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We present corrections to the expressions for surface-wave Green’s tensors shown in a recent paper by Malkoti et al .
Geodetic velocity data provide first-order constraints on crustal surface strain rates, which in turn are linked to seismic hazard. Estimating the 2-D surface strain tensor everywhere requires knowledge of the surface velocity field everywhere, while geodetic data such as Global Navigation Satellite System (GNSS) only have spatially scattered measurements on the surface of the Earth. To use these data to estimate strain rates, some type of interpolation is required. In this study, we review methodologies for strain rate estimation and compare a suite of methods, including a new implementation based on the geostatistical method of kriging, to compare variation between methods with uncertainty based on one method. We estimate the velocity field and calculate strain rates in southern California using a GNSS velocity field and five different interpolation methods to understand the sources of variability in inferred strain rates. Uncertainty related to data noise and station spacing (aleatoric uncertainty) is minimal where station spacing is dense and maximum far from observations. Differences between methods, related to epistemic uncertainty, are usually highest in areas of high strain rate due to differences in how gradients in the velocity field are handled by different interpolation methods. Parameter choices, unsurprisingly, have a strong influence on strain rate field, and we propose the traditional L -curve approach as one method for quantifying the inherent trade-off between fit to the data and models that are reflective of tectonic strain rates. Doing so, we find total variability between five representative strain rate models to be roughly 40 per cent, a much lower value than roughly 100 per cent that was found in previous studies (Hearn et al .). Using multiple methods to tune parameters and calculate strain rates provides a better understanding of the range of acceptable models for a given velocity field. Finally, we present an open-source Python package (Materna et al .) for calculating strain rates, Strain_2D, which allows for the same data and model grid to be used in multiple strain rate methods, can be extended with other methods from the community, and provides an interface for comparing strain rate models, calculating statistics and estimating strain rate uncertainty for a given GNSS data set.
We use a newly developed 2-D elastic reverse time migration (RTM) imaging algorithm based on the Helmholtz decomposition to test approaches for imaging the descending slab in subduction zone regions using local earthquake sources. Our elastic RTM method is designed to reconstruct incident and scattered wavefields at depth, isolate constituent P- and S- wave components via Helmholtz decomposition, and evaluate normalized imaging functions that leverage dominant P and S signals. This method allows us to target particular converted-wave scattering geometries, for example incident S to scattered P , which may be expected to have dominant signals in any given data set. The method is intended to be applied to dense seismic array observations that adequately capture both incident and converted wavefields. We draw a direct connection between our imaging functions and the first-order contrasts in shear wave material properties across seismic discontinuities. Through tests on synthetic data using either S → P or P → S conversions, we find that our technique can successfully recover the structure of a subducting slab using data from a dense wide-angle array of surface stations. We also calculate images with a small-aperture array to test the impact of array geometry on image resolution and interpretability. Our results show that our imaging technique is capable of imaging multiple seismic discontinuities at depth, even with a small number of earthquakes, but that limitations arise when a small aperture array is considered. In this case, the presence of artefacts makes it more difficult to determine the location of seismic discontinuities.
PyLith, a community, open-source code for modelling quasi-static and dynamic crustal deformation with an emphasis on earthquake faulting, has recently been updated with a flexible multiphysics implementation. We demonstrate the versatility of the multiphysics implementation by extending the code to model fully coupled continuum poromechanics. We verify the newly incorporated physics using standard benchmarks for a porous medium saturated with a slightly compressible fluid. The benchmarks include the 1-D consolidation problem as outlined by Terzaghi, Mandel’s problem for the 2-D case, and Cryer’s problem for the 3-D case. All three benchmarks have been added to the PyLith continuous integration test suite. We compare the closed form analytical solution for each benchmark against solutions generated by our updated code, and lastly, demonstrate that the poroelastic material formulation may be used alongside the existing fault implementation in PyLith.
