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At least 109 records · Page 6

Enhanced Microseismicity During Production Pumping Cessation at the San Emidio Geothermal Field (Nevada, USA) in December 2016

Abstract Tectonic activity, geothermal fluids, and microseismic events (MSEs) tend to occur in similar locations as a result of spatiotemporal changes in the subsurface stress state. To quantify this association, we analyze data from a dense seismic array deployed at the San Emidio geothermal field, Nevada for 1 week in December 2016 to coincide with a 19.45‐hr shutdown of all injection and production pumping operations. 123 MSEs were detected, of which 101 occurred during the shutdown. The spatial association of the MSEs with the production wells suggests a causal relationship between the production cessation and the MSEs. Here we performed a detailed analysis to investigate reservoir material properties, distribution of seismically activated faults, and local stress state. We determined the hypocenters, magnitudes, and focal mechanisms for the MSEs, P‐wave tomographic velocity model, and local stress tensor. The results show that most MSEs occurred near the production wells. Magnitudes fall between −2.2 and 0.0 with larger events located closer to the production wells. Most MSEs occurred within a westward‐dipping normal fault zone in the reservoir associated with anomalously low P‐wave velocity values. The focal mechanism and stress inversion results show predominantly normal faulting with the maximum horizontal stress oriented north‐south. We suggest that the MSEs during shutdown were triggered on pre‐existing, small‐scale, critically stressed fault patches in the reservoir as the pore pressure increased around the production wells when the production pumping ceased. We interpret the larger MSE magnitudes closer to the production wells as a result of higher pore pressure increase.

15 GEOTHERMAL ENERGY↗

Geothermal Play-Fairway Analysis of Washington State Prospects: Final Report

The Washington State Geothermal Play-Fairway Analysis overcomes the exploration challenges posed by dense vegetation, glacial deposits, and extreme precipitation. The geothermal play-fairways we target are locations where heat, permeability, and saturated porosity are present in sufficient volume to provide adequate heat exchange at depths accessible by modern drilling technology. The three study areas lie along the Cascade Range magmatic arc and are near Mount Baker, Mount St. Helens, and the Wind River Valley. The seven-year project is divided into three phases. In Phase 1 we build on a previous statewide assessment of geothermal resources and develop an initial modeling approach. The results are a series of favorability, uncertainty, and risk maps for three targeted study areas. Based on these initial results, we collect new geologic and geophysical data to further refine our modeling and reduce exploration uncertainty in Phase 2. We improve the modeling method to handle the new data and update the favorability, uncertainty, and risk maps. We also update the conceptual geothermal resource models. In Phase 3 we validate our modeling approach by drilling two temperature-gradient holes and collecting and analyzing core, image logs, and new geochemistry. Our modeling approach improves on an earlier statewide method through a more-rigorous and detailed assessment of heat and permeability. Permeability potential is assessed through geomechanical modeling of the deformation that can generate and maintain reservoir porosity and permeability. Metrics to inform heat potential include temperature-gradient wells, which are sparse in Washington; proximity of Quaternary volcanic vents and young intrusive rock; spring temperature; and reservoir temperature inferred from geothermometry. We weight the individual components using an expert-guided approach known as the Analytical Hierarchy Process. During Phase 2 we also develop a fluid-filled fracture model, and an infrastructure model that helps to delineate areas which are more favorable for geothermal development based on proximity to transmission lines, elevation, land ownership and use restrictions, and availability of process water. New geologic and geophysical data is collected during Phase 2 in each of our three main study areas. At Mount Baker and north of Mount St. Helens we conduct 1:24,000-scale geologic mapping and lidar analysis to better constrain the location and character of surface faults; detailed mapping in the Wind River Valley was completed just prior to the start of this project. Ages of intrusive rocks are determined with 40 Ar/ 39 Ar geochronology, though all of our samples are Miocene or older. We collect ground based gravity observations (a total of 1,580 new stations) in all of our study areas and ground-based magnetic lines (a total of 93 km) at Mount Baker. These data are combined with existing gravity and aeromagnetic data and used to constrain fault locations and geometry. Two to three cross sections are constructed at each study area using the mapped surface geology and forward-modeling of the gravity and magnetic data; these cross sections form the basis for our updated conceptual models. We collect magnetotelluric surveys at Mount Baker and Mount St. Helens and these data are inverted to form a resistivity model from the surface to about 10 km depth; each model shows conductive zones that can be interpreted as upwelling geothermal fluids. At Mount St. Helens we deploy a passive seismic array and use the newly detected events to refine the location of the Saint Helens seismic zone. We also employ ambient-noise tomography to develop a detailed seismic-velocity model for the study area and use this model to help constrain our cross sections and conceptual model. Based on the new data collected during Phase 2—and our updated models—we develop a campaign of temperature-gradient holes and core analysis to validate our modeling in Phase 3. Drill hole MB76-31 is located near Little Park Creek, 11 km west-southwest of the summit of Mount Baker, and is 1,471 ft deep. About 410 ft of core from the lower portion of the hole—and image logs from ~175 ft below ground surface to the bottom—are collected and analyzed. Water samples are collected and processed for geothermometry. Drill hole MSH17-24 is located along upper Schultz Creek, 16 km north-northeast of Mount St. Helens and has core from 470 ft to the bottom at 1,053 ft. We did not collect image logs due to borehole stability concerns, but water samples are collected and analyzed for geothermometry. Repeat temperature-gradient measurements are made at both sites and thermal conductivity is measured from core samples. At MB76-31, the equilibrated temperature gradient of 64°C/km and calculated heat flow of 141–159 mW/m 2 is more than twice the regional average. Detailed mapping and analysis of the core, coupled with correlation to the image logs, indicates a history of permeability generation consistent with our predictions of high permeability. Because the site has high favorability in the Phase 2 model, we consider the results a positive validation of the modeling. At site MSH17-24, the equilibrated temperature gradient of ~15°C/km and calculated heat flow of 41–43 mW/m 2 are similar to regional. Geochemical analysis of the water samples indicates a meteoric source without any geothermal component. Detailed outcrop-based mapping of fault exposures near the drill site and analysis of image logs from nearby boreholes indicates a history of permeability generation consistent with our predictions. Because the site has low favorability in the Phase 2 model, we consider the results a positive validation of the modeling. Together, the two sites provide a reasonably positive validation of the Phase 2 modeling and should encourage future use of this modeling approach.

