Physics of the Earth and Planetary Interiors, 10 (1975)12—48
©Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands
PARAMETRICALLY SIMPLE EARTH MODELS CONSISTENT WITH GEOPHYSICAL DATA
A.M. DZIEWONSKI*, A.L. HALES and E.R. LAPWOOD**
Research School of Earth Sciences, Australian National University, Canberra, A. C. T. (Australia)
Submitted September 3, 1974; revised version accepted December 3, 1974
We present a set of three parametric earth models (PEM) in which radial variations of the density and velocities are represented by piecewise continuous analytical functions of radius (polynomials of order not higher
than the third). While all three models are identical below a depth of 420 km, models PEM-O and PEM-C are
designed to reflect the different properties of the oceanic and continental upper mantles, respectively. The third
model PEM-A is a representation of an average earth.
The data used in inversion consist of observations of eigenperiods for 1064 normal modes, 246 travel times of
body waves for five different phases and regional surface-wave dispersion data extending to periods as short as 20
seconds. Agreement of the functionals derived for the PEM models with the appropriate observations is satisfactory.
In particular, the fit of free-oscillation data is comparable to that obtained in inversion studies in which constraints
imposed on the smoothness of structure were not as severe as in our study.
Our density distribution for all depths greater than 670 km is consistent with the Adams-Williamson equation to
within 0.2% maximum deviation, and these minute departures result only from the limitations imposed by the parametric simplicity of our models. We also show that the velocities in the lower mantle are consistent with the coin—
plete third-order finite-strain theory to within 0.2% for Vp and 0.4% for VS (r.m.s. relative deviations). The derived
pressure derivatives of the velocities are very similar to those obtained for corundum structures in laboratory experiments.
We conclude that any departures from homogeneity and adiabaticity within the inner core, outer core or lower
mantle must be very small, and that introduction of such deviations is not necessary on the basis of the available
observational evidence.
1. Introduction
Hales et al. (1974) suggested the construction of a
spherically symmetrical earth model in which the
radial variations of seismic parameters are expressed
by piecewise continuous analytical functions such as
low-order polynomials in the earth radius. This suggestion was related to the activities of the Standard Earth
Model Committee of the I.U.G.G. under the chairmanship of Professor K.E. Bullen. This representation is
consistent with the parametric simplicity required in
any reference earth model.
*
**
Permanent address: Department of Geological Sciences,
Harvard University, Cambridge, Massachusetts, U.S.A.
On leave from: Department of Applied Mathematics and
Theoretical Physics, University of Cambridge, Cambridge,
England.
If such a model were to be used as a basis for comparison with other earth models, or their functionals,
then representation of the seismic parameters by
piecewise continuous analytical functions would have
several advantages over a model defined at a number
of discrete points. These advantages are principally
related to the fact that the seismic parameters and
their derivatives could then be calculated exactly for
any desired value of the radius without resort to
numerical interpolation which always involves certain
subjective assumptions. Also, the functionals of the
model such as the travel times of body waves and their
derivatives would always vary smoothly as a function
of distance on a particular branch of a travel-time
curve
.
The remarks above descnbe the genesis of our approach to construction of an earth model, but the
more important reasons for undertaking this effort
PARAMETRICALLY SIMPLE EARTH MODELS
are closely related to basic problems of the structure
of the deep interior of the earth, such as the evidence
for radial inhornogeneities in the chemical or mineralogical composition of the matter in the lower mantle
and the cores. To be detected by seismologists, such
inhomogeneities must be expressed by a change in
seismic velocities or in density, and could be represented by a perturbation in the derivative of a seismic parameter with respect to radius or by a first-order discontinuity, depending on the method of inversion. If, on the
other hand, the material within the lower mantle, or
the cores, were homogeneous, the absolute values of
derivatives of the velocities and density with respect
to the radius should decrease smoothly and monotonically with increasing depth. Furthermore, if the temperature gradient below a depth of 600
700 km were
adiabatic, then the density gradient should satisfy the
Adams-Williamson equation.
It has been suggested in several studies based on
dT/dz~measurements using seismic arrays that there
are numerous perturbations in the gradient of cornpressional velocities in the lower mantle (for reviews
see Hales and Herrin, 1972; Wright and Cleary, 1972
or Wiggins et al., 1973). The discrepancies between
the results obtained at different sites suggest, however, that the perturbations in the dT/d~curves might
result from regional and local variations in the crustal
and upper-mantle structures,
Free oscillations, or normal modes, of the earth can
be assumed to be free of such local effects because of
their large wave lengths. Recently, Jordan and Anderson (1974) reported an earth model (Bl) consistent
with free-oscillation and travel-time data; Gilbert and
Dziewonski (1975) derived earth models (1066A and
1066B) using over one thousand normal-mode data.
All of these models show minor perturbations in the
gradients of seismic parameters that are inconsistent
with the hypothesis of homogeneity of the lower
mantle and the cores. The perturbations are particularly
noticeable in the outer core of model Bl and for the
compressional velocity in the lower mantle of the
models of Gilbert and Dziewonski. The inversion
methods used, although different in each study, require that the perturbation added to the starting model
be as smooth as possible. Thus, it would appear that
these variations in gradient must be real,
This conclusion may be invalid if the precision of
the data, or subsets of data, used in the inversion were
overestimated.
—
13
Gilbert (1971b) has shown that if we average observations of the normal-mode eigenfrequencies obtamed for a well-distributed network of stations and
sources, the effect of lateral heterogeneities would be
eliminated, and that the average eigenfrequency obtamed in this way would correspond to the degenerate eigenfrequency of a radially symmetric, nonrotating, average earth. However, the seismic stations
are certainly not evenly distributed on the surface of
the earth, and the bulk of our normal-mode data
comes from only two events (the Alaskan earthquake
of 1964 and the Colombian earthquake of 1970). Thus
a possibility of bias exists. Recognizing this, Gilbert
and Dziewonski (1975) assigned a minimum relative
standard error of 0.05%, and they checked that the
overall distribution of residuals followed closely the
normal distribution.
However, it is assumed in the inversion that the
covariance matrix is diagonal, i.e. that errors are not
correlated. If bias exists this may not be true and the
precision of the significant earth data may be overestimated. Some indication can be found in table 7
of Gilbert and Dziewonski (1975) where deviations
between the calculated and observed periods show
the same sign for more consecutive modes than statistically probable.
The effect of the estimates of the precision of the
data on the measure of the perturbation added to the
starting model is particularly clear in the formulation
of Gilbert (1971a, eqs. 8—10). For a particular set of
data an artificial decrease in the estimates of standard
errors will, in general, result in an increased measure of
perturbation in the model, unless the starting model
fits the data perfectly.
The significance of the features in an earth model,
obtained by application of an inversion method in
which the wavelength of perturbations are dependent
on the adopted errors of observations, is questionable
once the possibility that errors are correlated has
arisen.
In this paper we adopt a certain hypothesis with
regard to the smoothness of the distribution of the
seismic parameters with the radius; the observed data
are inverted using all the degrees of freedom allowed
by the hypothesis; following the inversion, the residuals
are examined if they are acceptable, the hypothesis
must be considered plausible if they exceed the al’
lowable limits, the hypothesis must be rejected.
Our hypothesis, consistent with the approach pro—
—
14
A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
posed by Hales et al. (1974), is that distribution of the
velocities and density in the lower mantle and the cores
can be adequately described by a set of second- and
third-degree polynomials in radius, the absolute values
of their first derivatives with respect to the radius decreasing within each region smoothly and monotonically with increasing depth. The requirement that the
density distribution should follow the Adams-Williamson equation is considered as an additional option in
further constraining the number of degrees of freedom.
The density distributions in many of the recent earth
models are very close to satisfying this equation (cf.
Dziewonski, 1971a; Jordan and Anderson, 1974;
Gilbert and Dziewonski, 1975).
The most significant differences between two earth
models presented by Gilbert and Dziewonski (1975)
occur in the upper mantle; model 1066A is described
by a set of functions that are continuous from the
bottom of the crust to the core—mantle boundary;
model l066B has two first-order discontinuities at
depths of 420 and 670 km, as does model Bl of
Jordan and Anderson (1974) which was used as a
starting mode in derivation of 1066B. Yet both 1066models give equally good fits to the data, and their
parameters below a depth of 1000 km are practically
identical. One obvious conclusion is That the data used
in the inversion fail to distinguish between the continGROUP VELOCITIES OF RAYLEIGH
~3.5
0
/
A
uous and discontinuous structures in the upper mantle,
which is not surprising if one examines the averaging
lengths for this region of the earth computed by
Gilbert et al. (1973). Another conclusion is that assumptions made with respect to the details of the
structure of the upper mantle do not introduce a detectable bias in the results of inversion at greater
depths. This allows us to consider certain aspects of
the earth’s structure at depths less than 420 kin simultaneously with the already outlined problem related to
the deep interior of the earth.
