Geophys. J. Int. (2000) 143, 821±836
Inversion for mantle viscosity pro®les constrained by dynamic
topography and the geoid, and their estimated errors
Svetlana V. Panasyuk1 and Bradford H. Hager2
1
2
Harvard University, 20 Oxford Street, Cambridge, MA 02138, USA. E-mail: [email protected]
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Accepted 2000 July 14. Received 2000 July 10; in original form 2000 January 12
SUMMARY
We perform a joint inversion of Earth's geoid and dynamic topography for radial
mantle viscosity structure using a number of models of interior density heterogeneities,
including an assessment of the error budget. We identify three classes of errors: those
related to the density perturbations used as input, those due to insuf®ciently constrained
observables, and those due to the limitations of our analytical model. We estimate the
amplitudes of these errors in the spectral domain. Our minimization function weights
the squared deviations of the compared quantities with the corresponding errors, so that
the components with more reliability contribute to the solution more strongly than less
certain ones. We develop a quasi-analytical solution for mantle ¯ow in a compressible,
spherical shell with Newtonian rheology, allowing for continuous radial variations of
viscosity, together with a possible reduction of viscosity within the phase change regions
due to the effects of transformational superplasticity. The inversion reveals three distinct
families of viscosity pro®les, all of which have an order of magnitude stiffening within
the lower mantle, with a soft Da layer below. The main distinction among the families is
the location of the lowest-viscosity regionÐdirectly beneath the lithosphere, just above
400 km depth or just above 670 km depth. All pro®les have a reduction of viscosity
within one or more of the major phase transformations, leading to reduced dynamic
topography, so that whole-mantle convection is consistent with small surface topography.
Key words: geoid anomalies, inverse problem, inversion, mantle discontinuities, mantle
viscosity, topography.
INTRODUCTION
The deviations of Earth's gravitational potential from hydrostatic are due to lateral density contrasts, which are related to
thermal and/or compositional variations within the planet and
to the de¯ections of external and internal boundaries such as
the surface, the core±mantle boundary and others related to
compositional or phase changes. Over thousands of years, the
large-scale internal density anomalies cause the mantle to creep;
the consequent mantle ¯ow de¯ects the boundaries. The longwavelength, non-hydrostatic geoid is highly sensitive both to
the internal density distribution and to the radial strati®cation
of mantle viscosity (Richards & Hager 1984; Ricard et al. 1984).
Seismic tomographic imaging of the structure of the interior
(e.g. Dziewonski et al. 1977; Clayton & Comer 1984; Inoue et al.
1990; Grand 1994; Masters et al. 1996; Ekstrom & Dziewonski
1998) as well as geodynamic models of slab reconstructions
(e.g. Hager 1984; Ricard et al. 1993) allow us to estimate the
density distribution within the mantle. An analytical description
of mantle circulation driven by those anomalies can give an
estimate of the de¯ections of the equipotential surfaces and the
# 2000
RAS
mantle boundaries (e.g. Richards & Hager 1984; Ricard et al.
1984; Hager & Clayton 1989; Forte & Peltier 1991; Dehant
& Wahr 1991; Panasyuk et al. 1996). The resemblance of a
modelled geoid to the observed geoid ®eld is used as a measure
of the feasibility of a proposed mantle viscosity pro®le (e.g. Hager
& Richards 1989; Ricard et al. 1989; Forte et al. 1994; King
1995).
A robust viscosity structure inferred from the gravitational ®t
should not be sensitive to the ®tting criteria, for example, variance
reduction (e.g. Mitrovica & Forte 1997), degree correlation
(e.g. Ricard et al. 1989) or power spectrum (Cizkova et al.
1996), used in the inverse method. However, several distinct
viscosity pro®le families are found by the above-mentioned
inverse studies that all satisfy the observed geoid reasonably
well. In order to improve the determination of the viscosity
structure, the most recent studies carry out joint inversions,
where, in addition to gravity, simultaneous ®ts to other observables are performed, such as to seismic data (Forte et al.
1994), postglacial rebound data (Mitrovica & Forte 1997) or
dynamic topography estimates (Quinn & McNutt 1998). An
interdisciplinary approach, assembling results of observational,
821
822
S. V. Panasyuk and B. H. Hager
analytical and numerical studies, can bring us closer to understanding the mantle viscosity structure. However, in carrying
out such an approach, one ought to consider the reliability
of the information, that is, to include an error analysis of the
data (measured or modelled) and an estimate of the model
de®ciencies. Previous inversions for mantle viscosity structure
that considered errors used highly simpli®ed treatments, assuming
a diagonal covariance matrix with a single error estimate
independent of harmonic degree. The main effect of such a
simple error model is to effect the trade-off of goodness of ®t
against model roughness (Forte et al. 1994), the departure from
an assumed a priori model (Forte et al. 1991) or the relative
weighting of ®ts to different data sets (Mitrovica & Forte 1997).
Here we carry out a joint inversion for geoid and dynamic
topography, simultaneously accounting for more realistic
assumptions about errors. We identify three classes of errors,
those related to the density distribution (e.g. uncertainty in
the seismic tomography models), those related to insuf®ciently
constrained observables (e.g. dynamic topography derived
from the observed surface topography and bathymetry after an
uncertain correction for static topography such as the subsidence of the oceanic lithosphere and the tectosphere), and
those related to limitations of our analytical model (e.g. an
absence of lateral viscosity variations). We estimate the errors
for geoid and dynamic topography in the spectral domain
and de®ne a ®tting criterion. A minimization function weights
the squared deviation of the residual quantities with a corresponding error, so that the components with more reliability
contribute to the solution more strongly than the less wellconstrained ones. Following this approach, we decrease the
contamination of our results by errors.
of the viscosity pro®le and of signi®cant slow-down of computer calculations. These become a problem for a highly nonlinear inversion when thousands of runs are to be done, as in
the case we discuss below.
The constraint of constant viscosity within a layer comes
from the analytical method used to solve the matrix differential
equation, the so-called matrixant, or propagator technique. This
method requires an exponent matrix to be constant throughout the layer in order to provide permutability in subintervals
(Panasyuk et al. 1996). We modify the mathematical representation of the governing equations and the boundary conditions and the subsequent matrix differential equation to allow
for exponential variations of viscosity within the layers (see
the Appendix). The matrixant solution is still valid, because the
new exponent matrixes are permutable (to the accuracy of
the solution for compressible ¯ow).
We solve the resulting system of equations with respect to
ocean±mantle boundary de¯ection, da, and potential anomaly,
V1, at the ocean surface (see the Appendix). By de®nition, the
Green's function, V1(r, l), represents a gravitational potential
disturbance at the Earth's surface caused by a unit density
anomaly of degree l within a layer of unit thickness located at
radius r. To obtain the geoid anomaly ®eld, dN, or its spherical
harmonic expansion coef®cients, we convolve the potential
Green's function with the density perturbations, Dlm, over the
radius, and approximate the integral by the sum
…
1 omb
V1 (r, l)Dlm (r) dr
dNlm ~
gsur cmb
METHOD DESCRIPTION
where the summation is performed over I propagation layers of
thickness Dri centred at radius ri [the notation is consistent with
Panasyuk et al. (1996) unless stated differently]. The number, I,
and the distribution of layers are chosen to provide suf®cient
accuracy of the minimization function. The lateral density
anomaly is expanded in a spherical harmonic set,
X
D(r, h, r)~
Dlm (r)Ylm (h, r) ,
(2)
Forward, analytical model
We assume that mantle rocks creep slowly, subject to stresses
generated by non-hydrostatic density variations. Following
the now standard approach (Hager & O'Connell 1981), we
employ the equations of continuity and motion, the constitutive
equation, which relates stress and strain rate linearly, as for a
Newtonian viscous rheology, and Poisson's equation of gravity.
