Potency–Magnitude Scaling Relations and a Unified Earthquake Catalog for the Western United States Open Access

Magnitude is the most widely used and societally relevant earthquake source parameter. Larger earthquakes have overall greater damage potential, and thus an accurate characterization of earthquake size is of great interest to both seismologists and the public. Despite this fundamental importance, accurately measuring earthquake size is challenging because of the attenuation of seismic waves as they propagate from the earthquake rupture process at depth to geophysical sensors at the surface. Additional complications arise from the strong structural heterogeneities around faults and at shallow depths. These issues render it impossible to resolve fine details of earthquake ruptures, but robust information on the size of earthquakes can be obtained from analysis of low‐frequency seismic waves. In this article, we develop a unified earthquake catalog for the western United States, including uniform magnitude estimates derived from seismic potency—a fundamental source parameter (the product of fault area and average slip) that controls the amplitude of low‐frequency seismic waves.

Over the years, numerous techniques have been developed to measure earthquake size. In pioneering work, Charles Richter (1935) defined a measure of earthquake size by correcting for the systematic decay with distance of waveform amplitudes recorded on Wood–Anderson seismometers in southern California. The measurement, now called local magnitude (⁠$ML$⁠), is still widely used by regional monitoring networks due to its simplicity and applicability to small and frequently recorded earthquakes. Building on this work, Beno Gutenberg demonstrated how body‐ (Gutenberg, 1945a) and surface‐wave (Gutenberg, 1945b) amplitudes could be analyzed similarly but at greater distances to create body‐ and surface‐wave magnitude scales (⁠$Mb$ and $Ms$⁠) of particular relevance for larger earthquakes. For small earthquakes, it can sometimes be advantageous to use duration rather than amplitude as a basis for calculating magnitude, with coda duration magnitudes (⁠$MD$⁠) being the most popular such metric (Aki, 1969; Herrmann, 1975; Bakun and Lindh, 1977). For any magnitude scale, it is important to recognize that different monitoring agencies can have different data processing procedures or apply different empirical correction functions, which can lead to slight differences in the magnitude estimates depending on the authors; see Bormann and Dewey (2014) for a detailed review.

There is a slight difference in these two definitions of moment magnitude, as $Mw∼M−0.0333$⁠. In this work, we use equation (1) and the notation $Mw$ to define moment magnitude.

Direct estimation of $Mw$ is often challenging for smaller earthquakes due to the requirement for sufficient signal‐to‐noise ratios at the long periods used in waveform inversions. As a result, many earthquake catalogs contain a mixture of different magnitude types, with local and duration magnitudes (⁠$ML$ and $MD$⁠) dominant for small events and moment magnitudes (⁠$Mw$⁠) prevalent for large ones. This heterogeneity can be problematic, particularly for interpreting earthquake statistics, frequency–magnitude distributions, and scaling of ground motion with magnitude, all of which are central to seismic hazard analysis (e.g., Herrmann and Marzocchi, 2020).

The objective of this study is to develop a uniform, physics‐based, and self‐consistent catalog of event sizes for historic and modern earthquakes in the western United States. To do this, we leverage the data from the U.S. Geological Survey’s (USGS) Comprehensive Catalog (ComCat), for which many individual events have multiple magnitude estimates of distinct types, allowing us to develop empirical relations between different magnitude types. We validate these magnitude scaling relations with synthetic waveforms generated via the stochastic method (Boore, 2003), for which the earthquake size is known by definition and various measures can be readily extracted for comparison. Our work extends classic research on magnitude scaling relations (e.g., Thatcher and Hanks, 1973; Bakun, 1984; Hanks and Boore, 1984) but with a greatly expanded sample of data.

In this work, we use seismic potency $P0$⁠, which has units of volume (e.g., Ben‐Zion, 2003), as our target size parameter. The more commonly used seismic moment $M0$ is given by $P0$ multiplied by an assumed rigidity at the source. However, the definition of seismic moment can be ambiguous because rigidity can change discontinuously across faults, and elasticity breaks down in rupture zones, making the rigidity at the source ill‐defined for earthquakes. Moreover, observed ground motions can be fully parameterized in terms of motions at the boundaries of the source volume that propagate from there in the form of elastic waves. The failure processes and rigidity in the source volume are completely accounted for by their action at the boundaries, so the nominal rigidity used to define $M0$ is an extra parameter that can create ambiguity in source characterization (Ben‐Zion, 1989, 2001).

