Introduction

Earthquake source spectra provide valuable insights into earthquake rupture dynamics and energy release. Source spectra are parameterized by two source parameters, seismic moment and corner frequency, which control the spectrum's shape and scale (e.g., Abercrombie, 2021;Brune, 1970;Haskell, 1964). However, several factors can cause the source spectra of real earthquakes to deviate from the predictions of simplified source models. For instance, heterogeneities in local stress state, fault materials, and fault roughness result in complex slip patterns (e.g., Bürgmann et al., 1994) and introduce complexity to earthquake source spectra, challenging the assumptions of simple theoretical models and necessitating a more nuanced interpretation of seismic source data.

Stress drop is a key source parameter that is nominally related to the change in stress over the ruptured area, but is usually estimated indirectly by fitting a simplified model to the shape of the source spectra (Scholz, 2019). Despite large uncertainties caused by path effects, numerous studies have aimed to identify the controlling factors underlying variations in estimated stress drops (computed from corner frequency and moment, assuming a circular source) and their spatially coherent patterns, including scaling with magnitude and depth (e.g., Abercrombie, 1995;Allmann & Shearer, 2009;Bindi et al., 2021;Oth, 2013;Shearer et al., 2006Shearer et al., , 2022;;Trugman, 2022). While correlations between spatial variations of estimated stress drops and geophysical parameters such as b-values and strain rates have been investigated (Hauksson, 2015), the cause of this spatial heterogeneity remains largely unknown.

• We analyze the influence of fault geometry on earthquake source spectra using data sets from multiple regions and scales • Higher corner frequencies and higher source complexity are observed for earthquakes occurring in regions with complex fault geometries • Earthquakes on complex faults produce stronger high-frequency ground motions, with important implications for seismic hazards

Supporting Information:

Supporting Information may be found in the online version of this article.

The geometric complexity of fault systems is known to significantly influence earthquake rupture processes and the resulting ground motions (e.g., Chatterjee et al., 2024;Gauriau & Dolan, 2021;Lee et al., 2024;Ozawa et al., 2023), yet there has been limited research on how these complexities affect the spectral characteristics of seismic radiation. The complex geometry of fault systems may include multiplanar ruptures (Holden et al., 2017), fault damage zones (Mitchell & Faulkner, 2009), and bends, gaps, stepovers, splay faults, and branches observed in surface fault traces (Perrin et al., 2016;Rodriguez Padilla et al., 2021). Previous research has proposed that additional high-frequency earthquake ground motions may result from the additional smaller-scale elastic loading and unloading of complex fault zone structures (Tsai & Hirth, 2020;Tsai et al., 2021). Several observations support this hypothesis, including the correlation between fault complexity and corner frequency in Southern California (Chu et al., 2021), the isotropic nature of high-frequency radiation patterns captured at close distances by the dense Oklahoma LASSO array (Trugman et al., 2021), and the correlation between fault complexity and high-frequency ground motions of earthquakes (Chatterjee et al., 2024).

Building on previous research, we show how the structural complexity of fault networks contributes to the variability of earthquake source spectra and high-frequency seismic radiation, as illustrated in Figure 1. In the idealized Brune-Haskell model (Brune, 1970;Haskell, 1964), rupture along the fault plane produces a simple moment-rate function controlled by rupture velocity and rise time. In this frictional framework, complexities in spectra are caused by heterogeneity of slip and rupture propagation speed. We suggest that the complex fault system surrounding the principal slip surface can produce additional high-frequency seismic radiation beyond that of the Brune-Haskell model due to the complex motions of the various subfault blocks. Additional contributions to source spectra are expected within a frequency range that scales with the size of structural complexities of the fault zone (e.g., assuming discrete fault structures on the order of 4-20 m, expected frequencies are in the range of 5-20 Hz). For structural complexities that are a fraction of the full rupture length in size, these frequencies are generally near or slightly higher than the corner frequency predicted by the Brune-Haskell model (Tsai & Hirth, 2020;Tsai et al., 2021). Due to this contribution, complexities in earthquake source spectra could lead to higher estimated corner frequencies-and thus higher estimated stress drops-and larger deviations from the Brune source model for earthquakes occurring along fault systems of greater geometric complexity (Figure 1). We test this hypothesis by comparing earthquake source spectra from fault zones with complex geometries to those with simple geometries.