We consider also the gradual (due to pressure) and step-like
(due to solid±solid phase change) radial variations in rock
density throughout the Earth (Dziewonski & Anderson 1981)
and the corresponding variations in gravitational acceleration
(Panasyuk et al. 1996). In addition, we assume a uniform ocean
layer overlying a free-slip Earth surface, and azimuthal symmetry
of viscosity, and we consider possible softening of deforming
material during phase transformations (Panasyuk & Hager
1998).
We modify the previous analytical models in order to handle
a continuous radial variation of mantle viscosity, together with
step-like, discontinuous changes (see also Panasyuk 1998). The
latter are meant to simulate major discontinuities in viscosity,
which are expected to occur across phase change boundaries
(Sammis et al. 1977). In ambient mantle of a constant solid
phase, however, the effects of gradual pressure and temperature
changes will lead to a continuous variation of viscosity (Ranalli
1991). Under the old formulation, to simulate continuously
varying viscosity, the number of isoviscous layers within the
mantle had to be increased at the cost of overparametrization
&
I
1 X
V1 (ri , l)Dlm (ri )*ri ,
gsur i
(1)
lm
where the Dlm coef®cients can be obtained from a tomographic
model of seismic velocity anomalies, do/o,
d ln o do
Dlm (r)~
o:
(3)
d ln o o lm
The conversion factor, d ln r/d ln o, depends on the type of
seismic velocity (e.g. VP or VS), the temperature, the pressure and
the compositional state of the mantle, and usually on the type of
tomographic inversion. However, due to the large uncertainties
of these dependences, we consider only its radial variation,
approximated as constant within three layers: the 0±220, 220±670
and 670±2891 km depth ranges. The density anomaly ®eld can
also be derived from a geodynamic model tracking slabs and
reconstructing their trajectories within the mantle. In that case,
we consider the scaling factor as the density contrast between
the slab and the ambient mantle.
The forward solution described above provides us with the
predicted geoid anomaly at the surface, dN, and the dynamic
topography at the ocean±mantle boundary, da (with the density
jump across it given by Dra). These two ®elds are to be compared
#
2000 RAS, GJI 143, 821±836
Mantle viscosity inversion using geoid and topography
with the observed ones during the inversion. The gravitational
potential ®eld is provided by satellite geodesy (GEM-L2,
Lerch et al. 1983). To obtain the non-hydrostatic geoid,
we correct the observed geoid for J20=1072.618r10x6 and
J40=x1.992r10x6, according to Nakiboglu (1982) (assuming
zero uncertainty associated with the correction). The dynamic
topography ®eld is unavailable for direct measurement; therefore, we derive the surface undulations from the observed topography and bathymetry by correcting for static loads (Panasyuk
& Hager 2000). Since the resulting dynamic topography refers to
the de¯ection of the air±mantle boundary, da (with the density
jump Dre+Dra), we correct for the difference in density across
the boundaries and obtain the spherical harmonic coef®cients
for the dry-planet dynamic topography by convolving with the
density anomaly:
X
*oa
da(ri , l)Dlm (ri )*ri :
(4)
dTlm ~ e
a
*o z*o i
The predicted ®elds, dNlm and dTlm and the model parameters such as viscosity and velocity-to-density (or trajectoryto-density) conversion factor are used for setting up the inverse
problem discussed below.
Inverse problem
Traditionally, an inverse problem deals with the minimization
of a multivariable function that determines the ®tting criteria.
For example, some previous studies of the geoid utilized the
reduction of variance (e.g. Hager & Clayton 1989; King 1995)
or the increase in degree correlation (Ricard et al. 1989) or the
resemblance to the slope of the geoid spectrum (Cizkova et al.
1996). Inversions of the free-air gravity jointly with postglacial
(Mitrovica & Forte 1997) or seismic (Forte et al. 1994) data
consider scalar weights assigned to each type of data set to
control its relative importance. However, none of these studies
takes into account the errors associated with both forward and
inverse problems and makes a thorough error analysis.
We suggest using a minimization function that weights
the mis®t of a quantity by an error related to a three-fold
uncertaintyÐthe uncertainties of estimating, observing and
modelling this quantity. Such a treatment makes the ®t to
a well-determined parameter more important than the ®t to a
poorly resolved one. In the case of a joint inversion, when the
®t is performed to two or more quantities simultaneously,
the minimization function is determined for each quantity
separately, and their sum is minimized. Here we perform a joint
inversion for the viscosity pro®le based on the ®t to the geoid
and the dynamic surface topography, and we minimize a
function in the spectral domain,
2
2
f 2 ~fgeoid
zftopo
:
(5)
The ®tting criterion for any of these ®elds, say an F-®eld, is
fF2 ~
obs
mod 2
{Flm
]
1 X [Flm
,
2
nF lm
plm
(6)
where the error includes contributions from the different error
2
2
2
sources, s2=sdensity
+sobs
+smodel
, and the scaling factor nF
equals the number of lm coef®cients of the F-®eld for which the
error is de®ned. The errors associated with the density anomaly
2
distribution, sdensity
, re¯ect the uncertainties in the velocity-todensity conversion factor, the seismic anomalies and the location
#
2000 RAS, GJI 143, 821±836
823
2
, is
and density contrast of slabs. The second type of error, sobs
related to the uncertainty in an observed ®eld such as the geoid
2
or dynamic topography. The errors smodel
contaminate the terms
that are mostly affected by the incompleteness of the forward
model (for example, a short-wavelength signal when lateral
variations of viscosity are ignored). In cases when the errors are
related to poor spatial coverage, one could use a minimization
function in the spatial domain, where the errors are de®ned in a
similar way as above but as a function of position.
Once we estimate the errors and de®ne the observed ®elds,
we characterize the minimization function and proceed with an
inversion for mantle viscosity. To perform an inversion, we use
an algorithm based on a Sequential Quadratic Programming
method, where in order to determine the search direction, the
gradients and the second derivatives are estimated numerically.
The method analyses the second derivative matrix, the Hessian,
constructs a quadratic multiparameter function and determines
its minimum as a tentative solution. The Hessian is usually
modi®ed or updated until the inversion converges successfully
(de®ned by the custom-supplied tolerance level) or aborts
due to exceeding the control parameters (e.g. the number of
iterations).
In order to minimize the ®tting criterion function, the inversion
program is allowed to vary the viscosity pro®le and the density
scaling factor within the speci®ed strati®cation and value
range. To reduce the number of inversion parameters and yet
achieve fast convergence and good resolution, we conducted an
elaborate study, altering the number and the depths of viscosity
layers, choosing constant and exponential laws for viscosity
variations. As a result we de®ne nine parameters to describe
mantle viscosity optimally. This parametrization is based on
the assumption that the effective viscosity of the mantle can
change abruptly across (Sammis et al. 1977) and within (Sammis
& Dein 1974) the phase change regions; otherwise, it varies continuously under the in¯uence of temperature and/or pressure
for a constant-phase material (e.g. Ranalli 1991). The viscosity
jump across a transformation and the reduction of viscosity
within the region we describe by two parameters, a total of
four for the entire mantle: two for the 400 km and two for the
670 km phase boundaries. We also approximate the viscosity
variation associated with the thermal boundary layers near the
ocean±mantle and core±mantle boundaries with discontinuous
jumps in viscosity at 75 and 2600 km depth. Continuous
variations of viscosity are generally described as an exponential
function of activation energy Ea and volume Va, pressure p and
temperature T,
dm
Ea zpVa
g(z)~A n{1 exp
,
(7)
p
RT
with the proportionality term related to the stress s and grain
size d dependence of the creep mechanism (Ranalli 1991).