Hales (1974a) has questioned the validity of the approaches to derivation of a structure in the outermost
100—200 km used by Dziewonski and Gilbert (1972)
and Jordan and Anderson (1974) in their constructions
of an average earth model. He proposed that one should
consider two earth models one with an oceanic crust
and upper mantle (including a layer of water at the
surface) and the other with an average continental
structure. The observed periods of free oscillations
should then be compared with a weighted average of
the periods computed for each of the structures. Although there exists a satisfactory approach to derivation of an average earth model, as we shall show later,
the suggestion of Hales of derivation of two earth
models is worth following up for at least two reasons:
Fig. I shows computed group-velocity dispersion
—
WAVES
0
0
—
MODEL 5081E
MODEL
RI
0
0
AR A
OBSERVED OCEANIC
000
OBSERVED CONTINENTAL
0
3-0
ooo°°°
15
20
0
30
MODEL I-$BI
——
0
40
50
100
PERIOD
150
200
300
400
(s)
Fig. 1. Group velocities of Rayleigh waves computed for model HB1 of Haddon and Bullen (1969), 5.08M of Kanamori and Press
(1970) and Bl of Jordan and Anderson (1974) are compared with typical dispersion curves observed for oceanic and continental
paths.
PARAMETRICALLY SIMPLE EARTH MODELS
curves to periods as short as 15 seconds for three
widely quoted average earth models: HB1 of Haddon
and Bullen (1969), 5.08M of Kanamori and Press
(1970) and Bi of Jordan and Anderson (1974). The
group velocities of Rayleigh waves fall, for all these
models, outside the range of observed values for any
type of terrestrial structure. For example, the group
velocity for model 5.08M reaches a value of 4.20 km/
sec at a period of 37 sec, which is 4—6% higher than
the maximum observed in this period range for oceanic paths. It should, of course, be recognized that shortperiod data were not used in the construction of
model 5.08M and because of the basic differences between the oceanic and continental dispersion curves
at short periods an average earth model would be of
doubtful usefulness in this period range anyway. On
the other hand, the models of the crust and upper
mantle that are obtained by inversion of the shortperiod data are usually terminated at a depth of several hundred kilometers, and little consideration is
given to the requirement that if combined with a
model of the deep interior of the earth, such models
should produce periods of free oscillations that are
consistent with observations.
We present in this paper average oceanic and continental models that are consistent, in the sense of
weighted averages, with the free-oscillation data and
“pure path” dispersion data from 150 to 300 sec, and
also reproduce the characteristic pattern of dispersed
surface waves in the period range from 20 to 100 sec.
Such reference models could be used, for example, in
calculation of excitation parameters by earthquake
sources, or in application of the “residual dispersion
method” of Dziewonski et al. (1972). But, above all,
the requirement that the models should be consistent
with the data in a period range from 20 sec to over
3000 sec may be expected to impose additional constraints on the properties of each type of upper-mantle
structure,
Another problem that can be investigated by construction of distinct models for the continents and
oceans is that of the base-lines to the travel times of
body waves. Gilbert et al. (1973) and Jordan and
Anderson (1974) have pointed out that it is necessary to introduce corrections to the base-line values
of the travel times so as to assure their compatibility
with the normal-mode data. Hales (l974a) noted that
the estimates of the base-line corrections suggested
15
by those authors could be biased, because of the structure adopted in the uppermost region of their average
earth models. As the measurements and the reduction
of the travel-time data are almost entirely for landbased stations, it is logical that the travel-time data
should be inverted for an earth model with continental
crust and upper mantle. The base-line corrections
determined for this type of model should be both
more objective and also of more immediate application, as it would be not unreasonable to suggest that
such corrections be introduced to existing traveltime tables.
.
-
2. Inversion procedure and data selection
As usual, the first step in attempting to solve a nonlinear inverse problem by the linear estimation method
is to create a satisfactory starting model. Since our hypothesis requires that the final model should be represented by a set of low-order polynomials in the
radius, it is only natural to demand that both the
starting model and the perturbations determined
through inversion be expressed by functiOns of the
same type.
Hales et al. (1974) made specific recommendations
with respect to parameterization of the starting and
final models. While their suggestion that the lower
mantle and the cores be represented by smooth functions should be considered uncontroversial in view
of the objectives of this work, the reasons for representing the upper mantle by a series of regions
separated by abrupt discontinuities are not equally
compelling. One can produce a number of arguments
both for and against this form of representation of
the properties of the upper mantle. Our final decision in this respect was based on conceptual grounds
rather than on direct evidence, which is difficult to
obtain. We believe that our representation clearly
identifies the particular regions of the upper mantle
such as the lithosphere, the low-velocity zone and regions of phase transformations. Each of these regions is
of distinct significance in our present understanding of
large-scale geological processes, and for this reason we
think that they should be clearly identified. At the
same time, we must caution the reader against attaching too much significance to the specific values of
depths assigned to the discontinuities, or of the gradientswithin particular regions.
Our starting model was based on model 1066B of
16
A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
Gilbert and Dziewonski (1975), which resulted from
inversion of model Bi of Jordan and Anderson (1974)
with an augmented set of normal-mode data. In the
lower mantle and the cores, polynomials of the order
specified in table I of Hales et al. (1974) were fitted
by least squares to the specific parameters of model
1066B. The fit was quite close; in the lower mantle
the maximum deviation was of the order 0.5%. We
have found that the density in the lower mantle can
be adequately described by a polynomial of only
second order and, also, that the density gradient in
the lower mantle and the outer core follows very
closely the Adams-Williamson equation. Consequently, we have changed the parameters of density distribution of our starting model so that it follows exactly
the Adams-Williamson equation for all regions below
670 km depth. The density gradient in the outermost
part of the inner core of model 1066B is significantly
greater than adiabatic, but we have forced it to follow
the Adams-Williamson equation, as the local deviations
of the density from values predicted by the AdamsWilliamson equation are of the order of 0.5% and
beyond the resolving power of the normal-mode data
set for the density in this depth range. The core radii r1
and r2 (Hales et a!., 1974) of the starting model were
those of model 1 066B and these parameters were
allowed to vary in the inversion.
For the radii r~and r~of the upper-mantle discontinuities (Hales et al., 1974) we have adopted values
of 5701 km and 5951 km (depths of 670 and 420 km,
respectively). These two radii were held fixed in the
inversion. Linear relationships were assumed for the
parameters in the depth range from 420 to 670 km.
The zone from the surface to a depth of 420 km is the
region in which oceanic and continental-type models
may, according to our assumptions, have different
properties. Our crustal models were adopted according
to recommendations of Hales (1974b); the oceanic
model having 4 km of water at surface and 7 km
crust; the continental crust being represented by two
layers upper crust of thickness 20km, and lower
crust 15 km. The top of the low-velocity zone (LVZ)
was assumed for both models to be at 120 km depth
(but was reconsidered later and changed for oceanic
structure tu be consistent with the surface-wave evidence). The velocities in the overlying lid were assumed
to be 8.1 km/sec for Vp and 4.65 km/sec for V~.The
bottom of the LVZ was assumed to be at a depth of
—
220 km, and the velocities in the LVZ (constant) and
the region between 220 and 420 km (linear variation)
were chosen so as to represent as closely as possible
the properties of model 1066B in this range.
Calculations of Gilbert et a!. (1973) indicate that
attempts to determine independently densities for each
of the three regions between the crust and a depth of
420 km would not be justified by the resolving power
of the data set. Accordingly, we have represented
densities for this entire depth range by a first-order
polynomial fitted to the density distribution of model
1066B. This representation has enough degrees of freedom that if a result such as that presented by Press
(1969) were required by the data, it could be accomodated by an increase in densities immediately below
the crust.
Once the starting models are established, it is possible to compute their functionals such as eigenfrequencies or travel times of body waves and their first-order
variational parameters.
The data considered in the initial stage of our study
consisted of a set of 1064 normal-mode data presented
in table 7 of Gilbert and Dziewonski (1975) and a
set of travel times of body waves. We have chosen to
include in our inversion travel-time data for P, S, SKS
and PKIKP and differential travel times of SKKS—SKS.
The normal-mode data are considered to represent
the average properties of the earth and will be referred
to as gross earth data (GED), following the term first
used by Backus and Gilbert (1967). The travel-time
data, which according to our assumptions are to be
consistent with the continental upper mantle, represent “regional earth data” (RED). The subset of RED
will be expanded later by incorporation of regional
surface-wave dispersion data.
If an observed datum ‘y~is of GED type, then the
computed average functional is defined as:
g~=
~
+ ~ g,~
where g~andg~are the functionals computed for our
oceanic and continental models, respectively, and the
weighting factors 2/3 and 1/3 are meant to approximate the areal proportions of the earth’s surface covered by oceans and continents.
We develop two types of observational equations:
one for GED and the other for RED data. For GED
data we have:
PARAMETRICALLY SIMPLE EARTH MODELS
N
N
0
~
n 1
Ap~jXn +
n1
A~°1X~
+
Nc
+
~
~
n’l
A~X~=
where 6gf~=
g~N represents the number of unknown parameters (coefficients of polynomials) that
are common to both models, N0 and Nc the number of
parameters that are distinct for oceanic and continental models, respectively.
