Geophys. J. R. astr. Soc. (1983) 12,1-21
The structure of the inner core inferred from shortperiod and broadband GDSN data
George L. Choy
us Geological Survey, BOX 25046, MS 967,
Denver Federal Center, Denver, Colorado 80225, USA
Vernon F . cOrmier* Harvard University, Department of Geological Sciences,
Cambridge, Massachusetts 02138, USA
Received 1982 March 4 ; in original form 1981 November 5
Summary. Broadband displacement and velocity records of PKP phases are
formed by the simultaneous deconvolution of the instrument responses of
waveforms recorded by the short- and long-period channels of the Global
Digital Seismograph Network. Revisions of standard earth models are inferred
from the comparison of synthetic and observed waveforms. The synthetics
incorporate a Q-model designed for the bandwidth of the data and a source
model that accounts for the variations of waveform about the focal sphere
due to the effects of finiteness and complexity of the rupture process. The
interpretation of the data waveforms allows the position of the D-cusp to be
fixed at 121 f 1" and the C-cusp at 154 f 2" for a surface source. The cusp
positions are consistent with a P-velocity of 10.82 f 0.02 km s-' on the underside of an inner core boundary at 1216 km and a P-velocity of 10.30 f
0.05 kms-' on its upper side, resulting in a P-velocity jump of Aa! = 0.52 k
0.07 km s-'. Based on the agreement of synthetic and observed waveforms
that have been diffracted beyond the Ccusp there is no zone of small or
negative P-velocity gradient or of strong anelasticity at the bottom of the
outer core. Second-order features in the observed waveforms near the D-cusp
suggest that the S-velocity may be nearly zero at the top of the inner core.
Q, must increase with depth in the inner core in order to satisfy the pulse
width of the PKP-DF phase at distances greater than 150". In order to satisfy
average properties of the inner core inferred from travel times and normal
modes, these combined results require strong gradients in P-velocity, S-velocity
and anelasticity in the upper 200-300 km of the inner core.
Introduction
Significantly different velocity models of the transitional region between the outer and inner
cores of the Earth cannot be distinguished by data consisting of travel times, slownesses and
* Present
address: Massachusetts Institute of Technology, Earth Resources Laboratory, Cambridge,
Massachusetts 02142, USA.
1
2
G. L. Choy and V. F. Cormier
peak amplitudes of core phases (e.g. Engdahl 1968). Stronger constraints on the permissible
velocity gradients on either side of the inner core boundary can be obtained by comparison
of observed with synthetic seismograms. Time-domain modelling, however, has generally
been restricted to long-period body-wave data at frequencies less than 0.2 Hz (Miiller 1973;
Cormier & Richards 1977; Rial & Cormier 1980). This was chiefly because the computation
and analysis of body waves can be greatly simplified if the wavelengths can be assumed to be
sufficiently larger than the source size, so that directivity effects can be neglected. The
restriction t o long periods results in a decrease in the ability to resolve fine structure near the
inner core boundary. Until recently this restriction has been acceptable because the reported
scatter in short-period amplitudes of core phases (Engdahl 1968; Buchbinder 197 1) appeared
to preclude a meaningful analysis of short-period waveforms. Advances in source theory,
however, now permit the computation of synthetic body waves that incorporate the frequencydependent effects that arise from a finite source (e.g. Boatwright 1981). This paper
extends the analysis of PKP core phases to short-period and broadband waveforms by combining theory for a finite source with a synthesis method that describes the frequencydependent effects incurred by body waves propagating through the Earth.
When the observed pulse shapes of PKP are from the same earthquake and the effects of
both source directivity and propagation are considered, the rapid changes in waveform as a
function of take-off angle and azimuth are remarkably predictable and exhibit no significant
scatter in amplitude. The signatures in the broadband and short-period waveforms are considerably more complex than in the corresponding long-period (essentially low-pass filtered)
body waves. The sensitivity of these waveforms to Earth structure permits the resolution of
the positions of the C- and Dcusps to ? 2" or better.
Computation of PKP body waves
At teleseismic distances, the cone of take-off angles for the seismic rays describing the
Earth's response between source and receiver is very narrow, allowing the propagation effects
to be decoupled from the source effects. The total far-field pulse shape, u , may then be
written as
u (x, t ) = g(4 t )* s (x, t )
where g is the propagation operator for distance A, and s is the source operator radiated by a
source toward a receiver at x.
T H E PROPAGATION OPERATOR
The PKP core phase incurs strong frequency-dependent effects near the cusps of its traveltime curve (Fig. 1). Phenomena which are not correctly or conveniently described by ray
theory include the following: waveforms associated with the AB branch, which are phasedistorted when their ray paths touch an internal caustic surface (Choy & Richards 1975);
waveforms near the B caustic; waves that are diffracted beyond the cusp C; and the interference head wave that emanates from the cusp D (Cormier & Richards 1977). The full
wave approach to seismogram synthesis (Richards, 1973; Choy, 1977) correctly models the
frequency dependence of these effects. This method computes propagation operators that
remain valid for rays at near-grazing incidence to a discontinuity. It thus correctly describes
the effects of diffraction and the system of rays that arise from a cusp or caustic. Dispersive
attenuation is introduced analytically through the use of a depth-dependent complex velocity
function (Kennett 1975; Cormier & Richards 1976).