This study provides self‐consistent and physics‐based estimates of both potency‐ and moment‐based earthquake magnitudes in the western United States that can be used for various research topics. For example, Mueller (2018) described a methodology for compiling a uniform earthquake catalog for the 2023 United States National Seismic Hazard Map using magnitude conversion relations derived from Utsu (2002) based on a legacy compilation of global data. The conversion relations developed in this study could be easily incorporated for this purpose and are based on a much larger and updated compilation of data specific to the western United States. As shown subsequently, the data require conversion relations that are quadratic rather than linear as is often assumed.

We analyze earthquake magnitude estimates listed in the USGS’s ComCat. Our study region encompasses the continental western United States (longitudes: from −128.0° to −109.0°, latitudes: 31.0° to 49.5°) from January 1950 to June 2024. For each unique event, ComCat lists a preferred origin and magnitude estimate, and we focus on events with preferred magnitudes 2.0 and greater (314,925 events in total). The preferred magnitudes are highly variable in type (Fig. 1); most small events list $ML$ or $MD$ as the preferred magnitude, whereas $Mb$⁠, $Ms$⁠, and $Mw$ are increasingly prevalent for larger events (ComCat uses equation 1 and not equation 2 to define moment magnitude). The frequency–magnitude statistics depend on the type of magnitude used in the data compilation, as noted in previous studies (e.g., Shelly et al., 2021). This variability compromises any characterization of magnitude statistics; it is remarkable that mixing all these distributions, as is done in many studies, leads to an approximate power‐law distribution over most of the magnitude range.

Our purpose here is to unify these data via a magnitude scale based on seismic potency. To do this, we use the multiple magnitude estimates for individual events in ComCat, obtained from different sources or different methods, to develop statistical relations that connect different magnitude types. For example, to obtain a conversion relation between moment and local magnitude, one could focus on the subset of events with at least one magnitude estimate of both types. Our approach focuses on developing scaling relations for seismic potency $P0$ in log units (Ben‐Zion and Zhu, 2002; Ross et al., 2016) and then reporting uniformly $log10P0$ and related magnitudes. The scaling relations between $log10P0$ and other magnitude scales can be used to derive those magnitudes in situations where this is desirable. In general, there will be a different potency scaling relationship for each magnitude type. We present the method in detail for $ML$ versus $log10P0$ subsequently; scaling relations for other magnitude types are shown in the supplemental material available in this article.

We assume a nominal rigidity $μ$ of 36 GPa consistent with average crustal values used in regional moment tensor inversions for earthquakes in the western United States (e.g., Ichinose et al., 2003). The units of potency here are centimeters per square kilometer, a convenient choice because most earthquakes in the dataset have average slips and fault areas of the order of centimeters and square kilometers, respectively. The value of 11 on the right side of equation (4) accounts for the unit conversion from $M0$ in $dyn·cm$⁠.

Though there is considerable scatter from event to event, we observe a systematic increase in median seismic potency values (parameterized as $log10P0$⁠) with local magnitude $ML$ (Fig. 2). This trend is nonlinear but is well described by the quadratic model of equation (5), with the local slope of the fitted model (⁠$c1+2c2ML$⁠) increasing from ∼1.0 at $ML$ 3.5 to ∼2.0 at $ML$ 6.5. This steepening of the curve can be interpreted in terms of magnitude saturation: any instrumental measure of earthquake magnitude derived from a finite‐frequency band will fail at some event size to capture true increases in earthquake magnitude (Kanamori, 1977). The change in the local slope may also reflect at least partially a transition in the physics governing small and large events (Ben‐Zion and Zhu, 2002). It is difficult to comprehensively compare our model to previous studies on magnitude scaling relations because the underlying data and functional forms of the models differ. However, a slope of order 1.5 seems to be the general consensus for moderate earthquakes (Thatcher and Hanks, 1973; Bakun, 1984; Ben‐Zion and Zhu, 2002) and is also consistent with the definition of $Mw$⁠. Because our data are concentrated between $ML$ 3.5 and 6.0, the uncertainties in the model predictions become large outside this range (Fig. S1). In particular, although the potency–magnitude scaling fit well by quadratic function within this range of event sizes, extrapolation of the model beyond these bounds may not be viable. We discuss this issue in greater detail subsequently.

The USGS ComCat catalog features several different preferred magnitude types; therefore, it is desirable to generalize this analysis to include other magnitude scales. We repeat this same basic procedure for $MD$⁠, $Mb$⁠, and $Ms$ (Fig. S2), with corresponding model coefficients and uncertainties listed in Table 1 alongside those for $ML$⁠. Similar to the results for $ML$⁠, we observe that the scaling of $MD$ with potency is best described by a quadratic model that steepens (saturates) with increasing magnitude. In contrast, for $Mb$ and $Ms$⁠, the potency–magnitude scaling is well described by a linear model, though the $Ms$ relation is more precise (Fig. S2). The approximately constant slope (lack of saturation) for $Mb$ and $Ms$ is likely an observational bias because there are very few of the large (⁠$Mw$ 7–8) events in our dataset where changes in slope could become clear.