If X,~is a coefficient of an m-th power term in a
polynomial describing the distribution of a seismic
parameter between radii and
then:
—
r1~1
=
J”
G1(r) . r°~dr
rj
where G~(r)is the differential kernel for a particular
seismic parameter; for example, shear velocity. More
specifically G7 for the common parameters Xn is
determined as a weighted average of the differential
kernels obtained for oceanic and continental structures.
An bbservational equation for a regional datum,
is:
N
NR
~ AniXn + ~
A’~’.X~
— V1
n1
n1
‘f~= g1R
NR being the number of parameters for the relevant
region. The unknown Y~represents a correction to
the j-th subset of RED. For example, it may be a
base-line correction to the set of observed P-wave
travel times; in this case f~,
= 1. Later when we introduce regional surface wave data we shall have f~j
=
[(Ii + ~)w~]’,where w, is the eigenfrequency of the
i-th datum and 1, the angular order number,
Because the upper mantle of the earth is heterogeneous, the travel times measured at particular locations
show systematic differences in comparison with other
locations. To remove this effect a concept of “station
corrections” has been introduced. As the proper
world-wide averages of the travel times are not known,
the choice of “zero-correction” or the base-line level
is to some degree arbitrary. The free oscillations and
17
long-period surface-wave data can be expected to contam information that represents a better areal average
of the properties of the earth. To reconcile these two
types of data it may be necessary to introduce corrections to the base-line level of particular subsets of the
travel-time data, and some specific values have been
recommended by Gilbert et al. (1973) and Jordan and
Anderson (1974). Jordan and Anderson used in their
inversion differential travel times which should be to
a great extent free of the near-surface regional effects.
However, the observations of differential travel times
such as PcP—P, ScS—S, etc. are not nearly as numerous as those for phases such as P, S, SKS. Also, the
assumption that the differential travel times are free
from the effects of lateral heterogeneities may not
be justified. Unpublished observations of a nuclear
explosion by Hales and Nation indicate a difference
of 1 sec in the differential travel time of PcP—P observed at locations less than 100 km apart after a
correction for the epicentral distance has been applied.
For these reasons we decided that it would be best
to use the travel times (P, S, SKS, PKIKP) that are
more likely to represent average properties of the
earth’s interior and to allow in the process of inversion for an adjustment in base-line values such as is
required to assure compatibility of travel time and
free-oscillation data.
In selecting a set of observed P-wave travel times
we have considered the results of Cleary and Hales
(1966) and Herrin et al. (1968). The principal difference between these two sets is in the slope of the
respective travel-time curves, which amounts to traveltime differences of approximately 1 .5 sec over the
epicentral distance range from 30 to 90°.We decided
to use both sets of data in separate inversion runs and
to retain for further analysis the set that should prove
more compatible with the normal-mode data.
The S and SKS travel times were taken from the
study of Hales and Roberts (1970), and in addition we
have also used SKKS—SKS differential travel times of
Hales and Roberts (1971). The inversion study of
Gilbert and Dziewonski (1975) in which only normalmode data were used showed that their free-oscillation data are more compatible, regardless of the starting model, with the PKIKP data of Cleary and. Hales
(1971) than with those of Bolt (1968). Accordingly,
we have adopted the set of Cleary and Hales.
The branches AB and BC of the PKP phase occur
18
A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
on seismograms as later arrivals, and for this reason
may be known less accurately. Also there is some
question as to the extent of errors due to the approximations inherent in geometrical ray theory. The discovery by Haddon (1972), later elaborated by Cleary
and Haddon (1972), that the so-called GH branch is
related to scattering at the core—mantle boundary
puts in doubt the interpretation of published observations of the BC branch. Rays belonging to the AB
branch bottom in the depth range covered by the observations of SKS. For these reasons we decided that
incorporation of PKP travel times is unnecessary, if
not undesirable.
The initial inversion runs showed that without additional constraints the problem of base-lines of travel
times could not be resolved successfully. We found
it possible to satisfy the travel-time data without baseline corrections, the necessary increase (model l066B
would have required positive base-line corrections for
both P and S travel times) of the velocities of the con-
path composition should approach the global area!
composition.
There has been some concern over the discrepancy
between the results of Rayleigh wave “pure path”
analyses by Kanamori (1970) and Dziewonski (1971b).
Wu (1972) divided the earth into four types of region
oceans, continents, arcs and ridges and found that
the two sets of data give similar results. As the discrepancy between the results of Kanamori and Dziewonski
is clearly due to inadequate sampling, it would appear
that in a search for consistency one should reduce
rather than increase the number of types of region; for
a single “region” the average earth, the results of
Kanamori and Dziewonski are practically identical.
We have found that if the earth is divided into two regions oceans and continents, the sets of data of
Içanamori and Dziewonski also give similar results.
From the analysis of combined sets of data of
Kanamori and Dziewonski we obtained periods of
spheroidal oscillations from 0S21 to 0S61 for average
tinental upper mantle being compensated by a decrease
of velocities of the oceanic upper mantle. However,
we could not verify whether the resulting differences
in the velocities were not artificial. Nevertheless, as the
result of this experiment we were able to determine
that the P-wave travel-time data of Herrin et al. (1968)
were more compatible with the normal-mode data than
those of Cleary and Hales (1966) and thus the former
set was retained for further analysis.
The data that might provide an effective control
over distribution of shear velocities and, to some extent, compressional velocities for oceanic and continental regions are the so-called “pure path” dispersion
curves. The concept and the method of “pure path”
analysis was introduced by Toksoz and Anderson
(1966). In their original formulation the approach involved rather formidable assumptions that the dispersion is the same for each particular region (oceanic,
shield or tectonic)and that geometrical ray theory is
applicable to long-period waves. These assumptions
were necessary in the study of ToksOz and Anderson,
as they derived three “pure path” dispersion curves
from four great-circle path measurements.
As the number of available observations increases,
the concept of “pure path” dispersion can be replaced
by “average regional dispersion”. Also, the dependence
on applicability of the ray theory is less, as for a large
number of randomly distributed paths the average
oceanic and continental regions, and periods of
toroidal oscillations from 0T23 to 0T67 from the
Love wave data of Kanamori.
After these data were included in our inversion, it
became clear that the problem of base-line corrections
to the travel times cannot be eliminated by merely increasing average velocities in the upper mantle under
continents and decreasing them under the oceans.
The base-line corrections necessary, at this point of
our study, to satisfy the normal-mode, travel-time and
regional-dispersion data,were +3.5 sec for S travel
times of Hales and Roberts (1970) and +1.1 sec for P
travel times of Herrin et al. (1968).
The average r.m.s. error for 1064 normal-mode
data, 246 travel-time data and 32 regional-dispersion
data was 0.183%, which lies between the overall r.m.s.
errors obtained by Gilbert and Dziewonski (1975)
for their models 1066A and B being 0.177 and 0.186,
respectively.
It appeared that we had reached a satisfactory
agreement with observations. However, at this point
we calculated dispersion curves for our models for
periods from 150 to 15 sec. Our oceanic group-velocity dispersion curve has a maximum of 4.3 km/sec
at 50 sec; this is even further from the normally observed values than is the case for any of the dispersion curves shown in Fig. 1.
It became clear that major modifications in our
—
—
—
—
PARAMETRICALLY SIMPLE EARTH MODELS
19
oceanic structure were necessary to achieve at least
a qualitative agreement with the typical shape of
group-velocity dispersion curves observed for oceanic
paths.
curves is achieved. Translating the equation above into
terms of free oscillations, the group velocities will remain unchanged if the periods in a set under consideration are offset by 6,1T1 A’~~T1/(l + ~),whereA’ is
The principal problem in introducing short-period
dispersion data into an inversion study such as the
current one is that the proper spatial averages are not
known. Biased short-period data would not be cornpatible with either regional averages for long-period
waves or the gross earth data. Yet construction of a
model that would predict realistic values of group and
phace velocities is desirable for the reasons specified
in the introduction.
Fig. 5 of Bloch et al. (1969) compares a number of
phase-velocity curves for different continental regions:
three curves obtained for the central United States
(McEvilly, 1964), eastern Australia (Landisman et al.,
1969) and southern Africa (Bloch et al., 1969). The
three curves have nearly identical phase velocities for
these three widely separated regions. Comparison by
Knopoff (1972, see fig. 7) of Rayleigh wave space
velocities indicates that the dispersion curves for continental aseismic regions have values between those for
shields and tectonically active regions (rifts) and agree
closely with the three of Bloch et al. (1969). It may be
assumed, therefore, that the phase and group velocities for these paths are close to an average for continental regions.
Landisman et al. (1969) have also measured interstation group velocity, using the cross-correlation
method, for the same path across eastern Australia referred to above.
The use of observations of group velocities as the
basis for derivation of the data for inversion has the
following advantage. All phase velocity curves —C(w),
that satisfy the equation:
an arbitrary constant.
