Geophys. J . R.astr. SOC.(197.5) 40,85-106
Regional Attenuation of Short-Period P and S Waves
in the United States
Zoltan A. Der, Robert P. Mass6 and John P. Gurski
(Received 1974 July 19)*
Summary
Regional distribution of anelastic attenuation beneath the United States
was investigated using amplitudes and dominant periods of short period
P and S waves originating from deep focus earthquakes in South America
and the Circumpacific seismic belt, and recorded at LRSM (Long Range
Seismic Measurement) stations. The observed regional distribution
pattern shows high attenuation in the western United States, including
California, and a less pronounced higher attenuation region in the northeastern United States. This distribution pattern is similar to that reported
by Solomon & Toksoz for long period S waves, but differs from it
sufficiently to indicate lateral variations in the frequency dependence of the
average crust-upper mantle attenuation across the United States.
Introduction
As a result of geophysical investigations over the past decade, the existence of
lateral variations in the crust and upper mantle has been well established. However,
many questions concerning the exact lateral variation of the physical properties of the
Earth remain to be answered. Particularly sensitive to such lateral variations in Q
structure are the spectra of short-period body waves from teleseismic events.
In this paper the variation of short-period P and S wave amplitudes across the
United States is investigated. A thorough study of the attenuation of long period body
waves has been presented by Solomon & Toksoz (1970) which we will use as a basis
for comparison. This study showed that for long-period body waves there exist two
regions with high attenuation in the United States, one in the north-eastern part of
the country, the other in the west. Low attenuation characterizes most of the remaining United States east of the Rocky Mountains and a strip along the Pacific coast
including California and a great part of Oregon. We wish to compare this regional
attenuation pattern with short-period observations and test whether the attenuation
patterns are identical or differ indicating that patterns are frequency dependent.
Since digitization and Fourier analysis of all of our data would have been a formidable task we decided to measure a few simple parameters of the short-period waveforms instead, the maximum amplitudes and dominant wave periods of P and S
waves. It can be expected that anelastic attenuation would reduce the amplitudes of
these waves and increase the dominant period. It was found, however, that P-wave
period was difficult to measure and had very little value in the analysis; therefore it
* Received in original form 1974 April 3.
85
Z.A. Der, R. P. Mass6 and J. P. Gurski
86
was not used. Thus three parameters were left and we let the statistical analysis
decide their relative importance.
When working with short-period body wave data one can expect considerably
more scatter than for similar data involving long-period observations, due to the
sensitivity of such data to small scale inhomogeneities. Multipathing, radiation
patterns, scattering and local layering all affect short-period body wave amplitudes
and dominant periods. It is not possible to account or correct for all these effects
at the present time. In spite of this we hope to show that our data are dominated by the
regional distribution of anelastic attenuation and it is extremely improbable that the
other factors mentioned above could fortuitously cause the distribution pattern of
the variables measured. Many recent studies (for example Oliver & Isacks 1967;
Molnar & Oliver 1969) also indicate that simple measurements performed on shortperiod body waves can give meaningful and useful results. A final test of the validity
of the results is the comparison with other types of geophysical data.
The measurement of seismic wave attenuation has structural implications since
the upper mantle low-velocity channel is generally associated with the region of high
attenuation (low Q). Therefore the extent and thickness of the low-velocity channel
may possibly be indicated by the geographical variation in the attenuation of the
short-period body waves. If this is so, then such information could be used to determine if the large low-velocity channel beneath the Western United States thins
gradually towards the east as suggested by Green & Hales (1968) or if it abruptly
decreases in size as proposed by Mass6 (1973), or disappears entirely.
Variation of the attenuation with frequency is diagnostic of the attenuation
mechanism and, in this respect, data from this study provides new information for
the short-period end of the spectrum. The dependence of attenuation on frequency is
obvious for, as noted by Solomon, Ward & Toksoz (1970), short-period P and S
waves would not be observed at all if the Q values appropriate for long-period S and
surface waves were independent of frequency. Therefore, the Q of seismic waves
can be expected to increase slowly with increasing frequency from the values determined using long-period observations. This increase of Q with frequency has also
been found in the study of long-period S waves by Sat0 & Espinosa (1967) and
Archambeau, Flinn & Lambert (1969).
Attenuation of seismic waves in the upper mantle is usually attributed to the
presence of melt in the rocks (Jackson 1969; Walsh 1968). The frequency dependence
of attenuation is indicative of the percentage and viscosity of the melt, the shape of
inclusions, and the values of various other parameters of geophysical interest. Since
all of these parameters can change from area to area, the frequency dependence of
attenuation can also change. Therefore attenuation maps for various period ranges
can give a considerable amount of information about the upper mantle.