We aim to identify the relationship between geometric complexity of fault networks and high-frequency complexities in earthquake source spectra by using multiple independent geological and seismological observations. First, we expand upon our previous findings of a positive correlation between estimated stress drop and fault complexity in Southern California (Chu et al., 2021) by examining this relationship across a wider range of geological regions at a range of scales. We compare cataloged stress drop values with fault metric measurements in Southern California, Central Italy, and Japan to test the hypothesis that the spatial heterogeneity of stress drops arises from fault network complexity. Additionally, we extend this analysis to large global earthquakes, using variability in regional focal mechanisms as a proxy for fault network complexity, to explain the complexities observed in their well-resolved moment-rate functions. We find positive correlations between fault complexity and earthquake stress drops, with our robustness tests confirming that our findings are consistent across multiple length scales. These analyses highlight contributions from fault zone complexity that are not easily explained by traditional earthquake models by drawing connections between measurable aspects of fault geometry and seismological observations.

Data and Methods

We compare measurements of fault network complexity with high-frequency characteristics of earthquake source spectra across different regions. To quantify the geometrical complexity of fault networks, we use fault network metrics and the variability of focal mechanisms. To quantify the high-frequency complexity in earthquake source spectra, we use corner frequency (or the equivalent estimated stress drop) and moment-rate roughness (also known as the radiated energy enhancement factor, which characterizes the variability of high-frequency spectral residuals relative to a smooth ω -2 source model) of the earthquake moment-rate function. The choice of metrics and measurements depends on the resolution of available data and the scale of the earthquake processes, yet the underlying features they represent are similar. According to our hypothesis and previous findings, we expect positive correlations between the fault complexity metrics and the high-frequency measurements.

We choose our study regions based on the availability of well-documented earthquake stress drop catalogs and high-resolution surface fault trace maps, and thus calculate fault network complexity at multiple length scales in Southern California, Japan, and Central Italy. We use the misalignment and density of fault networks (Chu et al., 2021) to quantify the degree of fault complexity of mapped surface fault traces. These metrics are expected to correspond to the aspect ratio and size of discrete structures comprising the fault system, which determines the enhancement in high-frequency ground motions according to our proposed model. We assume that the fault network complexity at seismogenic depths correlates with the fault network complexity we quantify based on the surface fault traces. As shown in Supporting Information S1, the strong spatial correlation (cc = 0.75) between shallow (<5 km) and deeper (10-15 km) focal mechanism complexity (Cheng et al., 2023) supports this assumption. For the regional earthquakes, we use published stress drop catalogs in the three study regions to examine the spatial heterogeneity of magnitude-dependent corner frequencies. For Southern California, we use a subset of the catalogs by Shearer et al. (2022) for earthquakes from 1996 to 2019. We limit our analyses to 61,439 earthquakes within the Mw range of 2.0-4.0, at depths shallower than 8.5 km, and within 5, 10, or 15 km of the nearest surface fault trace. The time period of earthquakes in this study differs from the stress drop catalog used in our previous work (Chu et al., 2021;Shearer et al., 2006), which covered 1989-2001, but the spatial patterns are similar. For Japan, we use 1,501 earthquakes from the stress drop catalog of Oth (2013), ranging from Mw 2.7 to 6, at depths shallower than 30 km, and within 5-30 km of surface faults. For Italy, we use the stress drop catalog of Kemna et al. ( 2021) spanning 2016-2017, which includes the Visso, Amatrice, and Norcia earthquake sequences. We limit our analyses to 7,098 earthquakes with Mw ranging from 2.0 to 4.0, at depths shallower than 8.5 km, and within 5 or 10 km of surface fault traces. We consider only shallow crustal earthquakes occurring near surface fault traces since fault metrics based on surface measurements may not accurately reflect fault complexity at greater depths or farther from surface faults. In Southern California and Italy, we restrict our analysis to Mw 2.0-4.0 events in order to exclude earthquakes with insufficient low-frequency constraints and to mitigate bias from the apparent scaling of stress drop with magnitude in smaller events (Kemna et al., 2021;Shearer et al., 2022). For Japan, our thresholds are expanded due to the limited number of earthquake data and the spatial sparsity of both fault traces and earthquakes across a much larger study region.