The structure of the inner core
I
I
I
3
I
920
8801
870
110
120
130
140
150
A (deg)
p@re 1. A reduced travel-time curve describing the PKP phases. The first few components of the interference head wave which emanates from the D-cusp are shown. It consists of the infinite sum of all
pKmZKP (rn = 1, 2, . . .), which are body waves transmitted into the inner core and internally reflected a
number of times before being transmitted back out. The dashed extension of the BC branch describes the
body wave diffracted beyond the Ccusp which propagates along the top of the inner core. It produces
arrivals far beyond the distance predicted by ray theory.
TH E S O U R C E O P E R A T O R
The source operators cannot generally be assumed to be independent of azimuth or take-off
angle. Waveforms with broadband content (0.02-5 Hz) can vary systematically about the
focal sphere as a result of source geometry and complexity for deep earthquakes as small as
mb 5.5 (Choy & Boatwright 1981). To obtain the source function, the method of Choy &
Boatwright was followed, whereby a suite of waveforms from the same earthquake is
examined to determine the effects of source directivity. The source operators of their method
were generated using the models of Boatwright (1981), which provide a realistic and causal
description of rupture and healing for simple earthquakes.
By using deep seismic sources in this study, it was possible to isolate that part of the
Waveform that is sensitive to interactions with core structure. Crustal structure, for example,
need be considered only for the receiver end of the propagation path. Because PKP body
waves have very steep angles of incidence in the upper mantle, their waveforms are also
hsensitive to variations in the velocity model of the upper mantle. At these steep angles of
incidence, the effects of the receiver crust on the vertical component of motion are generally
n e g i b l e although they could be easily incorporated if necessary with the matrix method of
Haslrell(l962).
For attenuation in the mantle, a Q-model which is appropriate for the frequencies used to
W e r a t e our synthetics (0.2-5 Hz),the AFL model (Archambeau, Flinn & Lambert 1969;
hndquist & Cormier 1980; Choy & Boatwright 1981), was used. Note that the source
ipodel that was derived for the source that was studied predicts the waveforms of P and p P
the distance range 30" Q A G 90". This indicates that, even though the attenuation and
4
G. L. Choy and V. F. Cornier
source operators may have some trade off, the convolution of the two effects correctly
yields the mantle response.
Earth models
Earth structure has been inferred through the use of synthetic seismograms. Starting earth
models are perturbed until the best fit is obtained in the comparison of synthetic with
observed waveforms. The parameters that were varied in the search through model space
included the size of the jump in P-and 5’-velocity across the inner core boundary, as well as
the velocity and attenuation gradients on both sides of the boundary. A representative
subset of the models that were tested is shown in Fig. 2, where velocity is plotted as a function of radius normalized to the radius of the inner core. The validity of the models is
evaluated in greater detail in the next section.
The PEM model (Dziewonski, Hales & Lapwood 1975) has a P-velocity jump across the
inner core boundary of 0.83 km s-l, while the PREM model (Dziewonski & Anderson 1981)
reduces that jump to 0.68 km s-’. The KOR5 model (Qamar 1973) differs substantially from
these models by having the smallest velocity contrast, the smallest velocity at the top of the
inner core, a pronounced velocity gradient in the top few hundred kilometres of the inner
core, and a very weak gradient at the bottom of the outer core. By taking combinations of
these models, it is possible to sample the effect on the waveforms of a wide spectrum of
velocity gradients and contrasts. Two variants of these models referred to in this paper are
PEM’, which uses the PEM velocity for the outer core but the KOR5 velocity for the inner
core; and PREM’, which uses the PREM velocity for the outer core but the KORS velocity
for the inner core.
All the models assume a sharp inner-core boundary. The sharpness is supported by observations of short-period PKiKP at short distances (Engdahl, Flinn & Mass6 1974; Bolt 1977)
12.0
T
h
U
z
\
11.0
E
s5
x
+
.-
Figure 2. The P-velocity profile in the vicinity of the inner core boundary as a function of radius
normalized to the radius of the inner core for some of the earth models that were investigated.
The structure of the inner core
5
Table 1. Travel-time residuals in seconds of observed arrivals for earth models used in Figs 5- 10. The PKP
travel times of PEMS' and PREMS' are identical to PEM' and PREM', respectively. The PKPCdiff travel
time is computed using a constant dT/dA for the diffracted portion of the propagation path. The residuals
for PKP-AB are relative to a reference feature on the waveform (the first trough in the short-period waveform) because the body wave is phasedistorted in propagating through the Earth (Choy & Richards 1975).
Phase
Station
Residual of earth model
Observed arrival time
PEM
PKP-DF
PKPBC
PKP-Cdiff
PKP-AB
M A10
CTAO
SHIO
CHTO
GUM0
SHIO
SHIO
CHTO
12h
12
12
12
12
12
12
12
35m
35
35
36
35
36
36
37
02.2'
15.2
52.0
02.7
31.8
02.1
25.1
06.2
0.6
1.8
1.0
- 1.8
-0.6
PREM
PEM'
1.5
2.5
1.2
2.1
0.2
0 .o
0.9
0.35
-0.5
- 0.4
0.15
1.1
PREM'
0.9
1.9
0.8
-3.2
- 0.5
0.9
1.9
KOR5
- 0.8
-
1.7
1.2
1.1
and by the good fit to the data which were obtained without the need of anything more
than a simple boundary. An inner core radius of 1217.1 km was taken for PEM and its
variants while a radius of 1221.5 km was used for PREM and its variants. From observations
of PKiKP and PcP, Engdahl et al. (1974) obtained an inner core radius of 1227.4 k 0.6 km.
