Earthquake magnitudes are widely relied upon measures of earthquake size. Although moment magnitude (Mw) has become the established standard for moderate and large earthquakes, difficulty in reliably measuring seismic moments for small (generally Mw<4) earthquakes has meant that magnitudes for these events remain plagued by a patchwork of inconsistent measurement scales. Because of this, magnitudes of small earthquakes and statistics derived from them can be biased. Furthermore, because small earthquakes are much more numerous than large ones, many applications, such as seismic hazard modeling, depend critically on analysis of events characterized by magnitudes other than Mw. To assess this problem, we apply coda envelope analysis to reliably determine moment magnitudes for a case study of small earthquakes from northern Oklahoma and southern Kansas. Not surprisingly, we find significant differences among ML, mbLg, and Mw for M ∼2–4 earthquakes examined here. More troublingly, we find that relations designed to convert other magnitudes to Mw, which are relied upon for important applications such as seismic hazard analysis, often increase rather than decrease this bias for our dataset. In our case study, we find that converted magnitudes can result in a systematic bias sometimes exceeding 0.5 magnitude units, a difference that typically corresponds to a factor of ∼3 in seismicity rate. Moreover, we find a correspondingly large bias in Gutenberg–Richter b-values, controlled primarily by inaccurate magnitude scaling in the conversion relationships. Although this study focuses on a relatively small geographic area, we can expect that similar issues exist with varying severity in other regions. Therefore, magnitudes of small earthquakes and their associated statistics, including seismicity rates and b-values, should be treated with caution.

Summary Robustness of source parameter estimates is a fundamental issue in understanding the relationships between small and large events, however it is difficult to assess how much of the variability of the source parameters can be attributed to the physical source characteristics or to the uncertainties of the methods and data used to estimate the values. In this study we apply the coda method by Mayeda et al. (2003) using the Coda Calibration Tool (CCT), a freely available Java-based code (https://github.com/LLNL/coda-calibration-tool) to obtain a regional calibration for Central Italy for estimating stable source parameters. We demonstrate the power of the coda technique in this region and show that it provides the same robustness in source parameter estimation as a data-driven methodology (GIT, Generalized Inversion Technique), but with much fewer calibration events and stations. The Central Italy region is ideal for both GIT and coda approaches as it is characterized by high-quality data including recent well-recorded seismic sequences such as L'Aquila (2009) and Amatrice-Norcia-Visso (2016–2017). This allows us to apply data-driven methods such as GIT, and coda-based methods that require few, but high-quality data. The dataset for GIT analysis includes ∼5000 earthquakes and more than 600 stations, while for coda analysis we used a small subset of 39 events spanning 3.5 < Mw < 6.33 and 14 well-distributed broadband stations. For the common calibration events, as well as an additional 247 events (∼1.7 < Mw<∼ 5.0) not used in either calibration, we find excellent agreement between GIT-derived and CCT-derived source spectra. This confirms the ability of the coda approach to obtain stable source parameters even with few calibration events and stations. Even reducing the coda calibration dataset by 75 per cent we found no appreciable degradation in performance. This validation of the coda calibration approach over a broad range of event size demonstrates that this procedure, once extended to other regions, represents a powerful tool for future routine applications to homogeneously evaluate robust source parameters on a national scale. Furthermore, the coda calibration procedure can homogenize the Mw estimates for small and large events without the necessity of introducing any conversion scale between narrowband measures such as local magnitude (ML) and Mw which has been shown to introduce significant bias (e.g. Shelly et al., 2022).

<p>It is well known that the use of different methods (e.g., spectral fitting, empirical Green’s functions) for compiling catalogs of source parameters (e.g., seismic moment, stress drop) can results in significant inconsistencies (Baltay et al., 2022). In this study, we present the application of coda-wave source parameters estimation by the Coda Calibration Tool (CCT) to different tectonic settings for closer analysis of the regional variations.  CCT implements the empirical methodology outlined in Mayeda et al., (2003), which provides stable source spectra and source parameters even for events recorded by sparse local and regional seismic networks (<em>e.g</em>., Morasca et al., 2022). The main strength of the method is the use of narrowband coda waves measurements, which show low sensitivity to source and path heterogeneity. Additionally, we use independent ground-truth (GT) reference spectra for which apparent stresses are independently calculated through the coda spectral ratio (Mayeda et al., 2007), to break the path and site trade-off.  The use of GT spectra eliminates the need to assume source scaling for the region, reducing the impact of <em>a-priori</em> model assumptions on the interpretation of scaling laws of source parameters and their variability. The CCT is a freely available Java-based code (https://github.com/LLNL/coda-calibration-tool) that significantly reduces the coda calibration effort and provides calibration parameters for future use in the same region for routine processing.</p> <p>Recently, several studies applied CCT in very different tectonic contexts, including (1) earthquakes in tectonically active regions (<em>e.g</em>., central Italy, Puerto Rico, southern California, Utah); (2) induced earthquakes in southern Kansas and northern Oklahoma; and (3) moderate-sized earthquakes in stable continental regions such as in Eastern Canada and the United Kingdom. There is excellent agreement between coda-derived M<sub>w</sub> in all regions and available M<sub>w </sub>from waveform modelling. In some cases, such as central Italy and Ridgecrest, the validation process also involved the comparison with estimates from different empirical techniques, such as spectral decomposition approaches applied to data sets sharing common events with CCT. Overall, there is a general consistency in the scaling laws obtained for different source parameters (<em>e.g.,</em> seismic moment, corner frequency, radiated energy and apparent stress), with earthquakes in the UK and Canada having similar and higher apparent stresses than Utah, central Italy, Puerto Rico and southern California, while the induced regions are characterized by the lowest values. In conclusion, the application of a consistent methodological framework and the robustness demonstrated by the results of the seismic coda analysis allow comparison of source scaling relationships for different tectonic settings over a wide range of magnitudes.</p>