Attenuation information for long-period P and S waves and long-period surface
waves is available from a number of previous studies (Solomon & Toksoz 1970;
Solomon 1972; Anderson & Archambeau 1964; Anderson, Ben-Menahem & Archambeau 1965). Variations of short-period attenuation have been used to study structure
in subduction zones (Oliver & Isacks 1967), but studies of short-period P and S wave
attenuation over wide areas are few. Studies such as those by Molnar & Oliver
(1970) and Sutton, Mitronovas & Pomeroy (1967) involve guided waves which tend
to give at best an average attenuation value over the path for only the crust and
uppermost mantle immediately below the MohoroviEiC discontinuity, possibly
modified by variations in the waveguide.
Data
Five deep earthquakes have been selected for analysis in this study. Epicentral
Regional attenuation of short-period P and S waves
87
Table 1
Events studied
Date
Region
1962 Sept. 29
1963 Nov. 11
1964 March 18
1964 June 30
1964 Nov. 28
Argentina
Peru-Brazil Border
NW Kuriles
Sea of Okhotsk
Western Brazil
Latitude Longitude
27.0s
9.1 S
52.5N
46.6N
7.7s
63.6W
71 $ 4W
153.6E
144.6E
71.2W
Magnitude Depth
(km)
6.5
575
4.9
585
5.6
400
5.5
383
5.4
626
Origin
time
15:17:47.7
19:54:09.4
04:37:26.9
20:08:28.5
16:41:33.4
data for these events are given in Table 1. Three of these earthquakes occurred in
South America and two on the far side of the Circumpacific belt of earthquakes.
This distribution of epicentres helps to minimize the azimuthal effects due to source or
near-source structure radiation patterns since it is unlikely that sources in the two
regions would behave the same way. Deep earthquakes with foci below most of the
asthenosphere were chosen since presumably energy from these events can freely
radiate into the high Q portion of the mantle and thus most attenuation will occur in
the ascending part of the ray path. Short-period recordings at all available stations
which were part of the Long Range Seismic Measurements Program (LRSM)
within the continental United States have been analysed. Maximum peak-to-peak
amplitudes of short-period S waves and the dominant periods of the waves at the
points of amplitude measurements were read on all available seismometers components. P amplitudes were read on the vertical component. Where no wave could
be seen, the noise background was read to establish an upper amplitude limit for the
wave. The amplitudes read were corrected for the instrument magnification at the
dominant period of the waves. The dominant period of S waves was defined as the
mean of the readings on all components where the wave could be identified. An
approximate distance correction for both P and S waves was also applied based on
the formula A - exp (-0.0148P). This correction was derived from the smoothed
Gutenberg body wave magnitude B factors, and was used to reduce the amplitudes to
the equivalent amplitude at the arbitrarily chosen distance of 50". The amount of the
distance correction is minor compared to the amplitude variations observed, and
never exceeds a factor of 1.5. The S-wave amplitudes were then determined by taking
the square root of the sum of the squared component amplitude readings. Table 2
shows the amplitudes corrected for distance effects and the dominant P- and S-wave
periods for each event as observed at various stations. The location of stations used
in this study is plotted in Fig. 1.
It was not possible to measure all of the parameters at each station. P-wave
amplitudes were frequently clipped when good quality short period S waves were
observed; there is a case when only P-wave ampIitude was measured. Noise background measurements at the time of the expected arrival time of a phase are meaningful
if the noise level is low since they establish an upper amplitude limit for that phase.
Such limits, however, are not well suited for statistical analysis and have to be considered separately.
Figs 2-1 1 show the original data plotted in two ways, in the first group S amplitudes are plotted against S-wave periods, in the second group P-wave amplitudes are
