Apparent Source Time Functions Research Articles

Sensitivity of the second seismic moments resolution to determine the fault parameters of moderate earthquakes

Second-degree seismic moments provide a simple description of the spatiotemporal extent of the earthquake source. Finite source attributes such as rupture length, width, duration, velocity, and propagation direction can be estimated by computing second-degree seismic moments without the need for a predefined rupture model. This is achieved by analyzing the properties of apparent source time functions (ASTFs) obtained from seismic signals recorded at different stations after eliminating instrument responses and path effects. In this study, to define the limits of its application in the analysis of small earthquakes and to evaluate the sensitivity and reliability of the results to uncertainties due to observations and prior knowledge, we modeled a synthetic seismic source and examined how potential uncertainties in hypocentral depth, velocity model, focal mechanism, source duration, and number of recording stations can affect the inversion results. An accurate ASTF is essential to obtain robust results and our findings show that the mean values of the key source parameters, i.e., fracture size, source duration, and rupture velocity, are generally well reproduced in all sensitivity tests, with some exceptions, within the standard deviation. We also demonstrate that large uncertainties in the hypocentral depth and inaccurate velocity models introduce a significant bias, especially in rupture size and average centroid velocity, indicating the strong influence of ray path calculation in the inversion process. These resolution limits must therefore be taken into account when interpreting the results obtained with this technique.

The 2016 Menyuan Earthquake: The Largest Self‐Arrested Crustal Earthquake Ever Observed

AbstractThe 2016 Mw 5.9 Menyuan earthquake occurred near the Haiyuan fault system in the northeastern Tibetan plateau. Although some kinematic properties of this event have been investigated, more dynamic properties still need to be clarified. First, we use the empirical Green's function method to obtain this event's apparent source time functions (ASTFs), which present a bell shape with a duration of 3s and corresponding smooth spectra. We then conduct dynamic rupture simulations, which are well constrained by the seismic and geodetic observations, to explore more in‐depth source properties. The results show that the characteristics of ASTFs can be explained by the self‐arresting rupture model, which implies that the rupture propagation of this event may spontaneously terminate before reaching the barriers. This result suggests that the initial conditions when this earthquake nucleated may have controlled the rupture process. Our work contributes to understanding the stopping mechanism of some moderate earthquake ruptures.

Rupture Process of the 2017 Mw 5.5 Pohang, South Korea, Earthquake via an Empirical Green’s Function Method

ABSTRACTThe Mw 5.5 earthquake occurred in Pohang, South Korea on 15 November 2017, which is known as a “runaway earthquake,” and was triggered from a critical-state fault as a result of fluid injection. As such earthquakes rarely occur, spatiotemporal slip distributions were investigated via the finite-fault inversion based on the empirical Green’s function in this study. The rupture process can be divided into three steps: first, slip initiated and propagated only to the southwest from the hypocenter during the initial 0.6 s; in the second step from 0.6 to 2.4 s, the slip occurred to the southwest and northeast parts, in which the maximum seismic moment was released; in the third step from 2.4 to 6.0 s, slip occurred around the edge of the fault plane farther from the hypocenter, particularly in the deep part in the northeast direction. In each step, the seismic moment was released as approximately 6%, 59%, and 35%, respectively. The first step can be interpreted not as a part of the rupture process of the mainshock but as the immediate and distinct foreshock. Overall, most of the slip distributed southwest is consistent with the results of the directivity analysis using apparent source time functions. Although the average stress drop (~1 MPa) of the Pohang earthquake is considerably lower than that (~20 MPa) of the Mw 5.5 Gyeongju earthquake that naturally occurred in the vicinity of the Pohang, it is difficult to attribute it only to the fluid injection effect. Through this study, we improve our comprehension of the seismic source physics and mechanisms of the Pohang earthquake by analyzing the spatiotemporal slip history, the directivity of rupture process, and the spatial distribution of the stress drop on the fault plane.