15 GEOTHERMAL ENERGY↗

Making Invisible Visible: Data-Driven Seismic Inversion With Spatio-Temporally Constrained Data Augmentation

Deep learning and data-driven approaches have shown great potential in scientific domains. The promise of data-driven techniques relies on the availability of a large volume of high-quality training datasets. Due to the high cost of obtaining data through expensive physical experiments, instruments, and simulations, data augmentation techniques for scientific applications have emerged as a new direction for obtaining scientific data recently. However, existing data augmentation techniques originating from computer vision yield physically unacceptable data samples that are not helpful for the domain problems that we are interested in. In this article, we develop new data augmentation techniques based on convolutional neural networks. Specifically, our generative models leverage different physics knowledge (such as governing equations, observable perception, and physics phenomena) to improve the quality of the synthetic data. To validate the effectiveness of our data augmentation techniques, we apply them to solve a subsurface seismic full-waveform inversion using simulated CO 2 leakage data. Our interest is to invert for subsurface velocity models associated with very small CO 2 leakage. We validate the performance of our methods using comprehensive numerical tests. Here via comparison and analysis, we show that data-driven seismic imaging can be significantly enhanced by using our data augmentation techniques. Particularly, the imaging quality has been improved by 15% in test scenarios of general-sized leakage and 17% in small-sized leakage when using an augmented training set obtained with our techniques.

58 GEOSCIENCES↗

High-resolution Bayesian spatial autocorrelation (SPAC) quasi-3-D Vs model of Utah FORGE site with a dense geophone array

SUMMARY We expand the application of spatial autocorrelation (SPAC) from typical 1-D Vs profiles to quasi-3-D imaging via Bayesian Monte Carlo inversion (BMCI) using a dense nodal array (49 nodes) located at the Utah Frontier Observatory for Research in Geothermal Energy (FORGE) site. Combinations of 4 and 9 geophones in subarrays provide for 36 and 25 1-D Vs profiles, respectively. Profiles with error bars are determined by calculating coherency functions that fit observations in a frequency range of 0.2–5 Hz. Thus, a high-resolution quasi-3-D Vs model from the surface to 2.0 km depth is derived and shows that surface-parallel sedimentary strata deepen to the west, consistent with a 3-D seismic reflection survey. Moreover, the resulting Vs profile is consistent with a Vs profile derived from distributed acoustic sensing (DAS) data located in a borehole at the FORGE site. The quasi-3-D velocity model shows that the base of the basin dips ∼22° to the west and topography on the basement interface coincident with the Mag Lee Wash suggests that the bedrock interface is an unconformity.