From observations ofPKiKP and PKZIKP, Bolt (1977) obtained a radius of 1216.0 f 1.0km.
Variation of the inner core radius within the range of these values does not affect our conclusions regarding the P-velocities at the inner core.boundary.
All the models that were tested used Q = 10000 for the outer core, which is consistent
with a variety of seismic evidence (Sacks 1971 ;Qamar & Eisenberg 1974; Cormier & Richards
1976). Unless otherwise stated, the P-wave anelasticity for the inner core was that given by
Corrnier (1981), Q = 285. Also, unless otherwise stated, all models assumed the S-velocity
of 3.5 k m s-l throughout the inner core as given by PREM.
Despite the wide variations in the velocity models, they cannot be distinguished by traveltime data. Note that the travel-time residuals for the PKP-DF arrivals of the data set for all
of these models are much smaller than the size of many station corrections (Table 1).
Data processing
Many methods that use synthetic waveforms to infer Earth structure convolve the synthetic
displacements with an instrument response prior to comparing them with the observed
records. The simulation of instrumentally filtered waveforms implicitly applies a frequencydependent weighting to the analysis by emphasizing those frequencies near the centre
frequency of the seismograph response at the expense of fitting spectral information near
the sidebands where the amplitude response is lower. The effect is to suppress the frequencydependent effects of propagation and source that are more evident with broadband records.
For example, the corner frequency of most teleseismically recorded earthquakes (mb > 5.5)
at a lower frequency than the sharp peak in the instrument response, usually at 1 Hz, of
'standard short-period seismographs. As a consequence, short-period records are usually not
representative of the energy radiated by earthquake sources. In fact, they are dominated by
&h frequencies which are probably more sensitive to scattering effects. A better procedure
that avoids the problems of instrument distortion and limited spectral bandwidth is to perform the inversion by the comparison of broadband records of displacement and velocity.
6
G. L. Choy and V. F. Cornier
The data (Figs 5-10) are presented in three ways. The original short-period records of
PKP are shown in the top trace. The next line shows the broadband displacement and
beneath it is the corresponding record of velocity. The ground displacements are obtained by
applying a simultaneous deconvolution of the instrument response to the long- and shortperiod channels of the GDSN instruments using the method described by Harvey & Choy
(1982). Velocity records are then obtained by taking the derivative of the displacements. In
contrast to the short-period record, a rise time and total pulse duration can generally be
associated with the displacement. Also, rapid changes in the displacement are emphasized in
the velocity record. For instance, the rise time of displacement becomes a pulse in velocity.
Without the insight provided by the broadband records, it would be difficult to assess the
significance of high-frequency features in the original waveforms.
Waveform analysis
Waveforms for analysis were chosen from a deep-focus earthquake that occurred in western
Brazil on 1978 July 1 1 at a depth of 643 km. The focal mechanism from the first motions of
P-waves (Fig. 3) is consistent with the moment-tensor solution determined by Dziewonski,
Chou & Woodhouse (1981). The source parameters of this event obtained by an inversion
with the source theory of Boatwright (1 981) using P and p P from stations ESK and TOL and
PKP from the GDSN stations are given in Table 2 . Eight PKP body waves from this event
azi plunge
T axis 241.5 20.3
P axis 48.7 69.3
B axis 149.9 4.2
Figure 3. Lower hemisphere focal mechanism of the western Brazil earthquake of 1978 July 11
(OT 12h 17m07.8S; 7.89" S , 71.415"W; depth 643km; mb5.8). Crosses are dilations and triangles are
compressions.
The structure of the inner core
7
Table 2. Source parameters of the western Brazil event of 1978 July
11 using the method of Choy & Boatwright (1981). Parameters are
scaled to a. rupture velocity of 0.7p = 3.78 km s-', where p is the
shear velocity at the source. Rupture length is measured along the
direction of greatest unilateral propagation from the hypocentre
to the fault perimeter. The rupture half-width is measured from
the hypocentre and normal to the direction of the rupture length.
Strike
Dip
Rake
Depth
Moment
Rupture length
Rupture half-width
Rupture area
Dynamic stress drop
150"
2 6"
- 27"
643 km
9.5 X loz4dyne cm
3.6 km
1.8km
45 km2
45 bar
were digitally recorded by the GDSN stations over a distance range of 127.2'-165.7". Fig.4
shows in plane view the ray paths of these body waves to the GDSN stations. The turning
points of these rays sample every important region of the outer and inner core.
W A VE F ORM S N E A R THE D - C U S P
The PKP-DF arrivals near 130" actually consist of PKiKP, a reflection at near-grazing incidence off the top of the inner core, and the interference head wave, which is the sum of all
PKmIKP (m = 1, 2, . . .), which propagates on the underside of the inner core boundary.
Thus, these pulse shapes are extremely sensitive to the velocity gradient at the top of the
inner core. PKP-DF was recorded by two stations near this distance, MA10 (A 127.2') and
CTAO (A 133.7'). Figs S(a) and 6(a) compare the observed waveforms (solid lines) at these
stations with the synthetic waveforms (dashed lines) generated by the PEM earth model.