Starting values of periods of free oscillations for the
continental regions were obtained by integrating the
group-velocity curve in the left part of fig. 16 of
Landisman et al. (1969). The integration constant was
adjusted so that the average phase velocities in the
period range from 18 to 85 sec are the same as the
average phase velocities for the three curves in the
figure of Bloch et al. (1969). The range of angular
order numbers of spheroidal oscillations extends from
~1O9 at 87.29 sec to 0S613 at 18.44 sec. Particular
modes were selected for inversion with approximately
equal steps on a logarithmic frequency scale.
Starting values for short-period free oscillations
for the oceans were obtained by combining the group
velocities observed by Landisman et al. (1969) for a
long (12790 km) oceanic path from an epicentre
near the west coast of Mexico to Riverview, Australia,
with the regional group velocities from 150 to 300 sec
obtained from a two-region analysis of results of
Dziewonski (1971b). Group velocities from 18 to 300
sec were derived by the least-squares fit of an 8-th degree polynomial to the two sets of data described above, and the integration constant was chosen so that
the phase velocities observed in a period range from
150 to 300 sec were satisfied. The resulting phase
velocities at short periods are relatively high; between
30 and 60 sec phase velocities are nearly constant and
are close to 4 km/sec, slightly above the range of values
indicated by Knopoff (1972).
The group-velocity curve used above appears to be
quite typical; it may be compared, for example, with
the results for the path Rat Island—Charters Towers,
Australia (fig. 10 in Landisman et al., 1969). Numerous other examples (cf. Goncz, 1974) support, in
qualitative terms, the validity of our choice.
Incorporation of short-period data required substan-
=
C(co)
-~—
C0
+
~—
w
~
J
u
+
A
w
have the same group velocities —u(w). Since we may
expect that a selected phase-velocity curve might deviate from the true average for a particular region, the
arbitrary constant A could be allowed to float, so
that the optimum level of consistency with the worldwide averages and long-period regional-dispersion
tial modifications in the structure of the oceanic hithosphere. Preliminary investigation revealed that two
types of models of the oceanic lithosphere are consistent with the data. The first is characterized by a
relatively thin (60 km, including the crust and water)
lithosphere with a shear velocity in the lid of 4.55
km/sec. The other type allows for a somewhat greater
20
A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
total thickness of the lithosphere (80 km), with the
mantle shear velocities being 4.5 km/sec to a depth
of 50 kin, and 4.65 km/sec in the depth range from
50 to 80 km; shear velocities in the low-velocity zone
are somewhat lower in this case.
The appealing feature of the model with a twolayer lithosphere is that it explains the observed velocities of the oceanic Sn-waves (Molnar and Oliver, 1969;
Hart and Press, 1973). Existence of a zone of increased
velocities would be consistent with an increase of the
abundance of garnet in that depth range: Observations
of P travel times for an oceanic path across the Gulf
of Mexico by Hales et al. (1970) indicate an abrupt
increase in velocities at a depth of 57 km. A depth of
80 km to the top of the low-velocity zone is more
consistent with the current mineralogical and petrological theories (cf. Green, 1973, fig. 3). A depth of
60 km would require a substantial increase in the
water content of the upper mantle, perhaps larger
than is possible according to the current evidence.
Despite the above arguments in favour of the twolayer hithospheric mantle we adopt a model of the
first type in this study for the sake of parametric
simplicity because we believe that the introduction
of additional complications in the structure should
be supported by a larger and more representative
body of data.
Our final decision regarding the data to be used in
the inversion concerns observations of the fundamental toroidal mode. Observations of Mendiguren (1973)
and Gilbert and Dziewonski (1975) indicate differences
of the order of 0.5% between.the periods of ~7’~observed for the Colombian earthquake and the Alaskan
earthquake (Dziewonski and Gilbert, 1972). Inconsistencies among the average periods of toroidal modes
and Love wave phase velocities have been noted by
Kanamori (1970) and Dziewonski and Gilbert (1972).
It is clear that we do not have at the present time
satisfactory world-wide averages for the periods of
the fundamental toroidal modes with angular order
numbers greater than 33—35. The fact that the “all
data” averages of Dziewonski and Gilbert (1972) are
biased with respect to the remainder of the presently
available data set of free oscillations is evident from
table 7 of Gilbert and Dziewonski (1975); the residuals for the modes 0T35 0T46, where all observations come from the paper by Dziewonski and Gilbert
(1972), are large and all of the same sign. Jordan and
—
Anderson (1974) were able to satisfy these observations, but their data set did not include the large number of toroidal overtone data which have recently become available (Brune and Gilbert, 1974; Gilbert and
Dziewonski, 1975). Although the overtone data are
not sensitive to the details of the upper-mantle structure, they are sensitive to the average properties of this
region. Because of the obvious bias of the modes from
0T34 to 0T46 we decided to remove these data from
the data set to be used in the final inversion. The same
decision was made in respect of the regional Love wave
data. Obviously, because of greater sensitivity of Love
wave data to lateral heterogeneities, a significant increase in the number of observations is needed before
the derived averages could be considered representative
of the average earth and even more, of an average continental or oceanic structure.
3. Inversion and parameters of regionalized earth models
The final data set used in inversion consists of 1051
gross earth data representing the observed free-oscillation periods from table 7 of Gilbert and Dziewonski
(1975) after 13 observations for 0T34 0T46 have
been culled for reasons explained in the previous section; the regionalized earth data for continental structure consisting of 246 travel-time observations and
periods for 43 spheroidal modes ~ with 1 ranging
from 24 to 613; and the data for oceanic structure
consisting of 43 periods of 0S1 with a range of 1 from
24 to 554. The regional-dispersion data were derived
from group and/or phase-velocity measurements, as
explained previously.
Our starting model defined at the beginning of the
previous section has undergone changes at all levels, as
in the process of testing the various subsets of data we
have perturbed the models to improve agreement with
observations. However, the changes for depths greater
than 670 km represented in all cases only a fraction of
1%.
We have performed a number of experiments related
to determination of the base-line corrections to the
subsets of travel-time data use. Despite relatively
numerous observations for spheroidal overtones dominated by compressional energy it does not seem possible to determine uniquely the base-line value for P
travel times. Corrections to the travel times of Herrin
—
PARAMETRICALLY SIMPLE EARTH MODELS
et a!. (1968) ranged from approximately 1 sec to 2 sec
depending on the set of data included. Fixing the baseline correction at, for example, 1 sec, does not change
substantially the overall fit to the data in comparison
with the case in which the base-line is allowed to assume an arbitrary value. In general, the present set of
free-oscillation data seems to prefer a base-line correction of the order of 2 sec. There is one important exception, however. The residuals for the fundamental
radial mode 0S0, which is not as subject to a bias due
to an uneven and incomplete distribution of sources
and receivers as other spheroidal modes, seem to correlate with the magnitude of the base-line correction.
For base-line corrections of the order of 2 sec the residual is of the order of 0.2%; it can be reduced to less
than 0.1% for base-line corrections of 1 sec. Model
1066A requires a 2.4-sec correction and the residual
for ~ is 0.23%; the correction for model 1066B is
1.6 sec and the ~ residual is 0.19%. With the baseline correction of 1.25 sec, finally adopted in this
paper, we were able to reduce the residual for ~ to
0.09%. If a correction of +1.25 sec is adopted for P
travel times, the PKIKP travel times of Cleary and
Hales (1971) require a correction of +0.19 sec.
The question of base-line corrections for the S
travel times
is relatively
less complicated.
Theclearly
data including
the regional
free-oscillation
periods,
prefer a base-line correction of +2.6 sec to the travel
times of Hales and Roberts (1970), for epicentral distances from 30°to 82°.The correction increases to
+3 sec if the data for distances up to 98°are included.
The magnitude of this correction (for continental
structure) appears to be quite reasonable.
Although the S and SKS travel times in the study
of Hales and Roberts (1970) should have a common
base-line correction, our results indicate that the correction for SKS should be only +0.85 sec. This discrepancy could be partly related to the difficulties in
determination of S travel times at distances greater
than 82°,beyond which the S phase is a secondary
arrival after SKS. The differential travel times SKKS—
SKS were assumed not to require any base-line correction (Jordan and Anderson, 1974).
It was noted earlier that without the constraints
imposed by the regional-dispersion data it was possible to achieve satisfactory fits to the travel-time data
without base-line corrections and also to satisfy the
gross earth data. Incorporation of the regional-disper-
21
sion data has imposed constraints on the degree to
which velocities can be changed in the oceanic upper
mantle to compensate for the higher velocities of the
continental upper mantle.
With the base-lines fixed as described above, we
performed the final inversion. Before presenting the
results we define the third type of earth model the
average earth.
Realizing that a model of an average earth may be
—
useful in some applications, we define such a model
as a weighted average of the oceanic and continental
models. The parameters for the starting average earth
model were derived from the final regionalized models.
Parameters for depths greater than 420 km, assumed
identical for both regional models, remain the same
for the average model and are not allowed to change.
The density distribution at shallower depths was also
not allowed to change as its gradient and the value of
density at 420 km were assumed to be the same as for
both regional models.
