Energy and Rupture Dynamics are Different for1
Earthquakes on Mature Faults vs. Immature Faults2
Rebecca M. HarringtonKarlsruhe Institute of Technology
Geophysical Institute
Emily E. BrodskyUniversity of California, Santa Cruz
Dept. Earth Sciences
3
May 3, 20114
1
Abstract5
We compare radiated energy per unit moment, and the seismologically ob-6
servable fracture energy per unit moment between earthquakes on immature7
fault surfaces in Parkfield, California, and Mount St. Helens Volcano, and on8
a mature fault surface, namely the San Andreas fault near Parkfield, Califor-9
nia. A comparison between the two populations indicates that earthquakes on10
immature fault surfaces exhibit more self-similar behavior. Energy to moment11
ratios remain roughly constant for immature faults, and decrease with magni-12
tude for mature fault surfaces. The decrease in the parameters for the mature13
population results from an unchanging fault area for earthquakes with catalog14
magnitudes smaller that M 3.0 on the San Andreas fault. A comparison of15
energy-moment scaling with other studies suggests that the proposed differ-16
ences in energy scaling between mature and immature faults is consistent with17
previously published data sets from mature faults in California, and immature18
faults in deep mines. In addition, the observed energy values fall within the19
expected range. The difference in source parameter scaling between immature20
and mature populations suggests that ordinary scaling relationships relating21
moment and area and an assumed constant stress drop are not valid for every22
earthquake population individually.23
2
1 Introduction24
Seismically observable earthquake source parameters such as duration (τ), seismic25
moment (M0) and the radiated seismic energy (ER) provide the only direct clues26
as to how faults rupture [Wesnousky , 2006; Kanamori and Rivera, 2004; Choy and27
Kirby , 2004; Wyss and Brune, 1968]. Quantities such as the seismic moment, and the28
radiated seismic energy can be measured directly from the earthquake source spec-29
trum. Other quantities, such as static stress drop (∆σ), or the seismically observable30
fracture energy (EG) must be derived via theoretical relationships related to rupture31
models. The scaling of earthquake source parameters with size provides clues as to32
whether rupture processes are independent of earthquake size, i.e. are scale invari-33
ant. A number of observations suggest that some parameters, such as static stress34
drop and the ratio of radiated seismic energy to moment (ER
M0), do not vary with size,35
leading to assumptions of scale invariance over a large magnitude range [Kanamori36
and Brodsky , 2004; Kanamori and Anderson, 1975; Abercrombie, 1995; Ide et al.,37
2003; Imanishi and Ellsworth, 2006; Abercrombie and Rice, 2005; Prieto et al., 2004;38
Shearer et al., 2006].39
The accepted values of roughly constant static stress drop range over 2 to 3 orders40
of magnitude for small earthquakes, calling into question the assumption of constant41
source scaling. Observations of a decreasing ER
M0ratio for small-magnitude earthquakes42
also seem to contradict the constant scaling assumption [Abercrombie, 1995; Prejean43
and Ellsworth, 2001; Venkataraman and Kanamori , 2004a; Mayeda et al., 2007]. The44
difficulties in accurately measuring radiated seismic energy for small earthquakes45
unfortunately makes the scaling issue even more ambiguous [Venkataraman et al.,46
2006b]. Certainly some of the scatter in measured energy values results from difficul-47
3
ties in accurately correcting for attenuation, as well as difficulties in accounting for48
all of the energy due to bandwidth limitations [Venkataraman et al., 2006a; Ide and49
Beroza, 2001]. However, some work suggests that heterogeneous properties in the50
faulting environment might influence the scaling of parameters such as ∆σ, and ER
M0.51
Such observations suggest that geophysical factors might also account for some of the52
range in scaling observations [Venkataraman and Kanamori , 2004a; Choy and Kirby ,53
2004]. One such physical factor which may contribute to the range in source obser-54
vations is fault surface roughness, which is modulated by fault maturity (cumulative55
displacement).56
Recent studies using laser techniques to image fault surfaces indicate that they57
become smoother with more cumulative displacement. Observations at the cm - km58
scale suggest that, similar to field observations at the km - 100 km scale, immature59
fault surfaces with less cumulative displacement are rougher than more mature faults60
(with more cumulative displacement) [Sagy et al., 2007; Sagy and Brodsky , 2009;61
Wesnousky , 1988]. The results of Sagy et al. [2007]; Sagy and Brodsky [2009] suggest62
that immature fault surfaces may be assumed self-affine in the direction of slip, i.e.63
surface roughness is scale invariant. However, mature fault surfaces become more64
smooth at smaller scales, and develop quasi-elliptical geometric asperities on the65
order of 10’s of meters for faults with cumulative slip on the order of 100’s of meters66
[Sagy and Brodsky , 2009]. Given the physical differences in surface geometry between67
mature and immature faults, it follows that earthquakes might rupture differently,68
depending on the faulting environment in which they occur.69
Ohnaka [2003] suggests that there are observable differences in dynamic source70
properties between rough and smooth faults in laboratory experiments. He develops71
4
a constitutive relation dependent on fault friction and seismologically observable frac-72
ture energy that governs fault rupture. The relation relates seismologically observable73
fracture energy to fault surface roughness, suggesting that the energy partitioning in74
an earthquake (and therefore the amount of radiated energy) depends on surface75
roughness as well. In this study, we examine whether dynamic properties between76
rough and smooth faults differ on a larger scale, by using a new compilation data set77
of earthquake recordings. The assumption that roughness correlates with maturity78
implies that the comparison between rough and smooth faults is also a comparison79
of how maturity affects the dynamic properties of earthquakes.80
Harrington and Brodsky [2009] showed that a group of earthquakes on the San81
Andreas fault have a duration-seismic moment scaling suggestive of an approximately82
constant fault area. Here we present new evidence using an expanded data set to sug-83
gest that fault maturity affects the seismically radiated energy (ER), the energy radi-84
ated away from the fault in an earthquake [Kanamori and Heaton, 2000; Abercrombie85
and Rice, 2005; Kanamori and Brodsky , 2004]. By comparing the energy-moment86
ratios (ER
M0, EG
M0) of earthquakes on the San Andreas fault to those on secondary faults87
in the Parkfield area, and on new fault surfaces in the Mount St. Helens edifice,88
we will show that the earthquakes on immature faults obey constant energy scaling,89
while those on mature faults exhibit a size-dependent energy scaling. In addition,90
we will present our results in the context of other studies, to show the more general91
features of our observations. Additionally, we will offer an interpretation of energy92
partitioning in the context of our observed energy scaling, as well as other studies of93
fault roughness and damage.94
The paper begins with the directly measured parameters: seismic moment and cor-95
5
ner frequency. After reviewing the methodology and geological context, we present96
measurements showing distinct trends in corner frequency and moment for mature and97
immature faults. We then proceed to interpret these results in terms of energy. We re-98
view a slip-weakening rupture model to infer energy terms from the observations, and99
then show how the ratios of radiated seismic energy to moment, and seismologically100
observable fracture energy to moment (ER
M0, EG
M0) for the mature population deviate101
from the constant scaling expected from the model based considerations. Finally,102
we compare our observations with studies which measure similar source properties in103
other locations, in order to show that they are a general feature.104
2 Earthquake Populations105
Making a comparison of dynamic source parameters between mature and immature106
earthquake populations requires a definition of maturity. A number of observations107
suggest that differences in dynamic source properties exist between earthquakes oc-108
curring on plate boundaries (interplate), and within plates (intraplate). Such com-109
parisons are based on proximity to the plate boundary, and do not directly relate110
source scaling to cumulative slip [Choy and Kirby , 2004; Choy et al., 2006; Kanamori111
and Anderson, 1975; Zhuo and Kanamori , 1987]. For example, subduction zone112
earthquakes are typically observed to have lower stress drop and radiated energy113
values [Allmann and Shearer , 2009; Venkataraman and Kanamori , 2004a; Choy and114
Boatwright , 2009]. The favored interpretation suggests that subduction zone events115
differ from other types of earthquakes because they occur in a mature faulting envi-116
ronment where the largest amounts of cumulative slip have accumulated. For faults117
6
which are not exposed at the Earth’s surface, the cumulative displacement on the118
fault surface serves as a measurable proxy for fault roughness. We therefore use it119
here to distinguish mature and immature faults, in order to make a comparison of120
similar sized earthquakes in both environments.121
Observational evidence from roughness measurements of exposed fault surfaces122
suggest that the transitional value of cumulative displacement separating immature123
from mature faults is ∼100-1000 m [Sagy and Brodsky , 2009; Sagy et al., 2007; Savage124
and Brodsky , 2011]. Observations of both mapped fault trace features, and appar-125
ent stress measurements (or the radiated energy-moment ratio) suggest that mature126
faults constitute those accommodating a few kilometers or more of slip [Choy and127
