## 1 Introduction

[2] Stress drop scaling has been a controversial issue since first proposed by *Aki* [1967]. Are earthquakes self-similar with the stress-related growth of small earthquakes the same as large earthquakes from the rupture initiation? Alternatively, is there some breakdown or divergence from constant-stress drop scaling over some magnitude range? Assumptions based on scaling are fundamental to our understanding of earthquake rupture physics and also to seismic hazard and seismic monitoring. Unfortunately, in the decades since *Brune* [1970] and *Madariaga* [1976] proposed their simple circular source models for calculating source dimension and stress drop from frequency amplitude spectra, it has become clear that resolving the source process from attenuation and other propagation effects in real data is far from straightforward. Many studies have been published finding inconsistent results; for example, see *Abercrombie and Rice* [2005], *Mayeda et al*. [2005], and *Kwiatek et al*. [2011] for reviews. Measurement uncertainties essentially increase with frequency as propagation effects become stronger and less well known. Seismic moment is relatively well resolved as it is measured at relatively low frequencies. Stress drop depends on the cube of the corner frequency, and so it is less certain both because it is a higher frequency measurement and because of the effects of cubing the uncertainties. Seismic energy does not suffer from this problem, but it is an even higher frequency measurement and so is at least as problematic.

[3] The fundamental problem in resolving earthquake source scaling is that it requires comparing low- and high-frequency measurements, and differences could result from the source, attenuation, and site effects [e.g., *Abercrombie and Leary*, 1993; *Ide et al*., 2003], or data quality (e.g., bandwidth, *Ide and Beroza* [2001]; *Viegas et al*. [2010]). In addition, in any one study, large earthquakes radiating maximum energy nearest to the low-frequency bandwidth limit are compared to small ones radiating maximum energy near to the high-frequency bandwidth limit, making systematic fitting biases possible. Attempting to resolve the controversy is not helped by the fact that different studies use different approaches to calculate the spectra, extract and model the source, and then calculate the source parameters from the measurements. In this study, I focus on comparison of direct and coda wave measurements, on the source model used for fitting, and on whether there is any difference between measurements when an earthquake is the larger or smaller one in a spectral ratio.

[4] Two principal methods are used to extract source information from seismograms. One is to model the propagation effects or to perform a combined inversion for both source and propagation effects, for example, *Prejean and Ellsworth* [2001], *Ide et al*. [2003], *Edwards et al*. [2008], and *Oth et al*. [2011]. The former approach was followed by *Abercrombie* [1995], but in that instance the attenuation and site effects were minimized by the deep location of the borehole seismometer.

[5] A popular alternative is the spectral ratio or empirical Green's function (EGF) method in which two collocated earthquakes of different magnitudes are assumed to experience identical propagation and site effects and so can be used to cancel out these effects empirically. This method varies from a simple frequency spectral ratio [e.g., *Izutani and Kanamori*, 2001] to calculating source time functions [e.g., *Mori and Frankel*, 1990; *Mori et al*., 2003], inversions using clustered events [e.g., *Ide et al*., 2003], and stacking of many events with varying spacing [e.g., *Shearer et al*., 2006; *Allmann and Shearer*, 2007].

[6] In addition to these studies and many others that use direct *P* and *S* waves, a growing number of studies have used coda waves to extract earthquake source information. Although coda waves have experienced much more attenuation, they have been shown to provide stable, spatially averaged results [e.g., *Mayeda et al*., 2003; *Mayeda et al*., 2007], which is important given that the number of stations available in source studies of smaller earthquakes is often very limited.