Though the gross features of the transient pulses are approximated by the synthetics, there
are significant differences with the observed data which can be resolved by modifying the
p i r e 4. A schematic cross-section of the Earth showing the ray paths of the PKP body waves for the
same distances as the GDSN stations. The distance range spanned by the data is from 127.2' to 165.7'.
Some ray paths sample only the outer core; some sample the deep interior of the inner core; other rays,
by interacting at near-grazing incidence with the inner core boundary, are sensitive only to the structure
immediately above and below the discontinuity. The interference head wave is represented by only the
first member of the infinite sum. PKIKP.
G. L. Choy and V. F. Cormier
8
velocity
velocity
-
3 sec
3 sec
(c)
Figure 5
(4
The structure of the inner core
9
short-period
MA10 ( 1 27.2")
velocity
P
(4
Figure 5. (a) The data (solid lines) from station MA10 (A 127.2') are shown in three ways. Top: the
original digitally-recorded short-period waveform of PKP-DF. Middle: the corresponding deconvolved
displacement pulse shape. Bottom: the broadband velocity waveform obtained by taking the derivative of
the displacement. The synthetic waveforms computed by using the PEM model for the corresponding
quantities are shown by dashed lines. The amplitude ratio of observed/synthetic velocity for this and all
the other body waves in Figs 5-10 are given in Table 3. (b) The synthetic waveforms (dashed lines) from
model PREM are compared with the data from MAIO. (c) The synthetic waveforms from model PREM'
(dashed lines) are compared with the data from MAIO. Though the agreement between synthetic and
observed records is substantially improved relative to waveforms from the PEM and PREM models, the
PREM' short-period and velocity records both show an inflection that is not seen in the data. Waveforms
from the PEM' model are essentially similar to the PREM' model for this station. (d) The synthetic waveforms from model PREMS'. The inflection in the short-period record that was seen in the synthetics of (c)
is now very weak. (e) The synthetic waveforms from model PEMS' are compared with the data from
MAIO. The models PEMS' and PREMS' generate synthetics that agree best with the data.
earth model. These differences include the disparity in the relative amplitudes of the troughs
and peaks of the displacement and velocity records. This disparity is manifested in the
narrow-band filtered records (i.e. the short-period response) as a mismatch in phase and
amplitude of the arrivals of the peaks and troughs. The synthetics generated by PREM
(Figs 5b and 6b) show a slight improvement, but the mismatch of the phase and amplitude
of the peaks and troughs persist, especially for the CTAO records. In the synthetic displacement of the MA10 record, the backswing following the initial trough overshoots the corresponding peak of the actual displacement. A significant improvement in the CTAO cornparison occurs when the PREM' model is used, in which the P-velocity at the top of the inner
core is that of KOR5 (Figs 5c and 6c). The backswing in the PREM' displacement for MA10
is also not as high as that seen in the PREM-generated displacement. The PKP-DF waveform
not sensitive to the gradient at the bottom of the outer core. The waveforms generated by
G. L. Choy and V. F. Cormier
10
PREM
velocity
,
3 sec
,
3 sec
The structure of the inner core
11
Figure 6 . (a) The PKP-DF data (solid lines) from station CTAO (A 133.7") are compared with synthetic
waveforms (dashed lines) computed by using model PEM. The displacement signal is riding on top of a
large microseism of approximately 6 s period. In this instance, only the short-period and velocity records
can be used to infer core structure. The peaks and troughs of the PEM waveforms disagree substantially
in amplitude and phase with the data. (b) The synthetic waveforms (dashed lines) from model PREM are
compared with the data from CTAO. There is still much disagreement between the synthetic and observed
waveforms. (c) The synthetic waveforms from model PREM' (dashed lines) are compared with the data
from CTAO. Though the agreement is substantially improved relative to the comparison using the PEM
and PREM waveforms, there are still some slight discrepancies. For instance, the initial slope of the
observed velocity is much steeper than the slope of the theoretical waveform. The PEM' waveforms are
essentially the same as these waveforms and are not shown. (d) The synthetic waveforms from models
PREMS' are compared with the data from CTAO. (e) The synthetic waveforms from the model PEMS'
are compared with the data from CTAO. This model agrees best with the data. While there appears to be
mme discrepancy in the short-period waveforms, the overall pulse shape of the velocity waveform, which
is broadband, is satisfied by the synthetic.
model PEM' (not shown), for instance, are virtually identical with that of PREM' although
these models have different P-velocities at the bottom of the outer core.
There are still rather small differences between observed and synthetic PREM' seismograms. The initial slope, for instance, of the observed CTAO velocity record is much steeper
than in the theoretical record. In addition to the backswing of the synthetic displacement
for MA10 which still slightly overshoots the observed displacement, an inflecton is seen in
the fust downward swing of the synthetic MA10 short-period response, which does not
Occur in the actual recording. Because no further improvement in matching the waveforms
could be made by perturbing P-velocities, tests were made by perturbing the S-velocities.
perturbations in density were not considered because body waves that graze a discontinuity
are generally not very sensitive to the density jump at the discontinuity. Muller (1973)
12
G. L. Choy and V. F. Cormier
showed that a variation in the density jump at the inner core of 1.O g ~ m produces
- ~
an
effect equivalent to a variation ins-velocity of only 0.3 km s-'. For perturbations in S-velocity
the best fit to the waveform is obtained with an S-velocity that is zero at the top of the inner
core. The synthetics from this model, PREMS' (Figs 5d and 6d), appear to resolve the
discrepancies that arose with the PREM' model. Model PEMS' which is PEM' with this modification of 5'-velocity in the inner core produces the same second-order improvement in the
waveform comparison (Figs 5e and 6e).