If the thickness of a zone such as, for example, the
lid or LVZ is d0 for the oceanic model and d~for the
continental model then the thickness dA for the average model is:
2d+1d
3 o
~
A
d -
.
the average vertical travel time is:
d~,
d~
—
tA
—
T
+ 1
and the average velocity:
—
1”A — dA/tA
This definition of an average model is consistent
with the normal-mode—body-wave analogy in which
the overtones of normal modes can be compared to
multiply reflected body waves (cf. Singli and BenMenahem, 1969; Dziewonski and Gilbert, 1972 and
1973). After a large number of reflections, the average
travel time through the uppermost regions of the earth
should be consistent with the representation proposed
above.
In the final inversion for an average earth model we
use only the gross earth data (free-oscillation periods
of Gilbert and Dziewonski, 1975) and only parameters
22
A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
in the upper mantle are allowed to vary, to compensate for possible errors introduced by our averaging
method,
Table I lists the coefficients of polynomials describing the distribution of density and velocities within
the appropriate ranges of radius for the oceanic, continental and average models called PEM-O, PEM-C and
PEM-A (parametric earth model) respectively. The radius in the polynomial expressions is normalized (R =
na, where a is the radius of the earth 6371 km).
Table II gives the parameters of the earth models
computed for particular values of radius using the
coefficients of Table I. Table II also contains other
parameters important in the studies of the properties
of the earth’s interior such as the incompressibility,
Lame constants, Poisson’s ratio, pressure and gravity.
Parameters for depths within the uppermost 420 km
are specified separately for the oceanic, continental
and average models.
The continental model is plotted in Fig. 2 and the
upper mantles of the oceanic, continental and average
models are compared in Fig. 3.
In Table III we compare r.m.s. residuals for groups
of modes with the same radial order number calculated
—
ioóo
3000
2000
4000
5000
DEPTH
Fig. 2. Earth
model
~km)
PEM-C computed using the parameters of Table I.
aobo
6000
PARAMETRICALLY SIMPLE EARTh MO1)ELS
23
TABLE I
Coefficients of the polynomials describing the parametrized earth models (PEM). The variable R is t1~enormalized radius: R
where a is the earth radius — 6371 km
Region
Innercore
Radius range (km)
0
3)
Density (g/cm
—1217.1
—
Outer core
1217.1
—
3485.7
13.01219
8.45292*R2
12.58416
1.69929*R
—
1.94128*R2
—
7.11215*R3
—
Lower mantle
3458.7
—
5701.0
—
—
Transitionzone
6.81430
1.66273*R
1.18531*R2
5701.0—5951.0
11.11978
—
7.87054*R
Be1owLVZ
5951.0—6151.0
7.15855
LVZ
6151.0—6311.0
AboveLVZ
6311.0—6360.0
Crust
6360.0
—
6366.0
Sediments
6366.0
—
6367.0
Ocean
Vp (km/sec)
—
11.24094
4.09689*R2
V~(km/sec)
—
10.03904
3.75665*R
_13.67046*R2
16.69287
6.38826*R
+ 4.68676*R2
— 5.30512*R3
3.56454
3.45241*R2
0
+
—
—
+
—
9.20501
6.85512*R
9.39892*R2
6.25575*R3
21.05692
_12.31433*R
15.04371
_10.69726*R
22.53683R
_ii.86483*R
Oceanic structure
7.87320
4.33450
7.90000
4.55000
2.85000
6.40000
3.70000
1.50000
2.00000
1.00000
6367.0—6371.0
1.03000
1.50000
0
Be1owLVZ
5951.0—6151.0
7.15855
17:63609R
_9.32106*R
LVZ
6151.0
—
6251.0
7.84750
4.45860
Above LVZ
6251.0
—
6336.0
8.02000
4.69000
Lower crust
6336.0
—
6351.0
2.92000
6.50000
3.75000
Upper crust
6351.0—6371.0
2.72000
5.80000
3.45000
28.48832
_20.90003*R
15.09536
_11.01544*R
7.89520
4.34060
7.93420
4.65400
—
3.85999*R
Continental structure
—
3.85999*R
Average structure
Below LVZ
5951.0—6151.0
LVZ
6151.0—6291.0
Above LVZ
6291.0
—
6352.0
Lower crust
6352.0
—
6357.0
2.90200
6.50000
3.75000
Upper crust
6357.0
—
6368.0
2.80200
6.00000
3.55000
Ocean
6368.0
—
6371.0
1.03000
1.50000
0
—
7.15 855
3.85999*R
=
na,
24
A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
~
01
~
~
~
~0
PS
* 0•••SU•
W*•*••
~.:
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•0000S*•0
P *0000**•
SOPS*~•~•S
* P0
~0
PP•
——————————
-EJ
SB
o
~
,‘.RIs
.0
N~RC~’~—RE.-—.tV--.P.,Q’C
~
.00*
~.Rn
......
S**0**..S*.00060t*..,00000
~
~
.~
A~
SB
a
—
.i~
u~,g
(.J
~
eva ~
p*
•0.p000p.
IPSO
P S *
~
P0*
*0*5
P’~C~
•****•
.-‘~O4E~C~V~V
SO..,...
*
~t~a~t%c~———--
—-A———
a~$~
I
—
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RE
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Rfl
S •*SS**
0**S*S*•
S •5000•5005
•P500*000
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~
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W
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r~p,j.-,,. ~
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0.
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0~0~ lMfl ~C
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cv*nz~.-.a,q-4~’0G
•.**
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a.—.
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u,u,
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IA~ ~~~
-~ ~.
~ ~~~‘RCR~
I
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‘-
zBcA.n’oa~~
~
O~
~
.
~
a~
0
I).4.-o~OL~B0J
**•5550000005100555•*
a~ a.a~aa.aca~oaa~a-.,-..,c~oic
____
•.S*000.S*
>.%
I
LB
ttl
0.t1)
~
~ao.cvu.a
~
aaa
a.~a.aa
~
*3
~
~
~
>—~-.
05050*000
555.050..*.
S~c~OR0,0Ba
P.
0*
CvevvCMO. ..-.————
.alo,o,a,a,o,
.~
‘0
T
‘0 At
o
.~
CLB
IL)
3.’
5,
—
a.
~
~“)csja~~
~
~
~•
~
~
..P.S...S
P
~a~s
-~CvnJrvev
‘0
o
0.
0
RE
I
1—I
p
——
S
RE
~
*
s~a. s s a.a.
RE.~4
*Rfl
~4
~
Z
~
SB
1
0
.L
C~
.4
o_~
-‘
,~
—
k
~
U
E~
~°“~
~
~
°
RE ~
~A0A::
~
—
_,
ev~i-~”,~
‘~~
~4
i~ ~
~
PARAMETRICALLY SIMPLE EARTH MODELS
25
~
‘—LB
~
~
‘‘IL
~
.
~
.4”.
0.1
I4~’ttP’.
-o
~
IL
.-~
.
.
— —I-..
a
~P’.,CRC
~ ~ ‘~U~ O~ ~
—
is.
.4
I
——a.a~~’c
~
aa~a—’cD,”~~
~
ev~-CVC’
—
00*5*005000*5*0
SOS
.4
~
—
H
.4
~
~
0OS5~
~
~
~
~
-~
H
—~
0
H
4~,O~a.qi~I
~
~
C)
~q~4’BCv~Cva.’q
~—‘It’~~—~’
I
C~~)’) a~’a
~
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H
.4
H
H
~
~DI~flevCVev-.-~O~
cc)
C!)
0
s—~~~u’a ~~.-———.Inr
N.
to,
I—I
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is..’
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1.1
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IaJ
a.,,,
~
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~F)tI1t~**
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.4,,~F.e-O.a,a.a.Ia.a.a.S.~
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r~a.v)p...a.
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~u~-,fl’.a.s~
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-~
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a.~
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~~~)45a.a.
‘U ,~)evCvO~—-A—
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15’Ø
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~CI~)N.-~C~CCCCP.’.
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—
c~.eva ~‘—~————
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—C ‘-s——CMOS,.’. N. t
aa~a.—— ~
-I
UJ
~-B
~J
~
..(w)q W~Cp’-~
S
‘q,~)r)cvo4evev.=-A
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oi ~*esr)’U
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26
A.M. DZIEWONSKI, A.L. HALES AND ER. LAPWOOD
~-‘::~~
0
100
200
300
DEPTh
400
500
600
kn,)
Fig. 3. Upper-mantle models PEM-C (continental), PEM-O (oceanic), and PEM-A (average earth). For depths greater than 420 km
parameters for all three models are identical; see Fig. 2.
for model 1066B of Gilbert and Dziewonski (1975),
the weighted averages for our oceanic and continental
models, and our average earth model. The overall
standard deviation of our models is only slightly greater
(2 and 5%) than the corresponding value for model
l066B. The r.m.s. residuals of the weighted average
periods for the fundamental modes 0T1 and ~ are
than those for models 1066B and
PEM-O; undoubtedly because of the bias introduced by
the regional thort-period surface-wave data. The overtones 10S1, ~~S1and 14S1 show a markedly poorer
overall fit to the data as compared with model 1066B.
