.

physics.geo-ph↗

arXiv2026

Linearised versus Nonlinear Estimates of Uncertainty in Full Waveform Inversion

Seismic full waveform inversion (FWI) is a powerful technique to generate high resolution images of the Earth's interior. However, significant uncertainty exists in all FWI solutions due to imperfect acquisition geometries, inherent noise in the data, and nonlinearity of the forward problem. Probabilistic Bayesian FWI addresses this non-uniqueness by estimating the entire family of possible model solutions described by the posterior probability density function (pdf). The posterior pdf can be estimated using nonlinear inversion methods to quantify full uncertainties. Alternatively, by linearising the physics relating parameters and observations around the maximum a posteriori solution, the posterior pdf is usually approximated by a Gaussian pdf. This is referred to as the linearised method. In this work, we apply both nonlinear and linearised methods to 2D acoustic Bayesian FWI problems. We use a variational inference algorithm for the nonlinear case, in which a transformed Gaussian is optimised to approximate the posterior pdf. The results can be compared with those from a linearised, locally-Gaussian based method. We also apply Stein variational gradient descent for comparison. The results show that while both the linearised and nonlinear methods recover the posterior mean models accurately, they exhibit significantly different posterior uncertainty structures, especially around layer interfaces, due to the linearisation of wave physics. Linearised uncertainty estimates are shown to be significantly less accurate: they provide far less accurate fits to observed waveform data, and yield biased estimates of inferred meta-properties such as volumes of geological bodies. This work therefore motivates the application of fully nonlinear inversion methods in Bayesian FWI if accurate uncertainty estimates over parameters, or inferred or interpreted meta-properties are important.

physics.geo-ph↗

arXiv2026

The UK and Ireland Geophysical Array -- Concept and Design

Scientific exploration of the UK and Ireland's subsurface has made important contributions to scholarship and prosperity for people and the planet, including economic growth, sustainable use of natural resources, storage of greenhouse gases, and inspiring curiosity about the Earth beneath our feet. This article outlines a vision for an array of seismological instruments spanning the UK and Ireland, UKI Array, augmented by other types of geophysical sensors, to maximise the value offered by existing equipment pools. The mission is to research natural phenomena and structure in the deep and shallow Earth, to solve problems concerning hazards and resources, to connect scientists to schools and the broader public, and thus to inspire a new generation to learn about geophysics. The vision was created through a community driven process of engagement and participation. This paper describes the concept and design of the UKI-Array; a companion paper discusses related opportunities and potential applications.

physics.geo-ph↗

arXiv2026

Variational and Monte Carlo Methods for Bayesian Inversion of Dynamic Subsurface Flow Simulations Using Seismic and Fluid Pressure Data

In order to predict future performance of subsurface fluid reservoirs under possible operating scenarios, a dynamic, porous-medium flow simulation model must be tuned to include representative properties of the reservoir. Estimating subsurface reservoir properties given remotely sensed or borehole-based observations typically involves finding the solution to a challenging inverse problem. We compare Monte Carlo random sampling to variational inference methods which use optimisation to constrain parametrised uncertainties in nonlinear Bayesian inversions. We use them to estimate the posterior probability distribution of reservoir permeability given fluid pressure and seismic measurements. The methods include automatic differentiation variational inference (ADVI), Stein variational gradient descent (SVGD), and a Monte Carlo method called stochastic SVGD (sSVGD), all of which we benchmark against results from Metropolis-Hastings McMC. We also test an ADVI variant called physically structured variational inference (PSVI): in our implementation this method estimates only spatially-local correlations between model parameters based on the intuition that such correlations are strong in remote sensing problems in which data only inform about spatial-averages of local dynamics. We apply the methods to two- and three-dimensional inverse problems of carbon dioxide storage, inspired by the Endurance field, located in the UK North Sea. Results show that PSVI achieves a good balance between mean-field ADVI and full-rank ADVI in terms of accuracy of the posterior approximation and computational efficiency. SVGD and sSVGD offer more accurate approximations of the target posterior distribution, but at far higher computational cost. Between them, sSVGD outperforms SVGD, exhibiting better computational efficiency and mitigating the problems of mode collapse and spurious correlations.

physics.geo-ph↗

arXiv2025

Detecting Model Misspecification in Bayesian Inverse Problems via Variational Gradient Descent