Based on the goodness of fit using core models with a P-velocity in the inner core given
by KOR5 and a P-velocity in the outer core given by PEM or PREM, the absolute velocity
a t the top of the inner core must be substantially reduced from that given by standard earth
models such as PEM and PREM to 10.82 0.02 kms-'. In order to satisfy the travel-time
data from PKP-DF rays that traverse the deep interior of the inner core, there must be a
pronounced P-velocity gradient in the upper 100-300 km of the inner core. Based on fitting
second-order features in the waveforms, the S-velocity is nearly 0.0 km s-' at the top of the
inner core. In order to agree with the shear modulus determined from observations of normal
modes by Dziewonski & Gilbert (1973), the S-velocity gradient must also increase rapidly
with depth to 3.5 kms-'in the top 100-300km of the inner core. Qamar (1973)noted that
the S-velocity at the top of the inner core would be greater than 0.0 km s-' for the inner core
to be stably stratified. His 5'-velocity distribution was derived by using the P-velocity gradient
for the inner core given by the KOR5 model and by assuming that q = 1, where q is the
stratification parameter. The stratification parameter (e.g. Bullen 1965 ; Masters 1979)
describes the state of adiabaticity and chemical homogeneity in the inner core and is defined
by
dk 1 d@
+_
+_
q=---dP g d r
where k is the bulk modulus, p is the pressure, g is the gravitational acceleration, C#J is the
seismic parameter and r is the radius. A value of 1) = 1 implies stable stratification at the top
of the inner core. For 1) = 1 Qamar found that the S-velocity increases from 3.2 to 3.5 km s-'
in the upper 200 km of the inner core and has a small negative gradient with increasing depth
in the rest of the inner core. The small or zero S-velocity at the top of the inner core that is
found in this study violates stable stratification. In order t o satisfy the average S-velocity for
the inner core of 3.5 km s-l determined from normal mode data, a steep gradient in the Svelocity at the top of the inner core is needed. This means the bulk modulus of the inner
core would rapidly increase with depth, contrary to the effects that would be expected with
increasing pressure. For this reason and because the second-order features of the waveforms
are weak constraints, the conclusion that the S-velocity at the top of the inner core is very
small should be viewed as tentative.
WAVEFORMS N E A R THE CAUSTIC
The PKP-BC arrival at GUMO (A 143.9') is formed as waveforms from the AB and BC
branches coalesce to form the B caustic. As the turning point of this arrival is in the region
of the outer core where the velocity structure is smoothly varying and well determined from
travel times, the synthetics for GUMO generated by all our earth models are practically the
same. Fig. 7 shows the theoretical records for the PREM earth model. The large amplitudes
of PKP at this distance are useful for normalizing the amplitude ratio of observed and synthetic records. An amplitude is measured from the baseline to the maximum amplitude of
the first downward swing in the velocity records. If the ratio of observed/synthetic amplitude
The structure of the inner core
13
I
short-period
GUMO (143.9O)
displacement
~
3 sec
Figwe 7. The PKP-BC data (solid lines) from GUMO (A 143.9")are compared with synthetic waveforms
(dashed lines) computed using model PREM. As this distance is very close to the caustic, the waveform
amplitudes are very large and can be used to normalize the amplitude ratio of observed/synthetic waveforms at other distances. The amplitude, measured in the velocity record from the baseline to the maximum
value of the first trough, is normalized to unity at GUMO. Ratios for the other waveforms and models are
given in Table 2.
is set to one at GUMO, the amplitude ratio at all GDSN stations was less than three except
for SHlO where the ratio was 3 .? or less. The amplitude ratio of observedfsynthetic velocity
for this and all the other body waves shown in Figs 5-10 are given in Table 3 .
W A V E F O R M S N E A R THE C-CUSP
At SHIO (A 156.3') arrivals from three PKP branches can be seen (Fig. 8). The waveform of
the first arrival, PKP-DF, is sensitive primarily only to the velocity gradient at the top of the
Table 3. Amplitude ratios of observed/synthetic waveforms for the earth models used in Figs 5-10. The
amplitude is measured from the baseline to the first trough except for the PKP-AB phase. For PKP-AB,
the measurement is from the baseline to the first peak. To normalize amplitudes, the ratio of observed/
synthetic at GUMO is taken as unity.