As these series contain a number of modes sensitive to
the compressional velocity in the lower mantle it may
be that this particular group of the observed modes
should be closely scrutinized with respect to a possible bias in observations,
Table IV contains a list of the observed eigenperiods
and the corresponding values computed for the oceanic, continental and average earth models. The eigenperiods are greater, in general, for the oceanic structure
than for the continental one. This is patticularly clear
for the toroidal modes, and is undoubtedly associated
with the fact that the vertical two-way travel time for
the S-waves between the surface and 420 km depth is
186.3 sec for the PEM-C model and 187.5 sec for the
considerably greater
PEM-O model, where it is measured from the bottom
of the ocean (4 km).
The corresponding travel times for P-waves are
103.4 and 102.50 sec, but if the oceanic P travel times
were measured at the surface the latter estimate would
increase to 107.8 sec. However, the reflection coefficient at the ocean bottom is —0.85 and the effective
(in the sense of the body-wave— normal-mode analogy) P travel times are only slightly greater fot the
oceanic structure. This may be inferred from the
comparison of eigenperiods of modes dominated by
compressional energy, such as the radial modes.
The calculated phase and group velocities for modes
PEM-O and PEM-C are compared with the observations
in Figs. 4A and B. It will be noted in Fig. 4A that
there are systematic differences between the observed
velocity and those computed for the continental model. These arise because we allowed the constant A’ to
float in order to preserve consistency with the gross
earth data.
Comparison of the computed and observed travel
times is made in Figs. 5A—E. While the overall trend
of the P travel times computed for model PEM-C
agrees well with that of the 1968-tables of Herrin et
al. (1968), it may be observed that the details of the
deviations coincide very closely with those for the
PARAMETRICALLY SIMPLE EARTH MODELS
27
TABLE III
Comparison of r.m.s. relative residuals of the eigenperiods ofradial modes and particular overtones of toroidal and spheroidal modes
computed for model 1066B of Gilbert and Dziewonski (1975), weighted averages for PEM-O and PEM-C models and PEM-A model
Overtone
Number of
Relative error (%)
Overtone
modes
Allmodes
5o
1066B
regional
PEM-A
1049
0.182
0.191
0.186
12
0.129
0.123
0.120
31
58
38
39
34
32
32
34
25
28
23
2
2
2
2
2
2
2
2
65
55
51
44
30
42
48
0.103
0.064
0.062
0.110
0.212
0.210
0.356
0.332
0.403
0.436
0.461
0.383
0.315
0.197
0.243
0.232
0.131
0.242
0.198
0.057
0.105
0.174
0.073
0.063
0.065
0.089
0.170
0.079
0.086
0.101
0.210
0.202
0.353
0.351
0.400
0.437
0.452
0.378
0.347
0.238
0.223
0.252
0.134
0.199
0.244
0.131
0.126
0.112
0.098
0.082
0.082
0.113
0.130
0.102
0.072
0.103
0.223
0.199
0.327
0.344
0.399
0.444
0.469
0.382
0.328
0.209
0.232
0.226
0.131
0.272
0.192
0.069
0.115
0.091
0.071
0.112
0.089
0.112
n
2T1
3T1
7T1
8T1
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travel times of Cleary and Hales (1966). The P-waves
for model PEM-C reach the core—mantle boundary
at a distance of 94.5°.This is somewhat earlier than
is commonly accepted, but is in general agreement
with observations of amplitudes by Sacks (1966, fig.l).
The computed travel times for the PKIKP phase
(Fig. SB) follow very closely observations of Cleary
and Hales (1971); the r.m.s. error within the epicentral
distance range from 130 to 180°is below 0.2 sec. The
differences are greater for distances less than 130°,
but the amplitudes in this distance range are small and
observations are difficult.
The r.m.s. error of our fit of the S-wave travel
Number of
modes
39
19
19
22
31
15
21
12
13
13
13
12
7
12
9
7
9
8
7
6
5
4
2
3
1
1
1
1
Relative error
(%)
1066B
regional
PEM-A
0.106
0.098
0.099
0.120
0.102
0.065
0.164
0.094
0.098
0.116
0.103
0.175
0.065
0.138
0.105
0.123
0.090
0.105
0.060
0.188
0.177
0.139
0.102
0.050
0.258
0.072
0.221
0.091
0.139
0.107
0.131
0.187
0.112
0.131
0.146
0.202
0.126
0.117
0.114
0.173
0.091
0.108
0.120
0.134
0.093
0.090
0.062
0.119
0.164
0.156
0.139
0.091
0.329
0.032
0.289
0.095
0.118
0.096
0.149
0.163
0.106
0.110
0.148
0.187
0.136
0.123
0.116
0.185
0.089
0.157
0.118
0.129
0.092
0.097
0.064
0.120
0.163
0.152
0.138
0.088
0.327
0.033
0.287
0.095
times of Hales and Roberts (1970) is 0.75 sec for the
distance range between 30°and 82°while eq. 2 of
Hales and Roberts gives an r.m.s. error of 0.73 sec.
The vertical line in Fig. SC separates the distance
ranges in which the S-wave precedes and then follows
the SKS phase. The increase in the scatter of observations for distances greater from 82°is evident; it also
coincides with the occurrence of systematic differences
between the computed and observed travel times.
The computed travel times for the SKS phase (Fig.
SD) agree with the observations of Hales and Roberts
(1970) nearly as well as the polynomial representation
given by those authors in their eq. 5. Also the devia-
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PARAMETRICALLY SIMPLE EARTH MODELS
39
/
PHASE VELOCITIES OF RAYLEIGH WAVES
5~0
>-
4.5
—
o
O
—
54
-
>
I
— —
40
_~
0 0O~
—
000
—
o~
0
.‘
0
“0
—
PEM-OCEANIC
___
PEM-CON4TINENTAL
000
OBSERVED CONTINENTAL
‘0
~0
3~ A
-,
15
20
30
40
50
100
150
200
300
PERIOD (s)
Fig. 4A. Computed Rayleigh
wave phase velocities for models PEM-O and PEM-C. The systematic differences between the observed and computed continental phase velocities result from an adjustment required to satisfy the gross earth data; see the text
for details.
4.5
GROUP VELOCITIES OF RAYLEIGH WAVES
00 0
OBSERVED CONTINENTAL
0~
30
QQQo000~
B
I
15
-,
~
I
20
30
40
I
I
4
50
PERIOD
4
I
100
150
200
300
400
(s)
Fig. 4B. Comparison of group velocities of Rayleigh waves observed for continental and oceanic paths with those computed for
models PEM-C and PEM-O.
40
A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
P
~
—
P04-C
iii ~
I’
::~
/
:E~/
10
20
300060
70
80
90
~0
EPICENTRAL DISTANCE (DEOREES)
Fig. SA. Deviations of the P-wave travel times computed for Model PEM~Cfrom the “1968 Travel Time Tables” of Herrin et al.
Deviations for Jeffreys and Bullen (1940) and Cleary and Hales (1966) travel times are added for comparison. Note the similarity
between the short wave length features in the curve for PEM-C and the data of Cleary and Hales.
PKIKP
— —
— —
—
—
120
I
I
130
140
—
MODEL PEM-C
———
CLEARY & HALES
(1971)
I
150
EPICENTRAL
DISTANCE
160
170
180
(DEGREES)
Fig. SB. Deviations of the PKIKP travel times computed for model PEM-C and observations ofCleary and Hales (1971) from the
data of Bolt (1968).
74P~.s
C
11~:~oS
EPICENTRAL
DISTANCE
(DEGREES)
Fig. SC. Deviations of the S-wave travel times computed for model PEM-C from the Seismological Tables of Jeffreys and Builen
(1940). Observations and polynomial representation (eq. 2) ofHales and Roberts (1970) are added for comparison; a base-line
correction of +2.60 sec has been applied to the values of Hales and Roberts. The vertical line designates the epicentral distance
beyond which S-wave arrivals follow those of the SKS phase.
PARAMETRICALLY SIMPLE EARTH MODELS
41
SKS
8-
•
0
MODEL
HALES & ROBERTS
HALES & ROBERTS
•••
-
.~
—
— ——
PEM-C
1970
1970
EQ 5)
OBSERVATIONS
D
80
90
100
110
EPICENTRAL DISTANCE
120
130
(DEGREES)
Fig. SD. SKS travel times computed for model PEM-C plotted against the Jeffreys-Bullen travel times. Observations and polynomial representation (eq. 5) of Hales and Roberts (1970) are added for comparison; a base-line correction of +0.85 sec has been
applied to the values ofHales and Roberts.
1
SKKS
-
SKS
8 E
EPICENTRAL
DISTANCE
(DEGREES)
Fig. 5E. Deviations of the differential travel times SKKS—SKS computed for model PEM-C from the values predicted by eq. 3 of
Hales and Roberts (1971).
tions of the computed differential travel times SKKS—
SKS from the values predicted by eq. 3 of Hales and
Roberts (1971) are not excessive, in view of the scatter
of the original observations shown in their fig. 1.