We also analyze large global earthquakes using metrics that similarly capture the characteristics of earthquake spectra and fault complexity. While spatially coherent patterns of heterogeneity are also observed in global stress drop catalogs (Allmann & Shearer, 2009;Denolle & Shearer, 2016), complexities in the source spectra of large (>Mw 5.8) global earthquakes are better resolved by moment-rate functions (Mache et al., 2022;Vallée et al., 2011;Ye et al., 2018). The temporal evolution of moment-rate functions allows us to quantify the highfrequency content of seismic radiation not only using the magnitude-scaled corner frequency, but also by evaluating the complexity of moment-rate functions. We use the SCARDEC database of moment-rate functions (Vallée et al., 2011) to fit the moment-rate functions to a single corner-frequency model to estimate the momentnormalized corner frequencies and also calculate the earthquake moment roughness, which quantifies the complexity of moment-rate functions (Ye et al., 2018).

High-resolution fault traces are unavailable on a global scale, but we use the variability of nearby focal mechanisms, which is expected to reflect differences in the orientations of fault segments, to serve as a reliable proxy of fault complexity instead. The mean value of the pairwise Kagan angle distribution (Kagan, 1991; quantifies the angular difference between two focal mechanisms) indicates the average rotation angle between focal mechanisms within an area, which is indicative of the fault misalignment. We use the Global CMT catalog (Ekström et al., 2012) to select moment tensors within a distance range of 30-130 km from SCARDEC earthquakes, depending on the event magnitude, to calculate the pairwise Kagan angle distribution.

We test the robustness of our analyses by examining aggregated correlation coefficients between the faultcomplexity metric and high-frequency characteristics of earthquake source spectra across different parameter sets. For regional stress-drop analyses, we varied the radii for calculating fault-metric and distance thresholds for fault-earthquake proximity. For the global SCARDEC analysis, we varied the minimum distance to include CMT moment tensors (15 and 30 km) and minimum CMT-count thresholds (4, 6, 8, and 10). See Supporting Information S1 for details.

Finally, we note that different methods for estimating stress drop are available for some of the regional earthquake catalogs. For Southern California, we analyze the catalog using a single empirical correction function; two additional analyses in Supporting Information S1 include one using the catalog from our previous study (Chu et al., 2021;Shearer et al., 2006) and another using a catalog estimated with a multiple (i.e., spatially varying) empirical correction function (Shearer et al., 2022). The use of a multiple empirical correction function, combined with an assumed corner frequency of the smallest local earthquakes fixed at 15 Hz, is intended to correct for spatial variability in attenuation effects, but also may dampen or remove true variability in the source properties of the smallest events. In contrast, using a single empirical correction function may introduce artifacts due to inaccurate path corrections. For Central Italy, we use a stress drop catalog based on the spectral ratio method instead of a catalog estimate from single spectrum fitting and the cluster-event method (Ko et al., 2012). We use the spectral ratio method for its effectiveness in providing reliable stress drop estimates by correcting for path effects using empirical Green's functions, while retaining spatial variability.