Kirby , 2004; Choy et al., 2006; Malagnini et al., 2010; Wesnousky , 1988]. Therefore,128
we classify the events in our data set occurring on faults with displacement values129
less than 1 km as immature. The study analyzes seismograms from two areas: the130
Parkfield region, and the edifice of Mount St. Helens volcano. Both areas have dense131
enough instrumentation that direct analysis of the waveforms of small earthquakes132
is possible, and all earthquakes in this study have magnitudes less than 3.7. The133
paper separates these earthquakes into two populations: (1) earthquakes on a mature134
fault (namely, the San Andreas fault near Parkfield), and (2) earthquakes on imma-135
ture faults which meet the < 1 km displacement criteria (namely, secondary faults in136
Parkfield and faults in the edifice of Mount St. Helens).137
7
2.1 Mature Fault Earthquakes: San Andreas Fault at Park-138
field139
Nearly all of the earthquakes in Parkfield listed here are contained in the relocated140
catalog of Thurber et al. [2006] and/or the ANSS catalog (for events in 2006 and 2007141
post-dating publication of the relocated catalog) [Thurber et al., 2006; Waldhauser142
and Ellsworth, 2000, 2002] (Table 1). We use waveforms from the High Resolution143
Seismic Network (HRSN) stations (Figure 1). The HRSN consists of 13 3-component144
borehole stations with depths ranging from 63-572 m, an average depth of 236 m,145
and a sampling rate of 250 sps. The high-quality data permits unusually clear source146
spectral estimation for small events.147
The San Andreas fault near Parkfield is definitively mature with cumulative dis-148
placement values in the range of 300 km [Revenaugh and Reasoner , 1997]. Earth-149
quakes that occur on the active fault strand may therefore be classified in the mature150
population. Of the cataloged earthquakes, 25 events located on the San Andreas fault151
with magnitude values ranging from 1.2 to 3.9 and occur in sufficiently close clusters,152
and at similar source-station distances to the immature events to permit an empirical153
Green’s function analysis. We analyze additional events clustered on the active fault154
in the middle of the HRSN array to verify that we obtain the same quality of results155
for the events not recorded with complete azimuthal coverage.156
Given that displacement often migrates to newly activated strands in a mature157
fault zone, the assumption of a 300 km offset on the active strand may be an overes-158
timate [Thurber et al., 2006; Simpson et al., 2006; Savage and Brodsky , 2011]. Events159
from the mature population may occur on the youngest, active strand, which has160
likely sustained less offset than the cumulative total. Studies of SAFOD fault cores161
8
suggest that the currently active strand has accommodated well over 100 km of offset162
[Titus et al., 2005; Schleicher et al., 2010]. Geological and geodetic data therefore163
suggest that the youngest fault strands in the area may easily exceed the ∼1 km164
displacement criterion designating a fault mature.165
2.2 Immature Fault Earthquakes: Mount St. Helens Edifice166
and Secondary Faults at Parkfield167
The earthquakes at Mount St. Helens (MSH) consist of 40 events on a newly formed168
fault surface created during the 2004 dome-building eruption (Figure 2). The events169
are located at the base of an extruding solid rock spine in the crater of the volcano in170
February and March of 2005, when the primary station used in our analysis, MIDE,171
was operational (seismic moment and corner frequency values listed in Table 2). Mo-172
ment magnitude calculations ranging from 0.4 to 1.4 result from using spectral ratios173
multiplied by the long period amplitude of the earthquake spectra for the empirical174
Green’s function event used at MSH (detailed in the seismic moments and magnitudes175
section). The source time functions used here all result from spectral deconvolution176
of a single empirical Green’s function. We therefore infer that relative moment values177
are precise, and that errors in absolute moment stem primarily from the moment178
estimation of the empirical Green’s function. The earthquakes at Mount St. Helens179
are not cataloged, and therefore not located, as the numerous events resulting from180
the ongoing eruption in 2005 prevented many events from being analyzed. We find181
co-located earthquakes pairs by cross-correlation of events over both low- and high-182
gain channels of Pacific Northwest Seismographic Network (PNSN) stations MIDE183
and NED. Only two seismic stations were both simultaneously operational in Febru-184
9
ary and March of 2005, and located close enough to the crater to record the small185
earthquakes associated with the solid rock spine extrusion in a large enough dynamic186
range necessary for our analysis. Stations MIDE and NED are short-period, vertical187
component seismometers peaked at 1 Hz, and sample rates of 100 sps. Additional188
stations on the volcanic edifice recorded larger events, and were used to estimate the189
shear wave velocity at shallow depths (Figure 2).190
Earthquakes more than 5 km from the main trace of the San Andreas fault at191
Parkfield are inferred to be on secondary faults, and of these, 11 are sufficiently192
clustered to permit analysis (Figure 1). These events occur well outside of the San193
Andreas system on small faults with virtually no expression of a surface trace. The194
three easternmost earthquakes in Parkfield do occur in the vicinity of the Kettleman195
Hills fault (∼ 5km) [Lin and Stein, 2006]. However, we do not exclude small events196
that happen on larger faults from the immature population, as long as the total fault197
displacement is less than 1 km. In that context, we argue that the three easternmost198
events may also be classified as immature, as the displacement on the faults in the area199
is orders of magnitude less than that on any active strand of the SAF, and likely within200
the 100s of meters range according to cross-sectional plots of the sedimentary structure201
[Lin and Stein, 2006]. The remaining immature events shown in Figure 1 occur202
on secondary faults at distances 30 km or more from the Coalinga, New Idria and203
Kettleman Hills faults. We therefore infer from the large distance that the remaining204
events are not associated with any of the previous large earthquakes that occurred on205
the larger faults (distance cited is perpendicular from strike) [Lin and Stein, 2006].206
The immature events at Parkfield and Mount St. Helens have different types of207
focal mechanisms and occur in different lithological settings. While most of the events208
10
on the San Andreas fault have strike-slip focal mechanisms, seven of the 11 events in209
the immature population in Parkfield have reverse faulting focal mechanisms. Focal210
mechanism solutions do not exist for the earthquakes recorded at Mount St. Helens,211
due to the small magnitudes. Some previous studies have found differences in source212
properties correlated with varying faulting style, implying that faulting style may213
influence source scaling characteristics [Allmann and Shearer , 2009; Venkataraman214
and Kanamori , 2004a]. Here we take a different approach by examining the behavior215
as a function of a less-commonly studied parameter: maturity. If systematics are216
revealed as a function of maturity, then the accompanying physical properties of the217
fault, e.g., roughness, are at least as viable an explanation for differences in behavior218
as any previously studied.219
3 Source spectral calculation and fitting220
Our analysis of dynamic source parameters is based on two fundamental parts: es-221
timation of seismic moment and spectral corner frequency using direct observations222
from the spectra, and energy calculations based on a slip weakening rupture model.223
We detail the direct source spectral observations in this section, and continue with224
the more model dependent energy parameters in section 4.225
Accounting for high-frequency attenuation (particularly in the volcanic edifice) is226
crucial to investigating source parameter scaling, particularly for small earthquakes227
where accurate corner frequency estimations are crucial. We use an empirical Green’s228
function approach to eliminate attenuation effects caused by the path the seismic229
waves travel from the source to the receiver, as well as the site effects at the station,230
11
and instrument response. In both Parkfield and at Mount St. Helens, we use co-231
located event pairs with a magnitude unit or more difference in size to deconvolve232
the path, site, and instrument response from earthquake recordings. The event co-233
location requirement insures a nearly identical source-receiver geometry, permitting234
instrument, and travel path attenuation effects to cancel effectively. The magnitude235
difference requirement permits the assumption that the source-time function of the236
smaller event is sufficiently close to a delta function over the bandwidth of interest.237
The resulting time series after the deconvolution step consists of the earthquake source238
time function. The source time function indicates moment rate of the earthquake as239
a function of time isolated from the other distorting effects listed above. Using the240
source time functions, we calculate the source time function spectra, and estimate241
the the long-period spectral ratio, and spectral corner frequency.242
First, we deconvolve an empirical Green’s function (eGf) from the entire waveform243
of the larger co-located event in order to obtain the source-time function. Figure 3244
shows two examples of eGf’s and their larger respective co-located events from both245
Parkfield and Mount St. Helens (left and right columns respectively). The example246
shows the Parkfield event recorded on the three-components of station VCAB, and247
the Mount St. Helens event on the single component station MIDE. Calculations of248
the source time functions for the Mount St. Helens events result from a water-level249
deconvolution (spectral division) of the eGf from the larger event. We calculate the250
