## Refined Determination of Epicenters (Mineral, VA)

## Contact

## Description

This catalog of earthquake source parameters was derived by the USGS National Earthquake Information Center (NEIC) as part of a relocation study of the Mw 5.8 Mineral, Virginia earthquake of August 23, 2011 and its aftershocks using a calibrated multiple event relocation method (McNamara et al., 2014). The purpose of this relocation analysis is to produce a set of reference hypocenters (with associated phase arrival times and empirical reading errors) for this source region that are especially well constrained, subject to minimal bias from unknown velocity structure and having realistic estimates of uncertainty based on the measured statistical properties of the data set.

## Method

Refined estimates of the epicenters, focal depths and origin times of earthquakes in the Mineral sequence were estimated using a method based on the Hypocentroidal Decomposition (HD) algorithm introduced by Jordan and Sverdrup (1981), but extensively developed for application in calibrated relocation studies, i.e., relocation studies that are specialized to provide minimally biased estimates of hypocentral parameters and realistic estimates of their uncertainties. The program MLOC implements this method, which is described here.

## Data

The data set of seismic phase arrival times used in this analysis includes picks from temporary seismograph stations that were installed by several research groups around the epicentral area; these picks were made by NEIC staff. Phase arrival times from permanent seismic stations from the ComCat (https://earthquake.usgs.gov/earthquakes/map/) and the bulletin of the International Seismological Centre (ISC) were also used. Arrival time picks from the ISC were not reviewed, but all arrival time data was subject to evaluation for outlier readings using empirically-derived estimates of reading error, as described in the relocation methodology, above.

## Velocity Model

The global 1-D travel time model ak135 (Kennett et al., 1995) fits the Mineral data well at local and near-regional distances, but the predicted travel times of direct crustal phases (Pg, Sg) in the distance range 40-60 km are a little late. Travel times for teleseismic phases were calculated with ak135 but for local and regional phases we calculated theoretical travel times with a model very similar to ak135 but with slightly higher velocities in the crust.

Depth (km) | Vp (km/s) | Vs (km/s) |
---|---|---|

0-20 | 5.95 | 3.60 |

20-37 | 6.45 | 3.85 |

>37 | 8.10 | 4.5 |

The HD algorithm used for the relocation analysis is limited in the number of events that can be relocated simultaneously by rapidly increasing computational time. The practical limit is ~200 events, about half the number of events in the Mineral sequence that were recorded well enough to be relocated with high accuracy. We therefore divided the sequence chronologically into three subclusters, which were relocated independently. Calibration of the Mineral subclusters was accomplished through the method of direct calibration, in which the hypocentroid, which establishes the location of the cluster in absolute terms, is estimated from arrival time readings at short epicentral distances to minimize the biasing effect of unknown velocity structure. For the Mineral subclusters the distance limit for data used to estimate the hypocentroid was 0.6°. Basic parameters for the three subclusters, including the number of P and S readings used to estimate the four hypocentral parameters of the hypocentroid are given in the table.

# Events | Mag Range | Dates (2013) | # Data | |
---|---|---|---|---|

minerala | 135 | 5.8-0.1 | 2013/8/23 - 2013/9/5 | 2610 |

mineralb | 148 | 3.0-0.2 | 2013/9/5 - 2013/10/31 | 3357 |

mineralc | 114 | 3.8-1.8 | 2013/11/2 - 2012/5/2 | 2085 |

## Calibrated Relocation of the Mineral Cluster

The HD algorithm used for the relocation analysis is limited in the number of events that can be relocated simultaneously by rapidly increasing computational time. The practical limit is ~200 events, about half the number of events in the Mineral sequence that were recorded well enough to be relocated with high accuracy. We therefore divided the sequence chronologically into three subclusters, which were relocated independently. Calibration of the Mineral subclusters was accomplished through the method of direct calibration, in which the hypocentroid, which establishes the location of the cluster in absolute terms, is estimated from arrival time readings at short epicentral distances to minimize the biasing effect of unknown velocity structure. For the Mineral subclusters the distance limit for data used to estimate the hypocentroid was 0.6°. Basic parameters for the three subclusters, including the number of P and S readings used to estimate the four hypocentral parameters of the hypocentroid are given in the table.