[7] There has been relatively little work on quantifying the uncertainties in source parameter results. Most studies simply measure the inter-station variation but take no account of modeling assumptions, fitting constraints, limited-frequency bandwidth, etc. Others simply assume a factor of 10. *Sonley and Abercrombie* [2006] and *Edwards et al*. [2008] both demonstrated that there are significant trade-offs between attenuation and source effects when attempting to resolve both from recorded spectra. When using an EGF approach, the main question is how well the propagation effects cancel out. *Mori and Frankel* [1990] and *Viegas et al*. [2010] showed that the pulse width increases (and corner frequency decreases) by 30% or more as spacing between earthquakes exceeds 400 m. *Kane et al*. [2011] found that uncertainties from using different EGFs (~30% in stress drop) were larger than between stations for closely located stations. *Prieto et al*. [2006] investigated using multiple EGFs to increase the frequency bandwidth. For coda waves, *Mayeda et al*. [2007] found that the results were significantly less affected by location and focal mechanism differences between the pairs of earthquakes used to calculate spectral ratios. *Prieto et al*. [2007] investigated the uncertainties from simply calculating the amplitude spectrum.

[8] The EGF approach has been used to calculate source parameters of both the collocated earthquakes in a pair [e.g., *Mayeda and Malagnini*, 2009a, 2009b] and of groups of closely located events [e.g., *Shearer et al*., 2006; *Allmann and Shearer*, 2007]. *Hough and Dreger* [1995] and *Viegas et al*. [2010] compared the uncertainties in source parameters for the small and large earthquakes in pairs and found that those of the smaller earthquake are significantly less well resolved. This probably is partially because of limited signal at the higher frequencies needed to resolve the smaller source, but it also makes sense when the basis for the EGF assumption is considered. For earthquakes with collocated centroids (or hypocenters), each point of the small earthquake source is within one fault length of the large earthquake, but each point on the large earthquake source is not within one fault length of the small earthquake. Thus, it is possible that the source parameters of the smaller earthquake cannot be resolved well from spectral ratios.

[9] There are many different ways of calculating stress drop (Δ*σ*), making it hard to compare studies directly. Almost all spectral source studies calculate a corner frequency, and so by using this measured parameter I can calculate a consistent stress drop for comparison. Stress drops calculated using the *Brune* [1970], *Sato and Hirasawa* [1973], and *Madariaga* [1976] models are based on the product of the seismic moment (*M*_{0}) and the cube of the corner frequency (*f*_{c}), and the *Eshelby* [1957] solution for a circular crack. They use different constants, but it is simple to account for the differences. Apparent stress, which is proportional to the ratio of energy and seismic moment, is calculated in two different ways: by integrating the velocity-squared source spectrum to estimate energy [e.g., *Abercrombie*, 1995] or by simply extrapolating from the corner frequency measurement using the relationships between moment, corner frequency, and energy for a model spectrum [e.g., *Mayeda and Malagnini*, 2009a, 2009b]. As long as the corner frequency values are calculated (and they usually are to apply model corrections for limited bandwidth [e.g., *Ide and Beroza*, 2001]) then it is possible to use them to obtain a corresponding stress drop to compare directly with the other studies. In this paper, I recalculate all mentioned stress drops (unless otherwise stated) using published corner frequency values and the *Madariaga* [1976] model, assuming *Eshelby* [1957]:

where *k* is 0.21 for *S* waves, and *β* is the *S* wave velocity.

[10] In this study, I calculate source parameters for aftershocks of the Wells, Nevada, earthquake using direct waves and compare them to published results using coda waves [*Mayeda and Malagnini*, 2009a]. I begin by refitting the coda wave spectral ratios using the same fitting approaches I use for direct waves to investigate fitting consistencies. I then calculate direct wave parameters using the same earthquake pairs as coda waves (main shock divided by each of six large aftershocks), and lastly I calculate direct wave parameters using smaller EGFs for the six large aftershocks. To quantify uncertainties, I develop further the approach of *Viegas et al*. [2010] to select only spectral ratios that meet clear objective criteria and then use a grid search around the preferred fit. I find that the source spectral ratios from the coda and direct waves are similar for the same pairs of earthquakes and that differences in spectral modeling approaches produce results within calculated uncertainties. I also find that the source spectra and corresponding source parameters depend systematically on whether the earthquake of interest is the larger or smaller within the spectral ratio.