We assume that outside the phase and thermal boundaries the
total radial variation of the under-exponent functions is close
to linear, and the pre-exponential term changes weakly with
depth. Then the viscosity can be approximated with a single
exponential within each layer of constant phase. Under this
assumption we prescribe a viscosity parameter above each inner
boundary: 400 and 670 km phase changes, and 75 and 2600 km
thermal boundaries. The viscosity of the mantle at 2500 km
depth is taken as a reference value, with the other eight values
permitted to vary during the inversion. Note that all nine
parameters used to describe the viscosity pro®le are pinned to a
824
S. V. Panasyuk and B. H. Hager
particular depth level, and only their values can be changed.
Such an imposed in¯exibility on the viscosity strati®cation is
based on the fairly hard constraints on the depths of phase
change regions and on the existence of thermal boundary layers
in the convecting mantle. The ranges for viscosity parameter
variations allow but do not require reduction of viscosity
within the phase transformations or discontinuous changes
across them.
The density conversion factor is kept uniform in the upper
and lower mantle, with these two values inverted for during
each solution. We chose this simple parametrization for two
reasons. First, the amplitude of the conversion factor (and even
its sign) is still highly ambiguous (explanation follows). The
possibility of an erroneous estimate of this factor for the whole
mantle increases because it is used as a multiplication term
between the kernels and the seismic anomaly data (eqs 1 and 3).
Therefore, varying the conversion factor spatially allows the
alteration of the geoid/topography kernels and/or the density
anomaly signal directly, creating numerous, mainly arti®cial,
density variations within the mantle.
Although mineral physics experiments (e.g. Karato 1993;
Chopelas 1992) provide some constraints on the value and
variation of the d ln r/d ln o factor, the conversion from velocity
to density anomalies is not obvious. Besides being dependent
on pressure, temperature, composition and melt fraction in the
crust and mantle, the factor also depends on the characteristics
of a particular tomographic inversion, for example, the types
of seismic waves involved, the reference earth model and the
method of inversion used. The most poorly constrained region is
the top part of the upper mantle, where the effects of chemical
composition (e.g. continental tectosphere, Jordan 1988) in combination with thermal variations and anelasticity obscure
the interpretation of seismic anomalies. The signal visible to the
seismic waves near the surface has contributions from static
surface features such as the crust and lithosphere as well as
from features participating in convection. Sharp heterogeneities
become smeared out over larger horizontal scales and smeared
out over depth. To reduce the contamination of our results by
the high uncertainty of the signal from the top part of the upper
mantle and to avoid double counting the dynamic features, we
make two assumptions. First, we account for the crustal, tectospheric and lithosphere static load in our model of dynamic
topography (Panasyuk & Hager 2000). Second, we assume that
within the errors considered, the ®rst 220 km beneath the surface
does not contribute to the density anomaly signal (although
this does not stop the top layer from participating in the mantle
¯ow). In addition to the seismic anomaly models, we also consider a geodynamical model that identi®es the slab trajectories
based on the locations of earthquake epicentres. Assuming that
the slabs consist of cold (and presumably dense) material, we
assign a conversion factor (similar to velocity-to-density) that
equals the density contrast between the slab and the ambient
mantle. These two types of density distribution models, seismic
tomography and slab recovery, provide us with a wide spectrum
of input density models.
Error analysis
The intricate part of the approach is the way in which one
estimates the errors. For statistically well-represented problems
such as seismic tomography, which deals with thousands of
arrival times of events, yet is often of poor spatial coverage,
there are error analysis methods (e.g. Tarantola 1997) that can
be used to help to determine a solution. However, there is
no unique choice of method and no completely objective way
of determining errors. Alternatively, having a dozen density
distribution models, several analytical and numerical studies
of convection and a few measurements of observed ®elds,
we suggest here an empirical approach to error analysis to be
applied to the inversion for mantle viscosity problems.
2
Uncertainty in density anomaly distribution, sdensity
Recent developments in seismic tomography, including a growing database and computerized methods of data processing,
have made it possible to produce several detailed models of
seismic velocity anomaly distribution inside the mantle (e.g.
Masters et al. 1996; Ekstrom & Dziewonski 1998; Grand 1994;
van der Hilst et al. 1997). The models give similar estimates
of the structure of the lower mantle signal; however, the
discrepancy among them grows in the more heterogeneous
upper mantle. The different depths of seismic wave resolution,
in combination with the variety of tomographic methods
and data sets, are responsible for the disagreements. The conversion of seismic anomaly to density perturbation introduces
additional errors. Although some authors perform an error
analysis, there is still a de®ciency in such analysis for many
models, and there is no straightforward way to account for all
errors.
Instead of evaluating the uncertainty in individual density
anomaly models, we estimate the discrepancies in the geoid
and dynamic topography ®elds predicted by our viscous ¯ow
approach when a number of density anomaly models are used
as input data. For this study we consider 22 density anomaly
models. 11 of these are derived entirely from the following
seismic tomography models: (1) Ekstrom & Dziewonski (1998)
(VS); (2) Su et al. (1994) (VS); (3) Liu et al. (1994) (VS);
(4) Masters et al. (1996) (VS); (5) Li & Romanowicz (1996) (VS);
(6) Grand (1994) (VS); (7) Ishii & Tromp (1999) (VS); (8) Ishii &
Tromp (1999) (VP); (9) Masters et al. (1996) (VP); (10) Karason
& van der Hilst (1999) (VP); (11) Boschi & Dziewonski (1999)
(VP). The other 11 are modi®cations of each of the above, such
that the upper mantle signal is replaced with a geodynamic
model of slab locations (Hager 1984). There are many differences in the way the tomography models were built and in the
range of data that were used. For example, in regions with poor
coverage such as the Southern Hemisphere, a regional model
that inverts for a signal within blocks would not resolve that
area at all, whereas a global inversion using polynomials would
assign a value despite poor data coverage. Since our analysis
is spectral, for models given on a spatial grid, we de®ne the
spherical harmonic coef®cients of the data ®eld by numerical
integration on a sphere at each depth where data are de®ned. To
handle the gaps in the block-type models during the numerical
integration, we zero them out. [In contrast, one could also invert
for spherical harmonic coef®cients using the least-squares technique (e.g. Panasyuk & Hager 2000).] To ensure that the density
models provide a consistent representation of the interior
structure that drives convection, we perform a test of the compatibility among the models (the description follows) and
estimate the errors, based on averaging over the models.
To determine the compatibility among the models, we
complete the inversion several times for each of the 22 density
models, each time starting from randomly chosen initial values
#
2000 RAS, GJI 143, 821±836
Mantle viscosity inversion using geoid and topography
for the viscosity parameters and the density conversion factors.
The initial parameter range allows roughly one order of
magnitude viscosity variation (light grey shading in Fig. 1a).
The conversion factor in the upper mantle is varied initially
near zero (positive and negative) for the seismic models and
between 100 and 200 kg mx3 for the hybrid models. In the
lower mantle the d ln r/d ln o factor varies between 0.1 and
0.2. The range of viscosity variations that is allowed during
inversion exceeds the initial range by several orders of magnitude (dark grey shading in Fig. 1a). The d ln r/d ln o factor can
change between x2 and 2 in the upper mantle, and between 0
and 2 in the lower mantle. The slab density contrast is allowed
to vary from 40 to 300 kg mx3. The ®tting criterion used at this
stage of our analysis is the reduction of the geoid and dynamic
topography variances,
X
X
obs
mod 2
obs
mod 2
[dNlm
{dNlm
]
[dTlm
{dTlm
]
1
lm
lm
X
X
F~
z
:
(8)
obs 2
obs 2
5
[dNlm
]
[dTlm
]
lm
lm
Note that the topography variance reduction is lessened
®ve times relative to that for the geoid. We apply this scaling
because the dynamic topography is more poorly constrained
than the observed geoid.