source-time functions of the Parkfield events using a projected Landweber deconvo-251
lution method. The method is a regularizing, iterative approach in the time domain252
that enforces a positivity constraint on the solution. The approach is outlined in253
detail in Lanza et al. [1999]. As a check on the method, we compare the Landweber254
12
source time functions to those obtained with a spectral deconvolution, finding nearly255
identical results (Figure 4). The Landweber approach has the added advantage of256
expanding the frequency band over which the signal to noise spectral ratio exceeded257
a factor of two for the Parkfield data, where the source-receiver distances are much258
larger than those at Mount St. Helens. Figure 4 shows the source time functions for259
each of the event pairs depicted in Figure 3.260
Second, we calculate the source spectra of each event using a multi-taper spectral261
calculation. The spectra are shown in the right column of Figure 4. The dashed262
lines represent the least-squares fit to a Brune spectral model, and the solid gray263
lines represent the noise spectra (Figure 4). Noise spectra were calculated from a264
time window directly preceding the source pulse. We used time windows of 256 data265
points, and 128 data points for the Parkfield and Mount St. Helens data respectively,266
corresponding to a time window of ≈ 1 second, for both the source-time function and267
noise spectral calculations.268
We calculate the Parkfield earthquake source spectra using the RMS spectra de-269
termined using the individual station component spectra. We stack the available270
source time spectra determined at each station in order to have a single, spatially271
averaged source time function spectrum for each event. In addition to the co-location272
and magnitude criteria, we choose events in the mature populations such that events273
located north and south of the HRSN array have source-receiver distances similar to274
events occurring on secondary faults. We also choose events in the middle of the array275
to check that we obtain similar source time function calculations when azimuthal cov-276
erage differs. For full details of the source time function spectral calculation, we refer277
the reader to Harrington and Brodsky [2007] and Harrington and Brodsky [2009].278
13
We model the source spectra for each event using a Brune spectral model with279
spectral falloff n = 2 [Brune, 1970; Abercrombie, 1995].280
Ω(f) =Ω0
(1 + ( ffc
)n)(1)281
We calculate a least-squares fit to the spectral data using the Brune spectral model282
in order to calculate the values for the corner frequency (fc) and the long period283
spectral amplitude (Ω0). The attenuation term typically included in Equation 1 is284
omitted here, as the spectra have already been attenuation corrected through the eGf285
deconvolution.286
3.1 Seismic moments and magnitudes287
Slightly different methods are appropriate for determining the seismic moment from288
the earthquake spectra in each location, due to the difference in earthquake relative289
locations and the available catalogs. We therefore describe the moment calculation290
separately for Parkfield and Mount St. Helens in the subsequent two subsections.291
3.1.1 Mount St. Helens292
The events at Mount St. Helens fall below the catalog completeness threshold, and293
are not cataloged. We calculate the moment of our eGf event using the long period294
amplitude of the earthquake spectrum, calculated using a time window of 128 samples295
starting 28 samples (i.e. 28 msec) before the P-wave arrival. We then calculate the296
moment using the long-period spectral amplitude, and the equation [Abercrombie,297
14
1995; Aki and Richards , 2002]298
M0 = 〈U2φθ〉Rpψ4πρα3rΩ0. (2)299
where ρ is the rock density (2.5×103 kg/m3), α is the P-wave velocity (1400 ms
), r300
is the distance to the station, 〈Uφθ〉 is the radiation pattern coefficient, and Rpψ is301
the free surface correction [Aki and Richards , 2002]. Although the events are not302
located, we can infer r based on the distance from the station to the active extruding303
spine. However, the events are shallow, and the source depths relative to the station304
are poorly constrained on the steep mountain. We therefore omit the free surface305
correction in the moment calculation for the events at Mount St. Helens, as we have306
no way of confidently estimating the incidence angle. Section 4.1.1 details the origin307
of the P-wave velocity value. The eGf event used for 36 out of our 40 events was308
extremely small (estimated magnitude of 0.07 ± 0.01 ), and therefore only recorded309
on a single station located an estimated ∼150 m from the source. We calculate310
the moments of the family of 36 events using the long period spectral amplitude311
of the eGf multiplied by the spectral ratios. The moment and corner frequency for312
the remaining four events recorded at station NED are calculated similarly to those313
events recorded at MIDE. We obtain a quantitative error estimation by applying314
the approach described in Prieto et al. [2007] for a single station to the individual315
deconvolved spectra. We estimate the errors of the source parameters using the316
standard deviation of a sub-set of parameter estimations. The sub-set is created from317
the single record by removing one of the Slepian tapers, and repeating the spectral318
estimation omitting a different taper for each estimation [Park et al., 1987]. The319
15
procedure varies slightly when multiple stations are available, in that the sub-set of320
parameter estimations is created using the multiple station recordings, rather than the321
multiple tapers. We refer the reader to Prieto et al. [2007] for a detailed description.322
3.1.2 Parkfield323
The earthquakes in Parkfield are not clustered at a single location, as are those at324
Mount St. Helens. Each source time function requires a separate empirical Green’s325
function (eGf), making calculating the relative seismic moment from a single event326
impossible. We therefore calculate seismic moments directly from the earthquake dis-327
placement spectra using a two-second window around the S-wave arrival, estimating328
the long period spectral amplitude (Ω0) with a least squares fit to a Brune spectral329
model. We obtain the seismic moment calculation from Ω0 by using Equation 2,330
substituting the shear wave velocity (β) for α, and dividing by the mean radiation331
pattern (Uθφ = 0.63 for S-waves) [Lay and Wallace, 1995]. The hypocentral locations332
of the Parkfield events are known, permitting an estimation of the incidence angle333
of the seismic wave at the station. We therefore include the free surface correction334
(Rpψ) for the Parkfield events.335
We check our work by using the standard relations between M0 and Mw to com-336
pare our calculated Mw and the catalog magnitude values (mainly Mc, with few ML337
values) [Hanks and Kanamori , 1979]. Figure 5 indicates that the Mw values calcu-338
lated here correspond well to catalog values, with some scatter found in the immature339
population. The ratio of Mw : Mc is 0.92 ± 0.04 for the mature (blue) events, and340
1.05 ± 0.13 for the immature events (red, Figure 5). Using the P-wave arrival for the341
Parkfield events instead of the S-wave arrival produces the same trend, however, with342
16
significantly more scatter in the values. The scatter in the immature event values343
(red diamonds) in Figure 5 likely results from the lack of azimuthal coverage by the344
HRSN stations (Figure 1). We repeated the moment calculations using additional345
waveforms from nearby stations with high-quality recordings in efforts to reduce the346
scatter via improved azimuthal coverage. The red squares indicate the Mw calcula-347
tions using the additional stations (∼5-10 extra stations for each event), suggesting348
that they help reduce the scatter. The moment values calculated with the additional349
stations are therefore used in the analysis.350
We estimate the errors for moment, corner frequency calculations, and energy351
ratios using the multiple station jackknife error estimation described in Prieto et al.352
[2007]. The method is similar to the single station jackknife error estimation described353
above. It works as follows: given a set of N stations, one calculates a parameter (such354
as the spectral corner frequency) from the station averaged spectra. The calculation355
is performed N times, each time with one station removed from the average. The356
standard deviation of the N parameter calculations provides the jackknife error for the357
given parameter. The errors for the energy values are estimated in a similar manner.358
The energy-moment ratios are model based parameters dependent on the moment359
and corner frequency values (discussed in section 4.1.1). We estimate the errors for360
energy values using the N estimations of M0 and fc to calculate N estimations of361
energy ratios, the standard deviation of which constitute the energy errors.362
The stability of the result for a particular choice of eGf is mainly dependent on363
the slight differences in hypocentral location, and the source-station geometry. For364
example, if two events are not perfectly co-located, the effect on the source time365
function will be less apparent on a station with travel path in line with the two366
17
events compared to a station with travel path normal to the event separation distance.367
Therefore, the multiple station jackknife error estimation is particularly well suited to368
quantify the stability of the source time function solutions, and their resulting spectra369
for our data set in Parkfield. While we do not have multiple stations available for370
the events at Mount St. Helens, given the unique geometry of the extruding spine,371
the short source receiver distance, and the similarity of the waveforms within event372
pairs, we estimate that difference in co-locations would not be greater than a scale of373
meters, and that the solutions are also stable as suggested by the single-station error374