Geochemistry & Geophysics↗

Analysis of Seismic Wave Amplification in Sedimentary Basins Using 3D Wavefield Simulations: Nenana Basin, Central Alaska

We investigate seismic wave amplification in Nenana basin, central Alaska, using 3D seismic wavefield simulations. We quantify frequency-dependent amplification by comparing synthetic seismograms among four different seismic velocity models: (a) a tomographic model, (b) a tomographic model with the uppermost 6.5 km replaced by a 1D basin profile, (c) a tomographic model with an embedded ellipsoidal basin, and (d) a tomographic model with an embedded realistic basin. For each model we perform wavefield simulations accurate up to 2 Hz for a set of 10 different earthquakes, which provide a range of frequencies and incident angles for waves interacting with the basin. Average amplification ratios are four on the horizontal components and seven on the vertical component. The dominant influence of the amplification is the 3D geometry of the basin, not the slow-velocity profile near the surface. Our synthetic amplification ratios are in general agreement with amplification estimates obtained from 14 stations that recorded the same set of earthquakes. Our approach offers a general strategy for documenting frequency-dependent basin amplification for a region with realistic basin structures and local earthquakes.

Geosciences↗

Modeling Induction Stirring and Particle Tracking in Molten Uranium

Two independent numerical models have been developed to simulate the behavior of carbon impurities in molten uranium metal. Informed by experimental parameters, one model was created using the commercial software Star-CCM+ and compared with another developed using open-source codes, including OpenFOAM, Finite Element Method Magnetics (FEMM) and a First Passage Kinetic Monte Carlo (FPKMC) approach. The target experimental system features a 404g uranium metal charge containing an average carbon concentration of 139 ppm which was melted in a vacuum induction furnace at 1400° C then resolidified. The microstructures of the uranium and its impurities before and after melting have been characterized and reported separately. Prior to simulating the uranium-carbon system described, the numerical models were validated using a previously published nonradioactive experimental system to ensure agreement with expected output values. Focus has been placed on modeling velocity fields under induction stirring and impurity particle trajectories.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Modeling Subsurface Explosions Recorded on a Distributed Fiber Optic Sensor

Fiber optic distributed acoustic sensors (DAS) are becoming a widely used tool for seismic sensing. Here we examine recordings of two subsurface chemical explosions, DAG-1 and DAG-3, each of which was about one metric ton (TNT equivalent), that were recorded from a helical fiber installed in two boreholes 80 m away from the source location. Several clear phases including the initial P wave, a weak S wave, and a surface reflected P wave are observed on the helical DAS data. We estimate a velocity model using arrival times measured from the fiber. The DAS waveform data were compared with colocated accelerometers at specific depths in both frequency and time domains. Furthermore, the spectra of the DAS data matched spectra estimated from the accelerometer records. Comparisons of observed waveform shape between the accelerometer records and the fiber measurements (strain-rate) show reasonable agreement except for the data near the event depth. The DAS data and the accelerometer agreed in relative amplitudes but we had difficulties in matching absolute amplitudes, possibly due to errors in metadata. Synthetic strain-rate waveforms were calculated using a 2D wavenumber algorithm and matched the waveform shape and relative amplitudes. In general, DAS is effective at recording strong ground motions at high spatial density. Comparison of the synthetic seismograms with observed data indicate that the waveforms are not consistent with a pure isotropic explosion source and that the observed S waves originate from very near the source region.

58 GEOSCIENCES↗

Multi-Scale Seismic Measurements for Site Characterization and CO2 Monitoring in an Enhanced Oil Recovery/Carbon Capture, Utilization, and Sequestration Project, Farnsworth Field, Texas

To address the challenges of climate change, significantly more geologic carbon sequestration projects are beginning. The characterization of the subsurface and the migration of the plume of supercritical carbon dioxide are two elements of carbon sequestration that can be addressed through the use of the available seismic methods in the oil and gas industry. In an enhanced oil recovery site in Farnsworth, TX, we employed three separate seismic techniques. The three-dimensional (3D) surface seismic survey required significant planning, design, and processing, but produces both a better understanding of the subsurface structure and a three-dimensional velocity model, which is essential for the second technique, a timelapse vertical seismic profile, and the third technique, cross-well seismic tomography. The timelapse 3D Vertical Seismic Profile (3D VSP) revealed both significant changes in the reservoir between the second and third surveys and geo-bodies that may represent the extent of the underground carbon dioxide. The asymmetry of the primary geo-body may indicate the preferential migration of the carbon dioxide. The third technique, cross-well seismic tomography, suggested a strong correlation between the well logs and the tomographic velocities, but did not observe changes in the injection interval.

