A Unified Mixed-Effects Framework for Earthquake Source-Parameter Scaling across Tectonics, Fault Mechanisms, Seismic Regions, and Finite-Fault Inversion Characteristics

Finite-fault inversion of tsunami waveforms requires prior knowledge of fault geometry, which is often subject to considerable uncertainty. Quantifying effects of such uncertainty is essential for evaluating the reliability of inversion results derived from inadequately constrained fault models. Although previous studies have examined tsunami sensitivity to fault geometry, they typically relied on forward modeling and assumed planar faults. Using the 2011 Tohoku earthquake as a case study, this paper investigates the impact of fault geometry uncertainty on the inverted slip distribution and tsunami predictions. We consider 2D non-planar fault geometry, defined by depth as a function of distance from the trench. 9592 fault depth profiles are generated by densely sampling the region of historical seismicity, representing the maximum uncertainty in the absence of other geophysical constraints. For each sampled geometry, we invert near-field tsunami observations for slip distribution and evaluate the waveform fitting quality. The results show that these fault samples introduce moderate (20–30%) relative variability in slip amount across most of the rupture area, and the overall slip pattern remains robust. Owing to the trade-off between fault slip and geometry, variations in fault geometry only slightly affect the tsunami predictions at most stations. Nevertheless, inaccurate fault geometry leads to noticeable misfits at several coastal stations, which cannot be compensated for by adjusting fault slip. In addition, fault samples yielding better waveform fits are generally closer to the true geometry. These findings suggest that, for shallowly dipping plate interfaces, fault geometry uncertainty has a limited impact on tsunami inversion. High-quality near-field tsunami observations may further help reduce this uncertainty when only historical seismicity is available as a constraint. More studies are needed for tsunamigenic faults with significant along-strike geometric variations and in distinct tectonic settings. Graphical Abstract

[1] Modern geodetic techniques, such as the global positioning system (GPS) and Interferometric Synthetic Aperture Radar (InSAR), provide high-precision deformation measurements of earthquakes. Through elastic models and mathematical optimization methods, the observations can be related to a slip distribution model. The classic linear, kinematic, and static slip inversion problem requires specification of a smoothing norm of slip parameters and a residual norm of the data and a choice about the relative weight between the two norms. Inversions for unknown fault geometry are nonlinear and, therefore, the fault geometry is often assumed to be known for the slip inversion problem. We present a new method to invert simultaneously for fault slip and fault geometry assuming a uniform stress drop over the slipping area of the fault. The method uses a full Bayesian inference method as an engine to estimate the posterior probability distribution of stress drop, fault geometry parameters, and fault slip. We validate the method with a synthetic data set and apply the method to InSAR observations of a moderate-sized normal faulting event, the 6 October 2008 Mw 6.3 Dangxiong-Yangyi (Tibet) earthquake. The results show a 45.0 ± 0.2° west dipping fault with a maximum net slip of ∼1.13 m, and the static stress drop and rake angle are estimated as ∼5.43 MPa and ∼92.5°, respectively. The stress drop estimate falls within the typical range of earthquake stress drops known from previous studies.

An inverse method to derive fault slip and geometry from seismically observed vertical stratigraphic displacements using elastic dislocation theory

Summary Fault geometry is a key factor in controling the mechanics of faulting. However, there is currently limited theoretical knowledge regarding the effect of non-planar fault geometry on earthquake mechanics. Here, we address this gap by introducing an expansion of the relation between fault traction and slip, up to second order, relative to the deviation from a planar fault geometry. This expansion enables the separation of the effects of non-planarities from those of planar faults. This expansion is realised in the boundary integral equation, assuming a small fault slope. It provides an interpretation for the effect of complex fault geometry on fault traction, for any fault geometry and any slip distribution. Hence the results are also independent of the friction that applies on the fault. The findings confirm that fault geometry has a strong influence on in-plane faulting (mode II) by altering the normal traction on the fault and making it more resistant to slipping for any fault geometry. On the contrary, for out-of-plane faulting (mode III), fault geometry has a much smaller influence. Additionally, we analyse some singularities that arise for specific fault geometries often used in earthquake simulations and provide guidelines for their elimination. To conclude this study, we discuss the limits of the infinitesimal strain theory when non-planar faults are considered.

Coseismic displacements and creeping along the Pernicana fault (Etna, Italy) in the last 17 years: a detailed study of a tectonic structure on a volcano

We study how the fault dip and slip rake angles affect near-source ground velocities and displacements as faulting transitions from strike-slip motion on a vertical fault to thrust motion on a shallow-dipping fault. Ground motions are computed for five fault geometries with different combinations of fault dip and rake angles and common values for the fault area and the average slip. The nature of the shear-wave directivity is the key factor in determining the size and distribution of the peak velocities and displacements. Strong shear-wave directivity requires that (1) the observer is located in the direction of rupture propagation and (2) the rupture propagates parallel to the direction of the fault slip vector. We show that predominantly along-strike rupture of a thrust fault (geometry similar in the Chi-Chi earthquake) minimizes the area subjected to large-amplitude velocity pulses associated with rupture directivity, because the rupture propagates perpendicular to the slip vector; that is, the rupture propagates in the direction of a node in the shear-wave radiation pattern. In our simulations with a shallow hypocenter, the maximum peak-to-peak horizontal velocities exceed 1.5 m/sec over an area of only 200 km^2 for the 30°-dipping fault (geometry similar to the Chi-Chi earthquake), whereas for the 60°- and 75°-dipping faults this velocity is exceeded over an area of 2700 km^2. These simulations indicate that the area subjected to large-amplitude long-period ground motions would be larger for events of the same size as Chi-Chi that have different styles of faulting or a deeper hypocenter.