EASOt-AP: An open-source MATLAB package to estimate the seismic moment, rupture radius, and stress-drop of earthquakes from time-dependent P-wave displacements

We developed a MATLAB package, to rapidly obtain the apparent EArthquake SOurce time function using the Average of near-source P-wave displacement records (EASOt-AP) and to estimate the source parameters of the seismic moment, the rupture radius, and the average static stress drop. The algorithm implemented in this package is based on a rapid and straightforward methodology that is recently developed by Zollo et al. (2021). To this purpose, EASOt-AP models a LPDT curve, which is the average P-wave displacement versus time in the logarithm scale. In this paper, the performance of the EASOt-AP has been shown by evaluating the strong motion data recorded in the near-source range of small magnitude events (2 = Mw ≤ 3) that occurred in the Irpinia region, southern Italy. The source parameters follow a self-similar, constant-stress-drop scaling with a relatively low average stress drop of about 01-0.2 MPa. This Tool is designed for an easy use, and all steps of data processing are performed completely within the numerical environment of MATLAB.

The 2020 Samos Mw7 earthquake: Source model depicting complexity and rupture directivity

The October 30, 2020, Mw7 Samos earthquake ruptured a north-dipping offshore normal fault, bounding the Samos basin; it accommodated ∼N-S extension and can be viewed as a modern manifestation of the basin evolution. It caused 118 fatalities, generated a tsunami, and caused a co-seismic uplift of 20 to 35 cm of the NW part of Samos Island. Using broadband, strong-motion, and geodetic data, we constrain the location and source geometry of the mainshock. A multiple-point source model suggests three sequential subevents providing 20 s of source duration. Our finite-fault kinematic model confirms the prevalence of large slip amplitudes (∼2.4 m) along the entire ruptured area and the up-dip and westward rupture propagation. This directivity is independently confirmed by Apparent Source Time Functions inferred from regional recordings using a herein developed new variant of the empirical Green's function method. Static GNSS displacements from inland stations yield a near-surface co-seismic slip of ∼1 m amplitude, contributing to any interpretation of the observed island uplift. The 2020 Samos event showed that in the spatially heterogeneous oblique transtentional regions in the back-arc Aegean region, normal faults bounding the basins are capable to rupture in M7 earthquakes, provoke tsunami generation, and constitute a constant threat to the nearby coastal areas of both Greece and Turkey.

Rupture Directivity of the 2021 ML 2.2 Gwangyang, Korea, Microearthquake: Toward Resolving High-Resolution Rupture Process of a Small Earthquake

Abstract Moderate-to-large earthquakes (M>6) frequently show clear rupture directivity. Recent studies revealed that a substantial percentage of small-sized earthquakes (M<∼5) display rupture directivity as well, owing to enhanced seismic monitoring. Is rupture directivity a common feature for earthquakes of all sizes? In this study, we investigated the rupture directivity of the 27 August 2021 ML 2.2 Gwangyang microearthquake that occurred at the southern tip of the Korean Peninsula by utilizing data from a recently expanded dense seismic network. The mainshock had two foreshocks and three aftershocks, which enabled us to define the fault plane from precise relative event locations and a well-constrained focal mechanism. Analysis of apparent source time functions obtained with the empirical Green’s function technique reveals that the mainshock ruptured unilaterally toward the east. A detailed analysis of the source pulse leaving the focal sphere in 3D indicates that rupture directivity is consistent with the fault geometry and slip direction. Our study demonstrates that we can resolve rupture directivity of unilaterally rupturing microearthquakes (M<∼3) given adequate azimuthal coverage, and that this will lead to an improved understanding of the seismogenic processes in regions of low seismicity worldwide.