# Events | Mag Range | Dates (2013) | # Data | |
---|---|---|---|---|

minerala | 135 | 5.8-0.1 | 2013/8/23 - 2013/9/5 | 2610 |

mineralb | 148 | 3.0-0.2 | 2013/9/5 - 2013/10/31 | 3357 |

mineralc | 114 | 3.8-1.8 | 2013/11/2 - 2012/5/2 | 2085 |

Estimates of relative locations (cluster vectors) in each subcluster were determined from arrival time data of stations at less than 10° epicentral distance. Only a few events in the cluster have significant amounts of data at greater distance, and we found that using those data introduced bias in the location of the mainshock epicenter, shifting it 1-2 km to the west, away from the aftershock pattern. The number of readings used to estimate the cluster vectors in each subcluster is slightly larger than the number of data used to estimate the hypocentroid.

The hypocentroid of each subcluster was estimated with readings out to an epicentral distance of 0.6°, to minimize the biasing effect of unmodeled crustal velocity variations. The calibration level of the hypocentroids of the subclusters is 0.1-0.2 km. The full uncertainty for hypocentral parameters for any given event is obtained by adding the uncertainty of the cluster vector for that event to the uncertainty of the appropriate hypocentroid. The full epicentral uncertainty, characterized by the longer semi-axis of the 90% confidence ellipse, is less than 1.5 km for 393 events out of 397 that were relocated.

Focal depths for all events were estimated during early runs with free depth solutions then held fixed in later relocations. Depths range from 0-11 km with a major peak between 5-7 km and a lesser one around 3 km, and with uncertainties of 0.3 to 3.8 km (median uncertainty 0.6 km).

Because the focal depths and origin times of the Mineral subclusters are considered to be calibrated in addition to the epicenters, the calibration class is “CH” and 393 events are classified as CH01 or better.

## Empirical Reading Error

The concept of "reading error" in earthquake location is normally understood as an estimate of the uncertainty of the reading of the arrival time ("pick") of a specific seismic phase on the seismogram of a specific earthquake. Seismic analysts rarely provide their own estimate of that uncertainty beyond a qualitative characterization as "emergent" or "impulsive", and earthquake location codes that employ a quantitative estimate of reading error, e.g., for inverse weighting, normally use an ad hoc value based on phase type. It is very important to realize that estimates of hypocenter uncertainty in any earthquake location algorithm depend on the accuracy of the uncertainties assumed for the data, as well as the proper treatment of other sources of uncertainty in the estimation process. This is a significant weakness of most earthquake location analyses.

“Empirical reading error” is a related concept, based on multiple event relocation, i.e., location analysis of a clustered group of earthquakes. The specific implementation discussed here is the one developed for use in the program mloc that is based on the Hypocentroidal Decomposition (HD) algorithm of Jordan and Sverdrup (1981). Seismic stations often observe the same seismic phase for multiple events in such a cluster. The resulting multiple observations of the same "station-phase" provide an opportunity to carry out a statistical analysis which leads to an estimate of the uncertainty of those readings that is based on the readings themselves, thus "empirical". It would be more correct to refer to this as an "empirical reading uncertainty" or even “empirical reading consistency”, but we follow the traditional seismological terminology. It is also important to note that this concept of empirical reading error includes contributions to the scatter of readings beyond reading error per se, i.e., the ability of the analyst to specify the “correct” time of onset time of a seismic phase arrival. For example it also absorbs differences in travel time to a station through a heterogeneous Earth from events that are not exactly co-located, as well as scatter arising from the different philosophies of arrival time picking used by different analysts, differences caused by picking from different channels or instrument responses, changes in station equipment, minor changes in instrument location, irregularities in timing systems, differences in the precision to which picks are reported, etc..

Empirical reading errors are estimated as the spread of the distribution of travel time residuals for a given station-phase (for example, the Pn phase at station TUC) for a specific cluster of earthquakes whose differences in location are typically, but not necessarily, small compared to the separation of the cluster and the station. The number of samples can range from two to several hundred. The analysis is done on the set of residuals obtained by subtracting a theoretical arrival time, based on some travel time model and the current hypocenter of the event, from each arrival time observation. Thus each arrival time reading of a given station-phase is assigned the same empirical reading error. Although this obviously falls short of the ideal of having a reliable estimate of the uncertainty of each reading, it is a significant improvement over the traditional methods for handling uncertainties in arrival time data. Because the arrival time readings are weighted inversely to their empirical reading errors in the location algorithm, the specification of reading errors has a major impact on the estimated hypocenters and their uncertainties.