Bayesian inference is optimal when the statistical model is well-specified, while outside this setting Bayesian inference can catastrophically fail; accordingly a wealth of post-Bayesian methodologies have been proposed. Predictively oriented (PrO) approaches lift the statistical model $P_\theta$ to an (infinite) mixture model $\int P_\theta \; \mathrm{d}Q(\theta)$ and fit this predictive distribution via minimising an entropy-regularised objective functional. In the well-specified setting one expects the mixing distribution $Q$ to concentrate around the true data-generating parameter in the large data limit, while such singular concentration will typically not be observed if the model is misspecified. Our contribution is to demonstrate that one can empirically detect model misspecification by comparing the standard Bayesian posterior to the PrO `posterior' $Q$, providing a novel and widely-applicable diagnostic tool for the standard Bayesian workflow. To operationalise this, we present an efficient numerical algorithm based on variational gradient descent. A simulation study, and a more detailed case study involving a Bayesian inverse problem in seismology, confirm that model misspecification can be automatically detected using this framework.

stat.ME↗

arXiv2025

Single and Multi-Objective Optimization of Distributed Acoustic Sensing Cable Layouts for Geophysical Applications

We present a systematic approach to optimise distributed acoustic sensing (DAS) fibre-optic cable layouts using global optimisation techniques. Our method represents cable geometries using splines, enabling efficient exploration of layouts while respecting physical deployment constraints. The use of evolutionary algorithms enables single and multi-objective optimisation, taking into account complex design constraints such as terrain, accesibility, exclusion zones and cable length, while allowing efficient parallelisation of the optimisation process. We demonstrate the approach on a real-world case study, optimising the layout of a DAS cable for monitoring slope stability in the Cuolm da Vi area of Switzerland. We adapt design criteria for seismic source location problems, and for ambient noise surface wave tomography, to account for the unique characteristics of DAS, such as directional sensitivity patterns. The results show significant potential for improvements in source location accuracy and surface wave tomographic resolution by optimising cable layouts, highlighting the potential of this approach for optimising DAS deployments in various geophysical applications.

physics.geo-ph↗

arXiv2025

A Semi-analytic but Biased Uncertainty Assessment Method using Sample Extensions, Analysed for Nonlinear Travel Time Tomography

Many geophysical problems can be cast as inverse problems that estimate a set of parameter values from observed data. Within a Bayesian framework, solutions to such problems are described probabilistically by the so-called posterior probability distribution functions (pdf's). To obtain robust inference results often requires millions of model parameter value samples to be drawn, and their simulation to be performed; this is a computationally expensive procedure. We investigate the concept of sample extensions as a means to improve efficiency when solving fully nonlinear inverse problems. A sample's extension is defined as the set of models or parameter values whose forward function values are directly accessible from a sample for which the forward function has already been evaluated, obviating the need for additional forward function evaluations. In a specific case of first-arrival travel time calculations used in seismic travel time tomography, we apply sample extensions to obtain a continuous region with non-zero hypervolume within parameter space, across all of which the forward function values are known given only a single forward simulation. We devise a deterministic sampling technique that identifies the most informative extensions by solving an optimisation problem. In an illustrative tomographic example that involves a single travel time datum, we find 51 optimal samples, and use them to construct an analytic approximation to the Bayesian posterior pdf. Additionally, we propose an extensions-based algorithm for real-world tomography scenarios and apply it to a synthetic 2D example. This study highlights two fundamental problems that make the method inefficient: (1) limited hypervolumes of extensions and (2) neglecting parameter correlations to simplify analytic calculations. Finding solutions to these problems defines possible directions for future research.

physics.geo-ph↗

arXiv2025

Direct-3D Variational Bayesian Surface Wave Inversion and Its Application to Ambient Noise Tomography beneath Great Britain

We present a new, variational, fully nonlinear, probabilistic ambient noise tomography method, which estimates subsurface structure and quantifies the corresponding uncertainties directly in three dimensions (3D) from inter-receiver seismic surface wave dispersion data. We use the method to invert for high resolution 3D seismic velocity models of the upper crust beneath Great Britain using seismic ambient noise data recorded around the region - a task that proved too high-dimensional and hence computationally demanding for Monte Carlo sampling to converge to a stable solution. We compare the inversion results from the new method to those obtained from two standard, indirect inversion methods, in which 2D (geographical) surface wave velocity maps and 1D (depth) shear velocity profiles are estimated in two separate, consecutive steps. The results show that the direct-3D scheme preserves better lateral continuity and produces better data fit than the two-step methods, and provides information about lateral correlations that is absent from the two-step solutions. The inversion results are consistent with large-scale geology of Great Britain, and for the first time provide seismologically-imaged evidence of the Great Glen Fault and other major tectonic faults. We therefore propose that direct-3D inversion schemes should be used where possible for surface wave inversion as they provide improved results at little additional computational cost.

physics.geo-ph↗

arXiv2025

The Roles of Low-Noise Stations, Arrays and Ocean-Bottom Seismometers in Monitoring UK Offshore Seismicity associated with Subsurface Storage of Carbon Dioxide