Energy & Fuels↗

Challenges and Potential of Waveform Modeling for Crustal Scale Predictions

Waveform modeling is crucial to improving our understanding of observed seismograms. Forward simulation of wavefields provides quantitative methods of testing interactions between complicated source functions and the propagation medium. Here, we discuss three experiments designed to improve under standing of high frequency seismic wave propagation. First, we compare observed and predicted travel times of crustal phases for a set of real observed earthquakes with calculations and synthetic seismograms. Second, we estimate the frequency content of a series of nearly co-located earthquakes of varying magnitude for which we have a relatively well- known 1D velocity model. Third, we apply stochastic perturbations on top of a 3D tomographic model and qualitatively assess how those variations map to differences in the seismograms. While different in scope and aim, these three vignettes illustrate the current state of crustal scale waveform modeling and the potential for future studies to better constrain the structure of the crust.

58 GEOSCIENCES↗

Regional Earthquake Ground Motion Simulations for Southern California With EQSIM: Insights From the 2008 Chino Hills, 2024 Highland Park, and 2021 Carson Earthquakes

This study presents physics-based, 3D simulations using the EQSIM framework for several earthquakes in the Los Angeles region. The primary objective was to assess the ability of deterministic physics-based ground motion simulations to reproduce the observed motions from historical events. The selected events included the mathematical equation M w 5.4 2008 Chino Hills, the mathematical equation M w 4.4 2024 Highland Park, and the mathematical equation M w 4.3 2021 Carson events. The simulated motions were evaluated by comparing the recorded and simulated seismograms, as well as the Fourier amplitude spectra, across multiple seismic stations. The SCEC 3D velocity model, CVM-S4.26.M01, was used to represent the regional geology, and ground motion simulations were carried out with a resolution of up to 5 Hz. The results indicate that the simulated motions captured the recorded motions up to approximately 4 Hz. While careful iterations regarding source parameters and corner frequencies were required, and, for the case of the Highland Park event, some of the near-source stations had relatively low accuracy, the present study established a positive step toward the utilization of physics-based simulations in practical applications. The computational efficiencies exhibited by EQSIM, especially on GPU clusters, further supported this assertion, as wall-clock times of simulations involving more than 10 billion grid points were as low as mathematical equation minutes. This permits ensemble simulations for a considered scenario event so that modeling uncertainties (e.g., source and geology) can be bracketed.

EQSIM↗

The Pantheon+ analysis: Improving the redshifts and peculiar velocities of Type Ia supernovae used in cosmological analyses

We examine the redshifts of a comprehensive set of published Type Ia supernovae, and provide a combined, improved catalogue with updated redshifts. We improve on the original catalogues by using the most up-to-date heliocentric redshift data available; ensuring all red shifts have uncertainty estimates; using the exact formulae to convert heliocentric redshifts into the Cosmic Microwave Background (CMB) frame; and utilising an improved peculiar velocity model that calculates local motions in redshift-space and more realistically accounts for the external bulk flow at high-redshifts. We review 2607 supernova redshifts; 2285 are from unique supernovae and 322 are from repeat observations of the same supernova. In total, we updated 990 unique heliocentric redshifts, and found 5 cases of missing or incorrect heliocentric corrections, 44 incorrect or missing supernova coordinates, 230 missing heliocentric or CMB frame redshifts, and 1200 missing redshift uncertainties. The absolute corrections range between 10 -8 ≤ Δz ≤ 0.038, and RMS(Δz) ~ 3 × 10 -3 . The sign of the correction was essentially random, so the mean and median corrections are small: 4 × 10 -4 and 4 × 10 -6 respectively. We examine the impact of these improvements for H 0 and the dark energy equation of state w and find that the cosmological results change by ΔH 0 = -0.12 km s -1 Mpc -1 and Δw = 0.003, both significantly smaller than previously reported uncertainties for H 0 of 1.0 km s -1 Mpc -1 and w of 0.04 respectively.