The estimate of spread of the residuals must be done with a robust estimator, i.e., one that is not sensitive to outliers, which are very common in arrival time data sets. The familiar statistic standard deviation is a very non-robust measure of spread. We employ the estimator Sn proposed by Croux and Rousseeuw (1992). Note that this has nothing to do with the seismic phase Sn. This measure of scale or spread has three desirable properties, 1) it requires no assumptions about the nature of the underlying distribution, 2) it requires no estimate of the central tendency (e.g., the mean or median) of the distribution, and 3) it reduces to the standard deviation if applied to a Gaussian distribution.

An important aspect of the relocation process consists of multiple cycles in which the current estimates of empirical reading error are used to identify outlier readings, which are then flagged so that they will not be used in subsequent relocations. In the following relocation, the new estimates of empirical reading errors will tend to be smaller because of the filtering of outliers and improvement in the locations of the clustered events. Therefore the process of identifying outliers is iterative and it must be repeated until convergence. In this context, "convergence" means that the distribution of residuals for a given station-phase is consistent with the current estimate of spread. As outlier readings are flagged, the distribution is expected to evolve toward a normal distribution with standard deviation σ equal to the empirical reading error. We generally continue this "cleaning" process until all readings used in the relocation are within 3σ of the mean for that station-phase, where σ is the current estimate of empirical reading error for the relevant station-phase. The procedures used to construct confidence ellipses and other estimates of hypocentral parameter uncertainty in mloc (and most other location algorithms) are based on the assumption that the residuals have a normal distribution. In the presence of outlier readings, the resulting uncertainty estimates will be biased.

Because of inverse weighting in the HD algorithm, it is necessary to impose minimum allowed values for empirical reading errors to prevent unrealistically small estimates from arising when the sample size is small and values are very near one another, or identical. We normally use 0.15 s as a minimum which is generally appropriate for the data sets obtained from standard earthquake catalogs, but smaller values can be permitted in special circumstances. For singlet station-phase arrival time data (only one observation) default values that are typical of many earthquake location algorithms (e.g., 0.5 s for teleseismic P) are applied. Singlet readings make no contribution to the estimate of relative locations in the HD algorithm, but they can be used to estimate the hypocentroid, in which case the reasonableness of the default value of empirical reading error must be evaluated for the particular data set in order to have confidence in the derived hypocentral parameter uncertainties.

In summary, the use of empirical reading errors in mloc allows us to treat the derived hypocentral uncertainties with considerable confidence. Although any multiple event relocation method could implement a similar analysis, we are not aware of any that do. Single event location methods are inherently handicapped by the lack of any way to investigate data uncertainty in a statistically robust way, although careful attention to the arrival time picking process can partially compensate. Failure to adequately characterize data uncertainties in the hypocenter estimation process leads to bias in the derived parameters and their uncertainties.

## Earthquake Location Accuracy

Here we provide a description of a system of codes used to characterize the accuracy of earthquake locations determined with mloc, a multiple event relocation program based on the Hypocentroidal Decomposition algorithm of Jordan and Sverdrup (1981), but extensively developed for application in calibrated relocation studies, i.e., relocation studies that are specialized to provide minimally biased estimates of hypocentral parameters and realistic estimates of their uncertainties.

The classification system described here is an extension of the well-known “GTX” system (e.g., Bondar et al., 2004). The primary extension is to generalize the single “class” of the GTX system to four classes that allow an accuracy code to be assigned to any hypocenter. A great advantage of this extra complexity is the ability to distinguish between the different ways in which constraints on location accuracy may have been derived. Moreover we extend the GTX system to carry information about the accuracy of the hypocentral parameters focal depth and origin time, rather than the epicenter alone. This new classification system takes its name “GCNU” from the first letters of the names of the four classes:

- G: ground truth
- C: calibrated
- N: network geometry criteria
- U: uncalibrated

The general form of a location accuracy code in the GCNU system is four characters, of which the first is one of the letters indicating accuracy class, as listed above. The second character carries information on which hypocentral parameters can be considered calibrated. The third and fourth characters are numeric and together provide a length scale in km for the accuracy of the epicenter (equivalent to the “X” term in the GTX system. There are several exceptions to these general rules, as noted below.