For each of the 22 density models, we select a viscosity
pro®le that provides the best combined variance reduction. The
pro®les form two distinct groups: one shows a low-viscosity
layer under the lithosphere and the other displays softening at
around 400 km depth (Fig. 1b). Apparently, the hybrid models
(slabs in the upper mantle and seismic tomography in the lower
mantle) produce viscosity pro®les of the ®rst group and the pure
seismic models give the second pro®le. To show the common
characteristics of the pro®les, we calculate the logarithmic
mean of the viscosity within each group and plot it in Fig. 1(b)
(the solid line with the squares centred at each step corresponds
825
to the hybrid models group and the line with the circles to the
pure seismic group). The standard deviation around the mean
is shown by the grey shade; dark/light are for the hybrid/pure
seismic groups, respectively.
In the next step we consider each of the selected 22 viscosity
pro®les as equally plausible. We hold them ®xed and perform
an inversion again for each of the density models within the two
groups. This time, using the same ®tting criterion, we allow for
the free adjustment of only the density conversion factor and its
jump between the upper and lower mantle. Such a choice of
®xed viscosity structure and free density conversion parameters
allows us to compare the self-tuned density models against each
other under otherwise equal conditions. After all inversions
have converged, we obtain two groups, each consisting of 121
sets of spherical harmonic coef®cients for geoid and topography ®elds, Clm, calculated using signals from the 11 ®xed
viscosity pro®les and the 11 different density distributions for
each group. To estimate the dispersion of the predicted geoid
and topography within each group, we calculate the geoid and
the topography means over the models (total of K=11) in the
spectral domain,
Sh Clm Tgroup ~
K
1 X
h k
C
K k~1 lm
(9)
and the standard deviation (std) for each model k and each
viscosity pro®le h (total of H=11),
X
h k2
k
p ~
(h Clm
{Sh Clm Tgroup )2 :
(10)
lm
The total of 242 standard deviations re¯ect the solution
sensitivity to the variation of the viscosity pro®le and to the
variation of the driving density model for each viscosity pro®le.
To differentiate among the models, we normalize the std for
each pro®le and model (eq. 10) by the mean within the group
initial and total range of viscosity
viscosity for error analysis
0
100
400
400
670
670
depth, km
0
100
2600
2600
(a)
−4
−2
0
2
log10(viscosity)
4
6
(b)
−6
−4
−2
0
log (viscosity)
2
10
Figure 1. Decimal logarithm of relative mantle viscosity versus depth (km). (a) Inversions are started from a randomly chosen viscosity pro®le
constrained by the light grey shading. During the inversions, the viscosity is allowed to vary within the dark grey shading area. (b) The logarithmic
mean of 11 viscosity pro®les for each density anomaly group chosen for the ®rst step of the error analysis (the solid line with the squares centred at each
step corresponds to the hybrid models group; the solid line with the circles corresponds to the pure seismic models group) and the standard deviation
around the mean (dark and light grey shading is for hybrid and pure seismic groups, respectively).
#
2000 RAS, GJI 143, 821±836
826
S. V. Panasyuk and B. H. Hager
for the same pro®le,
Sh p2 Tgroup ~
K
1 X
h k2
p :
K k~1
(11)
The results of the last normalization are shown in Fig. 2. The
abscissa shows the model number by order in the reference
list above. The ®rst set of numbers corresponds to the pure
seismic model group and the second set is for the hybrid group
(the upper mantle signal is replaced by slabs). The ordinate
corresponds to the normalized std, where light dots are for the
geoid and dark dots are for the dynamic topography. Each
point in the plot shows by how much the ®eld from a particular
model deviates from the mean ®eld in units of mean deviation
for a particular viscosity pro®le within the group. To generalize
the information over the range of viscosity pro®les, we plot an
std value averaged over 11 pro®les for each model and each
group. The wheels (for geoid) and the squares (for topography)
show the normalized standard deviation for each density model
for the viscosity assemblage (such as in Fig. 1b). The models
that produce ®elds within one standard deviation (below the
solid line in Fig. 2) form a self-consistent group.
To estimate the errors related to the variety of density models
used, we calculate the standard deviation around the meanover-models for each viscosity pro®le versus spherical degree
and harmonic number and display them in Fig. 3 by the dots.
To generalize the errors for a set of viscosity pro®les, we
calculate the mean-over-viscosity std as well,
Sp2lm T~
1
KH
K,H
X
k~1,h~1
k
(h Clm
{Sh Clm Tchosen )2
(12)
and show its values in Figs 3(a) and (b) for geoid and topography, respectively (the solid lines connect uncertainty in the
cosine spherical harmonic coef®cients; the dark line is for the
hybrid group and the light line is for the pure seismic group).
The markers denote the non-zero sine spherical harmonic
coef®cients uncertainty (squares for the hybrid group and
circles for the pure seismic group). The standard deviations
calculated as above re¯ect the sensitivity of the different
harmonics to the variations in viscosity pro®le (as long as
there is a low-viscosity zone in the upper mantle) and to the
variations in the density models (as long as the models are from
the chosen set). Later in the ®nal inversions we use these errors
(marked by the lines, squares and circles in Fig. 3) to account
2
2
for uncertainty in the density anomaly models, sdensity
=nslm
m.
2
Uncertainty in the observed ®eld, sobs
The undulation of the geoid is rather well measured by satellite
gravimetry. We correct the observed geoid for the relatively
small geoid anomalies that are not related to the viscous mantle
¯ow assumed in our approach. These are the gravitational
signals due to delayed postglacial relaxation (Simons & Hager
1997) and to isostatically compensated crust, tectosphere and
lithosphere (Panasyuk & Hager 2000). The root mean squared
(rms) amplitude of the observational error for the geoid is
taken from the estimate of the geoid due to the isostatic loads
(Panasyuk & Hager 2000) and is shown by the circles in
Fig. 4(a).
The other observable in the geoid±topography joint inversion,
dynamic topography, is not as well constrained as the geoid.
Since the topography due to mantle dynamics processes is
distorted by the crust and the lithosphere attached to it, there
are only imprecise estimates of its amplitude [for example,
¯ooding records related to the rise and fall of continents (Gurnis
1990)]. For the entire analysis we adopt the l=2±6 spherical
harmonic coef®cients of the dynamic topography that was
calculated assuming plate-like cooling of the oceanic lithosphere
by Panasyuk & Hager (2000), along with the corresponding uncertainties. The observational errors for the dynamic
topography ®eld are shown by the circles in Fig. 4(b).
relative standard deviation, geoid and topography
2.2
2
1.8
sigma/<sigma>
1.6
1.4
1.2
1
0.8
0.6
0.4
1
2
3
4
5
6
7
seismic models
8
9
10 11
1
2
3
4
5
6
7
8
9
10 11
hybrid models
Figure 2. Geoid (light dots) and topography (dark dots) standard deviations for each viscosity pro®le (total of 11 as in Fig. 1b) and each density
model (abscissa) normalized by the mean within each group (see text). The std averaged over the viscosity pro®les is shown by wheels (geoid) and
squares (topography) for each density model.
