Regional Attenuation Method Comparison for Northern California

Sean R. Ford, Douglas S. Dreger, Kevin Mayeda, William R. Walter (Lawrence Livermore National Laboratory), Luca Malagnini (Istituto Nazionale di Geofisca e Vulcanologia), and William S. Phillips (Los Alamos National Laboratory)

Introduction

Understanding of regional attenuation ($Q^{-1}$) can help with structure and tectonic interpretation, and correcting for the effects of attenuation can lead to better discrimination of small nuclear tests. Present threshold algorithms for event identification rely on $Q$ models that are derived differently, and the models can vary greatly for the same region. It is difficult to learn the cause of such discrepancies because the methods and parameterizations change for each analysis. In order to better understand the effects of different methods and parameterizations on $Q$ models, we implement four popular methods and one new method to measure $Q$ of the regional seismic phase, $Lg$ ($Q_{Lg}$), using a high-quality dataset from the Berkeley Digital Seismic Network (BDSN). With this knowledge, it will be possible to better assess the results of published attenuation studies, and future efforts can benefit from the outlined comprehensive analysis procedure.

Methods, Comparisons, and Sensitivity Tests

The dataset consists of 158 earthquakes recorded at 16 broadband (20 sps) three-component stations of the BDSN between 1992 and 2004. The wide distribution of data parameters allows for sensitivity testing to a given dataset. We calculate $Q_{Lg}$ by fitting the power-law model, $Q_0f^\alpha $, in Northern California using five different methods. The first two methods use the seismic coda to correct for the source effect. These methods can produce best-fit power-law parameters for specific stations. The last three methods use the spectral ratio technique to correct for source, and possibly site effects. These methods produce best-fit power-law parameters for specific interstation paths.

Figure 17.1: Spatial variability in the percent deviation from the method average power-law fit parameters ($Q_0f^\alpha $) for the (a) coda normalization method, (b) coda-source normalization method, (c) two-station method, (d) reverse two-station method, and (e) source-pair / receiver-pair method. The average models are $104f^{0.61}$, $172f^{0.57}$, $132f^{0.53}$, $121f^{0.52}$, $76f^{0.76}$, respectively
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\epsfig{file=sean06_1_1.ps, width=7.91cm}\end{center}\end{figure}

Coda normalization method (CNM)

CNM uses the coda as a proxy for the source and removes it from the $Lg$ spectrum (Aki, 1980; Yoshimoto et al., 1993). The amplitude is then least-squares fit as a function of distance in small frequency bands for each station, where the slope is related to path attenuation, $Q^{-1}$. $Q^{-1}$ at the center frequency of each band then reveals a power-law $Q$ model for each station.

Coda-source normalization method (CSM)

CSM uses the 1-D coda-source spectra previously calculated in the study by Mayeda et al. (2005) and removes it from the $Lg$ spectrum in small frequency bands (Walter et al., 2006). $Q^{-1}$ is calculated for each of these bands for each event-station path. In this application of CSM, all paths to a common station are fit to find a power-law $Q$ model for each station.

Comparison of CSM and CNM

Since both CNM and CSM give a result for each station, we compare these results by finding the percent deviation of each station from the average $Q_0f^\alpha $ produced for each method. Comparisons are mapped in Figure 17.1a-b when a solution was calculated for both methods. The average $Q_{Lg}$ model given by the CNM is 104$f^{ 0.61}$, while the CSM produces an average of 172$f^{ 0.57}$. The absolute difference in $Q_{Lg}$ models may be due to the absence of a site correction in the CSM. There is overall relative agreement in the two methods, with low $Q$ in the northern part of the study region and variable $Q$ in the Bay Area. This Bay Area variance may be due to paths crossing different tectonic regimes to reach these stations and forming an average fit. Stations MOD, FARB, and POTR appear to have a strong difference in measured $Q_0f^\alpha $. However, the fit to a power-law model is poor at frequencies $>$ 2 Hz, and the comparison for a power-law model may be flawed for these stations. Stations BKS and MHC are consistently lesser or greater, respectively, than the average for each method, but there is a large percent deviation for the two stations.

Two-station method (TSM)

TSM takes the spectral ratio of $Lg$ recorded at two different stations along the same narrow path from an event (Xie, 2002; Xie and Mitchell, 1990). We restricted the path to fall in an azimuthal window of 15$^{\circ }$The ratio removes the common source term and the amplitude is fit in the log domain so that the slope is $\alpha$ and the intercept is $Q$.