Station
Phase
PEM
PREM
PREM'
GUM0
PKP-BC
PKP-DF
PKP-DF
PKP-DF
PKP-C diff
PKP-AB
PKP-DF
PKP-AB
1
2.6
1.9
1
1.34
0.99
1
2.46
0.63
MA10
mAO
SHIO
CHTO
3 .O
0.9
PEMS'
PREMS'
KOR5
1
1
1.95
0.7 1
3 .O
3.1
2.8
1
2.2
1.5
3 .I
2.4
3.5
3.3
1.2
2.5
14
G. L. Choy and V. F. Cormier
PKP arrivals ot
KOR5
outer core
PKP arrivals at
SHlO ( 1 56.3')
1
PEM
outer core
The structure of the inner core
15
PKP arrivals a1
SHIO ( 1 56.3')
short-peri
SRO
--
rlisnlacement
4'
I\
I'
I
PREM
outer core
(c)
Figure 8. (a) The PKP waveforms (solid lines) at SHIO ( A 156.3") are compared to the synthetic waveforms (dashed lines) from a model using the weak KORS P-velocity gradient for the bottom of the outer
core. In (a-c), the observed and synthetic records are lined up such that the arrival times of the synthetic
and observed PKP-DF phases are aligned. Actual travel-time residuals are given in Table 1. The KOR5
model predicts a PKPCdiff arrival that is nearly twice as large as that seen in the data. (b) The PKP
waveforms (solid lines) at SHIO are compared to the synthetic waveforms (dashed lines) from a model
using the PEM P-velocity gradient for the bottom of the outer core. The theoretical PKPCdiff arrival is
slightly larger than the actual arrival. (c) The PKP waveforms (solid lines) at SHIO are compared t o the
synthetic waveforms (dashed lines) from a model using the PREM P-velocity gradient for the bottom of
the outer core. The theoretical PKPCdiff arrival is slightly smaller than the actual arrival. The actual
velocity gradient in the earth must lie between that of the PEM and PREM models.
inner core. Thus, it is relatively easy to model using the results of the previous analysis in
which the KORS velocity profile is used for the inner core. Similarly, the waveform of the
last arrival, PKP-AB, is also relatively easy to model as it is sensitive mostly to the upper part
of the outer core which has a smooth, well-determined velocity gradient. The middle arrival,
PQ-Cdiff, however, is an arrival not predicted by ray theory. It is diffracted a considerable
distance beyond the cut-off of the Ccusp by propagating along the top of the inner core
boundary. Thus, its waveformis very sensitive to the velocity gradient at the bottom of the
outer core.
By comparing the relative amplitude of the synthetic PKP-Cdfi arrival to the synthetic
PQ-DF and PKP-AB arrivals, the type of velocity gradient that exists at the bottom of the
.Outer core can be constrained. Three types of gradients were tested, corresponding to the
Outer-core velocities given by KORS, PREM and PEM. KORS has a weak, nearly zero gradient,
While the gradients of the other two models are succeedingly stronger. The synthetic records
w g the weak gradient (Fig. 8a) predict an extremely large PKP-BC arrival compared to the
Pm-DF mdPKP-ABarrivals. The synthetics for the other two models are shown in Fig. 8(b,c).
16
G. L. Choy and V. F. Cormier
The PEM gradient predicts a P K P - C J ~amplitude
~
that is slightly larger than that seen in the
data, while the PREM gradient predicts a slightly smaller arrival. The best earth model
probably has a velocity gradient between these two models. This restricts the velocity at the
t o p side of the inner core to 10.30 0.05 km s-'.
As ray theory does not account for diffraction beyond the Ccusp, observations of PKPCdZf at extended distances (e.g. Hai 196 1) have been erroneously interpreted as arising from
a low or negative P-velocity gradient at the base of the outer core. Neither the results of the
comparison shown here nor the analyses ofPKP-C,B by Rial & Cormier (1980) and Cormier
(1 981) support the existence of such a low P-velocity gradient.
_+
W A V E F O R M S T H A T SAMPLE T H E D E E P I N N E R C O R E
The PKP-DF arrivals that have been discussed thus far had turning points which grazed the
underside of the inner core boundary. In contrast, at 165.7", the PKP-DF arrival at CHTO
has traversed the deep interior of the inner core. Thus, its waveform is sensitive to the average
properties of the inner core. If a homogeneous Q, of 285 is used for the inner core, the
resulting synthetic records of displacement and velocity have much longer durations than the
data (Fig. 9a). The pulse durations of the synthetic records in Fig. 9(b) match the data much
better. They were generated by a model that has a thin, low-Q zone in the upper 100-300 km
of the inner core and a Q, of 1000 for the rest of the inner core. This model satisfies both
N\
n
Figure 9. (a) The PKP-DF data (solid lines) at CHTO are compared to synthetic waveforms from model
PEMS' with a Qa=285 in the entire inner core. The pulse widths of the synthetic broadband displacement
and velocity records are wider than the pulse widths of the data. (b) The PKP-DF data at CHTO are compared to the synthetic waveforms of model PEMS' with a Q, model that has a thin lowQ zone at the top
of the inner core and a high Q of 1000 in the rest of the inner core. This Q model is equivalent to that
proposed by Doornbos (1974). The pulse widths of the synthetic velocity and displacement pulse agree
much better with the data.
The structure of the inner core
17
n
short-period
PKP-AB at CHTO ( 1 65.7O)
,<
s
PREM
Figure 10. The PKP-AB arrival at CHTO ( A 165.7") (solid lines) is compared to synthetic waveforms from
the PREM model. It is a much later arrival than the PKP-DF phase, with a much smaller signal-to-noise
ratio.
the PKP-DF waveforms that were previously analysed for shorter distances as well as the
spectral ratios of PKP-DFIPKP-AB between 140"-150" (Doornbos 1974).
