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Article |
1 Department of Earth and Space
Sciences and
Center for Embedded Networked Sensing
University of
California, Los Angeles (UCLA)
Los Angeles, California 90095-1567
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Abstract |
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 Top Abstract IntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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2 source model. The S-wave attenuation was
separable into basin and bedrock contributions. In addition to the body-wave
analysis, S-wave coda were analyzed for coda Q and
coda-determined site effects. We find S- wave Q
(QS) in bedrock is higher than in the basin. High-frequency
QS is higher than low-frequency QS. Coda
Q (Qc) is higher than QS.
P-wave Q (QP) was not separable into
basement and bedrock values, so we determined an average value only. The corner
frequencies for P and S waves were found to be nearly the
same. The standard model fit over 97% of the S-wave data, but data from
six clustered events incident along the basin edge within a restricted range of
incidence and azimuth angles generated anomalous amplitudes of up to a factor of
5 higher than predicted. We test whether such basin-edge focusing might be
modeled by catastrophe theory. After ruling out site, attenuation, and radiation
effects, we conclude a caustic modeled as a diffraction catastrophe could
explain both the frequency and spatial dependence of the anomalous
variation. |
Introduction |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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To identify caustics it is necessary to examine other factors that may affect seismic amplitudes differentially, such as site effects, attenuation, and radiation pattern. Attenuation is separated into its near-station basin and basement contributions to more accurately account for its effect (Hough et al., 1988). We find that the radiation pattern is not evident in the spectral amplitudes we measured, possibly because of scattered secondary arrivals. We estimate site effects using relative coda amplitudes and find they are at a maximum where the basins are deepest, but they do not explain basin-edge effects. We conclude that the anomalous arrivals at the Los Angeles basin edge are due to seismic focusing.
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Data |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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More than 2000 events with ML >2 were recorded and cataloged during the experiment. Events were used in this study if the direct P or direct S wave was clearly distinguishable above the noise on at least half of the stations. We chose to study the 40 P-wave and 43 S-wave events that matched the criteria best. They ranged in magnitude from 2.1 to 4.8 (Table 1). Locations of the events and the stations are shown in Figure 1. Event 19 occurred 2 sec after event 18 (Fig. 2) and is not listed in the Southern California Seismic Network (SCSN) catalog. We put its location in the same place as event 18 because both P and S waves at all stations occur 2 sec after those waves for event 18 and the waveforms look quite similar.
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The data were corrected for (L4C) instrument response and converted to a flat velocity spectrum between 1 and 10 Hz. The seismograms were converted to spectral amplitudes by taking an fast Fourier transform (FFT) of a 2-sec window centered on the beginning of the first P wave and a 4-sec window centered on the beginning of the S wave. The FFTs were broken into two frequency bands for the inversion: 1– 5 Hz and 5–10 Hz. Average spectral amplitudes were used in each band to examine amplitude variation with frequency.
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Development of the Spectral Amplitude Model |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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| (1) |
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| (2) |
/2QC were fit to the
absolute values of the coda. QC is coda Q and
Aij is noise-corrected spectral amplitudes measured over 10
contiguous 4-sec windows starting at t0 = 80 sec
after the time of the rupture
(e.g., Steck et al., 1989).
The method of
Steck et al. (1989)
differs from ours by using a start time of twice the time from the earthquake to
the first S-wave arrival at the station. We chose to measure the same
time windows for all event and station combinations to have the same amount of
geometrical spreading at each site.
Noise-corrected spectral amplitudes are given by
|
| (3) |
is the
summation of the amplitudes squared from the FFT over a 4-sec window
from the coda for a given frequency range,
is the
noise-squared summation taken over the first 4 sec before the rupture in the
same frequency range, and N is the number of frequencies in the
FFT. If 
was less than 2.0 times
the data point was
not used because of a presumed poor signal-to-noise ratio. The high- frequency
(5–10 Hz) data in the tail of the coda (+100 sec after the rupture)
had very few points that met the signal-to- noise ratio criterion. Therefore,
only the low-frequency (1– 5 Hz) Aij was used for site
terms in equation (2). The noise-
corrected Aij was fit to
equation (2) to determine
aij and b. An initial fit was used to determine the
mean value of b, which in turn yielded QC. Values
of aij at each site were averaged to give
ai. The logarithms of the relative site factors were
obtained from the differences in ai. The resulting coda-
derived site factors are plotted in
Figure 3.