plotted against S wave amplitudes. Inspection of these figures immediately reveals
that stations in the western United States tend to have longer S wave periods and
lower wave amplitudes than those in the eastern US. It is possible to draw a line in
the figures which separates almost perfectly the eastern and western stations. Observations derived from stations in the north-eastern United States tend to occupy positions
88
Z. A, Der, R. P. Mass6 and J. P. Gurski
Table 2
(a) Argentina event
Station
ARWS
AYSD
BLWV
BUQB
CPCL
CTOK
CVTN
DHNY
DRCO
FMUT
FSAZ
GDVA
GVTX
HBOK
HLID
HNME
HTMN
KNUT
LCNM
MMTN
MNNV
MPAR
MUWA
MVCL
PTOR
SEMN
SJTX
SSTX
TFCL
TUPA
WINV
WNSD
Reduced
P amplitude
(w)
-
176.3
113.3
21.6
-
35.6
105.6
22.2
58.9
56.4
148.6
110.2
19.7
33.9
13.6
74,9
23.9
8.8
2.8
172.0
31.5
86.1
-
P period
S amplitude
(4
(mt4
6)
18.7
75.2
68.6
45.1
34.8
206.4
47.7
74.5
41.3
117.4
100.7
130.7
289.9
162.7
132.8
60.6
125.6
29.8
28.9
52.0
51.3
18.6
26.2
143.5
17.9
27.0
139.2
82.8
62.2
110.2
27.9
76.3
1.56
0.90
1.53
1.53
3.00
1.67
1.43
1.93
2.30
2.46
1.93
2.75
1.60
1.90
1.93
2.15
1.0
0.9
0.8
-
0.8
0.5
0.6
0.8
0.9
0.7
0.8
-
0.9
0.8
1.0
0.8
0.6
0.9
0.6
0.8
0.8
0.8
-
S period
1 .oo
2.40
1.83
1.63
2.96
1.56
3.60
3.53
3.70
1.06
2.13
1.80
2.56
2.33
1.73
0.96
(b) Peru-Brazil border event
Station
APOK
AZTX
BXUT
CPCL
CUNV
DRCO
DUOK
EBMT
EUAL
FRMA
GIMA
GVTX
HETX
HHND
HLID
KNUT
LCNM
LVLA
MNNV
MVCL
RYND
RTNM
SKTX
TDNM
WINV
Reduced
P amplitude
(mP)
27.6
21.4
16.7
3.3
9.4
5.1
23.9
21.6
55.7
33-2
24.8
45.3
6.7
2.3
<42.0
9.2
2.4
23.4
6.5
32.9
3.1
15.4
P period
6)
0.80
0.60
0.80
0.60
1 .oo
0.70
-
0.80
0.60
0.60
0.50
0.60
0.60
0.80
1 .oo
0.60
0.30
1.50
0.70
0.60
0.60
0.70
0.60
0.60
Reduced
S amplitude
(md
3.2
6.6
3.2
< 3.0
< 0.6
5.6
3.7
7.0
7.5
<23.8
< 10.0
< 6.8
< 20.6
-
< 6.1
< 2.8
<61.4
< 1.8
< 3.8
3.9
< 1.9
< 8.3
0.3
3.0
S period
(4
0.90
0.40
1.70
1
-
1 .oo
1.03
0.70
1.40
-
-
-
-
0.60
1.15
2.20
89
Regional attenuation of short-period S and P waves
(c) N W, Kuriles event
Reduced
Station
BLWV
BXUT
DHNY
DPCO
EBMT
EKNV
EUAL
FRMA
GIMA
GVTX
HNME
HHND
JELA
KMCL
KNUT
LCNM
LSNH
MNNV
PIWY
RKON
RYND
TKWA
TDNM
P amplitude
P period
Reduced
S amplitude
(md
(4
(md
6)
46.0
148.1
16.5
0.60
565.6
118.7
105.1
168.0
199.4
36.6
349.5
591.4
518.3
106.6
138.2
234.2
267.6
49.6
85.2
75.7
400.8
59.2
166.2
89.5
784.8
37.9
132.9
3.86
-
83.6
58.5
57.0
112.5
125.6
27.4
20.4
18.3
47.8
-
21 .o
-
36.2
I
0.70
0.70
-
-
0.80
1 .oo
0.70
0.70
0.60
0.80
0.80
0.60
0.50
0.60
-
0.90
0.70
S period
3.13
1.95
3.36
3.56
2.23
2.50
3.60
2.20
2-65
2.50
1.46
2.95
2.76
3.16
3.03
2.33
2.36
2.53
2.00
2.33
2.40
3.50
(d) Sea of Okhotsk event
Reduced
Station
BLWV
BRPA
DHNY
DRCO
EBMT
EUAL
FKNV
FOTX
FRMA
GIMA
GVTX
HHND
HLID
HNME
JELA
KNUT
LCNM
LGAZ
LSNH
MNNV
RKON
RTNM
RYND
SNAZ
WOAZ
WSND
P amplitude
P period
(mr)
(S)
27.4
37.4
60.2
16.4
44.7
95.7
21.6
31.3
69.3
36.0
57.6
53.1
(32.1
24.0
11.2
53.5
16.9
14.5
40.9
24.2
133.6
21.9
36.6
44.5
0.70
0.60
0.60
0.80
0.70
0.70
0.90
-
1.oo
1.20
0.80
0.80
0.70
0.40
1.00
0.80
0.80
0.60
0.70
0.60
0.80
0.60
0.80
0.80
0.90
Reduced
S amplitude
(mr)
38.1
30.9
19.9
24.7
22.9
47.7
< 2.4
32.8
< 13.4
< 8.4
151.7
< 13.7
2.2
17.3
<40.0
< 1.2
7.9
15.9
8.9
< 1.9
13.0
19.3
95.7
33.1
17.7
55.3
S period
(S)
2.40
2.36
2.35
2.96
1.96
0.80
2.80
-
1,45
2.30
-
2.53
2.40
2.00
1.36
3.00
2.00
3.00
2.40
2.00
90
Z. A. Der, R. P. Mass6 and J. P. Gurski
(e) Western Brazil event
Station
BLWV
BRPA
DHNY
DRCO
FOTX
GEAZ
GPMN
GVTX
HDPA
HLID
HNME
KNUT
LCNM
LGAZ
LSNH
MNNV
NLAZ
RKON
RTNM
RYND
SGAZ
VOIO
Reduced
P amplitude
P period
(nw)
6)
101.9
22.8
19.0
-
7.7
193.1
74.7
41.2
15.2
14.2
-
9.0
25.8
27-3
9.1
21.5
65.1
14.3
14.4
-
Reduced
0.50
-
0.50
0.60
1.00
0.60
0.50
0.60
0.40
0.50
0.80
-
0.50
0.60
0.60
0.80
0.60
0.50
0.60
-
0.60
S amplitude
(w)
47.5
11.7
<33.4
3.3
30.0
24.8
127.9
73.3
< 19.7
< 17.7
< 17-3
63.6
11.9
25.7
< 20.7
2.5
23.0
30.6
46.5
130.5
12.4
32.1
S period
6)
1 *30
1.40
-
1.43
1.30
2.66
1.86
1.63
-
2.50
1.36
3.13
-
2.00
2.65
1.13
2.86
1.65
2-40
1.20
intermediate between the western and the rest of the eastern stations. The observation
in California group with the rest of the western stations.