The Source Characteristics of the Mw6.4, 2016 Meinong Taiwan Earthquake from Teleseismic Data Using the Hybrid Homomorphic Deconvolution Method

The kinematic source rupture process of the 2016 Meinong earthquake (Mw = 6.4) in Taiwan was derived from apparent source time functions retrieved from teleseismic S-waves by using a refined homomorphic deconvolution method. The total duration of the rupture process was approximately 15 s, and one slip-concentrated area can be represented as the source model based on images representing static slip distribution. The rupture process began in a down-dip direction from the fault toward Tainan City, strongly suggesting that the rupture had a unilateral northwestern direction. The asperity with an area of approximately 15 × 15 km2 and the maximum slip of approximately 2 m were centered 12.8 km northwest of the hypocenter. Coseismic vertical deformation was calculated based on the source model. Compared with the results derived from InSAR (Interferometric Synthetic Aperture Radar) data, our results demonstrated that the location with maximum uplift was accurately well detected, but our maximum value was just approximately 0.4 times of the InSAR-derived value. It reveals that there are the other mechanisms to affect the vertical deformation, rather than only depending on the source model. At different depths, areas west, east, and north of the hypocenter maintained high values of Coulomb stress changes. This explains the mechanism behind aftershocks being triggered and provides a reference for predicting aftershock locations after a large earthquake. The estimated seismic spectral intensities, including spectral acceleration and velocity intensity (SIa and SIv), were derived. Source directivity effects caused damage to buildings, and we concluded that all damaged buildings were located within a SIa value of 400 gal. Destroyed buildings taller than seven floors were located in an area with a SIv value of 30 cm/s. These observations agree with those on damages caused by the 2010 Jiasian earthquake (ML 6.4) in Tainan, Taiwan.

The 2017 Ischia Earthquake (Southern Italy): Source Mechanism and Rupture Model From the Inversion of a Near-Source Strong Motion Record

With the aim to investigate the rupture complexity and the radiated wave field of the 2017, Mw 3.9, Ischia earthquake, south-west of Naples (Italy), we used finite-fault modeling to invert the near-source (< 1-km epicentral distance) horizontal-component velocity records of the accelerometric station (IOCA) and searched for the best-fit kinematic rupture parameters. This analysis showed that the rupture nucleated at about 600 m west of IOCA and 1.1-km depth, along a 1 km, NW-SE striking fault (i.e., thrust with right-lateral component), with a rupture velocity of about 0.7 km/s. The retrieved rupture model coupled with multipath reverberations effects related to a thin, low-velocity near-surface volcanic sedimentary layer, well explains the observed long ground motion duration and the large amplitudes recorded all over the island. Finally, the apparent source time function (STF), obtained from inverse modeling using a theoretical Green’ function (GF), is validated by implementing an empirical GF (EGF) analysis.

On the Source Parameters and Genesis of the 2017, Mw 4 Montesano Earthquake in the Outer Border of the Val d’Agri Oilfield (Italy)

On October 27, 2017, an Mw4 earthquake occurred close to the municipality of Montesano sulla Marcellana, less than 10 km external to the concession of the largest European onshore hydrocarbon reservoir—the Val d’Agri oilfield (Southern Italy). Being a weak event located outside the extended monitoring domain of the industrial concession, the relevance of this earthquake and the possible links with the hydrocarbon exploitation were not extensively discussed. Actually, the analysis of shallow seismic events close to subsurface exploitation domains plays a significant role in the definition of key parameters in order to discriminate between natural, triggered, and induced seismicity, especially in tectonically active regions. The study of weak-to-moderate earthquakes can improve the characterization of the potentially destructive seismic hazard of this particular area, already struck by M &amp;gt; 6.5 episodes in the past. In this work, we analyze the source parameters of this Mw4 earthquake by applying advanced seismological techniques to estimate the uncertainties derived from the moment tensor inversion and identify plausible directivity effects. The moment tensor is dominated by a NW–SE oriented normal faulting with a centroid depth of 14 km. A single ML2.1 aftershock was recorded and used as the empirical Green’s function to calculate the apparent source time function for the mainshock. Apparent durations (in the range 0.11–0.21 s, obtained from S-waves) define an azimuthal pattern, which reveals an asymmetric bilateral rupture with 70% of the rupture propagation in the N310°W direction, suggesting a rupture plane dipping to the SW. Our results tally with the activation of a deeper fault segment associated with the Eastern Agri Fault System close to the basement as the origin of the Montesano earthquake. Finally, the Coulomb stress rate induced by depletion of the oilfield is calculated to quantify the trigger potential estimated for the Montesano earthquake yielding relatively low probabilities below 10%. Our analyses point toward the conclusion that the Mw4 event was more likely due to the local natural tectonic stress, rather than induced or triggered by the long-term hydrocarbon extraction in the Val d’Agri oilfield.