## Ground Truth: the G Class

This class has only two instances, both of which have only three characters. The GT0 nomenclature is reserved for traditional (or literal) ground truth, events for which all four hypocenter coordinates are known a priori at levels of accuracy which are negligible for the purpose at hand. For epicenter and focal depth these uncertainties are typically less than about a hundred meters. At a typical crustal P velocity of 6 km/s 100 meters represents 0.015 s, so origin time should be known to several hundredths of a second in order to be compatible. These limits may not be suitable for some engineering purposes or specialized source studies. The designation “ground truth” has traditionally been reserved for nuclear tests and carefully engineered chemical explosions. It is possible to obtain this level of accuracy with natural seismic sources that are especially heavily instrumented at close range but it is still preferable to use the C class in such cases.

There is a need for a somewhat relaxed ground truth category, because even though the hypocentral parameters of a man-made explosion may be given a priori, the uncertainties may not meet the stricter requirements given above. This may be the case because of inadequate record keeping or the difficulty in carrying out suitably accurate surveying or timing prior to the availability of GPS technology. The GT1 category is meant for such cases. This still implies near-certain knowledge of location within a kilometer or so, with comparable uncertainty in origin time (several tenths of a second). Industrial explosions and even some nuclear tests may not meet this standard. Such events ought to be treated in the calibrated (C) class of events, as discussed below, rather than being assigned ground truth status with inflated scale lengths.

No length scale greater than 1 should be used in this class. If the uncertainty is greater than that it is not ground truth.

## Ground Truth: the C Class

In contrast to the ground truth class, where the concern is primarily with the scale of random error in the hypocentral parameters, the class of calibrated events is dominated by concern that the estimation process which has been used to determine hypocentral parameters may have introduced significant bias. Therefore we are very much concerned about minimizing bias and understanding which hypocentral parameters may be treated as effectively bias-free. Obviously we also desire to estimate the hypocentral parameters such that the formal uncertainties (driven by uncertainty in the data), usually expressed in Gaussian terms, are as small as possible; this will be handled similarly to the “X” in the GTX formulation, discussed below in the section “Scale Length”.

A very important point about the calibrated class of events, is that it includes only events for which the epicenter (at least) has been determined in such a way as to minimize bias. Although a bit unsatisfying in a logical sense, this policy reflects the reality that the seismological community overwhelmingly thinks of ground truth or GT events (using the popular current nomenclature) as referring only to the epicenter. The other important point to be made is that this class requires an actual location analysis, not just the application of some set of network geometry criteria such as those presented by Bondar et al. (2004). In other words, application of network geometry criteria to estimate location accuracy may be a precursor to calibration analysis, but is not a substitute for it.

Given that we do not know the Earth’s velocity structure to sufficient accuracy, the only way to reduce bias for an event that was not engineered is to keep path lengths through the unknown Earth structure as short as possible. In other words only near-source data should be employed for estimating calibrated parameters. “Near source data” is not restricted to seismological stations at short epicentral distance, although that is by far the most common case. Mapped surface faulting, treated with all due geological sensitivity, may serve as near source data for the purpose of constraining an epicenter, as may InSAR or other types of remote sensing analyses, since the ultimate signal (e.g., surface deformation) is not subject to bias from unknown Earth structure. InSAR analysis, through determination of the distribution of rupture on a fault plane, may be used to reduce bias in focal depth. Waveform modeling (even at regional or teleseismic distances) may similarly provide useful near-source constraint on focal depth through analysis of the interference of direct and near-source surface-reflected phases.

Unfortunately, there is no methodology for obtaining usefully-calibrated hypocenters for deep earthquakes because every available data type must propagate through an excessive volume of material with insufficiently well-known velocity. The exact definition of “deep” in this context must be evaluated on a case-by-case basis, but it probably includes any event deeper than about 100 km. If uncertainties in velocity structure (and their effect on raypath geometry) are honestly propagated into the uncertainties of the derived location parameters, then the issue will be resolved by the increasing uncertainty of the location, leaving aside the question of bias.

The nomenclature for the calibrated class is based on the following practical considerations about the calibration of the various hypocentral parameters:

## Epicenter

Bias in epicentral coordinates can be minimized by means of seismological analysis (typically a location analysis), as well as by other means, including geological and remote-sensing analyses and a priori knowledge of human-engineered sources that may be too weak for ground truth status. It is quite common for the epicenter to be the only hypocentral parameter of an event that can be usefully constrained with minimal bias.