Seismogenic fault of the 2021 Ms 6.0 Luxian induced earthquake in the Sichuan Basin, China constrained by high-resolution seismic reflection and dense seismic array

Fault geometry and mechanics of marly carbonate multilayers: An integrated field and laboratory study from the Northern Apennines, Italy

The Ms 5.5 earthquake struck on 24 October 2023, in Subei County, Gansu Province, China, occurring along the eastern segment of the Altyn Tagh fault. It raises the question of whether this earthquake is linked to the ongoing shortening slip rate along this segment or triggered by other seismic events. Analyzing the fault geometry of the Subei earthquake and understanding the significance of the weakening activity rate for seismic hazards in neighboring regions is crucial. The surface deformation from small- and medium-sized earthquakes (magnitudes less than Mw5.5) is often subtle, and the coseismic deformation detected by interferometric synthetic aperture radar (InSAR) is vulnerable to atmospheric disturbances, leading to significant measurement errors. Moreover, inaccuracies in the regional crustal velocity structure can cause errors in earthquake localization based on seismic data. These challenges complicate the establishment of a rupture model for seismogenic faults and hinder the inversion of fault slip models. To overcome these limitations, we employed the time-series InSAR stacking method and aftershock relocation to determine the fault geometry of the Subei earthquake. A two-step inversion method was utilized to ascertain both the fault geometry and slip distribution. Our modeling indicates that the 2023 Subei earthquake had a thrust mechanism with a component of strike-slip. The rupture did not reach the surface, with the maximum fault slip measuring 0.45 m at a depth of 2.5–3.5 km. The fault dips westward, and the moment magnitude is calculated at 5.4. This earthquake is associated with the ongoing weakening of the left-lateral strike-slip rupture along the Altyn Tagh fault in the Subei region. Furthermore, retrograde thrust tectonics significantly contribute to the absorption of accumulated stress during this process.Our findings highlight the potential of utilizing time-series InSAR images to enhance earthquake catalogs with geodetic observations, offering valuable data for further studies of the earthquake cycle and active tectonics. This approach is also applicable in other tectonically active regions, enhancing understanding of seismic hazards and risk assessment.

Injection-induced fault slip and associated seismicity in the lab: Insights from source mechanisms, local stress states and fault geometry

SUMMARYTeleseismic waveforms contain information on fault slip evolution during an earthquake, as well as on the fault geometry. A linear finite-fault inversion method is a tool for solving the slip-rate function distribution under an assumption of fault geometry as a single or multiple-fault-plane model. An inappropriate assumption of fault geometry would tend to distort the solution due to Green’s function modelling errors. We developed a new inversion method to extract information on fault geometry along with the slip-rate function from observed teleseismic waveforms. In this method, as in most previous studies, we assumed a flat fault plane, but we allowed arbitrary directions of slip not necessarily parallel to the assumed fault plane. More precisely, the method represents fault slip on the assumed fault by the superposition of five basis components of potency-density tensor, which can express arbitrary fault slip that occurs underground. We tested the developed method by applying it to real teleseismic P waveforms of the MW 7.7 2013 Balochistan, Pakistan, earthquake, which is thought to have occurred along a curved fault system. The obtained spatiotemporal distribution of potency-density tensors showed that the focal mechanism at each source knot was dominated by a strike-slip component with successive strike angle rotation from 205° to 240° as the rupture propagated unilaterally towards the south-west from the epicentre. This result is consistent with Earth’s surface deformation observed in optical satellite images. The success of the developed method is attributable to the fact that teleseismic body waves are not very sensitive to the spatial location of fault slip, whereas they are very sensitive to the direction of fault slip. The method may be a powerful tool to extract information on fault geometry along with the slip-rate function without requiring detailed assumptions about fault geometry.