79 ASTRONOMY AND ASTROPHYSICS↗

Machine Learning‐Assisted Microearthquake Location Workflow for Monitoring the Newberry Enhanced Geothermal System

Abstract Enhanced geothermal systems (EGS) offer a sustainable energy source but face challenges in accurately locating microearthquakes induced during reservoir stimulation. Locating these microearthquakes provides reliable feedback on the stimulation progress. Current deep learning methods for locating earthquakes require extensive data sets for training, which is problematic as detected microearthquakes are often limited. To address the scarcity of training data, we propose a practical workflow using probabilistic multilayer perceptron (PMLP) which predicts microearthquake locations from cross‐correlation time lags in waveforms. Utilizing a 3D velocity model of Newberry site derived from ambient noise interferometry, we generate numerous synthetic microearthquakes and 3D acoustic waveforms for PMLP training. Accurate synthetic tests prompt us to apply the trained network to the 2012 and 2014 stimulation field waveforms. To enhance the accuracy of source localization, we carefully handpick the P‐arrival times. Predictions on the 2012 stimulation data set show major microseismic activity at depths of 0.5–1.2 km, correlating with a known casing leakage scenario. In the 2014 data set, the majority of predictions concentrate at 2.0–2.9 km depths, consistent with results obtained from conventional physics‐based inversion, and align with the presence of natural fractures from 2.0 to 2.7 km. We validate our findings by comparing the synthetic and field picks, demonstrating a satisfactory match for the first arrivals. By combining the benefits of quick inference speeds and accurate location predictions, we demonstrate the feasibility of using realistic synthetic data set to locate microseismicity for EGS monitoring.

15 GEOTHERMAL ENERGY↗

Relocating microseismicity from downhole monitoring of the Decatur CCS site using a modified double-difference algorithm

SUMMARY The injection of CO2 at the Decatur carbon capture and storage site has generated significant microseismic activity, which occurs in distinct spatial clusters up to approximately 2.2 km from the primary injection well. Accurate and precise event locations are vital for the characterization of the microseismicity to help understand the reservoir response to the CO2 injection, whilst enabling the identification of minor faults and fractures below the resolution of conventional active seismic imaging. However, microseismic monitoring of fluid injection sites, such as Decatur, is often performed using a network of borehole sensors often from a single well. While these downhole sensors have excellent detection capabilities, their poor azimuthal coverage limits the ability to precisely determine event locations. We have developed a modified double-difference relocation algorithm suitable for both 1-D and 3-D velocity models, and which incorporates differential back azimuth observations to allow the benefits of the original double-difference algorithm to be applicable to a downhole microseismic monitoring setting. Applying the modified double-difference algorithm to the microseismicity at Decatur, we have successfully relocated 4293 events. The relocation included over 59 million observations for 757 285 event pairs, split across seven geographic regions. Despite the majority of observations being recorded in only two boreholes, with an almost identical azimuthal coverage, the results have shown to be reliable with significantly reduced residuals and low uncertainties associated with the final locations. We have analysed the residuals in terms of their association with each geographic region, data type, station and individual events, to fully appreciate their influence in the inversion and the fit of the data to the final set of event locations. For each region, the relocated seismicity has become less diffuse with improved clustering, and with newly visible linear features often orientated in a NE–SW direction. These results show the potential improvements that can be made to microseismic event locations recorded by a borehole network with a limited and variable azimuthal distribution.

58 GEOSCIENCES↗

Time-lapse seismic data inversion for estimating reservoir parameters using deep learning

Geologic carbon sequestration involves the injection of captured carbon dioxide ([Formula: see text]) into subsurface formations for long-term storage. The movement and fate of the injected [Formula: see text] plume is of great concern to regulators because monitoring helps to identify potential leakage zones and determines the possibility of safe long-term storage. To address this concern, we design a deep-learning framework for [Formula: see text] saturation monitoring to determine the geologic controls on the storage of the injected [Formula: see text]. We use different combinations of porosities and permeabilities for a given reservoir to generate saturation and velocity models. We train the deep-learning model with a few time-lapse seismic images and their corresponding changes in saturation values for a particular [Formula: see text] injection site. The deep-learning model learns the mapping from the change in the time-lapse seismic response to the change in [Formula: see text] saturation during the training phase. We then apply the trained model to data sets comprising different time-lapse seismic image slices (corresponding to different time instances) generated using different porosity and permeability distributions that are not part of the training to estimate the [Formula: see text] saturation values along with the plume extent. Our algorithm provides a deep-learning assisted framework for the direct estimation of [Formula: see text] saturation values and plume migration in heterogeneous formations using the time-lapse seismic data. Our method improves the efficiency of time-lapse inversion by streamlining the large number of intermediate steps in the conventional time-lapse inversion workflow. This method also helps to incorporate the geologic uncertainty for a given reservoir by accounting for the statistical distribution of porosity and permeability during the training phase. Tests on different examples verify the effectiveness of our approach.