Effective seismic monitoring of subsurface carbon dioxide storage (SCS) sites is essential for managing risks posed by induced seismicity. This is particularly challenging in offshore environments, such as the Endurance license area in the North Sea, where the UK's permanent land-based seismometer network offers limited monitoring capability due to its distance from the expected locations of seismic events. A Bayesian experimental design framework is used to assess enhancements of the network with a low-noise onshore station located at around 1~km depth in Boulby mine, the onshore North York Moors Seismic Array, an optimally-located additional on-shore monitoring site, and ocean bottom seismometers (OBS). We quantify the expected information gain about seismic source locations and introduce a practical method to incorporate signal-to-noise dependent detectability and velocity model uncertainty. We show that the Boulby station or an onshore array primarily lower the detection threshold for small-magnitude events (M=0-2), but offer limited improvement in location accuracy. An optimally-located additional land-based seismometer or local array provides little additional benefit. OBS deployments yield significant improvements in location accuracy due to their proximity to potential seismicity. Optimised networks of two to three OBS stations are effective for Endurance, while three to five OBS stations offer robust monitoring across North Sea carbon storage licence areas off England's east coast. Velocity model uncertainty remains a key limiting factor for location precision across all configurations. We conclude that deploying OBS networks is the most promising strategy for enhancing microseismic monitoring capabilities at offshore SCS sites, though potentially more expensive.

physics.geo-ph↗

arXiv2024

Near-real-time design of experiments for seismic monitoring of volcanoes

Monitoring the seismic activity of volcanoes is crucial for hazard assessment and eruption forecasting. The layout of each seismic network determines the information content of recorded data about volcanic earthquakes, and experimental design methods optimise sensor locations to maximise that information. We provide a code package that implements Bayesian experimental design to optimise seismometer networks to locate seismicity at any volcano, and a practical guide to make this easily and rapidly implementable by any volcano seismologist. This work is the first to optimise travel-time, amplitude and array source location methods simultaneously, making it suitable for a wide range of volcano monitoring scenarios. The code-package is designed to be straightforward to use and can be adapted to a wide range of scenarios, and automatically links to existing global databases of topography and properties of volcanoes worldwide to allow rapid deployment. Any user should be able to obtain an initial design within minutes using a combination of generic and volcano-specific information to guide the design process, and to refine the design for their specific scenario within hours, if more specific prior information is available.

physics.geo-ph↗

arXiv2024

Inconsistency and Acausality in Bayesian Inference for Physical Problems

Bayesian inference is used to estimate continuous parameter values given measured data in many fields of science. The method relies on conditional probability densities to describe information about both data and parameters, yet the notion of conditional densities is inadmissible: probabilities of the same physical event, computed from conditional densities under different parameterizations, may be inconsistent. We show that this inconsistency, together with acausality in hierarchical methods, invalidate a variety of commonly applied Bayesian methods when applied to problems in the physical world, including trans-dimensional inference, general Bayesian dimensionality reduction methods, and hierarchical and empirical Bayes. Models in parameter spaces of different dimensionalities cannot be compared, invalidating the concept of natural parsimony, the probabilistic counterpart to Occams Razor. Bayes theorem itself is inadmissible, and Bayesian inference applied to parameters that characterize physical properties requires reformulation.

stat.ME↗

arXiv2024

Efficient Bayesian Full Waveform Inversion and Analysis of Prior Hypotheses in 3D

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, and apply it 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. The results obtained using prior information that models should be smooth show loop-like high uncertainty structures that are consistent with fully nonlinear inversion results presented previously. These structures disappear when smoothing is not imposed, so we conclude that they may be caused by smoothness constraints in tomographic problems. We further analyse a variety of prior hypotheses by constructing Bayesian L-curves, which reveal the sensitivity of the inversion process to different prior assumptions. To our knowledge, this is the first study that allows such prior hypotheses to be compared in probabilistic 3D FWI at feasible computational cost. This work shows that fully probabilistic 3D FWI can be performed and can be used to test different prior hypotheses, at a cost that may be practical, at least in small problems.

physics.geo-ph↗

arXiv2024

Variational Prior Replacement in Bayesian Inference and Inversion

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.

stat.ME↗

arXiv2023

Bayesian Inversion, Uncertainty Analysis and Interrogation using Boosting Variational Inference

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.

physics.geo-ph↗

arXiv2023

VIP -- Variational Inversion Package with example implementations of Bayesian tomographic imaging

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.

physics.geo-ph↗

arXiv2023

Bayesian Variational Time-lapse Full-waveform Inversion

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.

physics.geo-ph↗