4. Discussion
The comparisons in Table III show that our parametrically simple spherically symmetric earth models
fit the observational data about as well as the models
derived by other inversion procedures, for example
those of Jordan and Anderson (1974) or Gilbert and
Dziewonski (1975). It is our belief that models with
this type of parameterization are particularly well
suited for use as reference or comparison models.
However, we must emphasise that this model, like all
other models obtained by generalized inversion procedures, is dependent on the characteristics of the starting model on which the inversion is based, the averaging or smoothing procedure used in the inversion
and, of course, the observational data set selected
for inversion. We have been careful to describe in detail the choices with regard to the observational data
set and the starting model for our inversion. We believe the use of simple polynomial-type parameteriza-
42
AM. DZIEWONSKI, AL. HALES AND E.R. LAPWOOD
tion exposes as fully as any other procedure the limita-
tions on the model which have been imposed by the
need for averaging or smoothing. We think, therefore,
that this model is an adequate representation of the
real earth.
It should be recognized, however, that the averaging process may have obscured certain minor deviations which could conceivably have considerable geophysical significance. Such deviations may be revealed
by special studies of the travel times or the apparent
slowness of particular phases.
In the inner core of our model the shear velocity
varies from 3.44 to 3.56 km/sec confirming the earlier
estimate of 3.5 km/sec by Dziewonski and Gilbert
(1971). We are not aware of any data which suggest
possible geophysically significant deviations from the
PEM models of the cores. The differences of the velocities and radii of the PEM core model from those of
Masse’ et al. (1974) seem to us to be within the limits
of observational error,
In the lower mantle there are two regions where
deviations from the PEM model may be geophysically
significant. We show in Fig. 6 calculated values of
dT/d~for the model PEM-C. It is known (Hales et al.,
1968; Hales and Herrin, 1972; Wiggins et al., 1973;
Wright and Cleary, 1972), that dT/d~varies linearly
with ~ up to about 850. Thereafter there is a signifi-
cant departure from the linear relation. Support for
this deviation comes from the amplitude relations
(cf. Hales and Herrin, 1972, fig. 13). It is possible that
the PEM model does not adequately reflect the behaviour of dT/di~beyond 85°.
The other region of dT/d~where a geophysically
significant deviation might occur is between 670 and
1000 km corresponding to body-wave travel-time
arrivals from 25°to 40°. There is some indication both
from S travel times (Hales and Roberts, 1970) and P
travel times that dT/d~does not follow the linear relation in this range of i~. Careful studies of the second
arrival phases and comparison with synthetic seismograms (cf. Helmberger and Wiggins, 1971) should clarify the velocity distribution between 670 and 1000
km.
The choice of a starting model has its most significant effect between 220 and 670 km. There we have
chosen to use sharp discontinuities rather than a
smooth velocity distribution. It is clear from table 7
of Gilbert and Dziewonski (1975) that models 1066A,
which is relatively smooth, and 1066B, which has
sharp discontinuities, fit the observational data equally
well. Thus free-oscillation data do not permit discrimination between these two radically different models
of the transition zone in the upper mantle. It is possible to show also that the surface-wave dispersion for
8
7~
6
—
PLO-C
—— —
HERRIN
21
.1
(I9RB)
N
5
~
~1~
4%
30
40
50
60
70
80
90
EPICENTRAL DISTANCE (DEGREES)
Fig. 6. Comparison of the dT/d~values computed for model PEM-C with the corresponding values of Herrin et a!. (1968).
PARAMETRICALLY SIMPLE EARTH MODELS
43
these two models does not lie sufficiently far outside
TABLE V
the experimental error of both the group and phasevelocity determinations to make discriminations between the models 1066A and 1066B possible on the
basis of dispersion measurements.
The choice of a sharp discontinuity at 670 km is
supported by the evidence of Engdahl and Flinn (1969)
that there are clear short-period reflections of the
PKIKP phase from the underside of this discontinuity.
As was pointed out by Hales (1972) and Richards
(1972), these sharp reflections from the underside of
the discontinuity imply that the major part of the
change of the velocity must take place within a relatively short depth range, estimated by Richards at 4
km. Petrological considerations suggest that there may
be more than one phase transition in the 600—700
km depth range. The seismological evidence implies,
however, that at least one of these is sharp and that a
major part of the velocity change is discontinuous or
nearly so. For the 420 km discontinuity the evidence
for or against sharpness is not clear. The choice of a
single sharp discontinuity was made in the interest of
simplicity of the model. Thus again progress towards
more precise models of the velocity distributions betweeñ 220 and 670 km must wait for further precise
travel-time studies especially of the second-arrival
phases coupled with synthetic-seismogram comparisons.
It should, of course, be clear that the velocity distributions so derived must be in accord with the freeoscillation data. The lack of control of the details of
the velocity distribution in the upper mantle does not
imply that the free-oscillation data can be fitted by
any velocity distribution whatsoever. The average
velocities are constrained within relatively close limits
(Gilbert et al., 1973), and thus the free- oscillation
data serve as a control on the validity of the velocity
distributions derived in other ways. It is probable
also that once the velocity distributions in the upper
mantle have been derived using other methods, it will
be possible to obtain tighter constraints on the density distribution in this region from the free-oscillation data.
In the uppermost upper mantle the PEM model fits
the data reasonably well. It is clear, however, that it
may be somewhat too simple in this region to fit the
data exactly. A combination of travel-time and surfacewave dispersion studies may lead to more detailed
models for specific regions and ultimately to better
Deviations of the densities in the cores and lower mantle of
the PEM models (p) from the densities predicted by the
Adams-Williamson equation (PAW)
_____________________________
p — PAW (g/cm3)
Radius (km)
p (g/cm3)
knowledge of upper-mantle structure.
_____
______
0
300
13.012
0.002
12.993
0.000
600
1217
1217
1500
1800
2100
2700
3000
3300
12.937
12.139
11.984
11.789
11.558
0.000
—0.001
—0.015
0.000
0.006
0.005
101974
—01006
10.611
10.195
—0.007
0.000
3486
3486
9.909
5.550
0.0 17
—0.008
4100
5253
4400
4700
s.ioi
0.002
0.000
—0.001
—0.003
0.000
0.005
5000
5300
5600
121704
4.943
4.779
4.611
4.437
.
-________
We are confident, however, that any new models
found by travel-time studies, or in other ways, will
have averages of the densities and velocities over depth
ranges of 200—300 km in substantial agreement with
similar averages found from the PEM model.
We remarked earlier in this paper that our starting
model for density followed the Adams-Williamson
equation very closely. Since the changes in density
during the inversion were small, the departures from
uniformity in the final PEM model, i.e. from the AdamWilliamson densities, would be expected to be small.
This is confirmed by Table V, which shows the devialions of the PEM model from the Adams-Williamson
predicted densities. The maximum deviation is 0.2%.
The question of uniformity can also be examined
in terms of equations involving the velocities. Bullen
(1949) introduced 1 —g1(d~/dr)as a measure of
uniformity. Birch (1952) showed that 1 g1(dØ/dr)
dK
5/dP + a~r/gwhere a = coefficient of thermal expansion, r the excess of the actual temperature gradient
44
A.M. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
over the adiabatic, K
5 the adiabatic incompressibilit1
and 0 is the seismic parameter (0 = K5/p = V~— ~ l’~).
Using finite strain theory he showed:
1 —g
(aK~\ l2—49e
J = 3(1—7e) +aterminvolving~,
‘ 3P i T
1dØ
—~
dr
.
TABLE VI
Zero-pressure parameters derived from a least-squares fit of
finite-strain theory to the PEM models
.
—_________
Parameter
Case I
Case II
3486~r~5701 3831~r~5359
3.99 1
3.986
3
e being the strain and given by p/p0 = (1 — 2e)~and ~
a temperature-dependent
coefficient arising in the
finite-strain
equation of state.
3)
p(g/cm
?t(Mbar)
This equation shows that for ~ = 0, [1 —g~(dØ/dr)]
should be 4 for zero strain and decrease as ci increases.
Values of [1 —g1(d~/dr)] for the PEM models are
shown in the last column of Table II. They decrease
from 3.388 at 670 km to 2.808 at 2886 km. Reference
to Birch’s table 5 shows that these values are too low
to fit the case ~ = 0 and must correspond to a small
~2(Mbar)
K (Mbar)
positive value of ~
Later, Sammis et al. (1970) derived complete thirdorder Eulerian equations for the velocities in BirchMurnaghan finite-strain theory (Birch, 1939). For
hydrostatic compression these.equations are:
pV~= (1
—
2e)512 [X0+ 2/10
pV~= (1
—
2C)5/2
=
—3K0e (1
where ~,
i~ are
—
e(1 1k
+ 10/10
—
[~o e (o
—
2
P
—
+ ~
+ ii)]
(1 + 2e~)
2e)SI’
dependent on temperature and:
1.305
1.293
1.332
2.193
9.972
5.777
54.95
1.346
2:190
9.999
5.811
54.94
..L ~! (Mbar—1) 0.483
Vp a
1°
1 ~VS
0.477
~
0.324
0.314
r.m.s. error (%) Vp
0.239
0.232
0.247
0.107
V8
0.390
0.182
~
1.244
0.413
Vp (km/sec)
VS (km/sec)
~I
(km2/sec2)
-~-(Mbar’)
to r = 5359 km alone. The deviations for this calculation are
shown
Fig.errors
7. The
of the
curvein
does
notalso
change,
butinthe
areshape
reduced
because
the first case considered the largest deviations were
towards the ends of the curves. The parameters K
0,
=
3~+ 4
p0, etc. were not changed appreciably as can be seen
12K0
from Table
VI. The
systematic
nature of
the deviations
may arise
because
the third-order
strain
theory
We have fitted by least squares these three equations
to the lower-mantle values of V13~ V~,
p and P given
in Table II to find k~
~ ~ and K0 for fixed values
of p0. Analogous calculations using the Sammis et al.