#
2000 RAS, GJI 143, 821±836
Mantle viscosity inversion using geoid and topography
sigma
[m]
(a)
sigma
topo
10
50
9
45
standard deviation around mean
standard deviation around mean
geoid
8
7
6
5
4
3
35
30
25
20
15
10
1
5
2
3
(b)
40
2
0
[m]
827
0
4
5
6
spherical harmonic degree,l
2
3
4
5
6
spherical harmonic degree,l
Figure 3. Uncertainty in the density anomaly models, geoid (a) and topography (b), in metres, versus spherical harmonic degree and order. The dots
are the standard deviations for each viscosity pro®le (from assemblages as in Fig. 1b). The solid lines connect cosine spherical harmonic coef®cients
that are averaged over the viscosity pro®les (dark line is for the hybrid group and light line is for the pure seismic group). The symbols correspond to
non-zero sine spherical harmonic coef®cients (squares for the hybrid group and circles for the pure seismic group).
2
Uncertainty due to incompleteness of forward model, smodel
Our current model treats only spherically symmetric variations
of viscosity, and it assumes free-slip boundary conditions at
the surface. However, lateral variations in viscosity (LVV)
are undoubtedly present due to temperature, composition and
deviatoric stress variations within the mantle. Closer to the
surface, they become critically high due to the change of mantle
rheology from viscous to elastic, and to brittle within the plates.
The essentially rigid plates, driven by mantle ¯ow, act back
sigma
[m]
geoid
25% 26%
40%
41%
on the ¯ow. Having relatively weak boundaries, plates cause
toroidal ¯ow in the mantle. Thus, the density-driven poloidal
¯ow is coupled to the plate-driven toroidal ¯ow, and the
resulting total ¯ow creates the dynamic topography and determines the gravity ®eld. Therefore, our assumptions are not
entirely justi®ed, which prevents us from explaining deterministically the part of geoid and topography signal related
to lateral viscosity variations, coupled toroidal ¯ow and nearsurface plate-like behaviour. On an optimistic note, however,
we recognize that for long-wavelength ¯ow these effects are
sigma
(a)
topo
52%
60% 74%
54%
[m]
58%
(b)
72%
80
8
60
6
4
40
2
20
2
3
4
5
6
harmonic degree and order
2
3
4
5
6
harmonic degree and order
Figure 4. The total error rms (solid line), geoid (a) and topography (b), in metres, versus spherical harmonic degree and order. Three types of errors
are shown: the density model uncertainty (asterisks), the observational error (circles) and the modelling errors (triangles). The numbers on the top
show the percentage of the harmonic degree error relative to the total signal amplitude of the same harmonic.
#
2000 RAS, GJI 143, 821±836
828
S. V. Panasyuk and B. H. Hager
of second order compared to those due to changes in radial
viscosity structure. Thus, instead of elaborating our model at
this point, we attempt to estimate the errors in geoid and
dynamic topography due to ignoring such effects, based on
different studies of the above-mentioned phenomena.
In earlier investigations, the observed nearly equal poloidal/
toroidal energy partitioning at the surface (Hager & O'Connell
1978) was directly related to the plate-like surface division
(O'Connell et al. 1991; Olson & Bercovici 1991), and plates
were incorporated into models (Hager & O'Connell 1981;
Ricard & Vigny 1989; Gable et al. 1991; Forte & Peltier 1991).
The importance of the poloidal±toroidal coupling on dynamic
surface topography and the geoid was pointed out (Forte &
Peltier 1987) and later investigated analytically and numerically
(Richards & Hager 1989; Ricard & Vigny 1989; Christensen &
Harder 1991; Ribe 1992; Zhang & Christensen 1993; Cadek
et al. 1993; Forte & Peltier 1994). Although each approach
above analyses the effects of LVV and boundary conditions by
different means, there are some general points they agree upon.
For example, the effects are much stronger when the viscosity
of the mantle increases with depth compared to the isoviscous
case. The strongest effects on the ¯ow (and hence surface
deformation and geoid) are produced by the self-coupling
between the density source and the viscosity anomaly and by
the coupling at the doubled harmonics. The relative effect is
stronger for the higher harmonics. We note that evaluation of
the error amplitudes is very ambiguous; however, we attempt
an estimate for the low harmonics (l=2±6). Richards &
Hager (1989), based on the results of perturbation theory and
numerical models of mantle convection, concluded that the
longest-wavelength geoid anomalies (l=2, 3) are not seriously
contaminated. However, for the higher degrees (li4) the effect
could be signi®cant due to coupling between density heterogeneity and viscosity variation. They showed that the selfcoupled anomalous surface deformation and the anomalous
geoid increase as l2 and estimated an anomalous geoid, in
per cent, as a function of the load/viscosity mode number for
different strengths of viscosity variations. The size of the effect
increases linearly with the wavenumber for the geoid due to its
l-dependence on the load. We use this result as an estimate of
the geoid and topography errors:
pgeoid
model (l)!l
X
m
mod
dNlm
and
2
ptopo
model (l)!l
X
m
mod
dTlm
:
(13)
To account partially for the higher-order coupling, we doubled
the errors at l=2, 4, 6.
To show the contribution of each error type to the total
errors assigned to the geoid and dynamic topography during
the inversion, we plot individual errors together with the total
error in Fig. 4 (asterisks are for density model errors, circles are
for observational errors, triangles are for the model de®ciency
and the solid line is for the total error). For the geoid ®eld, the
density model-related errors dominate strongly (Fig. 4a). The
total error is of the order of 25 per cent of the observed geoid
rms for long wavelengths, increasing to about 50 per cent
for the shorter-wavelength signals. (Note that since both the
coef®cients and the errors in the coef®cients of the free-air
gravity are proportional to those for the geoid, the relative errors
for free-air gravity are identical to those of the geoid. Thus, our
inversion results do not depend on which representation of the
gravity ®eld we use.) For the topography ®eld the observational
errors de®ne most of the total error (Fig. 4b). The least
resolvable features of topography are at l=3, 6, where the
uncertainty reaches over 70 per cent. For other harmonics, the
errors vary between 50 and 60 per cent of the signal.
RESULTS OF THE INVERSION
We performed an inversion based on a simultaneous ®t to the
geoid and dynamic topography, de®ned by the criterion in eqs
(5) and (6). The observed geoid was obtained from the original
spherical harmonic expansion data (Lerch et al. 1983) by the
removal of the hydrostatic (Nakiboglu 1982) and isostatic
(Panasyuk & Hager 2000) corrections. The observed dynamic
topography was reduced from the surface topography and
bathymetry by correcting for the crust, tectosphere and platelike cooling oceanic lithosphere (Panasyuk & Hager 2000). The
errors associated with the geoid and dynamic topography
modelling were taken as in Fig. 4. We inverted for the viscosity
pro®le and the density conversion factor. The mantle viscosity
was parametrized in ®ve layers. Three layers were assumed to
consist of a constant-phase material with the viscosity changing
continuously by an exponential law. The other two layers
simulate the thermal boundary layers, the lithosphere and Dkk,
and were assigned a constant viscosity within and a jump
in viscosity at their borders. The viscosity within the phase
transformation regions (400 and 670 km depths) was allowed,
but not required, to drop in magnitude relative to the ambient
mantle. The density conversion factor was allowed to adjust its
(constant) value within the upper and lower mantle during all
runs of the inversion. To make sure that the ®nal results do not
depend on the initial values of the parameters, each time we
started the inversions from randomly (uniform distribution)
chosen initial conditions within the assigned range (see light
grey area in Fig. 1a). During the inversion, the parameters are
allowed to vary within a range of several orders of magnitude
(dark grey area in Fig. 1a). To gain a statistically signi®cant
result, we carried out 100 inversion runs for each of the density
models referenced above (11 pure seismic and 11 hybrid models).