Kuster (1972) and Cormier (1981) favoured a nearly constant Q, with depth in the inner
core based on spectral ratios of PKP-DFIPKP-AB observed at 150°-170". In these studies,
however, the scatter in the observed spectral ratios does not exclude a model in which Q,
increases with depth. A problem in the use of the PKP-DFIPKP-A3 spectral ratio in this distance range is that PKP-AB can be a poor reference phase. For distances greater than 150",
the mantle legs of PKP-AB graze the core-mantle boundary, a region that may have strong
lateral heterogeneity (Sacks, Snoke & Beach 1979), rough topography (Doornbos 1978) and
strong anelasticity (Anderson & Hart 1978).
PKP-AB was also recorded at CHTO. It is shown along with theoretical records from the
P
m
M model in Fig. 10. The gross features of the observed waveform have been reproduced,
but the low signal-to-noise ratio prevents any useful inferences on earth structure to be
drawn. The decay in the amplitude ratio of PKP-DFIPKP-AB between SHIO and CHTO
measured in the synthetic records agrees with the observed decay.
Comparison with inner core structure inferred from low-frequency data
LON G-PE RIO D B O D Y W A V E S
h e r s i o n s For core structure using long-period core phases have generally fixed the shear
velocity at the top ofthe inner core to 3.5 km s-l in order to agree with the average S-velocity
determined from free oscillation. Muller (1 973) showed that the S-velocity must be less than
18
G. L . Choy and V. F. Cormier
4 k m s-' to obtain agreement of the peak amplitudes between synthetic and observed waveforms. Using an S-velocity of 3.5 km s-' at the top of the inner core, Muller constrained the
P-velocity jump across the boundary to be 0.6 -0.7 km s-' from a study of peak amplitudes
of PKP at distances less than the D-cusp. In an inversion of the same data set, Cormier &
Richards (1977) found that scatter in the data could permit a jump in P-velocity as high as
0.8-0.9kms-'. Rial & Cormier (1980) also found that a model with a P-velocity jump of
0.8 kms-' combined with an S-velocity of 3.5 km s-l fit a phase observed at the antipode
consisting ofPKP-Cdiffand the infinite sum ofPKmIKP (m = 2 , 3 . . .). In summary, the results
of long-period body-wave modelling under the assumption of an S-velocity of 3.5 km s-' at
the top of the inner core give a jump in P-velocity from 0.6 to 0.9 km s-' across the inner
core boundary. The long-period data, consisting primarily of peak-to-peak amplitudes, can
be fit equally well with any value of S-velocity at the inner core boundary from 0.0 t o
3.5 kms-'. The effect of a lower S-velocity at the boundary is to lower the estimate of the
P-velocity jump (Muller, 1973). The P-velocity jump of Aa = 0.52 f 0.07 km s-' found in this
study lies slightly below the lower bound of A a = 0.6 km s-' found in the studies of longperiod amplitudes. But since these studies assumed an S-velocity of 3.5 km s-' at the top of
the inner core, the amplitude observations would also be consistent with a slightly lower A a
combined with a lower S-velocity at the top of the inner core.
,t
F R E E OSCILLATIONS
Masters & Gilbert (1981) obtained a Q, of 3500 at 0.0035 Hz from a study of spheroidal
modes and noted the difficulty, if attenuation is assumed to occur purely in shear, in satisfying the smaller values of Q, inferred from high-frequency body-wave data. For the inner
core, time parameters T M and 7, can be used to describe a relaxation band (74,7;) in
radian frequency over which anelasticity is nearly constant (e.g. Minster 1978). In order that
anelasticity occurs purely in shear and at the same time satisfies the normal mode and highfrequency observations, the lower limit of the relaxation band must have T $ > 0.1 5 rad s-'.
This places strong frequency-dependent attenuation in the centre of the passband (- 1 Hz)
of standard short-period seismographs, making it difficult to satisfy spectral-ratio data in the
range 0.2-2 Hz that indicate a low Q, in the inner core (e.g. Sacks 1971; Kuster 1972;
Doornbos 1974; Bolt 1977; Cormier 1981). The problem is that a relaxation spectrum of
intrinsic anelasticity does not allow Q-' to decrease with decreasing frequency faster than w.
This restriction does not apply to attenuation due to scattering. if the Q, of the inner core
was the result of backscattering, the dominant scale lengths of inhomogeneities would have
to be such that little mode-splitting occurs at 0.004 Hz,but such that backscattering of body
waves increases rapidly in the 0.2-2 Hz band. The scale length of inhomogeneities would
thus be similar to the 10 km value suggested by Aki (1980) to explain the peak at 1 Hz in the
seismic attenuation of the lithosphere. Body-wave estimates of attenuation may not be relevant to modelling normal modes if backscattering is a significant mechanism of attenuation.
Whether the Q, in the thin low-Q zone at the top of the inner core is the result of intrinsic
anelasticity or backscattering, it is not necessary t o apply a dispersion correction for the
velocity jump at the inner core boundary when body wave and free oscillation data are compared. The maximum correction is given by assuming the Doornbos (1974) Q, model and
that intrinsic anelasticity occurs in bulk dissipation. This gives a correction of -0.01 km s-' in
the P-velocity jump at 0.004 Hz compared to that at 1 Hz.
