Malcolm Sambridge

• Professor at Australian National University

The lack of versatile tools for Bayesian inference presents a significant challenge to researchers in geophysics, who often resort to developing bespoke codes to address specific classes of inverse problems. In this study, we present BayesBay, a Python package for generalized transdimensional and hierarchical Markov chain Monte Carlo sampling. Leve...

Bayesian inference methods are widespread in geophysics and over the decades have been extensively applied to inverse problems, where they are particularly represented through Monte Carlo methods. These are popular as they allow to easily quantify uncertainties and parameter trade-offs but on the downside usually require large computational effort....

Trans-dimensional Bayesian sampling has been applied to subsurface imaging and other inference problems across the Earth Sciences. A particular style of Markov chain Monte Carlo (McMC) method, known as reversible-jump has been used almost universally in such studies. This algorithm allows sampling across variably dimensioned model parameterizations...

We present a theoretical framework that links Fermat’s principle of least time to optimal transport theory via a cost function that enforces local transport. The proposed cost function captures the physical constraints inherent in wave propagation; when paired with specific mass distributions, it yields shortest paths in the considered media throug...

In this chapter, we survey some recent developments in the field of geophysical inversion. We aim to provide an accessible general introduction to the breadth of current research, rather than focussing in depth on particular topics. We hope to give the reader an appreciation for the similarities and connections between different approaches, and the...

Regularized least-squares tomography offers a straightforward and efficient imaging method and has seen extensive application across various fields. However, it has a few drawbacks, such as (i) the regularization imposed during the inversion tends to give a smooth solution, which will fail to reconstruct a multi-scale model well or detect sharp dis...

When working with hydrological data, the ability to quantify the similarity of different datasets is useful. The choice of how to make this quantification has a direct influence on the results, with different measures of similarity emphasising particular sources of error (for example, errors in amplitude as opposed to displacements in time and/or s...

Concerns raised by Okazaki & Ueda (2022) on the paper by Sambridge et al. (2022) are addressed. Two issues are discussed and some new numerical results presented. The first concerns whether the properties of the Wasserstein time-series misfit introduced in our earlier paper will translate to model space non-uniqueness in a seismic waveform inversio...

When working with hydrological data, the ability to quantify the similarity of different datasets is useful. The choice of how to make this quantification has direct influence on the results, with different measures of similarity emphasising particular sources of error (for example, errors in amplitude as opposed to displacements in time and/or spa...

The package pyprop8 enables calculation of the response of a 1-D layered halfspace to a seismic source, and also derivatives (‘sensitivity kernels’) of the wavefield with respect to source parameters. Seismograms, seismic spectra, and measures of static displacement (e.g. GPS, InSAR and field observations) may all be simulated. The method is based...

We propose a new approach to measuring the agreement between two oscillatory time series, such as seismic waveforms, and demonstrate that it can be employed effectively in inverse problems. Our approach is based on Optimal Transport theory and the Wasserstein distance, with a novel transformation of the time series to ensure that necessary normalis...

We propose a new approach to measuring the agreement between two oscillatory time series, such as seismic waveforms, and demonstrate that it can be employed effectively in inverse problems. Our approach is based on Optimal Transport theory and the Wasserstein distance, with a novel transformation of the time series to ensure that necessary normalis...

Monte Carlo methods are widespread in geophysics and have proved to be powerful in nonlinear inverse problems. However, they are associated with significant practical challenges, including long calculation times, large output ensembles of Earth models, and difficulties in the appraisal of the results. This paper addresses some of these challenges u...

The relationship between ∆³⁶S and ∆³³S in Archean sedimentary pyrites has been used to evaluate early geologic processes, including photochemical reactions in the anoxic atmosphere, biological activity and thermochemical alteration during sediment deposition. We have applied statistical methods to quadruple S isotope analyses of Archean sedimentary...

Available here: https://arxiv.org/abs/2110.06017

The core‐mantle boundary (CMB) is the most extreme boundary within the Earth where the liquid, iron‐rich outer core interacts with the rocky, silicate mantle. The nature of the lowermost mantle atop the CMB, and its role in mantle dynamics, is not completely understood. Various regional studies have documented significant heterogeneities at differe...