## Depth

Focal depth is more difficult to constrain than the epicentral coordinates. In the location analysis, it requires data at epicentral distances comparable to the focal depth itself, a few tens of kilometers for crustal events, a much stricter requirement than for the epicenter, which can be usefully constrained with stations 100 km or so away. This distance requirement can be ignored for waveform modeling, however, as well as for analyses of teleseismic depth phases, most famously emphasized by the EHB algorithm (Engdahl et al., 1998). Therefore the minimization of bias in focal depth can be part of the general location analysis, coupled with the estimate of a minimally-biased epicenter, or it can be constrained independently, even when the epicenter may be uncalibrated.

## Origin Time

Calibration of origin time is only fully possible when both the epicenter and focal depth can be calibrated. Unless it has been specified a priori for a human-engineered event it must be estimated from seismic arrival time data at the shortest possible epicentral distances, and any bias in the location parameters would propagate into origin time. It is quite common, however, to encounter case where the epicenter and origin time of an event can be constrained with near-source data (not necessarily for the event in question but through linkage to other events in a multiple event analysis), but the focal depth of the event cannot be usefully constrained, other than as an average depth for a cluster of events, some of which have well-constrained depths, or through regional seismotectonic considerations. In this case the origin time itself cannot be considered to be unbiased, but since it is reliably coupled to the assumed focal depth, the combined hypocentral coordinates can still provide valuable information on empirical travel times from a specific point in the Earth.

Given the above considerations there are three cases that need to be distinguished in the calibrated class of the nomenclature. In the following table, the asterisk indicates parameters that have been calibrated in some manner:

Epicenter | Focal Depth | Origin Time | |
---|---|---|---|

CH | * | * | * |

CT | * | * | |

CF | * | * | |

CE | * |

CH (“H” refers to hypocenter). All four hypocentral coordinates have either been inferred by means that yield minimally-biased estimates or constrained a priori (as in some human-engineered events that don’t quite qualify for GT1 status or better).

CT (“T” refers to travel time). Epicenter has been calibrated; depth has been fixed at some assumed value (e.g., the average depth of nearby events with constrained depths); the estimate of origin time is based on local-distance data, but relative to an uncalibrated depth. Neither the focal depth not origin time can be considered calibrated in themselves but the combination can be used to estimate empirical travel times from the specific point in the Earth. Such events are not quite as valuable as CH events but still have considerable value as input to model-building exercises or as validation events.

CF (“F” refers to focal depth). Epicenter and focal depth have been calibrated, but not origin time. An example could be an inSAR location for an event and depth calibrated either by an additional analysis of surface deformation to infer distributed displacement on a fault surface, or through waveform analysis. The estimate of origin time is not based on near-source readings. These events can be used in validation exercises where their epicenters are compared with locations done with ray-traced travel-times through a model.

CE (“E” refers to epicenter). The epicenter is calibrated. As with the CT class, depth has been fixed at some assumed (albeit reasonable) value. If the calibration of the epicenter has not been based on near-source seismic data (e.g., an InSAR location), the estimate of origin time must be based on regional or teleseismic arrivals and therefore cannot be considered calibrated, nor can it be used for estimate of empirical travel times. These events can be used in validation exercises where their epicenters are compared with locations done with ray-traced travel-times through a model.

## Network Geometry Criteria: The N Class

Events in the N class are not considered to be calibrated in the sense defined here, but the arrival time data set has been processed with some network criteria (e.g., Bondar et al. (2004), but others are developing similar criteria for different source regions) based on simple metrics such as number of readings and distribution of reporting stations, in order to provide an estimate of epicentral accuracy that is expected to account for systematic location bias. The assumption here is that 1) the data do not permit a calibration analysis because there are insufficient near-source data, or 2) that such an analysis has simply not yet been done (i.e., a bulletin has simply been scanned for candidate calibration events). If a careful relocation analysis has been done to standards that can arguably justify classification as a calibrated event, the C class should be used.

NE (“E” from epicenter). The epicentral accuracy has been estimated with an appropriate network geometry criteria. Focal depth and origin time are uncalibrated. Many so-called “GT Catalogs” are dominated by events in this category. Requires a scale length.

NF (“F” from focal depth). As NE but focal depth is calibrated. Requires a scale length.