estimation.375
3.2 Results376
Values of seismic moment and corner frequency estimated from the source spectra sug-377
gest that the earthquakes in the immature population exhibit constant stress drop378
scaling, shown by a M0 ∝ f−3c dependence (Figure 6) [Kanamori and Rivera, 2004].379
The earthquakes in the mature population have corner frequencies which remain380
roughly constant, a scaling which is consistent with an roughly constant source dura-381
tion [Harrington and Brodsky , 2009]. Although stress drop values are similar for all382
events analyzed here, the immature population at Mount St. Helens does not follow383
the same line of constant stress drop in Figure 6. The reason for the difference is that384
the shear wave velocity in the volcanic edifice is approximately four times slower than385
in Parkfield, resulting in slower rupture velocities, and subsequently longer rupture386
durations [Harrington and Brodsky , 2007].387
18
4 Interpretation in Terms of Energy388
Using the spectra along with the modeled M0, and fc values, we can calculate other389
source parameters such as the energy moment ratio (ER
M0), and the seismologically ob-390
servable fracture energy moment ratio (EG
M0). A comparison of all source parameters391
between populations permits us to make inferences regarding how the fault maturity392
affects their relative scaling. We first review the theoretical background for determin-393
ing the energy parameters in the theoretical basis subsection before presenting the394
inferred energy results based on the source spectral observations above.395
4.1 Theoretical basis396
4.1.1 Radiated Energy397
The radiated seismic energy (ER) can be calculated for a given earthquake by inte-398
grating the source velocity spectrum (I) observed at a single station [Venkataraman399
et al., 2002; Venkataraman and Kanamori , 2004b; Prieto et al., 2004].400
I =
∫ f2
f1
2πfΩ0
1 + ( ffc
)n
2
df (3)401
Ω0 is the long-period spectral amplitude observed at a given station, fc is the402
spectral corner frequency, and n is the spectral falloff. f1 and f2 represent the limits403
of integration determined by the bandwidth over which the signal to noise ratio is404
high. Equation 2 relates Ω0 to the M0 via the source-station distance and material405
properties. The radiated seismic energy (ER) in terms of I is then given by:406
ER = 〈U2θφ〉R2
pψ4πρβr2I, (4)407
19
where 〈U2θφ〉 = 2/5 is the mean S-wave radiation pattern over the focal sphere, Rpψ408
is the free surface correction term, β is the S-wave velocity, ρ is the rock density,409
r is the source-receiver distance, and I is a definite integral given by Equation A.6410
[Abercrombie, 1995; Boatwright and Fletcher , 1984; Aki and Richards , 2002].411
Ide and Beroza [2001] show that radiated seismic energy values are often under-412
estimated due to spectral integration over a limited bandwidth. They suggest that413
radiated energy may be better estimated by integrating the power spectral density414
of a Brune source spectral model from zero to infinite frequency, given that the in-415
tegral has a closed form. Following Ide and Beroza [2001] and using the analytical416
expression resulting from integration of the square of Equation 3 from zero to infinite417
frequency, we determine the analytical expression for the radiated energy in terms of418
the moment and corner frequency obtained from the Brune spectral model (derivation419
in Appendix A):420
ER =π2
5ρβ5M2
0 f3c . (5)421
We calculate the energy values presented here using Equation 5. Equation 5 is422
independent of geometrical spreading and attenuation, and is therefore appropriate423
for source time spectra. The source parameter calculations performed here are based424
on the S-wave values, because the S-wave energy accounts for more than 97% of the425
total radiated energy [Kanamori et al., 1993].426
We use a shear wave velocity value of 3750 m/s given in the velocity model of427
Thurber et al. [2006] for the depth range of our events (namely, 4 to 8.8 km, with428
most events having depths ranging from 5 to 8.5 km, Table 1) in Parkfield. Allmann429
and Shearer [2007] found an apparent static stress drop depth dependence when430
neglecting shear wave velocity depth dependence in Parkfield, suggesting that the431
20
assumption of a constant shear velocity may be problematic. However, the strongest432
parameter changes they found were at depths above 3 km, and below 12 km. The433
values of shear wave velocity remains fairly constant within the depth range used434
here.435
For Mount St. Helens, we use a value of β = 800 ms
for the loosely consolidated436
volcanic rock and ash in the edifice. The value of 800 ms
results from the upper-437
limit value calculations of the average S-wave velocity at depths similar to the data438
set determined from P-S wave arrival times of earthquakes occurring in the same439
location (and during the same time period). Many of the earthquakes shown here440
were too small to be recorded on many stations, however, a number of larger events441
with clear P- and S-wave arrivals were recorded on additional stations in the volcanic442
edifice. We were therefore able to calculate velocity estimations using the slightly443
larger events with similar travel paths to our data set. (Cataloged events during the444
same time period are also located at the base of the actively extruding spine). Using445
29 larger events and seven stations azimuthally distributed over the volcanic edifice,446
we assume a Poisson solid, and obtain an estimated shear velocity value of 670 ±447
130 m/s. We take the upper-limit as the S-wave velocity value, as it is closer to the448
larger values cited by tomographic studies [Moran et al., 2008]. We use the values449
calculated here rather than values cited by Moran et al. [2008], as the short source-450
receiver distances and the similar locations may represent a more reliable average451
shear velocity value than one obtained from velocity models lacking resolution on the452
scale of 10s of meters.453
21
4.1.2 Seismologically Observable Fracture Energy454
In addition to the radiated energy and static stress drop, we examine the scaling of455
fracture energy with earthquake size. The relationship between EG and ER can be456
understood by considering a slip-weakening earthquake rupture model which assumes457
no overshoot or undershoot, such as that described by fracture mechanics theory458
[Anderson, 2004]. In such a model, rupture initiates as the stress on the fault reaches459
some yield value. The initial stress on the fault surface drops until the fault slips460
some critical value (Dc), due to plastic yield in a zone near the crack tip [Freund ,461
1979; Kanamori and Heaton, 2000; Kanamori and Brodsky , 2004; Tinti et al., 2005].462
Assuming no under- or overshoot requires that once the slip dependent stress drops463
to the frictional value, σ1 (at D = Dc), sliding continues at the frictional stress level464
until rupture terminates (at D > Dc). Using such a slip-weakening model permits465
one to relate radiated energy and fracture energy in a straightforward way. One can466
show that ER is directly proportional to the sum of stress drop and seismologically467
observable fracture energy EG [Kanamori and Heaton, 2000; Abercrombie and Rice,468
2005; Kanamori and Brodsky , 2004]. Consider that the total energy release in an469
earthquake (∆W ) is partitioned into radiated energy, ER, seismologically observable470
fracture energy, EG, and the constant component of friction, Ef .471
∆W = ER + EG + Ef (6)472
22
Alternatively, ∆W can also be written in terms of the average stress, the fault area,473
and the average slip on the fault,474
∆W =σ1 + σ0
2AD, (7)475
where σ1 and σ0 are the final and initial stresses respectively. Assuming the frictional476
stress (σf ) on the fault surface is equal to the final stress (i.e. no overshoot or477
undershoot) implies the frictional energy can be written in terms of the final stress.478
Ef = σ1AD (8)479
Substituting Equation 8 into the in the Equation 6, produces the following relation:480
1
2∆σAD = ER + EG ⇒ EG =
1
2∆σAD − ER. (9)481
Using the equation for scalar moment,482
M0 = µAD, (10)483
permits writing Equation 9 in terms of the energy moment ratios,484
EGM0
=1
2
∆σ
µ− ERM0
, (11)485
where µ is the rigidity [Keilis-Borok , 1959; Kanamori and Anderson, 1975]. We use486
values of µ = 5× 109 Nm2 , and µ = 3× 1010 N
m2 for Mount St. Helens and Parkfield re-487
spectively, based on the shear velocity in a Poisson solid (µ = ρβ2) [Aki and Richards ,488
23
2002]. Substituting the analytical value for radiated energy from Equation 5, and the489
expression for the static stress drop on a penny-shaped crack, namely,490
∆σ =7
16M0
(fc
0.32β
)3
, (12)491
we have an expression for EG
M0in terms of seismic moment and corner frequency:492
EGM0
=
(1
2
1
µ
7
16
[1
0.32β
]3
− π2
5ρβ5
)M0f
3c . (13)493
4.2 Energy Results494
Equation 13 indicates that, ER
M0, and EG
M0are both proportional to M0f
3c . Therefore,495
if a group of events has self-similar scaling, a constant ratio for both parameters is496
expected, based on the model assumptions. Deviations from the constant scaling of497
either of these parameters for a particular group of earthquakes would imply that498
scale invariance is not a characteristic of that group of events.499
The trends in energy with earthquake size exhibited in Figure 7 suggest that the500
earthquakes in the mature population do not exhibit scale invariance. The group501
of earthquakes on immature faults, i.e. those on secondary faults in Parkfield, and502
in the Mount St. Helens crater, occur presumably on rougher fault surfaces, based503
on the maturity criterion defined in the introduction, and on fieldwork examining504
the roughness of exposed fault surfaces Sagy et al. [2007]; Sagy and Brodsky [2009].505
Similarly, the events on the San Andreas fault, i.e. the mature population, occur506
presumably on a smoother fault. The ratio ER
M0may be considered as the amount of507
energy radiated per unit fault area per unit slip in an earthquake. We focus on this508