The scaling of instrumental magnitudes with seismic potency (or more commonly, moment) is a classic problem in seismology that we revisit in this study, armed with a large data set that spans a broad range of event sizes. Hanks and Boore (1984) developed a conceptual model of the scaling of local magnitude and seismic moment in terms of the natural frequency of the Wood–Anderson seismograph (⁠$fs$⁠), the corner frequency of the earthquake (⁠$f0$⁠), and near‐surface attenuation (⁠$fmax$⁠). They identified three regimes: (1) for large earthquakes with $f0$ much less than $fs$⁠, the scaling should be $log10M0∼3.0ML$⁠, (2) for moderate earthquakes for which $f0$ is much greater than $fs$ but less than $fmax$⁠, the scaling should be $log10M0∼1.5ML$⁠, and for small earthquakes with $f0$ greater than $fmax$⁠, the scaling should be $log10M0∼1.0ML$⁠. Our results are qualitatively compatible with this model; the slope of the scaling for small earthquakes is about 1 and increases systematically with event size. However, the steepest possible scaling compatible with the data is ∼2.0–2.4 for large events (Fig. 2, Fig. S1), never approaching the hypothesized value of 3.0. Hanks and Boore (1984) noted a similar trend in their dataset but attributed it to lack of observations; revisiting this problem 40 yr later with a much‐expanded catalog makes this explanation less compelling.

To study this problem further, we generated synthetic waveforms from earthquakes of known potency using the stochastic method (Boore, 1983), for which each synthetic waveform is a random realization of the target spectrum that combines modeled source, path, and site effects on ground motion. Here, we use a seismological model emulating the work of Yenier and Atkinson (2015) that includes a non‐self‐similar, generalized double‐corner model of the source spectrum, a transition from body‐ to surface‐wave geometric spreading with distance, frequency‐dependent attenuation, and both site amplification and attenuation effects (see Text S1 for additional details). Hanks and Boore (1984) applied a similar approach but with a simplified parameterization of source, site, and path effects. A major advantage of the Yenier and Atkinson (2015) model parameterization is that it was calibrated to match ground motion observations of the California earthquakes that comprise most of our dataset, lending confidence to its application in this context (see also Fig. S3). The stochastic simulations are designed for class B/C site conditions; we neglect here the amplification of individual sites relative to this reference condition because we are mainly interested in magnitude scaling.

We generate synthetic waveforms for $Mw$ 2–7 earthquakes at a range of distances up to 300 km, convert the acceleration time series into Wood–Anderson displacement and derive $ML$ from the peak amplitude corrected for distance (Hutton and Boore, 1987). The results, aggregated across hundreds of thousands of simulations, are presented in Figure 3. The scaling of $log10P0$ with $ML$ is quadratic with comparable scaling coefficients, if slightly steeper, than the model coefficients obtained from observational data (Fig. 2). Although the stochastic method assumes a simplified representation of the earthquake process, it provides a useful conceptual basis for understanding the magnitude scaling of ground motion observed in nature. In particular, it is noteworthy that saturation effects for $ML$ can be replicated using existing seismological models of source, path, and site effects without modification.

An important outcome of this study is the capability to develop a uniform set of magnitude estimates derived from seismic potency for earthquakes in the western United States. To do this for our ComCat dataset, we estimate potency either directly from moment when possible or using the scaling relations listed in Table 1 applied to the ComCat preferred magnitudes. About 5% of events list a preferred magnitude type other than $ML$⁠, $MD$⁠, $Mb$⁠, $Ms$⁠, or $Mw$⁠. Nearly all of these are small events with “helicorder” magnitudes that we assume for these calculations approximate $ML$⁠. Rather than extrapolate our quadratic models for $ML$ and $MD$ beyond the support of the fitted data (i.e., below magnitude ∼3.5), we assume a transition from quadratic to linear scaling of $log10P0$ with magnitude for small earthquakes (see Table 1 for details). This assumption is consistent with the conceptual model of Hanks and Boore (1984) and also with the work of Ross et al. (2016), who observed a linear scaling for small earthquakes in the San Jacinto fault zone. Similarly, for $Mb<4$⁠, we first convert to $ML$ (Fig. S4) and apply the appropriate relation to estimate potency. Though beyond the scope of this study, future efforts could be dedicated to refining potency–magnitude scaling relations for small earthquakes.