Slip Partitioning in the 2016 Alboran Sea Earthquake Sequence (Western Mediterranean)

A MW=5.1 earthquake on January 21st 2016 marked the beginning of a significant seismic sequence in the southern Alboran Sea, culminating in a MW=6.3 earthquake on January 25th, and continuing with further moderate magnitude earthquakes until March. We use data from 35 seismic broadband stations in Spain, Morocco and Portugal to relocate the seismicity, estimate seismic moment tensors, and isolate regional apparent source time functions for the main earthquake. Relocation and regional moment tensor inversion consistently yield very shallow depths for the majority of events. We obtain 50 moment tensors for the sequence, showing a mixture of strike-slip faulting for the foreshock and the main event and reverse faulting for the major aftershocks. The leading role of reverse focal mechanisms among the aftershocks may be explained by the geometry of the fault network. The mainshock nucleates at a bend along the left-lateral Al-Idrisi fault, introducing local transpression within the transtensional Alboran Basin. The shallow depths of the 2016 Alboran Sea earthquakes may favour slip-partitioning on the involved faults. Apparent source durations for the main event suggest a ~21 km long, asymmetric rupture that propagates primarily towards NE into the restraining fault segment, with fast rupture speed of ~3.0 km/s. Consistently, the inversion for laterally variable fault displacement situates the main slip in the restraining segment. The partitioning into strike-slip rupture and dip-slip aftershocks confirms a non-optimal orientation of this segment, and suggests that the 2016 event settled a slip deficit from previous ruptures that could not propagate into the stronger restraining segment.

Semiautomated Estimates of Directivity and Related Source Properties of Small to Moderate Southern California Earthquakes Using Second Seismic Moments

AbstractWe develop a semiautomated method for estimating with second seismic moments the directivity, rupture area, duration, and centroid velocity of earthquakes. The method is applied to 41 southern California earthquakes with magnitude in the range 3.5–5.2 and provides stable results for 28 events. Apparent source time functions (ASTFs) of P and S phases are derived using deconvolution with three stacked empirical Green's functions (seGf). The use of seGf suppresses nongeneric source effects, improves the focal mechanism correspondence to the analyzed earthquakes, and typically allows inclusion of 5 to 15 more ASTFs compared with analysis using a single eGf. Most analyzed earthquakes in the Trifurcation area of the San Jacinto Fault have directivities toward the northwest, while events around Cajon Pass and San Gabriel Mountain tend to propagate toward the southeast. These results are generally consistent with predictions for dynamic rupture on bimaterial interfaces associated with the imaged velocity contrasts in the area. The second moment inversions also provide constraints on the upper and lower bounds of rupture areas in our data set. Stress drops and uncertainties are estimated for elliptical ruptures using the derived characteristic rupture length and width. The semiautomated second moment method with seGfs can be used for routine application to moderate earthquakes in locations with good station coverage.

Supershear rupture of the 28 September 2018 Palu earthquake inferred from Empirical Green’s Function analysis

We deconvolved 22 teleseismic waveforms of Mw6.1 earthquake with Mw7.5 Palu earthquake to obtain Apparent Source Time Functions (ASTFs). Analysis of those ASTFs resulted rupture directivity, rupture length and rupture velocity of the earthquake. The main earthquake ruptured to N 2270E direction along 80 km away from the epicenter at an average speed of 4.5 km/s. The rupture velocity which is greater than local shear-wave velocity of 3.5 km/s indicated supershear characteristic of the earthquake. An ASTF of SNZO station was used to infer 1-D slip distribution of this earthquake. Based on this ASTF we calculated earthquake moment of 1.45 x 1020 Nm that released in 30 seconds. The largest seismic moment released between 0-15 seconds and decreased before it stopped at 30 seconds. A Maximum slip of 8 meters occurred at a distance of 20 km to the south of the epicentre and decreased to around 4 meter near Palu City.