Plate-coupling estimates and previous seismicity indicate that portions of the Makran megathrust of southern Pakistan and Iran are partially coupled and have the potential to produce future magnitude 7+ earthquakes. However, the GPS observations needed to constrain coupling models are sparse and lead to an incomplete understanding of regional earthquake and tsunami hazard. In this study, we assess GPS velocities for plate coupling of the Makran subduction zone with specific attention to model resolution and the accretionary prism rheology. We use finite element model-derived Green's functions to invert for the interseismic slip deficit under both elastic and viscoelastic Earth assumptions. We use the model resolution matrix to characterize plate-coupling scenarios that are consistent with the limited spatial resolution afforded by GPS observations. We then forward model the corresponding tsunami responses at major coastal cities within the western Indian Ocean basin. Our plate-coupling results show potential segmentation of the megathrust with varying coupling from west to east, but do not rule out a scenario where the entire length of the megathrust could rupture in a single earthquake. The full subduction zone rupture scenarios suggest that the Makran may be able to produce earthquakes up to M w 9.2. The corresponding tsunami model from the largest earthquake event ( M w 9.2) estimates maximum wave heights reaching 2–5 m at major port cities in the northern Arabian Sea region. Cities on the west coast of India are less affected (1–2 m). Coastlines bounding eastern Africa, and the Strait of Hormuz, are the least affected (<1 m).
A Bayesian linear regression to determine the bias in the Nafe–Drake relationship between compressional velocity and density provides an improved model for the density structure of Kīlauea volcano, Hawaiʻi. In previous work, we combined the results of seismic tomography with the Nafe–Drake relationship between compressional velocity and density to explain the large values of gravity disturbances overlying the summits and rift zones of the island's volcanoes. These results were used to determine mechanisms for gravitational instability of the island flanks. Here, we use laboratory measurements of the relationship of velocity and density for a wide range of Hawaiʻi island rocks as a prior in a Bayesian regression, with seismic tomography, to refine the 3-D density structure for Kīlauea volcano. This refined structure shows dense bodies (3220 kg m −3 ) between 5 and 8 km below sea level that underly regions of magma storage, found from geodetic and geophysical studies, beneath the summit and East Rift Zone of Kīlauea volcano. Above these bodies, density isosurfaces surround and cradle sources of pressure change determined from geodetic models, both at the summit and along the East Rift Zone. Continued subsidence of the summit following the 2018 eruption is aligned with a bowl-shaped density structure, formed primarily by density isosurfaces between 2800 and 2900 kg m −3 at 4–6 km depth. These surfaces underly the ∼3 km depth at which dyke injection initiates, are largely aseismic, and from their density values are inferred to contain high concentrations of olivine. Taken together, these density structures are consistent with an olivine-rich mush with variable porosity that increases in density with depth and provides a mechanism to form olivine cumulates both at the summit and along the rift zones. This structural framework for Kīlauea volcano is consistent with melt and mush transport occurring over a large range of depths to accommodate the growth and spreading of the volcano.
This paper introduces cross-fade sampling, a computationally efficient Markov Chain Monte Carlo simulation method that uses a semi-analytical approach to quickly solve Bayesian inverse problems that do not themselves have an analytical solution. Cross-fading is efficient in two ways. First, it requires fewer samples to obtain the same quality simulation of the target probability density function (PDF). Secondly, it is much faster to evaluate the posterior probability of each sample than conventional sampling methods for simulating Bayesian posterior PDFs. Conventional methods require evaluating the prior probability (which describes your a priori constraints) and data likelihood (which describes the fit between the observations and the predictions of the model) for each sample model. However, cross-fading does not require evaluating the data likelihood, meaning that ‘big data’ can be fit with zero additional computational cost. Further, the cross-fading approach can be used to calculate the marginal likelihood associated with a model design, facilitating model comparison and Bayesian model averaging. Topics covered in this paper include derivation of the cross-fade approach and how it can be used to simulate Bayesian posterior PDFs and compute the marginal likelihood, discussion of the class of problems to which cross-fading can be applied (with examples from earthquake statistics, earthquake ground motion modelling, volcanic eruption forecasting, and finite fault slip modelling), demonstration of efficiency relative to existing sampling methods and discussion of how cross-fading can be used to account for prediction errors (i.e. epistemic errors) as part of the geophysical inverse problem.