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Standard Model: S Waves
The standard model for spectral amplitudes is based on the
2 source
(Aki, 1967;
Brune, 1970) which, in general,
has been found to fit spectra from local earthquakes
(e.g., Abercrombie, 1995). We
assume that geometrical spreading causes amplitudes to decay as inverse distance
traveled and that the distance traveled is proportional to travel time.
Attenuation is separated into basin and bedrock paths and into the two different
frequency components, with a different Q value assigned to each. A
nonlinear least-squares inversion was applied to the spectral amplitude data
using
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| (4) |
is the mean angular frequency for each band, and
0j is the corner angular frequency. The mean
frequency of the two bands was set as the average of the band (i.e., for the
1–5 Hz band = 2
x 3.3 Hz, and
for 5–10 Hz = 2 x 7.0 Hz).
Q is the quality factor. The index l = 1 or 2 denotes
basin or bedrock, respectively. The Tij values are times
between earthquake initiation and the first S-wave arrival. The total
travel time is Tij =
tij1 + tij2,
where tij1 and tij2
are the basin and bedrock travel times, respectively. Travel times were
determined by ray tracing through a 1D linearized version of the Southern
California Earthquake Center (SCEC) velocity model as discussed
below.
Corner frequency, 0, can be interpreted as the
ratio of rupture velocity, VR, to the fault length,
lj, multiplied by a constant, 
(e.g., Stacey, 1992).
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| (5) |
S, was found to be 0.88 ± 0.1. The corner
frequencies are shown in
Figure 4 and are similar to
other estimates for this magnitude range
(e.g., Abercrombie, 1995).
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Travel times were determined analytically using a linearized version of the
1D velocity profile beneath each station fit to the SCEC 3D Velocity
Model
(Magistrale et al., 2000)
(Fig. 5). For velocity,
v, increasing linearly with depth, z, as
v(z) = a + bz, the analytical
solution (Lay and Wallace, 1995)
for horizontal distance traveled is
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| (6) |
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| (7) |
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Figure 6a and b shows the spectral amplitude data for each earthquake plotted against stations 1–18 (numbers increasing from the San Gabriel Mountains to Seal Beach) for the low (Fig. 6a) and high-frequency (Fig. 6b) bands. Arrivals from a cluster of six earthquakes that occurred east of Whittier, approximately 15 km from station 11, are readily identified by the well-developed spikes in the amplitude pattern affecting stations 9, 10, and 11 on the northern edge of the Los Angeles basin and into the Puente Hills. Maximum values occur at station 11. These arrivals all come from a narrow range of back azimuths of 110° to 130° (Fig. 6a and b) and incident angles. All the events that were amplified had ray paths near the same location on the edge of the Los Angeles basin. Some events from the opposite direction also exhibited high amplitudes at station 11. The fit of the standard model (equation 4) to the data, with site factors Si estimated independently from coda (as described in the preceding section) is shown as the solid line in Figure 6. The misfit at stations 9–11 is remarkable. The largest of the S- wave amplifications occur at station 11. Seismograms for event 21 (Fig. 6a and b) were recorded only on the north– south component of that seismometer, because the east–west component was faulty during this period. Given this, it is natural to question the validity of the measurement. However, other events were recorded on the same seismometer while the eastern component was faulty that do not have the amplification. In addition, several other events exhibit a large amplification at that site from similar backazimuths and incidence angles when both components were working. Waveforms from the most focused event (event 21), at station 11 are shown in Figure 7a compared with those from a nonfocused event (Fig. 7b, event 3). The remarkable amplification at station 11 is evident in the raw data (Fig. 7a).
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Focal Mechanisms
A term to account for radiation patterns was applied to those events for
which focal mechanisms had been compiled
(Hauksson, 2000;
Table 1).
Equation (4) becomes
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| (8) |
,
f) is the
amplification due to the radiation pattern of an earthquake for a given station,
f is the forward azimuth, and is the
takeoff angle (Aki and Richards, 2003). The addition of the radiation term did
not improve the fit to the data; in fact, it increased the misfit of the
inversion. This could be caused by several factors. Determination of the takeoff
angles was based on a simple model of crustal velocities that could be
incorrect. Another possibility is that the fault-plane solutions for small
events might be inaccurate. Even if either of these possibilities applies, the
takeoff angles vary little over the array for a given event, causing little
change in the radiation pattern. The more likely explanation is that the
radiation pattern is not strong because of scattering in the frequency ranges
studied. Given that our data are average spectra of 2- and 4-sec data windows,
they probably include energy from both primary phases and scattered phases and
may lose the radiation pattern associated with first arrivals. Because our tests
indicate that it is unlikely that focal mechanisms could explain the anomalous
variation at stations 9–11, radiation patterns were not included in our
final analysis.