Rupture directivity, stress drop, and hypocenter migration of small earthquakes in the Yamagata-Fukushima border swarm triggered by upward pore-pressure migration after the 2011 Tohoku-Oki earthquake

We examined a relationship between rupture propagation directions and the distribution of fault strength by analyzing seismological data from the earthquake swarm on the Yamagata-Fukushima border, NE Japan. This earthquake swarm exhibits a distinct hypocenter migration behavior and was estimated to be triggered by upward fluid movement after the 2011 Tohoku-Oki earthquake. We utilized the dense nationwide seismic network in Japan to estimate apparent source time functions of >1500 small earthquakes (MJMA ≥ 2). We found clear directional dependences of the peak amplitude and the pulse-width in the apparent source time functions, suggesting the earthquake rupture directivity, for half of the earthquakes. Rupture directions mostly avoid the directions of the hypocenter migration. The difference between the microscopic and macroscopic propagations of rupture might be explained by the spatial variation in the fault strength affected by pore pressure along the fault; ruptures of each earthquake are hindered from developing toward the region with higher fault strength ahead of the pore-pressure front. Estimates of stress drop systematically increased on taking the effects of rupture directivity into account. We observed a temporal increase in stress drop from 3 MPa to 10 MPa during the pore-pressure migration.

Stress Drops on the Blanco Oceanic Transform Fault from Interstation Phase Coherence

Oceanic transform faults display a wide range of earthquake stress drops, large aseismic slip, and along‐strike variation in seismic coupling. We use and further develop a phase coherence‐based method to calculate and analyze stress drops of 61 M≥5.0 events between 2000 and 2016 on the Blanco fault, off the coast of Oregon. With this method, we estimate earthquake rupture extents by examining how apparent source time functions (ASTFs) vary between stations. The variation is caused by the generation of seismic waves at different locations along the rupture, which arrive at different times depending on station location. We isolate ASTFs at a range of stations by comparing seismograms of collocated earthquakes and then use the interstation ASTF coherence to infer rupture extent and stress drop. We examine how our analysis is influenced by various factors, including poor trace alignment, relative earthquake locations, focal mechanism variation, azimuthal distribution of stations, and depth phase arrivals. We find that as alignment accuracy decreases or distance between earthquakes increases, coherence is reduced, but coherence is unaffected by focal mechanism variation or depth phase arrivals for our dataset. We calibrate the coherence–rupture extent relationship based on the azimuthal distribution of stations. We find the phase coherence method can be used to estimate stress drops for offshore earthquakes, but is limited to M≥5.0 earthquakes for the Blanco fault due to poor trace alignment accuracy. The median stress drop on the Blanco fault is 8 MPa (with 95% confidence limits of 6–12 MPa) for 61 earthquakes. Stress drops are a factor of 1.7 (95% confidence limits 0.8–3.5) lower on the more aseismic northwest segment of the Blanco fault. These lower stress drops could be linked to reduced healing time due to higher temperatures, which reduce the depth of the seismogenic zone and shorten the seismic cycle.

Rupture Model of the M5.8 Pawnee, Oklahoma, Earthquake From Regional and Teleseismic Waveforms

AbstractThe 2016 M5.8 Pawnee, Oklahoma, earthquake is the largest earthquake to have been induced by wastewater disposal. We infer the coseismic slip history from analysis of apparent source time functions and inversion of regional and teleseismic P waveforms, using aftershocks as empirical Green's functions. The earthquake nucleated on the shallow part of the fault, initially rupturing toward the surface, followed shortly thereafter by slip deeper on the fault. Deeper slip occurred below the aftershocks and at greater depths than most induced seismicity in the region, suggesting that small‐ to moderate‐sized earthquakes may not occur on deeper parts of faults in Oklahoma because they are further from failure than shallower fault sections. Comparisons with models of pore pressure perturbations further suggest that the earthquake may have initiated within a region of higher pore pressure perturbation but was not confined to this zone. These observations inform source physics and understanding of maximum magnitudes.