For imaging of seismic discontinuities at depth, reverse time migration (RTM) is a powerful method to apply to recordings of seismic events. It is especially powerful when an extensive receiver array, numerous seismic sources, or both, permit adequate reconstruction of incident and scattered wavefields at depth. Reconstructing either the incident or scattered wavefield at depth becomes less accurate when relatively few recordings of seismic events are available. Here we explore an inverse scattering approach to imaging discontinuities based on an adjoint method, employing sensitivity kernels (Frechet derivatives) that represent jumps in material properties across seismic-discontinuity surfaces. When combined with ray-based requirements on scattering geometry, it constitutes a powerful approach to determining the locations and amplitudes of the discontinuities, recovering only those properties that can be resolved by a spatially limited source and/or receiver distribution. This is illustrated by synthetic examples with local sources followed by a field example in a subduction zone setting
The thermal structure of the continental crust plays a critical role in understanding its elastic and rheologic properties as well as its dynamic processes. Thermal parameter data sets on continental scales have been used to constrain the crustal thermal structure, including both the direct (e.g. temperature, heat flux and heat conductivity measured at the surface) and indirect (e.g. seismically derived Mohorovičić discontinuity (Moho) temperature, geomagnetically derived Curie depth) observations. In this study, we present a new continental scale crustal heat generation model with additional information from seismologically inferred crustal composition. Together with previous direct and indirect thermal parameter data sets in the conterminous United States, we use the new crustal heat generation model to construct a 3-D crustal temperature model under a newly developed Bayesian framework. Specifically, we first derive profiles of crustal heat generation based on an empirical geochemical relationship at 1683 locations where seismologically derived crustal composition information is available. Then for each of these locations, the average heat generation values in the upper, middle and lower crust are combined with other thermal parameters through a Markov Chain Monte-Carlo inversion for a conductive, vertically smooth temperature profile. The results, posterior distributions of temperature profiles, are used to generate a 3-D crustal thermal model with the uncertainties systematically assessed. The new temperature model overall exhibits similar patterns to that from the U.S. Geological Survey National Crustal Model, but also reduces possible biases and the model's dependence on a single thermal parameter.
As probabilistic tsunami hazard analysis (PTHA) focuses more on assessments for localized, populous regions, techniques are needed to identify a subsample of representative earthquake ruptures to make the computational requirements for producing high-resolution hazard maps tractable. Moreover, the greatest epistemic uncertainty in seismic PTHA is related to source characterization, which is often poorly defined and subjective. We address these two salient issues by applying streamlined earthquake rupture forecasts (ERFs), based on combinatorial optimization methods, to an unsupervised machine learning workflow for identifying representative ruptures. ERFs determine the optimal distribution of a millennia-scale sample of earthquakes by inverting the observed slip rate on major faults. We use two previously developed combinatorial optimization ERFs, integer programming and greedy sequential, to produce the optimal location of ruptures with seismic moments sampled from a regional Gutenberg–Richter magnitude–frequency distribution. These ruptures in turn are used to calculate peak nearshore tsunami amplitude, using computationally efficient tsunami Green's functions. An unsupervised machine learning workflow is then used to identify a small subsample of the earthquakes input to ERFs for onshore PTHA analysis. We eliminate epistemic uncertainty related to source distribution under traditional PTHA analysis; in its place, a quantifiable, less subjective and generally smaller uncertainty related to the input to ERFs is included. The Nankai subduction zone is used as a test case, where previous ERFs have been conducted. Results indicate that the locations of representative earthquakes are sensitive to choice of magnitude–area relation and to whether a minimum cumulative stress objective is imposed on the fault. In general, incorporating ERFs into PTHA provide a physically self-consistent method to incorporate fault slip information in determining representative earthquakes for onshore PTHA, eliminating a major source of epistemic uncertainty.