Although these groupings of station observation values are quite clear cut, there is
considerable scatter in the data and it is desirable to quantify these qualitative
observations by statistical analysis. There is also a lack of visible correlation between
the periods and amplitudes of S waves, and there is a poor visual correlation between
P- and S-wave amplitudes for the individual events. Nevertheless the direction of the
shift in the regional groups suggests that the dominant reason for the grouping is
anelastic attenuation although other factors mentioned above can also play a significant role. As we shall see the conclusions reached by inspection of plots of raw
data above and given by us in a previous report (Der, Mass6 & Gurski 1974) will be
confirmed by statistical analysis.
Statistical analysis
The purpose of the following statistical analysis is to test whether the regional
attenuation patterns are identical to or different from those given by Solomon &
Toksoz, to quantify the qualitative observations made in the previous section and
establish that they are significant in the statistical sense.
First of all we subdivide the stations into groups falling into the major regions,
California, rest of western US (RWUS), north-eastern US (NEUS)and the rest of
91
Regional attenuation of short-period P and S waves
i
92
Z . A. Der, R. P. Mass6 and J. P. Gurski
the eastern US (REUS). In addition we shall use the major subdivisions western
United States (WUS) and eastern United States (EUS). The subdivisions are marked
on the map in Fig. 1. Statistical tests were performed for two cases. In the first case
both P- and S-wave amplitude and S-wave period observations were considered and
in the second case only S-wave measurements were used. This was dictated by the
limitations of the data as described in the previous section. Since amplitudes measured
vary several orders of magnitude and behave erratically, we used the logarithms of
amplitude as variables in the statistical study. This assures also that the distribution
of variables is closer to multivariate normal, the distribution on which many of the
tests used are based.
First of all we wish to test whether the observations in the various subregions are
statistically different. In order to do this we set up the linear hypothesis in the form
pij
(1)
= /~+Ei+Rj+~ij
where p , j is an observation vector of the i-th event in the j-th region, /L is an overall
mean vector, E i is an event mean vector, R j is the region mean vector, and c i j is an
error term with mean zero. The overall mean is defined such that
C E i = C R j = 0.
i
(2)
i
For testing we use the Wilk's generalized variance ratio U (Anderson 1958) the
distribution of which is known presupposing that the populations tested are multivariate normal and that the two samples are drawn from the same population. The
test is often used in statistics for cases where the above conditions are not strictly
satisfied since the statistic in question is not very sensitive to moderate deviations
from normality and differences in covariance matrices in the populations tested. In
view of this we did not test the populations for normality, although as seen by visual
inspection the deviation of populations from normality cannot be great, but we tested
PTOR
MUWA
MVCL
0
v1
0
0
ic
2
VI
c$cL
3
MZNV
GiVA
'c
E
l
TFCl
FMUl
a
KtUT
HNMEo
-
2
TUPA
SJTXO
OHNV
HLIO
Y
fa
z
MPARO*ARWS
OHBOW
0
LCNM 0
WINV*
P
Y
SSTX.
MMTN
BUGBO
FSAZ
CTOKO
GVTX
OBLWV
0
CVTN
SEMN
WNSD
8
0
HTMN
AYSD
1
10
100
S WAVE AMPLITUDE I N inp
FIG.2. S-wave amplitudes and periods for the Argentina event.
1
U
93
Regional attenuation of short-period P and S waves
I
3
I
I
I
I
l
l
1
I
I
I
1 1 "
I
I
1
1
I
I
I
I l l '
-
-
-
-
WFV
-
BX!T
5
i
-
-
FROMA
YI
SKTX
0
-
-
DUiK
E~MT
El?
-
RAD
I
I
I
I
1
1
1
1
I
1
1
-
T
1
1
1
1
i
I
I
I
I
10
Frc.. 3. S-wave amplitudes and periods for the Peru-Brazil border event.