Results

We present our findings on the relationship between fault complexity and earthquake source spectra, represented as correlations of estimated stress drop with fault misalignment for small regional earthquakes, and correlations of corner frequency and REEF with mean Kagan angles for large global earthquakes.

In our calculation of fault misalignment, significant variations are observed in the study regions. The variation is largest in Southern California, where the misalignment ranges from almost 0 at near-parallel fault strands of the San Jacinto fault, to 0.8 at complex intersecting fault strands of the Superstition Hills and Ridgecrest sequences (Figure 2a). The variation in misalignment is smaller in Japan, mostly ranging from 0 to 0.6, where the fault map resolution is lower (Figure 2b). The inadequate mapping of faults at small scales introduces uncertainties in analyzing fault misalignment measurements across large-scale areas with relatively low resolution and spatially sparse faults, as is the case for Japan in our study. In Central Italy, misalignments at earthquake locations vary only from 0 to 0.4 due to limited spatial extent (Figure 2c). While fault density also characterizes a component of geometric fault complexity in addition to fault misalignment, we focus our analysis on fault misalignment due to the higher reliability of fault misalignment to downscaling in fault map resolution. Analysis of fault density is provided in Supporting Information S1. We also provide our estimated global mean Kagan angles, derived from the CMT moment tensors, in Figure 2d, with mean values ranging from about 10°to 80°.

We present representative results of our analyses for a given set of parameter choices (Figure 3) while evaluating additional uncertainties related to parameter choices through a robustness test across a range of parameters (Figure 4). When estimated stress drops are plotted against fault misalignment, we observe a generally consistent positive trend for all three regional study areas of Southern California, Japan, and Central Italy (Figures 3a-3c and 4a-4c) despite spanning different ranges of fault misalignment and stress drops. For Southern California and Italy, we find large correlation coefficients ranging from approximately 0.5-0.8 and 0.8-0.97, respectively, for the mean values of binned data, with small bootstrap errors of about 0.01 and 0.08, respectively. More importantly, we observe consistent positive correlations across many different choices of fault misalignment scale and earthquake proximity in these two regions. In Japan, however, the significance of the overall trend is marginal, with small correlation coefficients in the range of -0.4 to 0.3 and generally large bootstrap errors (∼0.2), primarily due to the large measurement uncertainties and the relative lack of data. However, for analyses with earthquakes closer to faults (10-15 km) and at larger characteristic scales (20-30 km)-which may be more robust for the sparse distribution of faults in our Japan fault database-we observe consistent small positive correlations (see upper right of Figure 4b). While the correlation coefficient quantifies the best linear relationship between fault misalignment and stress drop, the true relationship may be nonlinear. Uncertainty analyses for additional stress drop catalogs (Supporting Information S1) demonstrate similarly consistent positive correlations.

In the analysis of SCARDEC moment-rate functions and the CMT Kagan angle distributions, we observe a consistent positive correlation between the complexity of the moment-rate functions and the mean of the pairwise Kagan angles (for stress drops estimated from moment-normalized corner frequencies and REEF, respectively, as the moment-rate function complexity metric), as illustrated in Figures 3d, 3e, 4d, and 4e. The correlation is particularly prominent for stress drops, with a high correlation coefficient of about 0.9 (Figure 4d). The distinct positive trends in both Figures 3d, 3e, 4d, and 4e suggest that complexities of earthquake moment-rate functions are affected by structural geometric complexity as quantified by variability in focal mechanisms. Overall, our results indicate that complexity in fault geometry may explain for up to a factor of 3 in the observed trends of apparent stress drop.