All inversion runs successfully converged. However, in the
®nal analysis we include only those that used both the upper and
the lower mantle signal (that is, inverted for non-zero velocityto-density factors). The solutions formed three families of
viscosity pro®les (Figs 5a, b and c). Since the solutions scatter
within each family, we calculate the logarithmic average of the
viscosity pro®les (solid lines). The standard deviation of all the
participating solutions from the weighted average is shown by
the grey area around the solid line. Each group of solutions in
Figs 5(a), (b) and (c) can be distinguished by the depth of the
low-viscosity zone (LVZ) in the upper mantle, clearly identi®ed
by the peak in the geoid kernels (Figs 5d, e and f). Almost all of
the viscosity pro®les have a signi®cant viscosity drop within
one or both phase change regions and display an increase
in viscosity within the lower mantle down to 2600 km depth,
followed by a soft layer above the CMB.
To show the general trend in the geoid/topography kernels
of each family, we calculate a representative geoid/topography
kernel for each l. The standard deviation when plotted around
the representative line overlaps for most harmonics. To avoid
overlap, we plot only one-®fth of the std around each representative kernel, with the shading getting darker towards the
higher harmonics.
#
2000 RAS, GJI 143, 821±836
Mantle viscosity inversion using geoid and topography
Family 1, f
min
Family 2, f
= 1.12
min
(a)
100
Family 3, f
= 1.15
min
829
= 1.17
(b)
(c)
400
depth, km
670
2600
−4
−2
0
2
47% of 1046 solutions
−4
−2
0
2
−2
0
2
8%
(d)
100
−4
46%
(e)
(f)
400
depth, km
670
6
5
4
3
2
2600
−0.2
0
0.2
0.4
−0.2
0
0.2
0.4
(g)
100
−0.2
0
0.2
0.4
(h)
(i)
400
depth, km
670
6
5
4
3
2
2600
−0.8 −0.6 −0.4 −0.2
−0.8 −0.6 −0.4 −0.2
−0.8 −0.6 −0.4 −0.2
Figure 5. Results of the inversion form three families, shown in columns one, two and three. The decimal logarithm of relative mantle viscosity for
each family is shown versus depth (km) for the ®rst (a), second (b) and third (c) families, with the weighted average (solid line), the standard deviation
(shaded area) and the weighted minimization function, fmin. Geoid kernels for spherical harmonics l=2±6 corresponding to the three families of
mantle viscosity pro®les are shown in (d), (e) and (f). Surface dynamic topography kernels for spherical harmonics l=2±6 corresponding to the three
families of mantle viscosity pro®les are shown in (g), (h) and (i). (For clarity, only one-®fth of the standard deviation around the weighted average is
shown in (d)±(i).
The two popular families (almost half of all solutions each)
have an LVZ centred at 100 km depth (Fig. 5a) or at the
400 km phase change region (Fig. 5b). The geoid kernels peak
with a positive value at these depths (Figs 5d and e). The
dynamic topography kernels are nearly linear from the surface
to the LVZ depth and decrease gradually down to Da, from
where they diminish to zero at the CMB (Figs 5g and h). The
less populated family of viscosity pro®les (several per cent of all
solutions) has a sawtooth variation in viscosity through the
upper mantle, with an increase by a factor of two at the 670 km
phase change (Fig. 5c). The geoid kernels peak at 670 km
depth (Fig. 5f), and the topography kernels are almost linear
#
2000 RAS, GJI 143, 821±836
from the surface to the LVZ depth, and stay nearly constant
throughout the lower mantle down to Da (Fig. 5i). All three
families show transformational superplasticity at 670 km depth,
and over half the models show it at 400 km depth as well.
To analyse the spatial characteristics of the resulting geoid
and topography ®elds, we calculated mean ®elds within each
solution family and we display the results in Fig. 6 (the geoid is
on the left and the topography is on the right), starting with
the observed geoid and dynamic topography (Figs 6a and b).
All three solutions produce similar minimization function ®t
values, f1=1.12, f2=1.15, f3=1.17 (see Fig. 6). However, the
geoid variance reduction (VR) and degree correlation (DC) in
830
S. V. Panasyuk and B. H. Hager
dynamic geoid, l=2−6
(a)
VR = 100%, DC = 100%
Family 1, f
min
= 1.12
min
= 1.15
min
(d)
VR = 2%, DC = 33%
(f)
(e)
VR = 69%, DC = 89%
Family 3, f
= 1.17
VR = 7%, DC = 34%
(h)
(g)
VR = 68%, DC = 88%
(b)
VR = 100%, DC = 100%
(c)
VR = 74%, DC = 92%
Family 2, f
dynamic topo, l=2−6
VR = 7%, DC = 35%
Figure 6. The dynamic geoid (left column) and the dynamic topography (right column). The estimated ®elds are shown in (a) and (b) and saturated in
colour. The means within each solution family are plotted in (c) and (d), (e) and (f), and (g) and (h) for the ®rst, second and third families, respectively
(as in Fig. 5). The contour levels are 20 m (geoid) and 100 m (topography), zero level is white, and positive values are lighter. The ®tting criteria, fmin,
the variance reduction, VR, and the degree correlation, DC, are shown for each family solution.
the ®rst family (VR=74 per cent, DC=92 per cent) exceed
those of the second (VR=69 per cent, DC=89 per cent) and
the third (VR=68 per cent, DC=88 per cent) families, providing better resemblance to the observed ®eld. The ®t of the
topography ®elds is relatively poor for all solutions, with the
better ®t for the third family (VR=7 per cent, DC=35 per cent
versus VR=7 per cent, DC=34 per cent for the second, and
VR=2 per cent, DC=33 per cent for the ®rst families). The
amplitudes of both geoid and topography are underpredicted.
The maxima for the geoid and topography coincide closely,
although for the topography it is displaced further into the
Central Paci®c.
We also analysed the density conversion factors in the upper
and lower mantle for each density model in each solution
family. The results for several density models are shown in
Fig. 7, where each plot corresponds to a density model, with
the ®rst number in the title corresponding to the number in the
reference list above. The abscissa always represents the d ln r/
d ln o value in the lower mantle. The ordinate is for the
conversion factor in the upper mantle. For the hybrid models it
corresponds to the subducted slab density contrast (kg mx3)
relative to the ambient mantle density (Figs 7g±l). For the pure
seismic models, it is the d ln r/d ln o value (Figs 7a±f). Each
data point corresponds to a solution of one inverse run, where
the ®rst family solutions are shown by squares, the second
family by circles and the third family by asterisks. The total
number of solutions within each family for a particular density
model is printed in brackets above the plot (the ®rst, second
and third numbers correspond to the ®rst, second and third
families). Most of the pure seismic models lead to convergence
#
2000 RAS, GJI 143, 821±836
Mantle viscosity inversion using geoid and topography
1
[ 50 41 2 ]
(a)
2
[ 0 26 22 ]
(b)
3
[ 0 28 18 ]
(c)
5
[ 0 85 9 ]
(d)
7
[ 0 20 8 ]
(e)
8
[ 31 64 0 ]
(f)
1
[ 67 17 0 ]
(g)
2
[ 71 20 0 ]
(h)
3
[ 75 13 0 ]
(i)
9
[ 51 27 0 ]
(j)
7
[ 74 14 0 ]
(k)
8
[ 16 15 0 ]
(l)
831
dlnrho/dlnv
0.11
0.1
0.09
0.08
0.07
0.06
0.05
dlnrho/dlnv
0.11
0.1
0.09
0.08
0.07
0.06
0.05
kg/m3
70
65
60
55
50
kg/m3
70
65
60
55
50
0.06
0.08
V2D lower mantle
0.1
0.06
0.08
V2D lower mantle
0.1
0.06
0.08
V2D lower mantle
0.1
Figure 7. The conversion factor determined for the density models. The abscissa corresponds to the d ln r/d ln o value of the lower mantle. The
ordinate is the d ln r/d ln o value of the upper mantle for pure seismic models (a±f) and the slab density contrast (kg mx3) relative to the ambient
mantle density for the hybrid models (g±l). Solutions correspond to the ®rst, second and third viscosity families (squares, circles and asterisks,
respectively) as in Fig. 5.