HHND
0
FIG.4. S-wave amplitudes and periods for the north-west Kurile Islands event.
I
I l l
100
L
-
-
23
L
O
w
u
-
LI
t
n
-
o
P
-
5
-
s
2
M
Y
-
U
c,
>
HiO
RKON
1 -
I
E5AL
I
t
I
I
t
1
1
1
I
I
I
1
I
I I I I
t
I
,
,
,
I
,
FIG.5. S-wave amplitudes and periods for the Sea of Okhotsk event.
4
I
I
I
I
I
1
1
I
1
I
I
I
I I I I I
I
I
I
I l l l r
I
I
I
I
LOAZ
.
8 3 r
0
um , -
RTNM
6E&Z
KNUT
NU2
I
a
SGAZ
-
(5
H
u"z
2
w
9
MNWV
-
mco
1 -
I
1
I
1
I
1
I
1
I
1
I
I
I
I
10
I
I
I
100
S WkYE AlPLlTIlDE IN n p
Fro.6. S-wave amplitudes and periods for the western Brazil event.
I
I
U
95
Regional attenuation of short-period P and S waves
10OOL
---
100
t
I
I
I I I I l l
I
I
I
I I l l
--
---
:
-
--
t
A
-
MUWA.C~CL
-
U
>
Y
-
CVTN
LCNM.
.
I
--
w
3
P
-
TFC'
0 ODRCO
KNUT
i
0.
I l I t 1 1
-
E
0
I
I
0
MNNV
10-
--
-
-
MVC L
-
-
PTDR
1
1
I
I
I
1
I l l l l
I
I
I I I l l 1
10
I
I
I
I 1 1 1 1
100
1000
1000
SKTX
APDK
iVND EUiL
*FRYA
BXUT
WINV**
1
10
1
S WAVE AMPLITUDE I N mp
Fro.8 . P-and S-wave amplitudes for the Peru-Brazil border event.
100
96
2. A. Der, R. P. Mass4 and J. P. Gurski
DHNY
HHND
&NME
10
100
EUAL
1000
S WAVE AMPLITUDE IN rnp
10.000
FIG.9. P- and S-wave amplitudes for the north-west Kurile event.
moo
RVJND
z
EUtL
100
E
>
w
s
a.
10
1
1
100
10
S WAVE AMPLITUDE 111 mp
FIG.10. P-and S-wave amplitudes for the Sea of Okhotok event.
1000
GPfN
lily
GVTX
RKON
LGp
ORCO
NLAZ.
0
RT!M
SGAZ
0
LCiM
M5NV
1
1
1
I
,
$
,
I
l
l
CEtZ
I
I
I
I
I I I I
10
I
I
I 1 1 1 1 1
100
1000
S W A V E ACPLITUDE IN mp
FIG.11. P- and S-wave amplitudes for the eastern Brazil event.
the equality of the covariance matrices. The event effects E i are of no interest in our
case except for the desire to eliminate them. One naturally expects variations in the
spectral content, amplitudes, both absolute and relative, of P and S waves from
event to event.
Table 3 shows the results of the statistical tests. The tests show that the various
events are significantly different (in amplitudes and dominant period), that the four
major regions are significantly different and, in particular, that the mesurements in
the eastern US differ significantly from those in the western US. Measurements in
the eastern US are characterized by higher wave amplitudes and longer S-wave
periods. The differences of the means are the following for the three variables, about
0-5for the logarithm of P amplitudes, 0.4 for the logarithm of S amplitude and about
0.9s for S-wave period. This displacement of the population means is consistent
with the assumption that the ultimate cause of the differences is anelastic attenuation
and is not consistent, for instance, with radiation pattern effect. The greater difference
between the logarithms of P amplitudes is explainable by the fact that the frequency
of the P waves measured is considerably higher than that of the S waves. Although
the hypothesis of the identity of measurement in the EUS us REUS cannot be rejected
at a high confidence level the U statistic is close enough to the 95 per cent limit that
one can argue in favour of some difference. California data do not differ significantly
from the rest of the WUS measurements, this feature is thus different from the
regional distribution of Solomon & Toksoz. The degree of attenuation in the northeast is also much less than that derived from long-period waves. The north-eastern
station observation means are displaced from the means of the rest of eastern stations
by -0.05, -0-2 and 0.2 for the three above mentioned variables indicating higher
attenuation, but as it is mentioned above the statistical significance of this displacement is not high.