Infrasound (acoustic waves below 20 Hz) can be used to detect, locate and quantify activity in the atmosphere such as volcanic eruptions and anthropogenic explosions. Attempts to quantify volcanic eruption parameters such as exit velocity, plume height and mass flow rate using infrasound data depend strongly on assumptions of the acoustic source type. Infrasonic sources may produce omnidirectional or directional wavefields, while propagation effects, such as interaction with topography, can induce further wavefield directivity that is measured by field instrumentation. Limited sampling of these wavefields can hinder our ability to infer the underlying source, and thus our understanding of the eruption characteristics. Equivalent sources are often used to represent acoustic source mechanisms and resultant wavefields. In this study, we review equivalent acoustic sources as they pertain to infrasonic scale and wavelengths commonly encountered in very local ( < 5 km range) geophysical field deployments. We highlight the equivalent infrasonic bipole source that can be induced by ground-reflection of an elevated monopole; we are not aware of any prior infrasound studies that use the bipole source concept. We use analytical and numerical methods to explore source directivity of monopole, dipole and bipole ground-reflected sources at infrasonic frequencies as well as the additional directivity complications introduced by interactions with topography. We illustrate that for typical volcano-infrasound wavelengths, increasing height above the ground as well as increasing source frequency leads to increased wavefield directivity. Numerical modelling using a simple omnidirectional monopole source embedded in topography further illustrates that both horizontal and vertical infrasound directionality can be induced by topography at the distance scales appropriate for local volcano infrasound monitoring. Information summarized in this analytical and numerical exploration of infrasound directivity may be used to help guide future volcano-infrasound field deployments intended to estimate source parameters or quantify wavefield directivity. Analytic solutions for simple whole-space or half-space atmospheres provide useful formulations for planning or initially analysing geophysical field-scale experimental data; however, especially at very local distances from the source ( < 5 km), 3-D simulations are necessary to account for complex topography commonly encountered in volcano-infrasound applications.
There are two fundamental probabilities in the seismic phase picking process – the probability of the existence of a seismic phase (detection probability) and the probability of correctly identifying the phase arrival time (timing probability). The nearly ubiquitous approach in developing deep learning phase picking models is to use a kernel, such as a truncated Gaussian, to mask the labeled phase arrival time, and train a segmentation model. Once a model is trained, the times of the peaks in the output are taken as phase arrival times (picks) and the height of the peaks are taken as “probability” of the picks. Here, we show that this “probability” represents neither the detection nor the timing probabilty because this approach forces the output to follow the shape of the kernel. We introduce an approach using two models to estimate these two distinct probabilities. We use a binary classifier with a calibrated confidence to address the detection probability and a multi-class classifier to obtain a probability mass function to address the timing probability. This new approach makes the deep learning-based phase picking process more interpretable and gives us options to logically control seismic monitoring workflows.
We refined the Attenuated ProPagation of Local Earthquake Shaking (APPLES) ground-motion-based earthquake early warning (EEW) approach, and directly compare APPLES performance with that of the source-characterization-based U.S. ShakeAlert EEW system for a suite of historical earthquakes in the U.S. West Coast and Japan. APPLES is an extension of the Propagation of Local Undamped Motion (PLUM) algorithm in which observed shaking intensity at seismic stations is used to forward-predict intensity distributions to surrounding areas using an attenuation model derived from an intensity prediction equation. We test new configuration options within APPLES, such as using the second highest estimated ground motion rather than the maximum, to better match median ground-motion observations and reduce alerts for small magnitude earthquakes, both of which are key alerting priorities within ShakeAlert. We evaluate these configurations alongside ShakeAlert by comparing the ground-motion estimation accuracy and available warning times relative to station observations and ShakeMap distributions. Our preferred APPLES configuration produces accurate ground-motion estimates and corresponds better with median observations compared to ShakeAlert’s estimates. This preferred configuration substantially reduces alert issuance for M < 5.0 earthquakes compared to the previous APPLES configuration, and alert-release criteria can further restrict alerts to primarily M ≥ 5.5 earthquakes without requiring magnitude estimation. Prioritizing matching median-observed ground motions may reduce APPLES warning times compared to configurations that were tuned to avoid missed alerts (such as those that use the maximum estimated ground motions), which can lead to shorter warning times compared to ShakeAlert for the same alert threshold. However, station-based warning time assessments demonstrate that APPLES can outperform ShakeAlert for high target thresholds. APPLES is a simple, independent EEW approach that may improve the robustness of EEW for the West Coast of the U.S.