only to the second and third families of viscosity pro®les
(Figs 7b±e). However, two models (Ekstrom & Dziewonski
1998, VS; Ishii & Tromp 1999, VP) recognize the ®rst family of
viscosity pro®les (Figs 7a and f). The hybrid models converge
only to the ®rst and second families of viscosity pro®les. Thus,
the only viscosity pro®les seen by all density models are those
from the second family.
The conversion factor varies from model to model, as we
would expect. Generally, the d ln r/d ln o value is between 0.06
and 0.1 for the lower mantle. For the upper mantle the d ln r/
d ln o value is about the same as for the lower mantle, or slightly
higher. The slab density contrast varies around 60 kg mx3.
The three families provide similar ®ts to the observables.
Note, however, that the ®tting criterion we use has information
on both the geoid and the topography, including errors. It
differs from the variance reduction and degree correlation
generally used.
#
2000 RAS, GJI 143, 821±836
DISCUSSION
The geoid and the dynamic topography obtained during the
inversion reproduce many features of the observed ®elds (Fig. 6).
The spatial resemblance is typically good, although the amplitude of the signal is usually underpredicted. To understand the
general spectral characteristics of the predicted ®elds, we plot
their spherical harmonic coef®cients (l=2±6) versus those of
the observed geoid (dots in Fig. 8a) and dynamic topography
(Fig. 8b). The diagonal line represents a perfect ®t. Since the
ratio of the geoid signal to total error is greatest at longest
wavelengths, we focus on the l=2 component of the ®elds and
identify its coef®cients by the markers (see legend in Fig. 8b).
The error bars represent the uncertainties of the observed ®elds
(as in Fig. 4) versus the std of the modelled ®eld coef®cients
within each solution family (black lines are for the ®rst family,
dark grey for the second and light grey for the third). Note that
832
S. V. Panasyuk and B. H. Hager
geoid, m
topography, m
20
120
100
10
80
5
60
0
40
modeled
modeled
15
−5
−10
20
0
−15
−20
−20
−40
−25
−60
−30
−80
−30
−20
−10
0
observed
10
20
A20 A21 A22
−50
B21 B22
0
50
observed
100
Figure 8. The spherical harmonic coef®cients (l=2±6) of the modelled ®elds, geoid (a) and topography (b), versus those of the observed ®elds (as in Fig. 6).
The dots correspond to coef®cients (l=3±6), whereas markers denote l=2 components according to the legend (in b). The mean of the ®rst family solutions
is shown by black, that of the second by dark grey and that of the third by light grey. The error bars represent uncertainties of the observed ®elds versus the
std of the modelled ®eld coef®cients within each solution family. The diagonal line represents a perfect ®t.
since we consider a complicated ®tting criterion, the dot distribution in each panel does not resemble a typical non-weighted
least-squares distribution.
The reason for the consistently underpredicted geoid amplitude becomes apparent now. The common feature for all
viscosity inversions is the relatively small lm=20 harmonic
(Fig. 8a), the harmonic providing the biggest signal in the
observed dynamic geoid. It seems that the slope formed by
most of the markers in Fig. 8(a) is about half the desirable one.
However, a doubling of a conversion factor to satisfy the geoid
would not improve the overall ®t, because it would also enlarge
the topography. In contrast to the red spectrum of the geoid, the
dynamic topography has a ¯atter spectrum. Therefore, doubling
of the amplitudes for all its harmonics would worsen the ®t
signi®cantly. As a result, the inversions ®nd a compromise
between the conversion factor and the topography amplitudes.
Another striking feature of the spectral characteristics is
the elongated scatter of the predicted topography coef®cients
around the best-®t line (Fig. 8b). This would again suggest
that the characteristic amplitudes of the density anomalies are
too small. However, because of the poor correlation of the
modelled and observed topography, the inversion drives the
amplitude of the modelled topography towards zero. There are
possibly several reasons for the unsatisfactory topography
resolution. The main reason, we believe, is related to the density
heterogeneity driving the ¯ow. The surface topography for all
three solution families is mainly determined by the upper mantle
density structure (where tomography shows large anomalies
and kernels are maximal, as in Figs 5g, h and i). This is a region
in which the density anomaly distribution probably cannot be
obtained by the direct multiplication of the seismic velocity
anomaly by a conversion factor. A more elaborate analysis that
accounts for compositional differences and relaxation effects in
addition to ordinary thermal differences in the upper mantle
may be needed. Another possible reason is the absence of the
LVV of the modelled upper mantle.
The viscosity pro®le solutions obtained can be compared
with previously published results. However, the fact that our
forward method considers continuous variations in viscosity
and density should be kept in mind as a possible reason for
disagreement. One way to compare the results is to match the
shape of the geoid kernels at lower harmonics. This, for
example, allows us to see the similarities between our third
family (Fig. 5c) and the viscosity pro®le preferred by the joint
inversion by Forte & Mitrovica (1996). Our second family of
solutions identi®es the gradual increase in viscosity within the
transition zone (Fig. 5b), which could result if a strong garnet
phase controls the deformation, instead of weaker olivine and
pyroxene phases (Karato 1989; Jeanloz 1989). A stiff transition
zone viscosity pro®le has been previously found by King
(1995); however, the exact pro®les and the geoid kernels are
hard to compare due to the differences in forward modelling.
The ®rst family of geoid kernels (Fig. 5a) resembles those of the
low-viscosity asthenosphere models (Hager & Clayton 1989).
The ®rst family of viscosity pro®les is also somewhat similar
to those of Cizkova et al. (1996) and Cadek et al. 1998). The
familial similarity to the results of these studies is explained by
the fact that in our analysis we consider not a single density
anomaly model, but 22 models, which include P- and S-wave
seismic tomography and a slab reconstruction. As a result, we
are able to determine that the third type of viscosity pro®les
is seen only by the purely seismic-tomography-based models
(Figs 7±f), and the ®rst type of viscosity pro®les is seen only
by models with the slab-related signal (Figs 7g±l), with the
exception of the two pure-seismic models (Figs 7a and f). It is
interesting to note that the second family of viscosity pro®les
was found by all density models.
Besides the viscosity pro®le, we simultaneously invert for the
conversion factors for each of the density models. The slab to
ambient mantle density contrast is in general agreement with
values proposed earlier (Hager & Clayton 1989; Ricard et al.
1993). The seismic slowness-to-density anomaly conversion
#
2000 RAS, GJI 143, 821±836
Mantle viscosity inversion using geoid and topography
obtained during our analysis differs from model to model; however, it is generally smaller than the ones predicted by experimental (Karato 1993; Chopelas 1992) or analytical (Forte et al.