The core-mantle boundary (CMB) is the most extreme boundary within the Earth where the liquid, iron-rich outer core interacts with the rocky, silicate mantle. The nature of the lowermost mantle atop the CMB, and its role in mantle dynamics, is not completely understood. Various regional studies have documented significant heterogeneities at differe...

The D″ layer, regarded as a thermal boundary layer and a chemically distinct region above the core-mantle boundary (CMB), has been associated with the phase transition from bridgmanite (Bm) to post-perovskite (pPv) in the lowermost mantle. However, the composition of the lowermost mantle and thermal conditions where Bm-pPv phase-transition occurs i...

The model of cylindrical anisotropy in the inner core (IC) states that seismic rays traveling parallel to the Earth's rotational axis travel faster than those parallel to the equator. There have been continuing discrepancies in estimates of the strength and orientation of anisotropy, with some evidence suggesting that such a model may not be suppor...

For more than half a century, seismic tomography has been used to map the volumetric structure of Earth’s interior, but only recent advances in computation have enabled the application of this technique at scale. Estimates of surface waves that travel between two seismic stations can be reconstructed from a stack of cross-correlations of continuous...

In order to improve exploration success under cover the UNCOVER initiative identified high resolution 3D seismic velocity characterization of the Australian plate as a high priority. To achieve this goal government and academia have united around the Australian passive seismic Array project (AusArray) which aims to obtain a national half degree dat...

In recent years, the concept of 'compressive sensing' has promised a revolution in data collection. Whereas traditional data acquisition requires regular sampling of a time (or spatial) signal, compressive sensing advocates a randomised data collection strategy. Provided the target signal is 'sparse', i.e. has only a few non-zero Fourier components...

We develop a theoretical framework for framing and solving probabilistic linear(ized) inverse problems in function spaces. This is built on the statistical theory of Gaussian Processes, and allows results to be obtained independent of any basis, avoiding any difficulties associated with the fidelity of representation that can be achieved. We show t...

By starting from a general framework for probabilistic continuous inversion (developed in Part I) and introducing discrete basis functions, we obtain the well-known algorithms for probabilistic least-squares inversion set out by Tarantola & Valette. In doing so, we establish a direct equivalence between the spatial covariance function that must be...

A method of extracting group and phase velocity dispersions jointly for Love‐ and Rayleigh‐wave observations is presented. This method uses a spectral element representation of a path average Earth model parameterized with density, shear‐wave velocity, radial anisotropy, and VP/VS ratio. An initial dispersion curve is automatically estimated using...

Conceptual uncertainty is considered one of the major sources of uncertainty in groundwater flow modelling. In this regard, hypothesis testing is essential to increase system understanding by refuting alternative conceptual models. Often a stepwise approach, with respect to complexity, is promoted but hypothesis testing of simple groundwater models...

Obtaining slip distributions for earthquakes results in an ill-posed inverse problem. While this implies that only limited and uncertain information can be recovered from the data, inferences are typically made based only on a single regularized model. Here, we develop an inversion approach that can quantify uncertainties in a Bayesian probabilisti...

Following the linearized attenuation tomography from our previous study (Pejić et al., 2017, https://doi.org/10.1002/2016JB013692), we perform hierarchical transdimensional Bayesian tomography of the upper ≈400 km of the inner core, using 398 globally distributed t∗ estimates. The results are in good agreement with the ones obtained through lineari...

The use of Bayesian trans-dimensional sampling in 2-D and 3-D imaging problems has recently become widespread in geophysical inversion. Its benefits include its spatial adaptability to the level of information present in the data and the ability to produce uncertainty estimates. The most used parameterization in Bayesian trans-dimensional inversion...

Antarctica and Greenland hold enough ice to raise sea level by more than 65 m if both ice sheets were to melt completely. Predicting future ice sheet mass balance depends on our ability to model these ice sheets, which is limited by our current understanding of several key physical processes, such as iceberg calving. Large-scale ice flow models eit...