Spatial autocorrelation (SPAC), the azimuthal average of the normalized cross-correlation between equidistant station pairs deployed in a 2-D array, is widely used to image the subsurface structure. However, the rigorous estimate of subsurface structure and its uncertainties as a function of depth using SPAC data is challenging due to the nonlinear relation between the SPAC data and Earth structure as well as the trade-off between depth and velocity. Additionally, data noise is strongly correlated due to data processing (e.g. filtering, stacking from multiple time segments and azimuthal averaging). Most studies do not account for the correlated noise and fix the ratio of compressional-wave velocity ( V P ) to shear-wave velocity ( V s ) (i.e. V P / V s ratio) and the number of layers, both of which are typically unknown. To address these challenges, we develop a hierarchical trans-dimensional Bayesian inversion of fundamental mode of SPAC data that properly accounts for the correlated data noise, samples the V P / V s ratio and relaxes the number of layers (i.e. model parametrization) to be unknown in the inversion. We further examine the limitation of using only fundamental modes in the inversion. Our synthetic experiments show that the inversion recovers an incorrect model unless we sample the correlated noise and V P / V s ratio in the inversion. The inversion is then applied to SPAC data acquired at 19 sites across the Tacoma basin in Washington State to characterize the V s and the time-averaged V s over 30-m depth ( V s 30 ). Our results show that the V s 30 varies from ∼200 to 800 m s −1 . The V s 30 within the basin is higher in the middle and lower on the east and west sides. We find that these V s 30 values vary with geologic unit. The uncertainties for V s 30 are within 20 m s −1 in average except for the most eastern site TB28. Additionally, the uncertainties are greater for deeper depths beneath most of the sites as the sensitivity decreases as a function of depth. The Vs structure as a function of depth is also complex beneath some sites, possibly because the SPAC curves are affected by higher order Rayleigh modes that are not considered in the inversion. To better constrain the deeper V s structure, V s 30 and/or other average measures of V s over depth, additional constraints from complementary data, such as ellipticity or geologic data are needed. Moreover, our synthetic experiments show that higher order modes can have significant effect in the inversion results, particularly when there is a low-velocity layer.
From late-December 2024 to mid-March 2025, a 50-km-long dyke intrusion triggered over 300 earthquakes (magnitude 4 to 5.9) between Fentale and Dofen volcanoes along the Northern Main Ethiopian Rift. Dyke intrusions periodically occur along the Fentale–Dofen magmatic segment and are an expression of ongoing rift extension. Preliminary analyses using interferometric synthetic aperture radar revealed extensive ground deformation (up to 60 cm), which closely matched the temporal and spatial evolution of surface manifestations and earthquake locations from global catalogues. While global catalogues are critical for real-time monitoring, the precision of locations in remote and or sparsely instrumented regions can be low. In this investigation, we present surface-wave relocation results of the dyking episode that began near Fentale volcano in December 2024. We estimate relative locations using differential traveltimes measured from regional-to-teleseismic distance surface-wave observations of earthquakes reported by the U.S. Geological Survey. Relative relocations reduce the initial region of diffuse seismicity to a 50-km-long narrow band bounding the strike of surface manifestations and the zone of maximum surface deformation. We demonstrate the precision of surface-wave relocations over incremental time periods, capturing the progression of dyking from seismic onset through seismic migration and caldera subsidence. Results showcase the utility of surface-wave relocations in the characterization of dyking episodes and provide complementary insights into the current understanding of the Fentale–Dofen volcanic plumbing system.