The rupture extent of low frequency earthquakes near Parkfield, CA

The low frequency earthquakes (LFEs) that constitute tectonic tremor are often inferred to be slow: to have durations of 0.2 to 0.5 s, a factor of 10 to 100 longer than those of typical M_W 1-2 earthquakes. Here we examine LFEs near Parkfield, CA in order to assess several proposed explanations for LFEs’ long durations. We determine LFE rupture areas and location distributions using a new approach, similar to directivity analysis, where we examine how signals coming from various locations within LFEs’ finite rupture extents create differences in the apparent source time functions recorded at various stations. We use synthetic ruptures to determine how much the LFE signals recorded at each station would be modified by spatial variations of the source-station travel time within the rupture area given various possible rupture diameters, and then compare those synthetics with the data. Our synthetics show that the methodology can identify inter-station variations created by heterogeneous slip distributions or complex rupture edges, and thus lets us estimate LFE rupture extents for unilateral or bilateral ruptures. To obtain robust estimates of the sources’ similarity across stations, we stack signals from thousands of LFEs, using an empirical Green’s function approach to isolate the LFEs’ apparent source time functions from the path effects. Our analysis of LFEs in Parkfield implies that LFEs’ apparent source time functions are similar across stations at frequencies up to 8 to 16 Hz, depending on the family. The inter-station coherence observed at these relatively high frequencies, or short wave lengths (down to 0.2 to 0.5 km), suggest that LFEs in each of the 7 families examined occur on asperities. They are clustered in patches with sub-1-km diameters. The individual LFEs’ rupture diameters are estimated to be smaller than 1.1 km for all families, and smaller than 0.5 km and 1 km for the two shallowest families, which were previously found to have 0.2-s durations. Coupling the diameters with the durations suggests that it is possible to model these M_W 1-2 LFEs with earthquake-like rupture speeds: around 70% of the shear wave speed. However, that rupture speed matches the data only at the edge of our uncertainty estimates for the family with highest coherence. The data for that family are better matched if LFEs have rupture velocities smaller than 40% of the shear wave speed, or if LFEs have different rupture dynamics. They could have long rise times, contain composite sub-ruptures, or have slip distributions that persist from event to event.

Dissipative Intraplate Faulting During the 2016 Mw 6.2 Tottori, Japan Earthquake

AbstractThe 2016 Mw 6.2 Tottori earthquake occurred on 21 October 2016 and produced thousands of aftershocks. Here we analyze high‐resolution‐relocated seismicity together with source properties of the mainshock to better understand the rupture process and energy budget. We use a matched‐filter algorithm to detect and precisely locate &gt;10,000 previously unidentified aftershocks, which delineate a network of sharp subparallel lineations exhibiting significant branching and segmentation. Seismicity below 8 km depth forms highly localized fault structures subparallel to the mainshock strike. Shallow seismicity near the main rupture plane forms more diffuse clusters and lineations that often are at a high angle (in map view) to the mainshock strike. An empirical Green's function technique is used to derive apparent source time functions for the mainshock, which show a large amplitude pulse 2–4 s long. We invert the apparent source time functions for a slip distribution and observe a ~16 km2 patch with average slip ~3.2 m. 93% of the seismic moment is below 8 km depth, which is approximately the depth below which the seismicity becomes very localized. These observations suggest that the mainshock rupture area was entirely within the lower half of the seismogenic zone. The radiated seismic energy is estimated to be 5.7 × 1013 J, while the static stress drop is estimated to be 18–27 MPa. These values yield a radiation efficiency of 5–7%, which indicates that the Tottori mainshock was extremely dissipative. We conclude that this inefficiency in energy radiation is likely a product of the immature intraplate environment and the underlying geometric complexity.