We address the problem of unpolarized light spectroscopy of geological materials. Using infrared radiation, the aim of this technique is to learn about the absorbing species, such as hydroxyl. The use of unoriented samples leads to the need to perform a rigorous statistical analysis, so that the three principal absorbances of the crystal can be ret...

Most linear inverse problems require regularisation to ensure that robust and meaningful solutions can be found. Typically, Tikhonov-style regularisation is employed, whereby a preference is expressed for models that are somehow ‘small’ and/or ‘smooth’. The strength of such preferences is expressed through one or more ‘damping parameters’, which co...

Due to increased emissions of greenhouse gases, oceans are warming, causing sea level to rise as the density of seawater falls. Predicting the rates of steric expansion is challenging because of the natural variability of the ocean and because observations are insufficient to adequately cover the ocean basins. Here, we investigate the ability of on...

Cross-correlation of seismograms provides new information on the Earth both through the exploitation of ambient noise and specific components of earthquake records. Here, we cross- correlate recordings of large earthquakes on a planetary scale and identify a range of hitherto unobserved seismic phases in Earth’s correlation wavefield. We show that...

Antarctica and Greenland hold enough ice to raise sea level by more than 65 m if they were to melt completely. Predicting future ice sheet mass balance depends on our ability to model these ice sheets, which is limited by our current understanding of several key physical processes, such as iceberg calving. Large-scale ice flow models either ignore...

The mean land-surface temperature represents an important boundary condition for many geothermal studies. This boundary is particularly important to help constrain the models made to analyse resource systems, many of which are shallow in nature and observe relatively small thermal gradients. Consequently, a mean land-surface temperature map of the...

Geophysicists are often concerned with reconstructing subsurface properties using observations collected at or near the surface. For example, in seismic migration, we attempt to reconstruct subsurface geometry from surface seismic recordings, and in potential field inversion, observations are used to map electrical conductivity or density variation...

Antarctic and Greenland hold more than 99 % of all fresh water on Earth and, therefore, can significantly influence global sea level. Predicting future ice sheet mass balance depends upon ice sheet modelling, but it is limited by knowledge of a number of processes, some of which are still poorly understood. One such process is the calving of the ic...

The solidification of the Earth's inner core shapes its texture and rheology, affecting the attenuation and scattering of seismic body waves transmitted through it. Applying attenuation tomography in a Bayesian framework to 398 high-quality PKIKP waveforms, we invert for the apparent Qp for the uppermost 400 km below the inner core boundary (ICB) a...

This paper presents the application of a novel trans-dimensional sampling approach to a time domain airborne electromagnetic (AEM) inverse problem to solve for plausible conductivities of the subsurface. Geophysical inverse field problems, such as time domain AEM, are well known to have a large degree of non-uniqueness. Common least-squares optimis...

An enigmatic patchwork of seismologically observed low-seismic velocity zones (LVZs) has been reported at various scales and locations above the 410-km discontinuity and in the mantle transition zone (TZ). Geodynamical models suggest that this is possibly a distinctive feature of a convective mantle that preserves small-scale chemical heterogeneiti...

The concept of uncertainty in geophysical inversion is often confined to quantification of errors in parameters estimated from some data. A broader definition is to include uncertainty arising from the assumptions made in posing the inverse problem in the first place. These may include assumptions about the physics of the relationship between obser...

Teleseismic waves that propagate near-vertically beneath seismic receivers, reflect from the free surface, propagate downwards and are subsequently reflected again back to the surface. Thus a single station can be used to convert a time series to a reflectivity record by means of computing autocorrelation. For Antarctic stations deployed on ice, mo...

A new approach is presented for the reconstruction of time series and other (y,x) functions from observables with any type of stochastic noise. In particular, noise may exist in both dependent and independent variables, i.e., y and x, or t, and may even be correlated between these variables. This situation occurs in many areas of the geosciences wh...

This paper studies the initial sea surface displacement and its uncertainty after an earthquake based on tsunami waveforms. The spatial distribution is inferred with a Bayesian approach that provides probabilities that are interpreted as uncertainties of the displaced sea surface. The parameterization is nonlinear and treats apparent rupture veloci...