24
parameter, as it provides a tenuous observational link to the rupture process [Wes-509
nousky , 2006; Kanamori and Rivera, 2004; Choy and Kirby , 2004; Wyss and Brune,510
1968]. If we can assume that differences in fault surface geometry cause the observed511
differences in energy-moment ratio scaling (ER
M0) between immature and mature pop-512
ulations, then the scaling differences demonstrate how faulting environment might513
influence properties of rupture (Figure 7).514
Using a linear regression to fit both the mature and immature populations indi-515
cates a slope of 0.9 ± 0.16 and -0.2 ± 0.08 respectively in the variation of energy-516
moment ratios with earthquake size (Figure 7). The fit suggests a negligible variation517
of the energy ratio for the immature (rougher) fault surfaces, and a much stronger518
dependence of energy on moment for the mature (smoother) fault surface. If we519
consider the case where our shear wave velocity for Mount St. Helens has the mean520
estimated value rather than the upper-limit value (namely, a value of 670 ms
instead521
of 800 ms
), the fitted slope value changes from -0.2 to -0.3 for the immature popula-522
tion, i.e., the radiated energy does not perceptibly change with moment. Even with523
the lower estimate of shear velocity at Mount St. Helens, the energy dependence on524
moment is still significantly greater for the mature population.525
4.2.1 Comparison with faults from the literature526
The earthquakes in both our mature and immature populations occur in non-typical527
faulting environments. The events comprising the mature population occur near528
the creeping/locked transition of the San Andreas fault, and the majority of those529
comprising our immature population occur in a volcanic edifice, rather than a tectonic530
fault. We consider the possibility that our observations result from using earthquakes531
25
in a unique environment by comparing our energy-moment ratios and static stress532
drop values to those of other studies. The analogous immature events consist of those533
analyzed by a number of mining seismicity studies [Yamada et al., 2007; Kwiatek et al.,534
2011; Gibowicz et al., 1991; Urbancic and Young , 1993; McGarr et al., 2010], while535
the mature events consist of events occurring on faults within the Basin and Range536
faulting system in the western US, as well as those associated with the San Andreas537
[Abercrombie, 1995; Abercrombie and Rice, 2005; Mayeda and Walter , 1996].538
Figure 8 plots our observations shown in Figure 7 along with the results of a539
variety of other studies including both immature and mature populations for com-540
parison [Yamada et al., 2007; Kwiatek et al., 2011; Gibowicz et al., 1991; Urbancic and541
Young , 1993; McGarr et al., 2010; Abercrombie, 1995; Abercrombie and Rice, 2005;542
Mayeda and Walter , 1996]. We fit the combined immature and mature populations543
using a linear regression for each population. The combined data set preserves the544
general feature of more self-similar behavior exhibited by the immature data sets in545
comparison to the size dependence exhibited by the mature data set. In particular,546
Figure 8 suggests that the earthquakes analyzed here follow the same trend as other547
populations of mature and immature events elsewhere on more ”typical” faults in548
southern California and the western United States.549
The seismicity induced in mines occurs as a direct result of the excavation ac-550
tivity, rather than long-term fault movement driven by tectonic stresses. Induced551
earthquakes can be assumed to occur on relatively new fracture surfaces, with low552
cumulative fault slip. The induced events may therefore be classified as immature for553
the purposes of comparison. The events included in the immature population shown554
in Figure 8 include seismicity recorded in the TauTona, and Mponeng gold mines in555
26
South Africa [Kwiatek et al., 2011; McGarr et al., 2010; Yamada et al., 2007], and556
seismicity recorded in mines in Manitoba, and Ontario, Canada [Gibowicz et al., 1991;557
Urbancic and Young , 1993]. An additional advantage of using mining seismicity for558
a comparison of small earthquakes, is that the data quality is high, which is partic-559
ularly important for energy calculations. In all cases, earthquakes were recorded on560
instruments located at depths ranging from 50 - 3500 m, at source receiver distances561
on the order of ∼ 100 m. We apply the finite bandwidth correction given by Equation562
5 of Ide and Beroza [2001] to the data provided by Gibowicz et al. [1991]. All other563
data is taken directly as reported.564
By similar reasoning, the events analyzed in the preferred model (2) by Abercrom-565
bie [1995], and those re-analyzed in Abercrombie and Rice [2005] are analogous to566
the mature population, as they occur near the Cajon Pass section of the San An-567
dreas fault, which has a cumulative displacement similar to Parkfield. The results568
of Abercrombie [1995] use a Brune source model with a seismic QS = QP = 1000 to569
estimate the source parameters, rather than a spectral ratio approach. We assume570
that the comparison of our results is justified due to the high-quality borehole seismic571
data used, and the consistency of their results with the later analysis of Abercrombie572
and Rice [2005], which uses a spectral ratio approach similar to the one used here.573
Mayeda and Walter [1996] also use spectral ratio approach applied to earthquake574
coda spectra to determine the source function of a variety of events in the western575
US. The data include nearly all M > 5.0 earthquakes and selected aftershocks be-576
tween January 1988 and December 1994 (117 events total). They apply an empirical577
Green’s function correction to the coda spectra to obtain the source time functions,578
and calculate the seismic energy by integrating the moment rate spectra, with an579
27
extrapolation assuming an ω−2 falloff. We omit the Northridge aftershocks included580
in their data set in the comparison (seven events total). The mainshock occurred on581
a previously unknown fault, and as a result, the maturity of the faults on which the582
aftershocks occur remains ambiguous. The remaining events occur on well-developed583
faults with surface traces of ∼10 km or more in the San Andreas, and Basin and584
Range faulting systems.585
We fit the composite dataset in Figure 8 and recover a similar trends to those586
in Figure 7. A linear regression indicates that the energy is roughly scale invariant587
for the combined immature population (slope of 0.07 ± 0.009), while the energy size588
dependence is more pronounced for the mature population (slope of 0.3 ± 0.01).589
The similarities in scaling between our observations and the combined data suggest590
that energy-size dependence scales similarly for events in Parkfield and the western591
US, as well as between the secondary faults in Parkfield, Mount St. Helens, and592
mining related events occurring on other immature faults. The only feature needed593
to distinguish the mature and immature populations in the comparison with other594
studies is the cumulative fault displacement. We propose that the similar scaling595
exhibited by the broader comparison with other studies suggests that the observed596
scaling features of the immature and mature populations presented here may be a597
general feature of earthquakes outside of our study areas, and not the result of some598
unusual faulting environment in the creeping/locked transition of the San Andreas599
fault, or in the volcanic edifice.600
28
5 Implications for the Role of Fault Maturity in601
Earthquake Dynamics602
The comparison of energy-moment ratios (ER
M0,EG
M0) between immature and mature603
populations suggests that fault maturity affects the dynamics of rupture. More specif-604
ically, the energy relations in Figures 7 and 8 suggest that at least for smaller earth-605
quakes, less energy is radiated away, and less energy goes into forming new fractures606
on smoother, more mature faults compared to rougher, less mature faults. Other607
studies which compare the radiated energy of earthquakes between interplate events608
(inferred as mature) with intraplate events (inferred as immature) conclude that in-609
traplate earthquakes tend to radiate more energy that their interplate counterparts610
[Zhuo and Kanamori , 1987; Choy and Kirby , 2004; Choy and Boatwright , 2009]. The611
key difference in our study is that we differentiate immature and mature populations612
based on the amount of cumulative fault slip, rather than on focal mechanism or613
faulting style (e.g. strike slip, vs. thrust, or interplate vs. intraplate). The advantage614
of the approach is that it systematically compares the effect of maturity based on one615
fundamental physical parameter (displacement), which is more directly correlated616
with it.617
Harrington and Brodsky [2009] interpret the constant corner frequency scaling618
for the mature population as indicating that mature faults have source areas that619
change only nominally over the magnitude range of our data set. We consider the620
possible relationship between an unchanging source dimension and the decreasing621
energy-moment ratios observed here. The roughly constant fault area may be dictated622
by the asperity size, meaning that the smallest patch of the mature fault surface623
29
that may rupture in an earthquake may be limited by the elliptical bump size. The624
scale invariant roughness observed for immature faults would imply that earthquake625
rupture might result from asperity rupture at a variety of length scales. In such626
a scenario, one would expect to observe a population of earthquakes with a fractal627
distribution of rupture areas, leading to an observation of self-similarity, or invariant628
energy-moment ratios (ER
M0, EG
M0). In fact, we observe a nearly constant energy values629
for the earthquakes in the immature population (Figure 7). We speculate that in a630
large event where the rupture length would exceed the elliptical bump (i.e. asperity)631