Global catalog of earthquake rupture velocities shows anticorrelation between stress drop and rupture velocity

Application of the SCARDEC method provides the apparent source time functions together with seismic moment, depth, and focal mechanism, for most of the recent earthquakes with magnitude larger than 5.6–6. Using this large dataset, we have developed a method to systematically invert for the rupture direction and average rupture velocity Vr, when unilateral rupture propagation dominates. The approach is applied to all the shallow (z < 120km) earthquakes of the catalog over the 1992–2015 time period. After a careful validation process, rupture properties for a catalog of 96 earthquakes are obtained. The subsequent analysis of this catalog provides several insights about the seismic rupture process. We first report that up-dip ruptures are more abundant than down-dip ruptures for shallow subduction interface earthquakes, which can be understood as a consequence of the material contrast between the slab and the overriding crust. Rupture velocities, which are searched without any a-priori up to the maximal P wave velocity (6000–8000m/s), are found between 1200m/s and 4500m/s. This observation indicates that no earthquakes propagate over long distances with rupture velocity approaching the P wave velocity. Among the 23 ruptures faster than 3100m/s, we observe both documented supershear ruptures (e.g. the 2001 Kunlun earthquake), and undocumented ruptures that very likely include a supershear phase. We also find that the correlation of Vr with the source duration scaled to the seismic moment (Ts) is very weak. This directly implies that both Ts and Vr are anticorrelated with the stress drop Δσ. This result has implications for the assessment of the peak ground acceleration (PGA) variability. As shown by Causse and Song (2015), an anticorrelation between Δσ and Vr significantly reduces the predicted PGA variability, and brings it closer to the observed variability.

Multichannel Deconvolution for Earthquake Apparent Source Time Functions

Abstract Previous studies of earthquake apparent source time functions (ASTFs) removed propagation effects through seismogram deconvolution with a smaller earthquake known as an empirical Green’s function (EGF). We develop a multichannel deconvolution (MCD) algorithm for recovering ASTFs that does not require an EGF, but instead the availability of two or more earthquakes that share a common Green’s function. Under this condition, ASTFs satisfy U i * S j − U j * S i =0, in which U i and S i are the seismogram and ASTF for a given earthquake. This system can be augmented with a scaling equation and written as Ax = b , in which matrix A comprises the seismograms in a block‐Toeplitz structure and x contains the target ASTFs. We minimize an objective function for this linear system with a Newton‐projection algorithm that honors positivity, causality, and duration constraints. If the earthquakes have a suitable range in magnitude, EGF deconvolution may be used to estimate differences in the duration of the events and to obtain a starting model for the larger ASTF(s). We demonstrate the effectiveness of MCD using synthetic tests and apply it to seven M w ∼5 earthquakes from the Kamaishi sequence, Japan, related to the 2011 Tohoku‐Oki M w 9 event. We demonstrate that MCD is an effective way to recover earthquake ASTFs and that the details of rupture revealed by MCD ASTFs will be useful in furthering our understanding of the earthquake source. Electronic Supplement: Figures of seismograms and multichannel deconvolution (MCD) apparent source time functions (ASTFs) from the Kamaishi earthquake sequence.

A MATLAB Toolbox for Estimating the Second Moments of Earthquake Ruptures

ABSTRACT I present a set of routines in MATLAB for estimating the second‐degree moments of an earthquake’s rupture from far‐field body waves. The second moments describe the length, width, duration, and directivity of a rupture. The second‐moments approach is particularly useful when a seismic dataset is dense enough to resolve the primary finite‐source properties but the geodetic data needed for a well‐resolved finite‐fault inversion are not available. In particular for M w 3–6 earthquakes, this approach can be a useful way to estimate rupture area without the assumptions of typical corner‐frequency approaches. The provided software utilizes empirical Green’s function deconvolution to isolate the apparent source time function (ASTF) for each station and phase. The spatial variations in the duration of the ASTF are quantified and inverted for the second moments. The inverse problem is solved using MATLAB’s convex optimization routines for systems of linear matrix inequalities. An error analysis using the jackknife and bootstrap methods is included. An example M w 4.7 earthquake from the San Jacinto fault is used to demonstrate the method.