Non-linear inverse problems in the geosciences often involve probabilistic sampling of multimodal density functions or global optimization and sometimes both. Efficient algorithmic tools for carrying out sampling or optimization in challenging cases are of major interest. Here results are presented of some numerical experiments with a technique, kn...

Working with a large temporal dataset spanning several decades often represents a challenging task, especially when the record is heterogeneous and incomplete. The use of statistical laws could potentially overcome these problems. Here we apply Benford's Law (also called the "First-Digit Law") to the traveled distances of tropical cyclones since 18...

It has been proposed that volcanic hotspots and the reconstructed eruption sites of large igneous provinces (LIPs) are preferentially located above the margins of two deep mantle large low shear-wave velocity provinces (LLSVPs), beneath the African continent and the Pacific Ocean. This spatial correlation has been interpreted to imply that LLSVPs r...

Seismic tomography is a powerful tool for illuminating Earth structure across a range of scales, but the usefulness of any image that is generated by this method is dependent on our ability to quantify its uncertainty. This uncertainty arises from the ill-posed nature of the tomographic inverse problem, which means that multiple models are capable...

This paper outlines a methodology for the estimation of the environmental noise equivalent reflectance in aquatic remote sensing imagery using an object-based segmentation approach. Noise characteristics of remote sensing imagery directly influence the accuracy of estimated environmental variables and provide a framework for a range of sensitivity,...

Significance Several areas of earth science require knowledge of the fluctuations in sea level and ice volume through glacial cycles. These include understanding past ice sheets and providing boundary conditions for paleoclimate models, calibrating marine-sediment isotopic records, and providing the background signal for evaluating anthropogenic co...

INTRODUCTION Astronomy has space travel and supernovas, chemistry has goo and explosions, biology has creepy-crawlies and medical miracles, and the earth sciences have earthquakes and volca-noes. Each of the sciences has some way to engage and inspire the future generations of scientists. As researchers we are con-stantly seeking ways to communicat...

This paper develops a probabilistic Bayesian approach to the problem of inferring the spatiotemporal evolution of earthquake rupture on a fault surface from seismic data with rigorous uncertainty estimation. To date, uncertainties of rupture parameters are poorly understood, and the effect of choices such as fault discretization on uncertainties ha...

Over the last 25 years, several studies have tested for a link between geomagnetic field intensity and reversal frequency. However, despite a large increase in the number of absolute paleointensity determinations, and improved methods for obtaining such data, two competing models have arisen. Here, we employ a new tool for objectively analyzing pal...

Knowledge of Nubia/Somalia relative motion since the Early Neogene is of particular importance in the Earth Sciences, because it (i) impacts on inferences on African dynamic topography; and (ii) allows us to link plate kinematics within the Indian realm with those within the Atlantic basin. The contemporary Nubia⁄Somalia motion is well known from g...

Knowledge of past plate motions derived from ocean–floor finite rotations is an important asset of the Earth Sciences, because it allows linking a variety of shallow– and deep–rooted geological processes. Efforts have recently been taken towards inferring finite rotations at the unprecedented temporal resolution of 1 Myr or less, and more data are...

Non-linear inverse problems in the geosciences often involve probabilistic sampling of multimodal density functions or global optimization and sometimes both. Efficient algorithmic tools for carrying out sampling or optimization in challenging cases are of major interest. Here results are presented of some numerical experiments with a technique, kn...

Coda waves arise from scattering to form the later arriving components of a seismogram. Coda‐wave interferometry (CWI) is an emerging tool for constraining earthquake source properties from the interference pattern of coda waves between nearby events. A new earthquake location algorithm is derived which relies on coda‐wave‐based probabilistic estim...

Determining the scale-length, magnitude, and distribution of heterogeneity in the lowermost mantle is crucial to understanding whole mantle dynamics, and yet it remains a much debated and ongoing challenge in geophysics. Common shortcomings of current seismically derived lowermost mantle models are incomplete raypath coverage, arbitrary model param...