For many applications, this adjustment is likely to be small, and thus $MP$ and $Mw$ could be used interchangeably. Unlike instrumental magnitude scales, the $MP$ and $Mw$ magnitudes do not saturate and can be used in applications for which a uniform and physically consistent magnitude scale is desired. Alternatively, if one desires estimates of instrumental magnitudes like $ML$⁠, it is possible to invert the developed scaling relations to translate potency into the magnitude type of interest. For convenience, we provide conversion relations from $log10P0$ and $MP$ to different magnitude scales in Table 1. The catalog also includes clustering designations for each earthquake based on the nearest‐neighbor method (Zaliapin and Ben‐Zion, 2013), which can be useful for some applications.

It is illuminating to compare the frequency–magnitude distribution for $MP$ with the equivalent distribution of preferred magnitudes listed by ComCat (Fig. 4). The two distributions are similar for larger events but differ markedly for smaller earthquakes where the ComCat preferred magnitude systematically underestimates $MP$⁠. The apparent simplicity of the ComCat magnitude distribution (Fig. 1) obscures the more complex magnitude distribution revealed when this bias is corrected. This has important implications for calculating parameters like the b‐value, which assumes an underlying exponential distribution and depends on the mean magnitude above the completeness level of the catalog, so is more sensitive to the frequently occurring small events than the infrequent large ones. Assuming a conservative completeness magnitude of 3.5, the maximum‐likelihood estimate of the b‐value (Bender, 1983) is 0.90 (±0.02) when using the heterogeneous set of ComCat preferred magnitudes, compared to 1.07 (±0.02) when using the uniform magnitude $MP$⁠. Although the numerical difference between these b‐value estimates is relatively small, the consequences are important for any statistical parameterization of earthquake processes, including those used in hazard calculations.

Although potency is a desirable metric for parameterizing earthquake size, it is important to acknowledge several current limitations of our study. First, our potency scaling relations were derived from moment estimates in which the assumed rigidity was not documented. To our knowledge, all moment tensor inversion codes used operationally within the western United States assume an effective rigidity in the 30–40 GPa range. The difference between these two extremes translates into an epistemic uncertainty of ±0.06 log units of potency (±0.04 $MP$⁠), which could be remedied with clearer documentation in moment tensor solutions. Second, there is a scarcity of small‐magnitude data in ComCat with which to develop potency scaling relations. Dedicated studies focused on obtaining potency estimates for small earthquakes (e.g., Ross et al., 2016) could improve the resolution beyond our present means. Third, we neglect regional variations within a given magnitude type (e.g., estimates of $ML$ published by different monitoring agencies) because these differences are typically smaller and less systematic than the differences between magnitude types, but further studies could investigate this issue in greater detail. Finally, it is important to recognize that potency, even if accurately estimated, only captures the product of fault area and average slip. Earthquakes of equivalent size can produce different ground motions due to different stress drops, rupture velocities, directivity, or other dynamic effects that are not represented by seismic potency (or moment) measurements. We may never be able to characterize all properties of individual earthquakes with perfect accuracy. But we can better represent and quantify the important processes, statistics, and hazards related to earthquakes by better understanding scaling relations between physics‐based and instrumental magnitude measurements.

Earthquake catalog data for this study were obtained from the U.S. Geological Survey (USGS) Comprehensive Catalog (ComCat; https://earthquake.usgs.gov/earthquakes/search/) using the Python library libcomcat (https://code.usgs.gov/ghsc/esi/libcomcat-python). Stochastic method simulations were conducted using Python scripts written by the authors and benchmarked against ground motion simulation system (GMSS, https://github.com/Y-Tang99/GMSS1.0). The catalog produced in this study is archived on Zenodo (doi: 10.5281/zenodo.12554989). All websites were last accessed in July 2024. The supplemental material includes a detailed description of the stochastic method implementation used to generate synthetic waveforms, and four additional figures that support the results presented in the main text.

The authors acknowledge that there are no conflicts of interest recorded.

The article benefitted significantly from thoughtful and constructive reviews by G. Atkinson and R. Graves, as well as comments from Associate Editor S. Gibbons and Editor‐in‐Chief K. Koper. The authors thank C. Kreemer for discussions on seismic and geodetic potency scaling relations, and A. Patton and B. Savran for suggestions on the figures. The study was supported by the National Aeronautics and Space Administration (NASA) Award Number 80NSSC24K0736 and the Statewide California Earthquake Center (SCEC) Award Number 24169. SCEC is funded by NSF Cooperative Agreement EAR‐2225216 and USGS Cooperative Agreement G24AC00072‐00).