dimension by a significant amount, that fault slip and length may scale according to632
constant stress drop.633
The development of roughly constant source dimensions on the mature fault may634
result from a minimum asperity size asymptotically tending toward a fixed value.635
Observations of fault damage zones may also suggest that asperities trend toward a636
fixed size. Studies suggest that when a fault is first formed, the width of the damage637
zone increases with fault displacement. Such an effect occurs up to a certain offset,638
after which the rate of damage widening decreases [Savage and Brodsky , 2011; Chester639
and Chester , 2005]. Savage and Brodsky [2011] argue using observations of fracture640
density that the amount of displacement at which the damage rate decreases is ∼100641
m. Once cumulative displacement exceeds ∼ 100 m, the fault damage zone width is642
no longer determined by larger asperities, and the observed damage formation rate643
becomes roughly constant (see Savage and Brodsky [2011], Figure 4). They interpret644
the decline in damage formation dependence on asperity size as the result of slip645
being distributed throughout a wider patch on the fault surface, making it easier646
for the slipping patches to get around large asperities without damaging host rock.647
30
Consequently, earthquakes on less mature faults which accommodate more slip past648
asperities should have higher radiated energy values compared to their counterparts649
on mature fault surfaces. We claim that seismologically observable fracture energy650
values should be higher as well, if more asperities are being damaged. Figures 7651
and 8 indicate a higher ER, and EG per unit M0 for the immature population, and652
are consistent with such claims.653
6 Conclusions654
A comparison of ER
M0, and EG
M0between earthquakes on immature fault surfaces in655
Parkfield, California, and Mount St. Helens Volcano, Washington, and on a mature656
fault surface, namely the San Andreas fault near Parkfield, CA, indicates that the657
values are roughly constant for immature faults, while the values decrease with de-658
creasing magnitude for mature fault surfaces. Theoretical considerations based on659
a slip-weakening rupture model indicate a M0f3c dependence of static stress drop660
and energy ratios, which should remain constant for populations that exhibit self-661
similarity. The decreasing energy ratios and stress drop values with size observed662
for the mature population suggest that rupture properties are not self-similar for the663
mature population.664
The difference in source parameter scaling between immature and mature earth-665
quake populations suggests that the ordinary scaling relationships resulting from a666
commonly assumed constant stress drop are not generally valid for each earthquake667
population individually. The correlation of self-similar source parameter scaling with668
fault immaturity suggests that fault surface roughness may be a physical factor af-669
31
fecting fault rupture.670
7 Acknowledgments671
The Mount St. Helens seismic data used here were collected by the Cascades Vol-672
cano Observatory, and the Pacific Northwest Seismograph Network, and distributed673
by the Incorporated Research Institutions for Seismology (IRIS). The IRIS DMS is674
funded through the National Science Foundation and specifically the GEO Direc-675
torate through the Instrumentation and Facilities Program of the National Science676
Foundation under Cooperative Agreement EAR-0552316. Data on fault locations and677
displacements was collected from the U.S. Geological Survey, California Geological678
Survey, 2006, Quaternary fault and fold database for the United States, accessed679
Sept. 15, 2009, from USGS web site: http//earthquakes.usgs.gov/regional/qfaults/.680
Parkfield waveform data is provided by Berkeley Seismological Laboratory, Uni-681
versity of California, Berkeley and accessible through the Northern California Earth-682
quake Data Center (NCEDC) website. The projected Landweber deconvolution code683
was provided by Hiroo Kanamori, and the FDDECON deconvolution code was pro-684
vided by Thorne Lay. This work was supported by the National Science Foundation685
grant EAR-0711575, and the Alexander Von Humboldt foundation.686
32
References687
Abercrombie, R. E., Earthquake source scaling relationships from -1 to 5 M(L) using688
seismograms recorded at 2.5-km depth, J. Geophys. Res., 100, 24,015–24,036, 1995.689
Abercrombie, R. E., and J. R. Rice, Can observations of earthquake scaling constrain690
slip weakening?, Geophys. J. Int., 162, 406–424, 2005.691
Aki, K., and P. G. Richards, Quantitative Seismology, 2nd ed., Univ. Sci., Sausalito,692
CA, 2002.693
Allmann, B. P., and P. M. Shearer, Spatial and temporal stress drop variations in694
small earthquakes near Parkfield, California, J. Geophys. Res., 112, B04,305, 2007.695
Allmann, B. P., and P. M. Shearer, Global variations of stress drop for moderate to696
large earthquakes, J. Geophys. Res., 114 (B01310), 2009.697
Anderson, T. L., Fracture Mechanics: fundamentals and applications, 621 pp., CRC698
Press, 2004.699
Boatwright, J., and J. B. Fletcher, The partition of radiated energy between P and700
S waves, Bull. Seismol. Soc. Am., 74 (2), 1984.701
Brune, J. N., Tectonic stress and the spectra of seismic shear waves from earthquakes,702
J. Geophys. Res., 75, 4997–5009, 1970.703
Chester, F. M., and J. S. Chester, Fracture surface energy of the Punchbowl fault,704
San Andreas system, Nature, 437, 133–136, 2005.705
Choy, G. L., and J. Boatwright, Differential energy radiation from two earthquakes706
in Japan with identical M-w: the Kyushu 1996 and Tottori 2000 earthquakes, Bull.707
Seismol. Soc. Am., 99, 1815–1826, 2009.708
Choy, G. L., and S. H. Kirby, Apparent stress, fault maturity and seismic hazard for709
normal-fault earthquakes at subduction zones, Geophys. J. Int., 159, 2004.710
Choy, G. L., A. McGarr, S. H. Kirby, and J. Boatwright, An overview of the global711
variability in radiated energy and apparent stress, Geophysical Monograph, 170,712
43–57, 2006.713
Freund, L. B., The mechanics of dynamic shear crack propagation, J. Geophys. Res.,714
84 (B5), 2199–2209, 1979.715
33
Gibowicz, S. J., R. P. Young, S. Talebi, and D. J. Rawlence, Source parameters716
of seismic events at the underground research laboratory in Manitoba, Canada:717
Scaling relations for events with moment magnitude smaller than -2, Bull. Seismol.718
Soc. Am., 81 (4), 1157–1182, 1991.719
Hanks, T. C., and H. Kanamori, Moment magnitude scale, J. Geophys. Res., 84,720
2348–2350, 1979.721
Harrington, R. M., and E. E. Brodsky, Volcanic hybrid earthquakes that are brittle-722
failure events, Geophys. Res. Lett., 34 (L06308), 2007.723
Harrington, R. M., and E. E. Brodsky, Source duration scales with magnitude differ-724
ently for earthquakes on the San Andreas Fault and on secondary faults in Parkfield,725
CA, Bull. Seismol. Soc. Am., 99, 2323–2334, 2009.726
Ide, S., and G. C. Beroza, Does apparent stress vary with earthquake size?, Geophys.727
Res. Lett., 28 (17), 3349–3352, 2001.728
Ide, S., G. C. Beroza, S. G. Prejean, and W. L. Ellsworth, Apparent break in earth-729
quake scaling due to path and site effects on deep borehole recordings, J. Geophys.730
Res., 108 (2271), 2271, 2003.731
Imanishi, K., and W. L. Ellsworth, Source scaling relationships of microearthquakes732
at Parkfield, CA, determined using the SAFOD pilot hole seismic array, in Earth-733
quakes: Radiated Energy and the Physics of Faulting, edited by R. E. Abercrombie,734
A. McGarr, H. Kanamori, and G. Di Toro, Geophysical Monograph Series, pp. 81–735
90, AGU, 2006.736
Kanamori, H., and D. L. Anderson, Theoretical basis of some empirical relations in737
seismology, Bull. Seismol. Soc. Am., 65, 1073–1095, 1975.738
Kanamori, H., and E. E. Brodsky, The physics of earthquakes, Reports On Progress739
In Physics, 67, 1429–1496, 2004.740
Kanamori, H., and T. H. Heaton, American Geophysical Union Geophysical Mono-741
graph 120, chap. Microscopic and macroscopic physics of earthquakes, American742
Geophysical Union, 2000.743
Kanamori, H., and L. Rivera, Static and dynamic scaling relations for earthquakes744
and their implications for rupture speed and stress drop, Bull. Seismol. Soc. Am.,745
94, 314–319, 2004.746
34
Kanamori, H., J. Mori, E. Hauksson, T. Heaton, L. K. Hutton, and L. M. Jones, De-747
termination of earthquake energy-release and M(L) using Terrascope, Bull. Seismol.748
Soc. Am., 83, 330–346, 1993.749
Keilis-Borok, V., On estimation of the displacement in an earthquake source and of750
source dimensions, Ann. Geofis. (Rome), 12, 205–214, 1959.751
Kwiatek, G., K. Plenkers, and G. Dresen, Source parameters of picoseismicity752
recorded at Mponeng deep gold mine, South Africa: Implications for scaling re-753
lations, submitted to BSSA, 2011.754
Lanza, V., D. Spallarossa, M. Cattaneo, D. Bindi, and P. Augliera, Source parameters755
of small events using constrained deconvolution with empirical Green’s functions,756
Geophys. J. Int., 137, 651–662, 1999.757
Lay, T., and T. Wallace, Modern Global Seismology, 517 pp., Academic Press, San758
Diego, 1995.759
Lin, J., and R. Stein, Seismic constraints and Coulomb stress changes of a blind760
thrust fault system, 1: Coalinga and Kettleman Hills, California, USGS Open File761
Report, (1149), 17, 2006.762
Malagnini, L., S. Nielsen, K. Mayeda, and E. Boschi, Energy radiation from763
intermediate- to large-magnitude earthquakes: implications for dynamic fault weak-764
ening, J. Geophys. Res., 115 (B06319), 2010.765
Mayeda, K., and W. R. Walter, Moment, energy, stress drop, and source spectra of766
western United States earthquakes from regional coda envelopes, J. Geophys. Res.,767
101 (B5), 11,195–11,208, 1996.768
Mayeda, K., L. Malagnini, and W. R. Walter, A new spectral ratio method using769
narrow band coda envelopes: Evidence for non-self-similarity in the Hector Mine770
sequence, Geophys. Res. Lett., 34 (11), 2007.771
McGarr, A., J. B. Fletcher, M. Boettcher, N. Beeler, and J. Boatwright, Laboratory-772