We conduct an experiment to investigate whether linearity in the observed velocity gradient in the volume of the inner core sampled by the PICP ray paths beneath Central America is a robust approximation. Instead of solving an optimization problem, we approach it within the Bayesian inference. This is an ensemble approach, where model specification...

Geodynamical models and seismic observations suggest that the Earth's solid inner core rotates at a different rate than the mantle. However, discrepancies exist in rotation rate estimates based on seismic waves produced by earthquakes. Here we investigate the inherent assumption of a constant rotation rate using earthquake doublets--repeating earth...

In a linear ill-posed inverse problem, the regularisation parameter (damping) controls the balance between minimising both the residual data misfit and the model norm. Poor knowledge of data uncertainties often makes the selection of damping rather arbitrary. To go beyond that subjectivity, an objective rationale for the choice of damping is presen...

The Australian Seismometers in Schools program (AuSIS) has just completed year one of its initial four-year program. The year has been filled with excitement as we completed installing pilot instruments in schools, launched the program nationally and received over 110 "Expressions of Interest" from schools around Australia. The data quality has exc...

The field of seismology is rich with inverse problems. Seismologists are constantly seeking new ways to use seismic waveforms, and data products derived from them, to constrain subsurface structure in the form of Earth properties in 1-, 2- and 3 dimensions, as well as seismic sources in space and time. Every approach has its limitations and a virtu...

We present a method to quantify abrupt changes (or changepoints) in data series, as a function of depth or time. These changes are often the result of environmental variations and can be manisfested differently in multiple data sets, but all data can have the same changepoint locations. The method uses transdimensional Markov chain Monte Carlo to i...

Seismologists construct images of the Earth's interior structure using observations, derived from seismograms, collected at the surface. A common approach to such inverse problems is to build a single 'best' Earth model, in some sense. This is despite the fact that the observations by themselves often do not require, or even allow, a single best-fi...

SUMMARY A new approach for the 1D inversion of AEM data has been developed. We use a reversible jump Markov Chain Monte Carlo method to perform Bayesian inference. The Earth is partitioned by a variable number of nonoverlapping cells defined by a 1D Voronoi tessellation. A cell is equivalent to a layer in conventional AEM inversion and has a corres...

This paper applies parallel tempering within a Bayesian formulation for strongly nonlinear geoacoustic inverse problems. Bayesian geoacoustic inversion consists of sampling the posterior probability density (PPD) of seabed parameters to estimate integral properties, such as marginal probability distributions, based on ocean acoustic data and prior...

Interpolation of spatial data is a widely used technique across the Earth sciences. For example, the thickness of the crust can be estimated by different active and passive seismic source surveys, and seismologists reconstruct the topography of the Moho by interpolating these different estimates. Although much research has been done on improving th...

Understanding lithospheric plate motions is of paramount importance to geodynamicists. Much effort is going into kinematic reconstructions featuring progressively finer temporal resolution. However, the challenge of precisely identifying ocean-floor magnetic lineations, and uncertainties in geomagnetic reversal timescales result in substantial fini...

A meaningful interpretation of seismic measurements requires a rigorous quantification of the uncertainty. In an inverse problem, the data noise determines how accurately observations should be fit, and ultimately the level of detail contained in the recovered model. A common problem in seismic tomography is the difficulty in quantifying data uncer...

A new theory is described for the uptake of U in an open system applied to the dating of archaeological bones. Analytical solutions are obtained for the rate of radioactive decay of 238U, 234U and 230Th as a function of position for the case where both 238U and 234U diffuse across a bone, and where external supply of 234U is not in equilibrium with...

We report the first observational evidence that the complex rotational dynamics of the Earth's inner core appear to be in close relationship with the geomagnetic field. We infer from a newly observed collection of earthquake doublets that the Earth's inner core "shuffles", exhibiting both prograde and retrograde rotation in the reference frame of t...

We present a novel method for joint inversion of receiver functions and surface wave dispersion data, using a transdimensional Bayesian formulation. This class of algorithm treats the number of model parameters (e.g. number of layers) as an unknown in the problem. The dimension of the model space is variable and a Markov chain Monte Carlo (McMC) sc...