P waves
Unlike the S waves, no significant azimuthal concentration in
P-wave amplification was seen at station 11
(Fig. 8a and b). High
amplification was seen at station 6, which sits in the middle of the San Gabriel
basin, but this appeared to be explicable as a site effect. It is difficult to
tell whether a caustic interpretation might be applicable without an independent
estimate of P-site factors as we had for S waves using
S coda. S-wave site factors exhibit a maximum at station 6, so
site factor seems a reasonable explanation.
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After performing various inversions using equation (4) we found we were unable to resolve QP in the basin and in the bedrock because the least-squares uncertainties in the respective QP values are greater than the inverted values. We thus restricted the inversion to solve for low- and high- frequency values averaged over the complete path.
The fit to the 1126 P-wave data
(Fig. 8a and b) is reasonably
good. The inverted parameters include 40 event terms, 18 site factors, the
corner frequency factor, and the low- and high-frequency QP
values. The variance reduction is 81%. The Q values are low at 96
± 11 and 248 ± 33. The frequency factor
P = 1.0 ± 0.1 is very similar to the
S-wave value of S = 0.88 ± 0.1.
Although not a central part of our analysis because of the limited range of
frequencies used (1–10 Hz), the fact that both P and S
waves give very similar results suggests that when all factors are taken into
account the simple "Brune" model, which predicts
P = S, works quite
well.
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Diffraction Catastrophes |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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Caustics formed by finite-frequency monochromatic radiation generate Airy and
Pearcey diffraction patterns referred to as "diffraction
catastrophes"
(Berry, 1976). The maximum
amplitudes of diffraction catastrophes are frequency dependent
(Berry, 1976;
Rial, 1984). The amplitude of
the fold goes as 1/6, the cusp as
1/4, the swallowtail as
3/10, and the umbilics as 1/3. In
contrast, the amplitude of the point focus is proportional to
1 and the line focus to
1/2 (see
Davis et al., 2000 for
application to Santa Monica basin-edge data).
Development of the Caustic Model
In this section we develop an empirical diffraction catastrophe model for the
anomalous arrivals. The sharp peaks
(Fig. 6) suggested a 1/distance
fall-off in amplitude, similar to what is seen in Airy and Pearcey diffraction
patterns. When multiwavelength radiation is involved, such as in seismology, the
nodes of the Airy or Pearcey diffraction patterns will blur. In the
infinite-frequency case,
Chapman (1976) and
Richards (1973) derived a
1/distance fall-off in amplitude for seismic waves forming a caustic by using
the Wentzal, Kramers, Jeffreys, and Brillouin (WKJB) approximation to
the wave function in a vertical velocity gradient (i.e., triplications). A clue
that catastrophe theory might apply is that the amplitude increase with
frequency
(Fig. 6a compared with b) is
evident but fairly small, suggesting a small exponent, n, in the
n frequency dependence. We have observed that
earthquakes with different travel times showed much greater focusing (e.g.,
event 20), so we included a term to account for travel-time dependence of the
maximum amplitude. Taking these considerations into account we used a simple
empirical function to model the spectral amplitudes. We envision a
high-amplitude zone or bright spot appearing on the surface in the vicinity of
station 11 with amplitude determined by how close the travel time, incidence,
and azimuth angles are to a maximally focusing path. The amplitude decays from
the bright spot across the surface as 1/distance. The brightness has a power-law
frequency dependence n but also depends on the
earthquake being at the right distance as reflected in the travel time.
Equation (4) then becomes
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| (9) |
,) has been added to take into account
caustic effects from a restricted range of incident angles and
azimuths and is given by
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| (10) |
is the incidence angle, and is the backazimuth.
0 and 0 are values where the
caustic would be at a maximum (i.e., next to station 11). T0
is the travel time for which the amplitude is perfectly focused. We tried a fit
with and without this term, but by far the best results occur when it is
included. Its inclusion is empirical, however, and takes into account optimal
location of the source for focusing. The maximum of
equation (9) is infinity, but it
is assumed that [T0, 0,
0] represents a point offset both temporally and
spatially from station 11 and that diffraction would blunt the infinity.