Geochemistry & Geophysics↗

Utah FORGE Project 3-2417: DAS Microseismic Event Catalog from the 16A/16B Circulation Test, 2023

This preliminary data archive includes the relocated microseismic event catalog, 1D velocity model, and methods report from DAS acquisition conducted during the Well 16A and 16B circulation test (July 19th and 20th, 2023) at Utah FORGE. The methods report describes all processing steps, including real-time event detection, hierarchical clustering, joint velocity/hypocenter inversion, and relocation. The resulting work is accepted and will be presented at IMAGE 2024. This dataset was acquired by the FOGMORE R&D project (Fiber Optic MOnitoring for Reservoir Evolution), Utah FORGE R&D Project 3-2417.

15 GEOTHERMAL ENERGY↗

Utah FORGE: GES Well 16A(78)-32 and Well 16B(78)-32 Stimulation Seismic Event Catalogs

This dataset contains seismic event catalogs from the hydraulic stimulation of wells 16A(78)-32 and 16B(78)-32 at the Utah FORGE site in April 2024. The data was collected by Geo Energy Suisse (GES) using a variety of seismic monitoring technologies, including 3-component (3C) geophones and distributed acoustic sensing (DAS) systems. These technologies were deployed across several locations, including wells 16A, 16B, and Delano-1, with sensor arrays at multiple depths to capture microseismic activity during the stimulations. The catalogs provide both real-time and manually checked seismic event locations, with detailed parameters such as trigger conditions, velocity models, and data acquisition settings. The dataset includes information on the stimulation stages, event rates, and hydraulic injection conditions for each well, with a report detailing the data acquisition configuration and seismic event location methodologies. Users will need to reference the included report for a complete understanding of the sensor network, data processing techniques, and accuracy considerations.

15 GEOTHERMAL ENERGY↗

Relocation of the Seismicity of the Caucasus Region

Our objective is to improve the view of the seismicity in the Caucasus region using instrumental data between 1951 and 2019. To create a comprehensive catalog, we combine the bulletins of local agencies and the International Seismological Centre, and use an advanced single-event location algorithm, iLoc, to obtain better locations. We show that relocations with iLoc, using travel-time predictions from the 3D upper mantle velocity model, Regional Seismic Travel Time, improve the locations. Then, using the iLoc results as initial locations and the ground-truth events identified in the iLoc results as fix points, we apply Bayesloc, a multiple-event location algorithm, to simultaneously relocate the entire seismicity of the Caucasus region. Here we demonstrate that the simultaneous relocation of the seismicity with Bayesloc clarifies the location and geometry of major active structures accommodating ongoing convergence between the Arabian and Eurasian continents between the Black and Caspian Seas. Among our major findings is the confirmation of widespread seismicity in the mantle beneath the northern flank of the Greater Caucasus and central Caspian, resulting from north-dipping subduction of the Kura and South Caspian basins and the identification of a discrete band of crustal seismicity beneath the southern flank of the Greater Caucasus.

58 GEOSCIENCES↗

Geometric Interpretation of the Cluster Location Problem Part I: Theory

We present a new framing of the seismic location problem using principles drawn from differential geometry. Our interpretation relies upon the common assumption that travel times observed across a network are continuous, differentiable functions of source location. In consequence, travel‐time functions constitute a differentiable map between the source region and a Riemannian manifold. The manifold is said to be the image of the source region embedded in a generally high‐dimension travel‐time vector space. A cluster of events in the source region has an image of discrete points on the manifold, that, except in the simplest cases, cannot be viewed directly. However, it is possible to project the image of a cluster into a tangent space of the manifold for direct visualization. The projection operator can be computed directly from the data without a velocity model, but produces a distorted rendering of the cluster geometry. With a model we can predict the distortions and correct them to estimate cluster geometry. We develop these points with the simplest possible example, one for which direct visualization of the manifold is possible, using the example as an introduction to the relevant concepts from differential geometry in a familiar setting. The tangent space, a local linearization of the manifold, plays a key role. We develop a metric to estimate the limits of linearization, that is, to determine when the curvature of the manifold invalidates the linear assumption. We also examine the interplay of model error, inadequate network geometry, and pick error. We then generalize our results from the simple case to the general case of 3D source regions observed by general networks. Although we do suggest a new “project and correct” method for location, we do not develop it into a practical algorithm. In conclusion, our intention rather is to highlight new analytical methods grounded in differential geometry.

East Pacific Ocean Islands↗