2. A. Der, R. P. Mass6 and J. P. Gurski
98
Table 3
(a) Wilk's test f o r the equality of the means 3 variate case (P and S amplitudes $-S period)
Test
description
Event effect
Regional effect
(General)
EUS us WWS
NEWS vs rest
of EUS
California us
rest of WUS
U
of freedom
Degrees
0.18
0.43
3 4 63
3 3 63
Critical
value 95
per cent
0.70
0.75
0.60
0.90
3 1 63
3 1 63
0.87
0.87
Significant
Not significant
0.98
3 1 63
0.87
Not significant
Conclusion
Significant
Significant
(b) Wilk's test for the equality of the inems 2 variate case (S-wave amplitudes and
periods)
Test
description
Event effect
(General)
Regional effect
(General)
EUS us WUS
NEUS us REUS
California us
rest of WUS
Degrees
W
Critical
of freedom value 95
Conclusion
0.25
2 4 93
per cent
0.83
0.57.
2 3 93
0.86
Significant
0.72
0.93
0.98
2 1 93
2 1 93
2 1 93
0.93
0.93
0.93
Significant
Marginal
Not significant
Significant
The above analysis is based on the assumption that the populations being compared are multivariate normal with equal covariance matrices. We have tested the
covariance matrices for equality using the modified likelihood ratio test. The event
means have been subtracted prior to this analysis. The test statistic used (Morrison
1967, p. 153) for the comparison of K populations
where IS1 is the determinant of the pooled sample covariance matrix
s = ic=k l - nx insii
njis the number of samples in the i-th population, ISi\ is the determinant of the sample
covariance matrix for the i-th population. The statistic M multiplied with the factor
2p2+3p-1
c = - 6@+f)(k-1)
k
1
1
(z----)
ni
Eni
i=l
where p is the number of variables, is approximatelyxz distributed with-$@- l)p(p+ 1)
degrees of freedom if the covariance rnztrices are equal.
In our case k = 2 p = 2 or 3 depending how many variables are considered. The
test statistic computed in both the 2 and 3 variable cases yielded values which were
close to the expected value of the xz variable indicating that the hypothesis of the
equality of covariance matrices cannot be rejected, thus conditions for the validity of
Wilk's test are satisfied.
Regional attenuation of short-period P and S waves
99
Classijicatiopt of stations with respect to attenuation characteristics
The above results established that there are two major subdivisions in the United
States in which the parameters measured significantly differ. In the western United
States the amplitudes of the short period P and S waves are lower and the dominant
periods of short period S waves are larger. The first regional subdivision was made
on the basis of previous research and it is desirable to re-examine this subdivision.
Although both visual and statistical analysis shows that the first regional subdivision
between EUS and WUS achieves a very good separation between observation points
it is possible that some stations should be more properly assigned to a different
general region. It is also desirable to quantify the position of stations which are
intermediate in their characteristics such as those in the NEUS. The boundary of
primary interest is the one between EUS and WUS.
It would seem that the boundary obtained by the procedure outlined above would
be predetermined by the first subdivision and we would essentially get our initial
assumptions back. The reader can easily convince himself, however, that the boundary
obtained is inherent in the data, is a natural one, and shifting it would lead to insurmountable inconsistencies.
We use the linear discriminant function for the classification of observations into
several populations (Anderson 1958, p. 137). Based on the previous results we shall
have only two populations the parameters of which were derived from the total set of
observations in the EUS and the WUS, respectively. We assume that the costs of
misclassifications are equal, that is, the decision limit is halfway between the two
populations. The test assumes that the distributions two populations are approximately multivariate normal and that the variances of the two populations are equal.
As we have indicated above these assumptions are reasonable. The effect of events
were removed from each population by subtracting the corresponding terms prior to
the discriminant analysis. Subsequent to this the means and covariance matrices of the
two population samples were computed. These were substituted into the expression
for the discriminant function
-1
X'C ($
-1
-p2)-+(p(1)-p(2))'C
(p-p)
(6)
where the prime denotes transpose, x is the observation vector,
is the sample
covariance matrix, p("'-s are the mean vectors of the two populations after the event
terms were removed. Each individual observation was classified into one of the two
populations. The posteriori probabilities of that observation belonging to population
EUS (n = 1) were also computed using the formula
where N(p('"),C)is the normal probability density with mean p(") and covariance
matrix C. If all observations at a given station fall into the same population that
station is classified more reliably than a station where the observations contradict each
other. Moreover the probabilities themselves are also indicative of the quality of the
separation. We have averaged the posteriori probabilities for the various events at each
station. Although more rigorous procedures can be used this approach is adequate
for our purposes and more sophisticated approaches are unlikely to improve the
results. We shall refer to stations showing average probabilities between and not
including 0.40 and 0.60 as intermediate, less than or equal 0.40 as western or high
attenuation, larger or equal to 0.60 as eastern or low attenuation. Probabilities
given as zeros are actually less than 0-01.