Incidence angle ij and distance
Rij are measured at the depth where the caustic is
generated, that is, near the bedrock- sediment boundary. For LABPSE
data that is the depth where the S-wave velocity becomes 3.3 km/sec
(5.6 km under station 11). The unknowns in this equation are
T0, C0, 0,
0, and n. As we shall see this equation fits
the data remarkably well, but it must be kept in mind that we are fitting 34
high- amplitude data points (six events at stations 9–11 in two frequency
bands minus errant components) with five parameters, so the redundancy of data
to parameters is marginal. |
Test of the Caustic Model Using Spectral Ratios |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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Let Sij() be the S-wave
spectrum (at 51 frequencies) at station i for event j. The
spectral ratio Cj() for the caustic
events measured at station 11, relative to the average spectra of stations
1–6, is given by
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| (11) |
() for the caustic events (upper curve,
j = 17–22) and the noncaustic events measured at
station 11 (lower curve, j = 1–16, 23–43). We see
that the average spectral ratio for the noncaustic events is flat, which
supports our choice of stations 1–6 as a reference. For the caustic events
however, () increases with
frequency and a fit of
equation (10) gives an exponent
of 0.25 ± 0.06. The amplification is several times that of the noncaustic
events. This is what is expected if the source of the extra amplification is a
diffraction catastrophe cusp. The errors are large enough that a fold,
swallowtail, or umbilic cannot be ruled out, however. In the next section on
data inversion, we use n = 0.25. |
S Wave Inversion Results Including a Cusp |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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s. In some inversions, greater weight
was given to the peak caustic data (by a factor of 4) in the inversion. The site
terms were constrained to be those found in the coda analysis (see Coda
Q and Site Factors section) by adding the coda values as a
priori data in a nonlinear Bayesian inversion
(Jackson and Matsuura, 1985).
Without this constraint the inverted site terms exhibited similar spikes to the
caustics, but less strong. This occurred because arrivals from other azimuths do
not show caustic effects. They represent a compromise in fitting the caustic and
noncaustic data, but do a poor job for both. The inversion results and standard
deviations are listed in
Tables 1,2,3.
The variance reduction is 81%. The fit of the model is shown in
Figure 10a and b.
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For the caustic model with frequency dependence, the exponent was set to
n = 0.25 based on the spectral ratio analysis (see preceding
section), we obtain 0 = 114° and
0 = 78°. For comparison, the values for
the highest observed peak seen in the data (station 11, event 22) are
11,22 = 117° and
11,22 = 69°.
The Q values given in Table 2 show greater attenuation in the basin versus the bedrock, and higher attenuation of the low-frequency band than the high-frequency band. Table 2 also lists the attenuation values with the coda Q values and those obtained by Frankel (1991), showing that we obtained similar results where the coda Q was much higher than the direct-wave Q.
Finally, to test whether the inversion results were sensitive to the frequency bands chosen, we repeated the procedure for four bands (1–3, 3–6, 6–9, and 9–12 Hz). We obtained similar inversion results but the fits were very noisy especially at the outer bands where the signal-to-noise ratio is lowest.
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Discussion of Results |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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We find, as did Frankel (1991), that QC is twice as large as QS. Average values of QS in basin and bedrock, QP and QC from the inversion, are given in Table 2. QS was found to be greater by about a factor of 3 in bedrock than in the basin. High-frequency Q is greater than low-frequency Q for both P and S waves. We can make no direct comparison of QP and QS because QP was not separable into basin and bedrock. We compare our attenuation values with those given by Frankel (1991) for southern California (Table 2). We cannot directly compare our results with Mayeda et al.’s (1992) frequency dependence of intrinsic and scattering attenuation results because we analyze different parts of the waveform. However, with our less detailed study of attenuation, we can confirm the conclusions of Frankel (1991). Like Frankel (1991) we find that the coda decays at a slower rate than the direct wave front (i.e., QC > QS; see Table 2). In addition, high-frequency energy, which is trapped more effectively, decays at a slower rate than low-frequency energy (i.e., Qhigh freq > Qlow freq; see Table 2). Our results differ from Frankel’s (1991) in that QS values were relatively similar but our values of QC are greater, supporting the view that more scattering occurs in the Los Angeles basin. This probably occurs because the Los Angeles basin area has more unconsolidated material than the region to the east where his array was centered and therefore experiences more scattering.
Our observation that a diffraction catastrophe occurred at the edge of the Los Angeles basin and into the Puente Hills relies on ruling out site effects. We were alerted to this possibility when we noticed that arrivals at stations 9–11 were only amplified from critical azimuths, that is, those along the basin edge. We estimated site affects from coda decay amplitudes which showed an overall correspondence with basin depth, but site effects did not peak at stations where the largest amplitudes were observed. At the edge of the Los Angeles basin there is a horizontal positive velocity gradient out of the basin to the north, in addition to the positive velocity gradient with depth. We infer that multipathing caused by diving rays in addition to rays redirected horizontally gave rise to the caustic.