Discussion

There are significant uncertainties involved in quantifying both source spectra and fault complexity. Our analyses here have been performed at a larger scale (e.g., 10s of km for misalignment in California) compared to the length scales we believe to be most important for generating high-frequency ground motions, due to practical resolution issues. Nonetheless, if fault networks are fractal, then the misalignment measurements would be unchanged. Even if fault networks are not truly fractal, we expect the misalignment measurements at a smaller scale to correlate with those made at a larger scale. While measurements of fault misalignment are relatively reliable, uncertainties in individual earthquake stress drop estimates are substantial due to the difficulty of isolating earthquake source terms (Abercrombie, 2021), which necessitates the use of statistical estimates from a large data set to investigate spatial heterogeneity in stress drops. Uncertainties in focal mechanisms can be as large as several tens of degrees (Rösler & Stein, 2022), manifesting as uncertainties in our measurements of Kagan angles, while the inherent smoothing and bandwidth in the SCARDEC moment-rate functions introduce additional limitations to our analysis. Despite the large uncertainties, we emphasize that our analysis targeting four different data sets (Southern California, Japan, Central Italy, and global) consistently indicate a positive correlation between fault misalignment and complexity in earthquake source spectra at different geologic settings and length scales. The considerable scatter within the positive trend suggests that additional factors-such as regional tectonic stress states or fault zone properties (e.g., Boyd et al., 2017)-also contribute to variability in source characteristics; we do not attempt to quantify the influence of such factors. Attenuation also remains a known challenge; we cannot differentiate near-source path effects from true source terms. Strong attenuation in complex fault zones could dampen even higher corner frequencies than are reported; the trends we demonstrate could be even more significant if these near-source path effects were accounted for. Additionally, the observed correlation between fault complexity and estimated stress drops might not indicate a direct relationship, but, another underlying mechanism may mediate both observables. Although our study is driven by the hypothesis that the elastic loading and unloading of discrete fault structures lead to enhanced high-frequency radiation, other mechanisms, such as the uneven partitioning of slip within complex fault networks (Dolan et al., 2007), acceleration and deceleration of rupture speed due to geometric complexities of fault networks, and spatial variability in true stress states (Kilgore & Beeler, 2025) could also play a role. The analysis described here would therefore benefit from additional observations. Our findings suggest that the geometry of the fault system plays a substantial role in controlling the highfrequency content of earthquake source spectra. There have been differing interpretations on the influence of fault geometry on stress drop, and various studies have reported conclusions both consistent and inconsistent with our findings. For a few documented large earthquakes in California, Perrin et al. (2021) found that earthquakes along mature fault segments with simpler and more localized fault geometry have lower estimated stress drops, consistent with our observations and conceptual model. Using geometric proxies of fault maturity, Guo et al. (2023) found that the moment-scaled radiated energy in documented global large earthquakes is more prominent in medium-immature faults compared to very-immature or mature faults. Similarly, Milliner et al. (2024) found weak to no correlation between the geometrical complexity of mapped surface rupture and moment-scaled radiated energy, based on measurements of optical and radar pixel tracking. Given the challenges in obtaining a large number of reliable measurements for earthquake radiated energy, estimated stress drops and source spectra, as well as the lack of consensus about how to define geometric fault complexities, further study is necessary to reconcile these observations and identify the complex role of fault geometry, including its relationship with fault maturity.

Our findings have implications for seismic hazard analysis since we provide an explanation for why earthquakes of the same magnitude on certain faults may lead to increased ground motions. Along with other findings that report the influence of fault complexity on observed high-frequency seismic radiation (e.g., Chatterjee et al., 2024;Chu et al., 2021), our results suggest that fault complexity could be considered as a potential new independent factor in ground motion models.

Conclusions

In this study, we demonstrate consistent positive correlations between the complexity of earthquake source spectra and the geometrical complexity of nearby fault networks across multiple independent data sets. Specifically, we find positive correlations between estimated stress drop and fault misalignment in Southern California, Japan, and Central Italy, as well as positive correlations between the complexity of global moment-rate functions and the variability of regional focal mechanisms. While our model of elastic interactions between discrete fault structures during rupture may not be the only explanation, it provides a coherent interpretation of these consistent patterns. Our results thus contribute to the growing body of evidence that highlights the significance of fault network complexity in the earthquake rupture process.