The 2004–2008 eruption of Mount St. Helens exemplified the practical challenges of processing high seismic event rates and data volumes. This protracted multi-year eruptive sequence was accompanied by a near-continuous occurrence of long-period (LP, 0.5–5 Hz) seismicity, but network-based picks and locations have previously been performed for only a relatively small subset of the overall seismicity, and full-waveform source–mechanism inversions have previously been performed for just two event types during the eruption. We present a new workflow for the bulk detection, cluster picking, 3-D location, and full-waveform source–mechanism inversion of repetitive LP volcanic seismicity recorded by a local broadband seismic network. Our emphasis is on methods that: exploit waveform similarity (multiplets) commonly observed for volcanic LP seismicity swarms; can take into account 3-D topographic and velocity structure heterogeneity typical of volcanoes; can process high event rates with relatively low analyst burden; and do not require a priori catalogue picks or known waveform templates. We develop and illustrate the workflow using data from Mount St. Helens from 2005 June 29 to July 31, a time-window characterized by high-rate LP seismicity, including numerous lower amplitude small LPs, and for which relatively dense seismic network coverage is available. Approximately 40 per cent of seismicity (20 561 events out of 52 141 triggers) during 2005 June 29 to July 31 is repetitive enough to be grouped into 17 multiplets of at least 300 events. We present 3-D locations of the 17 multiplets, representing 20 076 events having picks from at least eight stations, along with full-waveform source mechanism inversions of the 17 separate multiplet stacks. Collectively, our seismic 3-D hypocentre distributions for 2005 June 29 to July 31 are compatible with sources around the margins of the upper conduit section that is associated with plug solidification. All location results indicate compact seismicity clusters, with the 17 multiplets spatially connected and collectively located at shallow depth, approximately ~ 775 m depth beneath the 1980s crater floor and directly beneath, or immediately adjacent to, the extruding dacite spine. The full-waveform source–mechanism inversions for the 17 multiplets generally indicate mechanisms dominated by volumetric components consistent with subhorizontal cracks along with minor contributions from subvertical cracks. The overall picture is that of a stack or plexus of subhorizontal cracks located at shallow depth beneath the extruding dacite spines, consistent with subhorizontal hydrofractures extending from the conduit margins into the surrounding wall rock; minor subvertical crack components of the mechanisms could possibly arise from surfaces parallel to the conduit wall, such as within the damage halo associated with degassing pathways.
The nucleation area of the series of M6 events in Parkfield has been shown to be characterized by low b-values throughout the seismic cycle. Since low b -values represent high differential stresses, the asperity structure seems to be always stably stressed and even unaffected by the latest main shock in 2004. However, because fault loading rates and applied shear stress vary with time, some degree of temporal variability of the b -value within stable blocks is to be expected. We discuss in this study adequate techniques and uncertainty treatment for a detailed analysis of the temporal evolution of b -values. We show that the derived signal for the Parkfield asperity correlates with changes in surface creep, suggesting a sensitive time resolution of the b-value stress meter, and confirming near-critical loading conditions within the Parkfield asperity.
The use of seismic noise interferometry to retrieve Green's functions and the analysis of volcanic tremor are both useful in studying volcano dynamics. Whereas seismic noise interferometry allows long-range extraction of interpretable signals from a relatively weak noise wavefield, the characterization of volcanic tremor often requires a dense seismic array close to the source. We here show that standard processing of seismic noise interferometry yields volcanic tremor signals observable over large distances exceeding 50 km. Our study comprises 2.5 yr of data from the U.S. Geological Survey Hawaiian Volcano Observatory short period seismic network. Examining more than 700 station pairs, we find anomalous and temporally coherent signals that obscure the Green's functions. The time windows and frequency bands of these anomalous signals correspond well with the characteristics of previously studied volcanic tremor sources at Pu'u 'Ō'ō and Halema'uma'u craters. We use the derived noise cross-correlation functions to perform a grid-search for source location, confirming that these signals are surface waves originating from the known tremor sources. A grid-search with only distant stations verifies that useful tremor signals can indeed be recovered far from the source. Our results suggest that the specific data processing in seismic noise interferometry—typically used for Green's function retrieval—can aid in the study of both the wavefield and source location of volcanic tremor over large distances. In view of using the derived Green's functions to image heterogeneity and study temporal velocity changes at volcanic regions, however, our results illustrate how care should be taken when contamination by tremor may be present.