## Everything Else: The U Class

All seismic events that do not fit into one of the GT, C or N classifications are considered uncalibrated. That does not mean that none of the hypocentral coordinates are calibrated, only that the epicenter is not considered to be calibrated. The following classifications are defined:

UE (“E” from epicenter). No hypocentral parameters are calibrated but there is a credible estimate of epicentral accuracy from a location analysis (confidence ellipse), leaving aside the question of systematic location bias. Requires a scale length.

UF (“F” from focal depth). As UE, but focal depth is calibrated. The subset of events in the EHB catalog (Engdahl et al., 1998) that carries depth estimates based on analysis of teleseismic depth phases would fall into this category, as would any event that has been the subject of a waveform modeling exercise that solves for focal depth. Requires a scale length.

U (uncalibrated). Simply a dot on a map. No credible information is available on location accuracy, epicentral or otherwise. No scale length is used.

## Scale Length

With the exception of the “U” category all classifications should carry a scale length, equivalent to the “X” in the GTX formulation. The ground truth (GT) class categories are defined with specific scale lengths, which refer to the uncertainty in both the epicenter and focal depth.

For the Calibrated (C) and Network Geometry Criteria (N) classes the scale length is related to the uncertainty in epicenter only. For the CH class one would have to refer to a more detailed description of the data set to learn anything quantitative about the uncertainty in focal depth. The scale length is an integer, in kilometers, related to the uncertainty of the epicenter. Network geometry criteria always yield a single value for scale length. For the C class, as discussed above, there is no consensus about how the 2-dimensional uncertainty in an epicenter should be reduced to a single number. Three possibilities that seem reasonable when dealing with an ellipse with semi-minor axis a and semi-major axis b are:

- Nearest integer to the semi-major axis length of the confidence ellipse: nint(b).
- Nearest integer to the average of the two semi-axis lengths: nint((a+b)/2).
- Nearest integer to the radius of the circle with the same area as the ellipse: sqrt(ab).

For a circular confidence region all three methods are equal. As the ellipticity of the confidence region increases, there will be substantial differences between the different scale lengths, but the first method will always yield the largest value. For a confidence ellipse with semi-axis lengths 1 and 5 km, for example, the scale length calculated with the three methods would be 5, 3, and 2 km, respectively. In analyses using mloc we use the first method, with the largest estimate of uncertainty.

Scale lengths larger than 9 are permitted, but they have rapidly diminishing value in the current research environment. When the scale length of confidence ellipses moves into double digits, one ought to begin to worry about the legitimacy of the assumptions underlying the statistical analysis. Such events may better characterized by one of the uncalibrated categories.

## Confidence Levels

As Bondar et al. (2004) pointed out, it is necessary to specify the confidence level that has been used in determining epicentral uncertainties, e.g., as a subscript in the form “GT905” to indicate that the confidence ellipse was calculated for a 90% confidence level. The concept of scale length is meaningless without it. Over more than a decade since the proposal was made, compliance on this point seems to be casual at best. It is admittedly awkward to include the subscript in computer output, and since the nomenclature is primarily intended to be carried in digital files it may be best to leave it out, but with the recommendation to clarify the issue in accompanying documentation. In the case of analyses done with mloc, the standard confidence level is 90%.

## References

- Bondar, I. K., Myers, S. C., Engdahl, E. R., & Bergman, E. A. (2004). Epicentre accuracy based on seismic network criteria. Geophysical Journal International, 156, 483–496. http://doi.org/10.1111/j.1365-246X.2004.02070.x
- Croux, C. and Rousseeuw, P.J. (1992). Time-efficient algorithms for two highly robust estimators of scale, Computational Statistics, 1, 411-428
- Engdahl, E. R., van der Hilst, R. D., & Buland, R. P. (1998). Global teleseismic earthquake relocation with improved travel times and procedures for depth determination. Bulletin of the Seismological Society of America, 88(3), 722–743
- Jordan, T.H. and K.A. Sverdrup (1981). Teleseismic location techniques and their application to earthquake clusters in the South-central Pacific, Bull. Seism. Soc. Am., 71, 1105-1130.
- McNamara, D.E., Benz, H.M., Herrmann, R.B., Bergman, E.A., Earle, P., Meltzer, A. Withers, M., and M. Chapman (2014). The Mw 5.8 Mineral, Virginia, earthquake of August 2011 and aftershock sequence: Constraints on earthquake source parameters and fault geometry, Bull. Seism. Soc. Am., 104, doi 10.1785/0120130058