To better constrain the structure of the Earth's interior, new theoretical developments on seismic wave propagation have emerged in recent years, and received increasing attention in global tomography. A recent focus has been to take into account the "Finite-Frequency" (FF) behaviour of seismic waves (e.g. wavefront-healing). We have chosen to use...

For the past forty years seismologists have built models of the Earth's seismic structure over local, regional and global distance scales using derived quantities of a seismogram covering the frequency spectrum. A feature common to (almost) all cases is the objective of building a single `best' Earth model, in some sense. This is despite the fact t...

Shuffling is a tribal dance recently adapted by teenagers as a street dance. In one of the most popular moves, the so-called "Running Man", a stomp forward on one foot, shifted without being lifted from the ground, is followed by a change of position backwards on the same foot. Here, we present strong observational evidence from a newly observed co...

The Australian Seismometers in Schools (SIS) programme is a four-year project (2011-2014) funded by the Education component of AuScope Australian Geophysical Observing System (AGOS). Over the next four years SIS will build a network of 40 seismometers installed in high schools across the nation to provide real-time monitoring of the Australian cont...

We present a method to quantify abrupt changes (or changepoints) in data series, represented as a function of depth or time. These changes are often the result of climatic or environmental variations and can be manifested in multiple datasets as different responses, but all datasets can have the same changepoint locations/timings. The method we pre...

1] Coda wave interferometry (CWI) can be used to estimate the separation between a pair of earthquakes directly from the coda recorded at a single station. Existing CWI methodology leads to a single estimate of separation and provides no information on uncertainty. Here, the theory of coda wave interferometry is revisited and modifications introduc...

The goal of exploration geophysics is to infer the nature of buried structure and, in particular, generate drill targets that lead to a mineral deposit discovery or reserve delineation. As a profession, we aim to turn geophysical data into geological information. Most geophysical techniques enable inferences to be made from airborne, ground-based o...

There is significant seismic activity in the region around Australia, largely due to the plate boundaries to the north and to the east of the mainland. This activity results in serious seismic and tsunami hazard in the coastal areas of Australia. Hence seismicity is and will be monitored in real time by Geoscience Australia (GA), which uses a netwo...

More than 100 years ago it was predicted that the distribution of first digits of real world observations would not be uniform, but instead follow a trend where measurements with lower first digit (1,2,...) occur more frequently than those with higher first digits (...,8,9). This result has long been known by mathematicians but regarded as mere mat...

Monte Carlo sampling, relying on random numbers, has been used in Geophysics for over 40 years, although the increase in computing power has seen a commensurate increase in applications in the last 15 years or so. This approach avoids the use of gradients, is robust to local minima, and so is suitable for nonlinear inverse problems which often have...

Wireline logs record the variation in a number of physical measurements, sometimes 20 or more different properties, with depth down a borehole. They are routinely correlated with, and/or interpreted in terms of, the rock stratigraphic record. Logs are also interpreted with the aim of inferring other useful physical properties not directly measured....

The geologic structure of southeastern Australia is a complex accumulation of multiple Palaeozoic orogenic belts that abut the Precambrian Australian craton. A lack of outcrop due to extensive Mesozoic-Cenozoic sedimentary and volcanic cover sequences have limited direct access to much of the Palaeozoic substrate, resulting in a variety of plausibl...

Trans-dimensional Bayesian inverse methods have been recently introduced in the geosciences to solve a variety of inverse problems. This class of algorithms treat the number of model parameters (e.g. number of layers) as an unknown in the problem. The dimension of the model space is variable and Markov chain Monte Carlo (McMC) schemes are used to p...

A longstanding issue in seismic tomography is discriminating between features that are real and required by the data, and those which are artefacts of the inversion process. Most inverse problems are under-determined and solved using an optimization based approach which generates an optimal model in some sense. In these circumstances all characteri...

We present a globally distributed data set of ∼400 000 frequency-dependent SH-wave traveltimes. An automated technique is used to measure teleseismic S, ScS and SS traveltimes at several periods ranging from 10 to 51 s. The targeted seismic phases are first extracted from the observed and synthetic seismograms using an automated time window algorit...