based maximum slip rates in earthquake rupture zones and radiated energy, Bull.773
Seismol. Soc. Am., 100 (6), 3250–3260, 2010.774
Moran, S. C., S. D. Malone, A. I. Qamar, W. A. Thelen, A. K. Wright, and J. Caplan-775
Auerbach, Seismicity associated with renewed dome building at Mount St. Helens,776
20042005, USGS Open File Report, A volcano rekindled: the renewed eruption of777
Mount St. Helens, 2004-2006, (1750), 27–60, 2008.778
35
Ohnaka, M., A constitutive scaling law and a unified comprehension for frictional slip779
failure, shear fracture of intact rock, and earthquake rupture, J. Geophys. Res.,780
161 (9-10), 2003.781
Park, J., C. R. Lindberg, and F. L. Vernon III, Multitaper spectral analysis of high-782
frequency seismograms, J. Geophys. Res., 92, 12,675–12,684, 1987.783
Prejean, S. G., and W. L. Ellsworth, Observations of earthquake source parameters784
at 2 km depth in the Long Valley caldera, eastern California, Bull. Seismol. Soc.785
Am., 91, 165–177, 2001.786
Prieto, G. A., P. M. Shearer, F. L. Vernon, and D. Kilb, Earthquake source scaling787
and self-similarity estimation from stacking P and S spectra, J. Geophys. Res., 109,788
2004.789
Prieto, G. A., D. J. Thomson, F. L. Vernon, P. M. Shearer, and R. L. Parker, Confi-790
dence intervals for earthquake source parameters, Geophys. J. Int., 168, 1227–1234,791
2007.792
Revenaugh, J., and C. Reasoner, Cumulative offset of the san andreas fault in central793
California, Geology, 25 (2), 123–126, 1997.794
Sagy, A., and E. E. Brodsky, Geometric and rheological asperities in an exposed fault795
zone, J. Geophys. Res., 114 (B02301), 2009.796
Sagy, A., E. E. Brodsky, and G. J. Axen, Evolution of fault-surface roughness with797
slip, Geology, 35 (3), 283–286, 2007.798
Savage, H. M., and E. E. Brodsky, Collateral damage: Evolution with displacement of799
fracture distribution and secondary fault strands in fault damage zones, J. Geophys.800
Res., 116 (B3), B03,405, 2011.801
Schleicher, A. M., B. A. van der Pluijm, and L. N. Warr, Nanocoatings of clay and802
creep of the San Andreas fault at Parkfield, California, Geology, 38 (7), 667–670,803
2010.804
Shearer, P. M., G. A. Prieto, and E. Hauksson, Comprehensive analysis of earthquake805
source spectra in southern California, J. Geophys. Res., 111 (B06303), 2006.806
Simpson, R. W., M. Barall, J. Langbein, J. R. Murray, and M. J. Rymer, San Andreas807
fault geometry in the Parkfield, California, region, Bull. Seismol. Soc. Am., 96,808
S28–S37, 2006.809
36
Thurber, C., H. Zhang, F. Waldhauser, J. L. Hardebeck, A. Michael, and D. Eberhart-810
Phillips, Three-dimensional compressional wavespeed model, earthquake reloca-811
tions, and focal mechanisms for the Parkfield, California, region, Bull. Seismol.812
Soc. Am., 96, S38–S49, 2006.813
Tinti, E., P. Spudich, and M. Cocco, Earthquake fracture energy inferred from kine-814
matic rupture models on extended faults, J. Geophys. Res., 110 (B12), B12,303,815
2005.816
Titus, S. J., C. DeMets, and B. Tikoff, New slip rate estimates for the creeping817
segment of the San Andreas fault, California, Geology, 33, 205–208, 2005.818
Urbancic, T. I., and R. P. Young, Space-time variation in source parameters of mining-819
induced seismi events with M ¡ 0, Bull. Seismol. Soc. Am., 83 (2), 378–397, 1993.820
Venkataraman, A., and H. Kanamori, Observational constraints on the fracture en-821
ergy of subduction zone earthquakes, J. Geophys. Res., 109 (B05302), 2004a.822
Venkataraman, A., and H. Kanamori, Effect of directivity on estimates of radiated823
seismic energy, J. Geophys. Res., 109 (B04301), 2004b.824
Venkataraman, A., L. Rivera, and H. Kanamori, Radiated energy from the 16 October825
1999 Hector Mine earthquake: Regional and teleseismic estimates, Bull. Seismol.826
Soc. Am., 92, 1256–1265, 2002.827
Venkataraman, A., G. C. Beroza, and J. Boatwright, A brief review of techniques used828
to estimate radiated seismic energy, Geophysical Monograph, 170, 15–24, 2006a.829
Venkataraman, A., G. C. Beroza, S. Ide, K. Imanishi, H. Ito, and Y. Iio, Measurements830
of spectral similarity for microearthquakes in western Nagano, Japan, J. Geophys.831
Res., 111, 2006b.832
Waldhauser, F., and W. L. Ellsworth, A double-difference earthquake location algo-833
rithm: Method and application to the northern Hayward fault, California, Bull.834
Seismol. Soc. Am., 90, 1353–1368, 2000.835
Waldhauser, F., and W. L. Ellsworth, Fault structure and mechanics of the Hayward836
fault, California, from double-difference earthquake locations, J. Geophys. Res.,837
107 (2054), 2002.838
Wesnousky, S. G., Seismological and structural evolution of strike-slip faults, 1988.839
Wesnousky, S. G., Predicting the endpoints of earthquake ruptures, Nature, 444,840
358–360, doi:10.1038, 2006.841
37
Wyss, M., and J. N. Brune, Seismic moment, stress, and source dimentions for earth-842
quakes in the california-nevada region, J. Geophys. Res., 73, 4681, 1968.843
Yamada, T., J. Mori, S. Ide, R. E. Abercrombie, H. Kawakata, M. Nakatani, Y. Iio,844
and H. Ogasawara, Stress drops and radiate seismic energies of microearthquakes845
in a South African gold mine, J. Geophys. Res., 112 (B03305), 2007.846
Zhuo, Y., and H. Kanamori, Regional variation of the short-period (1 to 10 second)847
source spectrum, Bull. Seismol. Soc. Am., 77, 514–529, 1987.848
38
A Analytical expression for radiated energy849
Following Ide and Beroza [2001], we re-derive the analytical expression for ER here850
leading to the expression for radiated energy given in Equation 5. Equation 1 of Ide851
& Beroza, 2001 is the expression for a velocity spectrum based on a Brune spectral852
model:853
ˆu(f) =M0
4πρα3r
f
[1 + (f/fc)2]=
Ω0f
[1 + (f/fc)2](A.1)854
with an assumed spectral falloff of n = 2. Note that the integrated spectral velocity855
in Equation 3 differs from that given in Ide and Beroza [2001] in that it contains the856
spectral amplitude measured at the station (Ω0) instead of M0. In order to obtain857
the seismic moment, the observable Ω0 must be multiplied by the source-receiver858
distance (r), as shown in Equation A.1. I in Equation 3 is therefore proportional to859
the radiated energy. The constant of proportionality is given by Equation 4.860
Calculating the integrated velocity I using the power spectral density of the veloc-861
ity spectrum measured at a single station as given by the expression in Equation 3,862
and assuming an ideal case of an unlimited bandwidth (i.e. integrating from 0 to863
infinite frequency),864
I =
∫ ∞−∞
ˆu2(f)df = 8π2Ω20
∫ ∞0
f 2
[1 + (f/fc)2]2df (A.2)865
I = 8π2Ω20f
4c
[arctan(f/fc)
2fc− f
2(f 2c + f 2)
]∣∣∣∣∣∞
0
. (A.3)866
If we define F (f, fc) as follows,867
39
F (f, fc) ≡−f/fc
1 + (f/fc)2+ arctan(f/fc), (A.4)868
then869
I = 8π2Ω20f
3c F (f, fc)
∣∣∣∣∣∞
0
= 2π2Ω20f
3c (π
2+ 0), (A.5)870
I = 2π3Ω20f
3c . (A.6)871
Substituting Equation A.6 into Equation 4 gives us:872
ER = 〈U2θφ〉R2
pψ8π4ρβr2Ω20f
3c . (A.7)873
Substituting the expression in Equation 2 relating Ω0 and r with M0 leads to the874
analytical expression for radiated energy in terms of moment and corner frequency875
given by Equation 5:876
ER =π2
5ρβ5M2
0 f3c . (A.8)877
40
Date Time, UTC Lat. Lon. Depth Magnitude Event ID fc (Hz)2004/09/05 07:05:04.86 35.7685 -120.319 8.369 1.5 21393520 14.42004/09/28 17:57:43.98 35.7784 -120.33 8.388 2.5 21400463 15.62004/09/28 19:40:23.6 35.7785 -120.33 8.649 2.6 21400518 16.12004/09/28 20:55:07.64 35.7818 -120.323 4.161 2.0 51148134 15.02004/09/28 21:49:23.56 35.7771 -120.329 8.575 2.7 21400488 7.42004/09/28 21:57:37.88 35.7983 -120.342 4.163 2.0 51148203 15.02004/09/29 18:59:59.92 35.7827 -120.334 8.328 1.8 51148847 15.22004/09/30 01:57:50.06 35.7812 -120.323 3.97 1.6 21400936 16.22004/10/07 14:32:03.64 35.781 -120.332 8.593 2.2 51150206 12.02004/10/18 06:22:57.07 35.7826 -120.334 8.192 1.8 51151460 14.92004/10/18 06:27:51.91 35.7822 -120.334 8.279 1.6 51151464 14.52004/10/19 15:52:17.48 35.7699 -120.321 8.619 1.9 51151568 14.62004/10/20 09:09:47.86 35.7978 -120.341 4.268 2.2 51151627 14.72004/10/29 03:32:43.73 35.781 -120.333 8.751 3.0 21415362 13.42004/11/03 14:42:54.09 35.7667 -120.318 8.324 1.6 21416910 4.52001/09/20 20:06:02.64 35.9347 -120.487 5.261 2.1 21194856 20.92004/09/28 18:37:16.76 35.9345 -120.487 5.216 1.6 51148006 19.32004/09/28 23:33:49.64 35.9347 -120.487 5.272 2.2 21400519 18.32004/10/02 23:25:08.72 35.9354 -120.487 5.349 2.3 51149613 18.52005/01/08 04:46:6.00 35.9346 -120.487 5.27 2.1 21432539 18.22004/02/04 14:29:57.39 36.0951 -120.66 7.518 1.2 21340072 24.82005/02/09 10:45:23.62 36.095 -120.66 7.518 1.6 21438412 20.42005/02/09 10:53:15.6 36.0953 -120.66 7.527 1.2 21438414 16.82006/03/11 08:25:42.98 36.0362 -120.596 4.82 2.2 21508986 13.52007/01/17 10:11:35.6 36.037 -120.595 4.83 2.1 51177776 17.92002/06/04 22:33:56.28 35.932 -120.676 10.163 2.1 21228776 23.52002/06/04 22:50:02.08 35.932 -120.676 10.168 2.1 21228784 16.62004/09/06 03:43:04.43 36.148 -120.653 4.398 2.6 21393628 6.12004/09/26 15:54:05.96 36.143 -120.666 4.48 3.2 21399972 4.82004/09/27 08:52:40.96 36.154 -120.658 4.821 2.5 21400160 9.72004/09/27 10:38:56.32 36.152 -120.658 4.32 2.0 21400192 12.22004/10/03 02:11:15.36 36.153 -120.658 4.347 1.5 21402827 13.72006/06/27 21:38:11.87 36.065 -120.192 16.95 3.9 21524551 8.72006/12/15 19:50:25.33 36.170 -120.298 9.96 3.2 51176831 9.72007/03/12 12:13:54.63 35.938 -120.691 8.89 1.4 40194471 31.62007/09/20 05:41:07.00 36.064 -120.194 14.34 2.0 40202213 8.8
Table 1: Parkfield earthquakes considered in this study, taken from the catalog ofThurber et al. [2006], and from the ANSS catalog. (Events in Mount St. Helens arenot cataloged). The 25 events located on the San Andreas Fault are grouped intothe three clusters from southeast to northwest (Figure 1). Each cluster is separatedby a single line. The 11 events located on secondary faults (bottom) are separatedby a double horizontal line. Duration is taken as the width of the source time pulse[Harrington and Brodsky , 2009], and corner frequencies are determined using thesource time function and a least squares fit to a Brune spectral model. Magnitudevalues are calculated according to the method described in the text.