The ill-posed nature of earthquake source estimation derives from several factors including the quality and quantity of available observations and the fidelity of our forward theory. Observational errors are usually accounted for in the inversion process. Epistemic errors, which stem from our simplified description of the forward problem, are rarely dealt with despite their potential to bias the estimate of a source model. In this study, we explore the impact of uncertainties related to the choice of a fault geometry in source inversion problems. The geometry of a fault structure is generally reduced to a set of parameters, such as position, strike and dip, for one or a few planar fault segments. While some of these parameters can be solved for, more often they are fixed to an uncertain value. We propose a practical framework to address this limitation by following a previously implemented method exploring the impact of uncertainties on the elastic properties of our models. We develop a sensitivity analysis to small perturbations of fault dip and position. The uncertainties of our fixed fault geometry are included in the inverse problem under the formulation of the misfit covariance matrix that combines both prediction and observation uncertainties. We validate this approach with the simplified case of a fault that extends infinitely along strike, using both Bayesian and optimization formulations of a static slip inversion. If epistemic errors are ignored, predictions are overconfident in the data and slip parameters are not reliably estimated. In contrast, inclusion of uncertainties in fault geometry allows us to infer a robust posterior slip model. Epistemic uncertainties can be many orders of magnitude larger than observational errors for great earthquakes (Mw > 8). Not accounting for uncertainties in fault geometry may partly explain observed shallow slip deficits for continental earthquakes. Similarly, ignoring the impact of epistemic errors can also bias estimates of near-surface slip and predictions of tsunamis induced by megathrust earthquakes.

<p>Earthquake fault ruptures are typically complex and can consist of en echelon segments, have bends, large step-overs, and be curved or warped at different spatial scales. Although surface fault ruptures can be mapped using a variety of geological and geophysical techniques, the subsurface topology of faults is challenging to estimate. One of the main options is to use geodetic data (InSAR and GPS) of coseismic surface deformation to estimate the subsurface earthquake fault geometry along with the distributed slip. The general practice is to assume a planar fault surface and estimate the strike and dip of a simple rectangular fault prior to the spatially-variable slip estimation. Using such simplistic fault geometry during source fault estimations of large earthquakes rarely captures all the crustal deformation details seen in the data and can cause biased estimation results of the fault slip. Here, we show how complex non-planar fault geometry can be estimated simultaneously with spatially-variable slip from geodetic data in a Bayesian framework, where our non-planar fault geometry parametrization approach allows for various undulations of the fault surface in both the along-strike and down-dip directions.</p><p>We exemplify this approach through synthetic tests considering a checkerboard-like slip pattern on a listric non-planar fault. The results show that fault slip can be over-estimated by about 50-100% when using pre-assumed planar fault geometry. In contrast, both the non-planar fault geometry and spatially-variable slip are better retrieved when using our estimation approach. We then apply this modeling approach to the 2011 M<sub>W</sub>9.1 megathrust Tohoku-Oki (Japan) earthquake. Here we use prior information like the location of the trench and earthquake hypocenters during the Bayesian estimation to reduce the extent of the model space. The resulting fault geometry shows variations in fault dip in both the along-strike and down-dip directions that compare well with Hayes’ slab1.0 model of the subduction interface. The estimated fault slip is also comparable to the results that pre-defined the fault geometry using the slab1.0 model. In the future, the proposed method could lead to more realistic source models of major earthquakes, aided by improving computational resources and spatial resolution of geodetic data.</p>

<p>For understanding the formation of mountain belts it is necessary to gain quantitative insights on fault and fracture mechanics on multiple scales. In particular, for addressing the role of fluids on larger processes, it is inevitable to constrain fault and fracture geometries at depth, as well as gain insights on how fluids influence fault mechanics. At least partly, the future of such analyses lies in exploiting large data sets, as well as in multi- and interdisciplinary research.</p><p>In this talk I will present results from variety of geological settings, including dilatant faults at Mid-Ocean Ridges, the Oman Mountains, the Khao Kwang fold-trust belt in Thailand, and the European Alps. I will show how multi-scale studies and the use of large data sets helps constraining fluid migration in mountain belts, fault geometries, as well as possible feedbacks between fluid flow and strain localization. Results are then applied to discuss the role of mechanical stratigraphy on structural style in foreland fold-thrust belts.</p>

The 2013 Lushan Ms 7.0 earthquake occurred on the Longmenshan thrust tectonic zone, a typical blind reverse-fault type earthquake that caused the death of nearly 200 people. The investigation of the fault geometry and fault slip distribution of this earthquake is important for understanding the seismogenic tectonic type and seismic activity mechanism of the Longmenshan Fault Zone. In this paper, for the fault geometry of the Ms 7.0 earthquake in Lushan, the geometric parameters of the planar fault are inverted based on the rectangular dislocation model using GPS coseismic displacement data, and on this basis, a curved fault steeply-dipping on top and gently-dipping at depth is constructed by combining the aftershock distribution. The GPS and leveling data are used to invert the slip distribution of the curved fault for the Lushan Ms 7.0 earthquake. The results show that the fault is dominated by reverse slip with a small amount of sinistral rotation, and there is a peak slip zone with a maximum slip of 0.98 m, which corresponds to a depth of ~ 13.50 km, and the energy released is 1.05 × 1019 N/m with a moment magnitude of Mw 6.63. Compared with the planar rectangular dislocation model, the curved fault model constructed by using triangular dislocation elements can not only better approximate the fault slip, but also better explain the observed surface displacement, and the root mean square error of the GPS and leveling data fitting is reduced by 1.3 mm and 1.9 mm, respectively. Both the maximum slip and moment magnitude of the fault based on the inversion of the curved structure are slightly larger than the results based on the planar structure.