Figure 17.2: 95% confidence ellipses for the power-law model parameters for station PKD calculated by the (a) coda normalization, (b) coda-source normalization, (c) two-station, (d) reverse two-station, and (e) source-pair/receiver-pair methods obtained by the original parameterization (black), and by varying the spreading exponent (red), bandwidth (green), distance (dark blue), and time window (light blue). The small cross is the best-fitting parameter estimate. The coda-based methods are for station PKD, and the spectral ratio-based methods are for the path from MHC to POTR.
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Reverse two-station method (RTSM)

RTSM uses two TSM setups where an event is on either side of the station pair in a narrow azimuthal window (Chun et al., 1987; Fan and Lay, 2003). The two ratios are combined to remove the common source and site terms and the amplitude is fit in the log domain so that the slope is $\alpha$ and the intercept is $Q$.

Source-pair / receiver-pair method (SPRPM)

SPRPM is basically the RTSM with a relaxation on the narrow azimuthal window requirement (Shih et al., 1994).

Comparison of TSM, RTSM, and SPRPM

Since TSM, RTSM, and SPRPM give a result for interstation paths, we compare these results by finding the percent deviation of each interstation path from the average $Q_0f^\alpha $ produced for each method. Comparisons are mapped in Figure 17.1c-e when a solution was calculated for all methods. The average $Q_{Lg}$ model given by the TSM is 132$f^{ 0.53}$, by RTSM is 121$f^{ 0.52}$, and by SPRP is 76$f^{ 0.76}$. Values of $Q_0$ are fairly uniform with greater than average and lesser than average values consistent across each of the respective methods. A notable exception is the path from MHC to SAO, where the TSM calculates a greater than average $Q_0$ and the other methods find a less than average $Q_0$. The mean value of $Q_0$ and $\alpha$ for SPRPM are very low for the region, and the power-law exponent, $\alpha$, varies widely among all methods. This may be due to the variance in the spectral amplitudes, and robust methods of spectrum estimation may reduce the variance.

Sensitivity tests

We investigated how the choice of parameterization affects the results. In each test, only one parameter was varied, and $Q_0f^\alpha $ was calculated with each of the methods. The varied parameters were geometrical spreading dependence ($r^{0.5}$ to $r^{0.83}$), measurement bandwidth (0.25-4 Hz to 0.5-8 Hz), epicentral distance of the data (100-400 km to 100-700 km), and the $Lg$ window (2.6-3.5 km/s to 3.0-3.6 km/s). The range of parameterization was chosen based on the values used in previous studies.

All methods were affected by a change in spreading exponent, where there is a systematic increase in both $Q_0$ and $\alpha$ as the spreading exponent increases. Also, when more of the spectrum below 1 Hz is sampled, $\alpha$ can change significantly. The methods that use a maximum $Lg$ amplitude in the time domain to measure $Q_{Lg}$, CNM and SPRP, are less sensitive to $Lg$ window choice than the other methods. However, CNM is affected by epicentral distance, which may be due to the fixed time that the coda is sampled for all distances. The RTSM is the most robust and resistant to changes in parameterization.

In order to better visualize the sensitivity of the methods to varied parameterization, we produce 1-D power-law $Q$ models for a common station, PKD, for the coda-based methods (Figure 17.2a-b), and a common path, MHC to POTR, for the spectral ratio based methods (Figure 17.2c-e). Error in these 1-D fits is calculated and we produce 95% confidence ellipses for each of the power-law model parameters (Aster et al., 2005).

Estimates of the power-law parameters, $Q_0$ and $\alpha$, have a complex relationship with parameterization choice. The greatest variance in $Q_0$ is given by the CSM ($\sim $200-500), while there is a large variance in $\alpha$ calculated with the SPRPM ($\sim $0.6-1.8). However, for all methods (with the possible exception of the RTSM) the 95% confidence region is large, and the range of the parameter estimates is greater than is given by previous 1-D $Q_{Lg}$ studies, which often only present one choice of parameterization.

Conclusions

There is lateral variability in $Q_{Lg}$ at 1 Hz and the power-law dependence on frequency in Northern California. The spatial variability is similar to that found by Mayeda et al. (2005), where there is high attenuation in the northern region of the study area and variable attenuation in the Bay Area. Trends in calculated power-law parameters are similar among the methods investigated in this study, though there is large variability in the absolute values for $Q_0f^\alpha $.

The choice of spreading exponent, distance, and measurement window has a large influence on the best-fit power-law parameter estimates. Unless the parameterization choice can be constrained from a priori information, regional attenuation studies should search the entire solution space in order to report useful power-law $Q$ models.

Acknowledgements

This research is sponsored by the Department of Energy through the National Nuclear Security Administration, Office of Nonproliferation Research and Development, Office of Defense Nuclear Nonproliferation.

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