(1970) equations were carried out by Anderson et al.
(1971).
The deviations of the calculated values from the
values given in Table II are a minimum for p0 = 3.991.
The parameters for this case are given in Table VI.
(Note that these parameters are for lower-mantle
temperatures and that density should be increased by
4% and 0 by 10%, approximately, for comparison
with laboratory measurements.) The deviations of
the velocities are plotted in Fig. 7. The deviations are
small, 0.23% for Vp, 0.38 for V~,but are systematic.
We, therefore, fitted the tabular values from r = 3831
is not adequate, or because of the temperature-dependence of the constants ~, 17, and ~ in third-order
theory, or by reason of minor inhomogeneity of the
material of the lower mantle. It will be studied further
in a separate paper.
The constants K0, ~ [(1/ V~)(a
V5/aP)] and
[(1/ V~)(a
Vp/al’)] are important for the identification
of the materials of the lower mantle. In so far as
and p0 are concerned our values confirm Birch’s
(1952) conclusion that the constants for “periclase,
corundum and rutile show that oxide structures can
possess the required tightness of binding, combined
with a suitable density”.
Table VII shows values of these parameters for a
number of minerals; it was prepared for us by Dr.
Robert C. Liebermann. It is clear that the logarithmic
PARAMETRICALLY SIMPLE EARTH MODELS
45
\\~,_~
DEVIATIONS OF FiNITE STRAIN
FROM PEM MODEL
I
MODEL
I
VFS
-0~O5’
-
v~M
1
5
I
600
CASE
I
1000
3000
2000
DEPTH
)km)
Fig. 7. Deviations of the compressional and shear velocities computed according to finite-strain theory from the appropriate
values for the lower mantle of the PEM models. For Case I the finite-strain theory equations were applied to the entire lower
mantle; Case II corresponds to a radius range from 3831 to 5359 km.
TABLE VII
Pressure derivatives for various minerals
Structure
Compound
t)
1/V[(a V)/(aP)TI, (Mbaf
Reference
Compressional
Shear
2SiO4
(Mg0 93Fe0 07)2SiO4
Fe2SiO4
1.249—
1.211
1.325
0.7 14
0.736
0.172
Kamazawa and Anderson (1969)
Pyroxene
(Mg0 8Fe0 2)Si03
2.644
1.093
Friliblo and Barsch (1972)
Garnet
almandite-pyrope
spessartite-almandine
grussularite
0.919
0.842
0.513
0.456
0.467
—0.022
Anderson et al. (1968)
Wang and Simmons (1974)
Halleck (1973)
Spinel
MgAI2O4
0.563
0.559
0.494
—0.008
Mg0 75Fe036Al190O4
MgO~2.6 A12O3
Chang and Barsch (1973)
Wang and Simmons (1972)
NiFe2O4
0.610
—0.008
Ti02
Ge02
Sn02
0.825
0.652
0.4.65
0.101
0.206
0.026
Manghnani (1969)
Wang and Simmons (1973)
Chang and Graham (1974)
Rocksalt
MgO
0.740
1.308
0.478
0.621
0.602
Corundum
CaO
A12O3
0.347
Spetzler (1970)
Anderson et al. (1968)
Anderson et al. (1968)
Fe203
0.591
0.151
Anderson et al. (1968)
Si02
1.325
0.172
Anderson et al. (1968)
Olivine
RutileS
a-quartz
Mg
—0.093
0.076
Kamazawa and Anderson (1969)
Chung (1971)
Anderson et al. (1968)
Liebermann (1972)
46
derivatives of the velocities,
AM. DZIEWONSKI, A.L. HALES AND E.R. LAPWOOD
[(l/V~)(aV~/3P)Jand
S. Conclusions
[(l/V~)(aV5/aP)],may
provide a tight control on the
lattice structures of the material predominant in the
lower mantle. The only single mineral in Table VII
which comes close to fitting these parameters is
corundum (A1203). Anderson et al. (1971) observed
that the laboratory measurements of the logarithmic
deviations of the velocities for Al203 and Fe203 give
the closest match to the values derived from finitestrain theory. Our values of [(l/V~)(3V5/aP)] are too
high to be compatible with Fe203.
In general one expects that the materials of the
lower mantle will consist of high-pressure transforms
of olivine (Mg,Fe)25i04 and pyroxene (Mg,Fe)SiO3.
One of the possibilities is disproportionation into penclase and stishovite. The logarithmic derivatives of the
velocities have not yet been determined fo~stishovite,
but would be expected to be similar to those for the
rutile structures in Table VII (R. Liebermann, personal
communication, 1974). Clearly it would be difficult to
match øo’ p0 and the logarithmic derivatives with such
a mixture because of the considerable differences in
[(l/v~)(av~/aP)]0 and [(I/V5)(aV5/3P)]0.
Although logarithmic derivatives of the velocities
for stishovite are not available, Mizutani et al. (l972a)
have determined the velocities for stishovite at pressures less than 10 kbar and room temperature. Their
value for the shear modulus, p, is 1.30 ±0.07 Mb almost exactly the same as the value for the shear
modulus of p for periclase given by Anderson et al.
(1968), namely, 1.288 Mbar. The values of p for
wtistite (FeO) quoted by Mizutani et al. (l972b), and
Akimoto (1972) are 0.51 and 0.55, respectively. Thus
any disproportionation of olivine or pyroxene into
MgO, SiO2, FeO will inevitably result in a zero-pressure
room-temperature value for p less than 1.30 Mbar.
This is lower than the value of 1.332 Mbar given in
Table VI which is for lower-mantle temperatures.
Thus it seems that unless the proportion of FeO to
MgO is improbably close to zero the material of the
lower mantle must consist at least in part of denser
structures (e.g., perovskite, calcium ferrite) though
admixture with oxide structures is possible (Ringwood.
1970). Similar conclusions with regard to the implications of the shear velocities found for the lower mantle
were reached by Mizutani et al. (1972a).
It has been shown that it is possible to construct a
parametrically simple earth model (PEM-A) which fits
the observed free-oscillation data satisfactorily. This
average earth model was derived from an inversion in
which separate models, PEM-O and PEM-C, of the
uppermost 420 km were used for the oceans and continents. Below a depth of 420 km all three models are
the same. These regional models are consistent with
principal features of oceanic and continental dispersion curves to periods of about 20 sec.
The seismic-body travel-time data are regarded as
applying principally to the continents because of the
preponderance of land-based stations. The continental
structure was therefore used for the inversion of the
travel-time data. If the regional surface-dispersion data
are not included in the inversion, satisfactory fits to
all data can be obtained without introducing baseline corrections. In the final model the incorporation
of regional-dispersion data made introduction of baseline corrections necessary. These corrections are + 1.25
for the Herrin et al. (1968) P travel times, +3.0 sec
for the Hales and Roberts (1970) S-times, +0.19 for
Cleary and Hales (1971) PKIKP times and +0.85 to the
SKS travel times of Hales and Roberts (1970). The
difference in the two-way vertical travel times for Swaves of PEM-O and PEM-C to 420 km depth is only
1.1 sec.
The densities and velocities in the lower mantle
and cores were found to be consistent with the AdamsWilliamson equation to less than 0.2%. Thus it can be
inferred that any departures from homogeneity and
adiabaticity within each of these regions must be
very small. It was shown also that the velocities in the
lower mantle were consistent with the complete firstorder finite-strain theory to within 0.2% for V~and
0.4% for V5. The logarithmic derivatives of the velocities with respect to pressure obtained from the application of finite-strain theory were very similar to those
for corundum structures.
Acknowledgements
This paper was written while two of the authors
47
(A.M. Dziewonski and E.R. Lapwood) were Visiting
Fellows at the Australian National University.
Dr. Robert C. Liebermann has read the manuscript
critically and made many helpful suggestions. We wish
to thank Dr. Liebermann, Professor A.E. Ringwood and
Dr. D.H. Green for their interest and helpful discussion.
We have pleasure in acknowledging our indebtedness
to Professor Freeman Gilbert for the use of his program
for the numerical solution of the normal-mode problem and to Dr. Bruce Julian for the use of his traveltime program.
One of us (A.M. Dziewonski) wishes to acknowledge
support under National Science Foundation Grant
GA-32320 and the Committee for Experimental Geology and Geophysics, Harvard University.
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