41
moment, M0 (Nm) corner frequency, fc (Hz)2.8e10 8.31.4e11 8.13.6e10 9.14.7e10 13.34.6e10 11.62.8e10 9.71.0e11 8.42.4e10 11.95.6e10 9.34.3e10 10.91.2e11 5.13.6e10 9.63.1e10 9.66.6e10 13.36.3e10 10.08.2e10 6.74.5e10 9.94.7e10 9.45.0e9 12.15.4e10 8.94.7e10 13.02.7e10 15.04.6e10 13.04.0e10 12.86.0e10 7.95.0e10 12.24.1e10 8.59.7e10 10.03.7e10 12.89.8e10 10.39.8e10 8.71.1e11 9.72.2e10 13.14.7e10 11.42.8e10 11.24.6e10 11.43.8e10 11.55.5e10 10.94.7e10 11.05.5e10 9.8
Table 2: Seismic moment and corner frequency values of Mount St. Helens earth-quakes calculated here. Events are not cataloged, and therefore have no origin timeor epicentral location. Empirical Green’s function moment value calculated via thelong-period spectral amplitude, and remaining relative moment and corner frequencyvalues originate from a least-squares fit to a Brune spectral model of the source timefunction (see text). Source time functions determined with a water-level spectraldeconvolution.
42
0 5 10 km
120.7° W 120.6° W 120.5° W 120.4° W 120.3° W 120.2° W
35.8° N
35.9° N
36.0° N
36.1° N
36.2° N
mature
immature
1966 and 2004 mainshocks
HRSN Stations
Figure 1: Parkfield study area. The map shows the location of the mature (circles),and immature (diamonds) events used in our study. Symbols are scaled accordingto earthquake duration as determined from the source time function pulse width.Source time functions for these earthquakes are obtained by projected Landweberdeconvolution of a co-located earthquake that is at least one magnitude unit smallerthan the events shown. The large stars represent the 1966 and 2004 earthquakeepicenters. Symbol shapes and colors follow the same convention in the subsequentfigures.
43
Figure 2: Mount St. Helens study area. Grayscale indicates elevation in feet.The map shows the location of cataloged earthquakes occurring from 2/26/2005 to3/8/2005 (the time of our data set). Note that events shown here are meant to give anindication of the location of cataloged seismicity during the time period of our datacollection. They do not correspond to the exact location of the events in our data set.(Earthquakes analyzed here are below the catalog threshold). S-P wave arrival times,as well as a lack of seismicity elsewhere in the volcanic edifice indicate that theirlocations are in the crater. Stations MIDE and NED of the Cascades Chain networkare shown as white triangles. Additional stations used for S-wave velocity estimationindicated by white squares (the 7th additional station used, JUN, is located to thenorth, outside the map boundary, at 45.15 N, 122.22 W).
44
−101
x 105
VCAB vertical
Main Event
−101
x 105
VCAB horizontal
0 5 10
−101
x 105
VCAB horizontal
−10000
1000
eGf
−10000
1000
0 5 10
−10000
1000
0 2 4 6−500
0
500MSH
MIDE vertical
time (s)0 2 4
−20
0
20MSH
vert
ical
time (s)
Figure 3: Examples of two earthquake-empirical Green’s functions (eGf) pairs fromstation VCAB in Parkfield (three components shown in the top three rows) andstation MIDE at Mount St. Helens (bottom row). eGf’s shown in the right column,and events for which the source-time function is calculated shown in left column. Themain event and eGF Mw values respectively for Parkfield are 2.1, and 0.8, and forMount St. Helens are 1.2, and 0.07.
45
0 1 2 3 4
0
2000
4000VCAB vertical
0 1 2 3 4
0
2000
4000VCAB horizontal
0 1 2 3 4
0
2000
4000VCAB horizontal
Am
plitu
de R
atio
0 1 2 3 4
0
1000
2000MIDE vertical
time (s)
100
101
102
100
101
102
103
spec
tral
am
p. r
atio
VCAB
100
101
100
101
102
103
spec
tral
am
p. r
atio
frequency (Hz)
MIDE
Figure 4: Left: source-time functions calculated using the event pairs shown in Fig-ure 3. Three components for station VCAB shown in top three rows. The source-timefunctions in Parkfield determined using a Projected Landweber deconvolution, andthose at Mount St. Helens determined using a water level deconvolution. Water leveldeconvolution in Parkfield depicted as light gray line in top three panels on the leftto show consistency with the Projected Landweber deconvolution (see text). Source-time function from MIDE vertical component shown on the bottom. Right: Sourcespectra calculated from the three-component, RMS-spectra at station VCAB (top)and MIDE bottom. The least-squares fit to a Brune spectra shown with dashed line,and the noise spectra shown with a thin, gray line.
46
0.5 1 1.5 2 2.5 3 3.5 4 4.50.5
1
1.5
2
2.5
3
3.5
4
4.5
Mc (catalog)
Mw
(ca
lcul
ated
)
matureimmatureimmature (extra stations)1:1
Figure 5: Moment magnitude values (MW ) calculated in this study compared withcatalog magnitude values (predominantly duration magnitudes Md) [Thurber et al.,2006]. The average ratio of Mw : Mc is 0.92±0.04 for the mature (blue) events,and 1.05±0.13 for the immature events (red diamonds). The scatter in immatureevents likely results from a lack of azimuthal coverage from HRSN stations (Figure 1).The red squares indicate Mw calculations based on moment values estimated usingadditional stations (see text).
47
109
1010
1011
1012
1013
1014
1015
100
101
102
moment (Nm)
f c (H
z)
constant ∆ σM
0 ∝ f
c−0.03
Figure 6: Seismic moment vs. corner frequency estimated using a least-squared fit toa Brune spectral model. Red diamonds and squares indicate the immature populationin Mount St. Helens and Parkfield respectively. Blue circles represent the maturepopulation in Parkfield. Filled symbols indicate seismic moments estimated here usinglong-period spectral ratios, and open symbols indicate moment values calculated usingconverted catalog values for comparison. Error bars not visible on filled symbols aresmaller than the symbol size. The dashed line indicates a least squares fit to themoment-corner frequency trend for the mature population, and the fit indicates thatfc is independent of M0 to within error for the mature population. The solid linerepresents lines of constant stress drop (∆σ = 2 MPa, β = 800 m/s for MSH, and∆σ = 10 MPa, β = 3750 m/s for Parkfield). Immature events in Parkfield and MSHfollow two separate lines of constant stress drop because the shear wave velocity atMSH is roughly four times slower than at Parkfield.
48
1e−9
1e−8
1e−7
1e−6
1e−5
1e−4
1e−3
ER/M
0
109
1010
1011
1012
1013
1014
1015
1016
1e−8
1e−7
1e−6
1e−5
1e−4
1e−3
EG
/M0
immature (Parkfield)immature (MSH)mature (Parkfield)immature (catalog moment)mature (catalog moment)slope =0.9 ± 0.16slope = −0.2 ± 0.08
Figure 7: Ratio of radiated seismic energy vs. seismic moment (ER
M0, left axis) and
ratio of seismically observable fracture energy (EG
M0, right axis) vs. earthquake size.
Earthquakes on immature faults are represented by filled diamonds, and squares.Earthquakes on a mature fault are represented by filled circles. Error bars not visibleon filled symbols are smaller than the symbol size. Values of moment taken from thecatalog are shown for comparison as open symbols. A comparison between valuesfrom earthquakes on mature, vs. immature faults indicates that the ratio remainsroughly constant for earthquakes on immature, or rougher faults, (as expected forconstant stress drop scaling) and decreases with moment for earthquakes on mature,smoother faults.
49
100
102
104
106
108
1010
1012
1014
1016
1018
1020
1022
1e−10
1e−9
1e−8
1e−7
1e−6
1e−5
1e−4
1e−3
ER/M
0 (B
rune
mod
el)
Moment (Nm)
1e−9
1e−8
1e−7
1e−6
1e−5
1e−4
1e−3
EG
/M0immature (Parkfield)
immature (MSH)Yamada et al., 2005Kwiatek et al., submittedUbancic et al., 1993Gibowicz et al., 1991McGarr et al., 2010matureAbercrombie 1995 (Model 2)Abercrombie & Rice, 2005Mayeda and Walter, 1996slope = 0.3 ± 0.01slope = 0.07 ± 0.009
Figure 8: Figure 7 shown together with the data points [Yamada et al., 2007; Kwiateket al., 2011; Gibowicz et al., 1991; Urbancic and Young , 1993; McGarr et al., 2010;Abercrombie, 1995; Abercrombie and Rice, 2005; Mayeda and Walter , 1996] for com-parison, suggesting that our results are a general feature. Events comprised largely ofmining induced seismicity are representative of an immature population, while eventsin the western US (San Andreas and Basin and Range faulting systems) are repre-sentative of a mature population (see text). We estimate the size dependence of thecombined immature and mature populations using a linear regression. The estimatedslope values shown in the legend are for the combined data sets.
50