Table 4 shows the averaged posteriori probabilities that the given station belongs
to the low attenuation (EUS) population in the three and two parameter cases. The
Z. A. Der, R. P.Ma& and J. P. Gurski
Table 4
3 variable case
Averaged
Station
name
APOK
ARWS
AYSD
AZTX
BLWV
BRPA
BUQB
BXUT
CPCL
CTOK
CVTN
DHNY
DRCO
DUOK
EBMT
EKNV
EUAL
FOTX
FMUT
FRMA
FSAZ
GEAZ
GDVA
GIMA
GPMN
GVTX
HBOK
HHND
HLID
HNME
HTMN
JELA
KMCL
KNUT
LCNM
LGAZ
LSNH
MNNV
MMTN
MPAR
MVCL
MUWA
NLAZ
PIWY
PMWY
PTOR
RKON
RTNM
RYND
SEMN
SGAZ
SJTX
SKTX
SNAZ
SSTX
TDNM
TFCI,
TKWA
TUPA
VOIO
WINV
WOAZ
WNSD
2 variable case
Averaged
posteriori
Number of
posreriori
probability
events used
1
-
-
probability
0.70
0.42
0.97
0.97
0.64
0.60
0.77
0.16
0.02
0.94
0.78
0.63
0.16
0.80
0.82
0.93
1
0.50
0.80
-
0.93
0.65
0.61
0.96
0.19
0.03
0.71
0.87
0-13
1
4
1
1
1
1
1
3
4
3
-
-
0.62
0.59
1
1
-
-
0.02
0.33
0.93
0.99
0.84
0.95
0.99
0-79
1
1
1
1
3
1
1
2
-
-
0.25
0.07
0.15
0.18
0.34
1
1
1
4
2
2
2
0.55
0.02
-
-
0.79
0.01
0.01
0.09
0.28
0.46
1
1
1
1
1
1
1
2
2
2
1
1
1
0.00
0.94
0.07
0.93
0.05
0.94
0.86
0.12
0.08
0.22
1
1
1
-
-
0.75
0.48
0.88
1
-
1
1
0.31
0.93
0.60
0.45
0.48
0.75
0.09
0.23
0.96
0.93
0.76
0-85
0.98
0.58
0.48
0.97
0.49
0.12
0.23
0.30
0.18
0.67
0.15
0.69
0.37
0.05
0.00
0.09
0.63
0.62
0.00
0.87
0.11
0.93
0.85
0.07
0.66
0.82
0.20
0.73
0.06
0.20
0-22
0.46
0.93
0.49
0.38
0.95
Number of
events used
1
1
1
1
4
a
1
2
1
1
1
3
4
1
3
1
3
2
1
2
1
1
1
1
1
3
1
1
2
3
1
1
1
3
4
2
2
3
1
1
1
1
1
1
1
1
3
2
4
1
1
1
1
1
1
1
1
1
1
1
1
1
2
101
102
Z.A. Der, R. P. M a d snd J. P. Gumkt
Regional attenuation of short-period P and 5' waves
103
table shows that the classification does not change for many stations. GIMA is
reclassified as an eastern (low attenuation) station and FRMA shows low to intermediate attenuation. Figs 12 and 13 show the geographical distribution of low (L),
high (H) and intermediate (I) stations for the three- and two-parameter cases. The
boundaries between the regions of contrasting attenuation are also marked. These
boundaries were sketched using all data considered, including noise background
readings; which in all cases agree with the rest of the data quantitatively analysed.
The coefficients of the individual parameters in equation (6) are indicative of the
relative importance of those parameters in discriminating between the two basic
populations (EUS us WUS). These are shown in Table 5. In the three parameter
case the most important parameter is the P-wave amplitude, the least important is
the S amplitude. In the two parameter case the S-wave period is somewhat less
significant.
The good agreement between the two- and three-parameter cases suggests that the
causes of attenuation for P and S waves are related since in the three parameter case
the P-wave amplitudes are the most heavily weighted and in the two parameter case
only S waves are considered. Magnitude and P-wave delay measurement results of
other investigations quoted later in this paper also correlate well with the geographical
distribution patterns for the two-parameter case shown. The probable reason for the
similar P- and S-wave behaviour is a complex shear modulus which influences both
P and S waves. Complex compressibility would affect only P waves. Our data are
not suitable to show this since the degree of attenuation of a given frequency component of S waves cannot be estimated by measuring the maximum amplitude only
and the dominant period changes considerably from region to region.
Discussion
The results of the present study demonstrate that in spite of the great variability
of short-period observations, short-period waves can be used to obtain a fairly
coherent picture of regional variations in attenuation within the upper mantle and
crust. In particular, the combination of the amplitude and period measurements
provides quite consistent indicators of regional attenuation. The least effective parameter used is the S-wave amplitude, the most effective is the P amplitude. S-wave
measurements alone are sufficient to outline the same features.
Observations of short-period waves presented in this paper indicate considerable
attenuation in the western United States. The general geographical pattern of the
distribution of attenuation is consistent with similar observations of long-period body
waves (Solomon & Toksoz 1970). There seems to be a difference in the short-period
and long-period attenuation patterns in California, where our short-period observations indicate high attenuation, but long-period S-wave observations of Solomon &
Toksoz (1970) indicate low attenuation. Our data also indicate the possible existence
of a second, but less pronounced, attenuating region in the north-eastern United
States.