To be consistent with body-wave results, it is more important that a normal-mode model
satisfy the position of the D- and C-cusps. There is no indication that scattering in the inner
core or near its boundary affects our conclusions for the positions of the C- and D-cusps as
The structure of the inner core
19
determined by waveform fitting. Strong scattering would have introduced random fluctuations
in phase and amplitude between stations that could not be satisfied by a single earth model.
There is no evidence of strong amplitude fluctuation in the broadband data summarized in
Table 3 nor of strong phase fluctuation in the waveforms used to infer the position of the
Dcusp. The strong amplitude and phase fluctuations often reported in short-period waveforms are partly the result of the peaked response of the instrument which emphasizes a
narrow band of frequencies around 1 Hz that are sensitive to small inhomogeneities rather
than the energy radiated by the seismic source. The greater stability of broadband waveforms and their relative insensitivity to varying receiver structure has been confirmed by a
comparison of broadband and short-period waveforms recorded across the stations of the
Graefenberg array (Choy 1982). That the amplitude and phase discrepancies that remain
between our synthetics and the actual waveforms are so small is probably because the broad
range of frequencies under consideration averaged out the effects of small inhomogeneities
(e.g. near the station) which may have differed among the propagation paths. In particular,
the waveforms at MA10 and CTAO which were used to determine the position of the D-cusp
were matched very well despite having traversed propagation paths with quite different
azimuths that sampled widely separated regions of the inner core boundary. The position of
the C-cusp was determined in part by matching amplitudes of three arrivals from the PKP
triplication at a single station. These amplitudes are consistent with amplitude data of body
waves having different ray paths that are compiled in several other studies.
A small shear modulus at the top of the inner core is favoured by second-order features in
the waveforms. A shear velocity profile having a near-zero value at the top of the inner core,
however, cannot be constructed to be stably stratified or to satisfy the bulk modulus
distribution expected for a solid inner core. Some of these restrictions on the S-velocity distribution are removed in the model of the core proposed by Anderson (1 980), in which the
inner core boundary marks a liquid-glass transition, and in the model of Bukowinski &
Knopoff (1976), in which the inner core boundary marks an electronic transition in iron.
A small shear modulus in the upper part of the inner core could mean that the normal
modes sensitive to the inner core actually sample an inner core of effectively smaller radius
than the inner core inferred from body waves. The high Q obtained by Masters & Gilbert
would then be relevant to modes for which energy is concentrated in a region of high shear
modulus and high Q at radii from 0 to 1000 km. As a consequence, it would become important to consider the trade-off noted by Dziewonski & Gilbert (1973) between the value of
the inner core radius and the value of the average shear velocity. For example, if the normal
mode data are really appropriate to an inner core having an effective radius of lOOOkm
rather than 1216 km, then the average shear velocity would have to be reduced from 3.5 to
3.0kms-l. This reduced value would have to be used in any construction of an S-velocity
profile of the inner core that is derived from knowledge of the average S-velocity in the inner
core, the size of S-velocity jump across the inner core boundary and various assumptions
about the stratification parameter.
Conclusions
Revisions of standard earth models have been determined from a comparison of synthetic
with observed PKP waveforms. The sensitivity to inner core structure of the broadband
and short-period waveforms that have been modelled permit the C- and D-cusps of the traveltime curve of PKP to be determined to less than 2". Based on eight recordings of PKP from
a deep focus earthquake, the D-cusp has been fixed at 121 f 1" and the C-cusp at 154 f 2"
20
G. L. Choy and V. F. Cornier
for a surface source. The revised cusp positions are consistent with a P-velocity of 10.82 f
0.02 kms-' on the lower side of the inner core boundary at a radius of 1216km and a Pvelocity of 10.30 f 0.05 kms-' on its upper side. The jump in P-velocity across the inner
core boundary is Acw = 0.52 f 0.07 km s-'. Based on the agreement of synthetic and observed
waveforms near the C-cusp, there is no zone of small or negative P-velocity gradient nor a
zone of high anelasticity at the bottom of the outer core (Bullen's 1965 F region). Secondorder features in observed waveforms near the D-cusp suggest that the S-velocity at the top
of the inner core may be substantially less than 3.5 km s-'. This result is consistent with longperiod amplitude data measured at distances less than the D-cusp. Q, in the inner core must
increase with depth in order to satisfy the observed pulse width of PKP-DF at 165.5". In
order to satisfy the average properties of the inner core that have been inferred from travel
times and normal modes, these results require strong gradients in P-velocity, S-velocity and
anelasticity in the upper 200-300km of the inner core. Waveforms provide the firmest
evidence for the position of the travel-time cusps.
The cusp positions are a constraint that models based on normal modes must satisfy.
Body-wave data, however, cannot resolve with any confidence details of the S-velocity profile
in the inner core. Nor can they resolve whether the frequency dependence of anelasticity
occurs in shear and/or bulk dissipation or whether the losses are due to back-scattering. The
results obtained from waveform modelling in this study indicate that the process of inner
core solidification produces the most evident seismic effects on body waves with turning
points that bottom in the upper 200-300 km of the inner core, rather than at the lowermost F-region of the outer core.
Acknowledgments
This research was partially supported by a grant from the National Science Foundation
EAR 79-04167. The authors wish to express their gratitude to Dr John Boatwright, who provided invaluable help in the inversion for source parameters, and to Drs Raymond Buland,
Guy Masters and Stuart Sipkin for critically reviewing the manuscript.
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