By inspection of
Figure 6a and b we see that the
anomalous amplitudes increase by about 20% when frequency doubles, accounting
for the frequency exponent n = 0.25 ± 0.06 in
equation (10). This value
corresponds to a cusp catastrophe (n = 0.25) but the range of
0.19 to 0.31 admits other possibilities. The lowest theoretical exponent
(n = 0.1667) is the fold catastrophe, which has amplitude
variation described by the Airy function given by
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| (12) |
Coda site factors, in general, gave good estimates of S- wave site factors and averaged out azimuthal effects (Fig. 3). Using this method to determine site factors, however, assumes, that the scattering model is common to all earthquake-site pairs and that the only difference in the coda is the local amplification at the site. The very low coda-derived site factors at nonbasin stations 1–3 in the San Gabriel foothills, compared with those of the joint inversion, suggest that the scattering is less and the coda-derived site terms are biased low. Using a combination of both direct-wave site factors Si and coda values in the constrained inversion eliminates contamination from both a nonuniform scattering model and caustic effects.
We do not understand why the strong basin-edge amplification was only seen in the S waves and not the P waves, but we suppose that the spatial variation in the P/S velocity ratio is enough to eliminate the caustic in the P waves. This contrasts with the observation in Santa Monica data (Gao et al., 1996) that focusing occurred for both P and S waves. We also note that the Santa Monica data were interpreted in terms of point foci of lens theory (Davis et al., 2000). Our work suggests that an interpretation in terms of diffraction catastrophes is probably more applicable (J. F. Nye, personal comm., 2001). Both exhibit a 1/distance fall-off, the main difference being that the exponent in the frequency dependence is n = 1/6 to 1/3 for a diffraction catastrophe and n = 1 for a lens. This and the greater stability of catastrophes make them more likely candidates for explaining bright spots in a natural system.
Other mechanisms that might have given rise to observed high amplifications include trapped waves in the Whittier fault system that runs along the Puente Hills or highly azimuth-dependent resonant site effects. These effects appear to be less likely but are not easily ruled out with the available data set. Complete understanding of this phenomenon will require installation of 2D seismic arrays along basin edges. Because seismic waves become uncorrelated at distances greater than a wavelength, the station spacing of an ideal array would be on the order of 100 meters rather than the several kilometer spacing as used here.
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Conclusion |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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2 source model and 1/distance geometric
spreading. After taking these factors into account we found evidence for a
caustic on the northern edge of the Los Angeles basin from a group of
earthquakes incident along the basin edge. We determined QS in different spatial regions (bedrock and basin) by weighting the attenuation factor by the time the ray was in bedrock or basin as determined by ray tracing through the SCEC velocity model. We found differences in Q during different temporal sections of the waveform (P, S, direct waves, and coda) and attributed them to the effects of scattering. We found that the standard model worked well for 97% of the data except for specific path effects due to basin-edge structures from a narrow range of backazimuths. Stations on the northern end of the Los Angeles basin (stations 9–11) exhibit a basin-edge effect for S waves that can be accounted for by the addition of a caustic modeled as a diffraction catastrophe. Our interpretation that the anomalous basin arrivals form a caustic is based on the following observations: (1) site effects, radiation pattern, and attenuation were unable to explain the anomalous arrivals; (2) the spatial decay in amplitude corresponds to an inverse distance variation; (3) the frequency dependence fits that expected from a diffraction catastrophe; and (4) the basin edge has both vertical and horizontal gradients that can lead to multipathing to generate a caustic. Diffraction catastrophes may be quite common in regions of lateral heterogeneity and may account for anomalous damage such as that from the 1967 Caracas and 1994 Northridge earthquakes. Seismic array experiments using 2D networks with stations at 100-m spacing are needed to fully resolve the phenomenon. Earthquake engineers need to be aware of the potential consequences of diffraction catastrophes on the amplitudes of ground shaking because standard models may underestimate localized amplification by a factor of up to 5.
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Appendix |
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TopAbstractIntroductionDataDevelopment of the Spectral...Diffraction CatastrophesTest of the Caustic...S Wave Inversion Results...Discussion of ResultsConclusionAppendix |
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Manuscript received November 18, 2004
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