Solomon (1972) modelled the general attenuation in the United States by super-
Table 5
Relative importance of parameters for discrimination
(coeflcients of discriminantfunction)
P-wave amplitude
S-wave amplitude
S-wave period
3-parameter
case
4.229
1,644
-2.386
2-parameter
case
3.439
-3.028
104
Z. A. Der, R. P. Mass6 and J. P. Gurski
position of two relaxation mechanisms, one prevailing at shallow depths, the other at
greater depth. Although present observations do not allow the verification of this
model, this implies that attenuation of short-period and long-period waves can vary
independently resulting in different geographical patterns.
The geographical attenuation pattern presented here also correlates well with
P- and S-wave travel-time residuals which are late throughout the western United
States including California (Cleary & Hales 1966; Doyle & Hales 1967; Hales et al.
1968; Hales & Roberts 1970; Julian & Sengupta 1973). P wave amplitude measurements by Evernden & Clark (1970) and magnitude residuals (Marshall 1974, private
communication) also show the same pattern.
In the Texas panhandle (north-west Texas) the location of stations permits the
definition of a fairly sharp boundary between the high and low attenuating regions.
This boundary lies between AZTX, SKTX on one side and RTNM as evidence by the
contrasting behaviour of the short period waves at these stations. (See Table 4.)
The observations do not permit the exact definition of the location or nature of the
boundary further north. On the basis of the two-parameter case one could argue in
favour of relatively lower attenuation in the Wyoming, Idaho and northern Utah
and Nevada region. The three-parameter analysis on the other hand classifies PIWY
as a high attenuation station. We have sketched in a tentative boundary based on
both maps.
Our map is in agreement with studies of upper mantle conductivity using geomagnetic variations (Gough & Porath 1970; Porath & Gough 1971; Gough 1973).
The boundary line of Gough (1973) falls somewhat to the west of ours, and some
regions of high attenuation, California and Colorado plateau for example, have low
conductivity. It must be remembered, however, that measurements of conductivity
reflect the variations in the depth of the upper boundary of the conducting region while
body waves sample the total thickness of the upper mantle, thus no complete correlation between the two types of measurements can be expected. The same general
remarks apply to heat flow measurements also (Blackwell 1971). The results of
Archambeau et a!. (1969) also indicate a variation in the depth of low velocity material
in the mantle of the western United States.
The region of higher attenuation in the north-east does not show up on the maps
since the associated probabilities fall into the low attenuation classificaticin. As
shown above the observations in this region are different from the REUS data only
at about the 90 per cent confidence level. The typical probability of the north-eastern
stations in Table 4 is around 0.65 while most of the rest of eastern stations are
characterized by higher probabilities.
A few stations (for example MPAR, ARWS, GDVA, JELA) seem to be anomalous
with respect to the behaviour of the surrounding stations. All of these are based on the
readings of one event only and may reflect nothing more than the statistical uncertainties of the data. Considering the fact that we are dealing with short-period data
these apparent anomalies are few. We emphasize again that the classification of a
single station especially if based on a single event is substantially less certain than the
regional shift of populations.
Differences in the attenuation patterns of short-period waves relative to longperiod waves indicate that the average attenuation mechanism changes as a function
of geographical position, although it cannot be ascertained whether the attenuation
of short- and long-period waves occurs at different depths. Mapping relative
attenuation as a function of period and geographical position can be an important
tool in exploring the upper mantle. Variation of attenuation also influence the
detectability and character of various phases and thus have a direct bearing on studies
dealing with source spectra and discrimination of earthquakes and explosions.
Attenuation of the direct arrival of short-period seismic waves through low Q
regions as demonstrated in this paper coupled by non-minimum time arrivals through
Regional attenuation of short-period P and S waves
105
the high Q lithosphere can account for the changes in the complexity of signals as
proposed by Douglas et al. (1973). The complexity of signals was not investigated in
this paper, but it seems to be a worthwhile research topic for the future.
Acknowledgments
We wish to thank Drs R. R. Blandford, S. S. Alexander and E. A. Flinn for reviewing the paper and offering valuable suggestions.
We thank Dr R. H. Shumway and Mr H. Husted for assistance in running the
statistical analysis programs. Dr R. H. Shumway substantially contributed to
the paper in advising us in the selection of statistical procedures used and providing
valuable advice and criticism throughout this work.
We thank the anonymous reviewer for his valuable criticisms. Following his
suggestion we substituted the statistical analysis for a more qualitative evaluation of
measurements thereby greatly strengthening the conclusions of this paper.
This research was supported by the Defense Advanced Research Projects Agency,
Nuclear Monitoring Research Office, under Project VELA and accomplished under
the technical direction of the Air Force Technical Applications Center under Contract
FO 8606-74-C-OOO6.
Teledjwe Geotech,
Alexandria, Virginia 22314.
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