1994; Hager & Clayton 1989; Cadek et al. 1998) studies. There
are many reasons for these differences. When comparing to the
experimental results, one has to remember that the factors
obtained during the inversion depend on the types of seismic
waves involved, the earth reference model and the method
of inversion. The analytical studies adopt a different parametrization of the conversion factor, continuous or step-like
pro®les and a different modi®cation of the seismic/slab models
such as a combination of signals from seismic tomography, slab
reconstruction and dynamic topography of internal boundaries.
However, the main reason for the smaller scaling factor is that
it is necessary to satisfy the relatively small amplitudes of
the surface topography. In that matter, our study is the ®rst
global inversion for mantle viscosity that incorporates dynamic
topography in addition to the gravitational constraints.
833
change, with gradual stiffening to the surface and to 670 km
depth. The third family displays a gradual reduction of viscosity
to 670 km depth, with strong softening at both phase change
regions, 400 and 670 km depth.
A characteristic feature of all viscosity pro®les is a one to two
orders of magnitude reduction of viscosity within the major
phase transformations, at 670 km depth, with two families
having reductions at 400 km as well. Such a viscosity pro®le
leads to a reduced dispersion of the geoid and the dynamic
topography kernels in the upper mantle: the different wavelength kernels essentially overlap at the location of the softest
layer. This leads to a strong positive correlation of the modelled
geoid with the density anomalies at the base of the upper
mantle, and to a relatively small amplitude of calculated surface
dynamic topography.
ACKNOWLEDGMENTS
CONCLUSIONS
The main goal of this study was to analyse the gravitational
constraints on the mantle viscosity pro®le using a variety of
models for the Earth's inner structure and additional constraints applied by the surface topography. We suggest a new
approach in viscosity inverse studies, which takes into consideration uncertainties in the observables and in the input
density data, and the de®ciencies of forward modelling. This
method uncovers the reasons for the inexact resemblance to
the geoid ®eld, which are mainly due to the uncertainties in
measurement and interpretation of the Earth's density anomalies.
This careful error analysis allows us to reduce the impact of the
most erroneous information on the results of the inversion.
To improve the quality of the forward model, we modi®ed
the formulation to handle more realistic, continuous variations
in radial viscosity, allowing for discontinuous jumps where
appropriate (Appendix). To perform a joint inversion in a selfconsistent way, we utilize a model of the surface dynamic
topography with associated spectral errors and also account for
errors associated with the correction for the geoid anomalies
produced by the isostatically compensated crust, lithosphere
and tectosphere. To gain statistical con®dence, we performed
the inversion using a variety of density models (based on
harmonic and block-type, S and P seismic velocity tomographic
models, in combination with a slab reconstruction model)
starting from 100 randomly distributed initial conditions.
The inversion revealed three distinct viscosity pro®le families
that provide almost equally good ®ts (Figs 5a, b and c). All
three solution groups identify a similar viscosity pro®le in the
lower mantle: one order of magnitude stiffening from 670 to
2500 km, followed by a three orders of magnitude reduction in
viscosity of the 300 km thick boundary layer at the CMB. The
small standard deviations around the mean pro®les show the
apparent robustness of this viscosity pro®le within the lower
mantle. The main distinctions among the families lie within
the upper mantle: the depth of the layer that has the lowest
viscosity is either right under the lithosphere or at 400 km
depth or at 670 km depth. The ®rst family identi®es a soft
asthenosphere followed by a stiff transition zone. The second
family is characterized by a soft layer around the 400 km phase
#
2000 RAS, GJI 143, 821±836
We thank Tom Jordan for inspiring the error analysis and Rick
O'Connell for suggesting the spectral analysis shown in Fig. 8.
Financial support was provided by NSF Grants EAR95-06427
(MIT) and EAR97-06210 (Harvard).
REFERENCES
Boschi, L. & Dziewonski, A.M., 1999. `High' and `low' resolution
images of the Earth's mantleÐimplications of different approaches
to tomographic modeling, J. geophys. Res., 104, 25 567±25 594.
Cadek, O., Ricard, Y., Martinec, Z. & Matyska, C., 1993. Comparison
between Newtonian and non-Newtonian ¯ow driven by internal
loads, Geophys. J. Int., 112, 103±114.
Cadek, O., Yuen, D.A. & Cizkova, H., 1998. Mantle viscosity inferred
from geoid and seismic tomography by genetic algorithms: results for
layered mantle ¯ow, Phys. Chem. Earth, 23, 865±872.
Chopelas, A., 1992. Sound velocities of MgO to very high compression,
Earth planet. Sci. Lett, 114, 185±192.
Christensen, U. & Harder, H., 1991. 3-D convection with variable
viscosity, Geophys. J. Int., 104, 213±226.
Cizkova, H., Cadek, O., Yuen, D.A. & Zhou, H.-W., 1996. Slope of the
geoid spectrum and constraints on mantle viscosity strati®cation,
Geophys. Res. Lett., 23, 3063±3066.
Clayton, R.W. & Comer, R.P., 1984. A tomographic analysis of mantle
heterogeneities, Terra Cognita, 4, 282±283.
Dehant, V. & Wahr, J.M., 1991. The response of a compressible,
non-homogeneous earth to internal loading: theory, J. Geomag.
Geoelectr., 43, 157±178.
Dziewonski, A.M. & Anderson, D.L., 1981. Preliminary reference
Earth model, Phys. Earth planet. Inter., 25, 297±356.
Dziewonski, A.M., Hager, B.H. & O'Connell, R.J., 1977. Largescale heterogeneities in the lower mantle, J. geophys. Res., 82,
239±255.
Ekstrom, G. & Dziewonski, A.M., 1998. The unique anisotropy of the
Paci®c upper mantle, Nature, 394, 168±172.
Forte, A.M. & Mitrovica, J.X., 1996. A new inference of mantle
viscosity based on a joint inversion of post-glacial rebound data
and long-wavelength geoid anomalies, Geophys. Res. Lett., 23,
1147±1150.
Forte, A.M. & Peltier, W.R., 1987. Plate tectonics and aspherical earth
structure: the importance of poloidal-toroidal coupling, J. geophys.
Res., 92, 3546±3680.
834
S. V. Panasyuk and B. H. Hager
Forte, A.M. & Peltier, W.R., 1991. Viscous ¯ow models of global
geophysical observables, 1 Forward problems, J. geophys. Res., 96,
20 131±20 159.
Forte, A.M. & Peltier, W.R., 1994. The kinematics and dynamics
of poloidal-toroidal coupling in mantle ¯ow: the importance of
surface plates and lateral viscosity variations, Adv. Geophys., 36,
1±119.
Forte, A.M., Peltier, W.R. & Dziewonski, A.M., 1991. Inferences of
mantle viscosity from tectonic plate velocities, Geophys. Res. Lett.,
18, 1747±1750.
Forte, A.M., Woodward, R.L. & Dziewonski, A.M., 1994.
Joint inversions of seismic and geodynamic data for models
of three-dimensional mantle heterogeneity, J. geophys. Res., 99,
21 857±21 877.
Gable, C.W., O'Connell, R.J. & Travis, B.J., 1991. Convection in
three dimensions with surface plates; generation of toroidal ¯ow,
J. geophys. Res., 96, 8391±8405.
Grand, S.P., 1994. Mantle shear structure beneath the Americas and
surrounding oceans, J. geophys. Res., 99, 11 591±11 621.
Gurnis, M., 1990. Bounds on global dynamic topography
from Phanerozoic ¯ooding of continental platforms, Nature, 344,
754±756.
Hager, B.H., 1984. Subducted slabs and the geoid: constraints on
mantle rheology and ¯ow, J. geophys. Res., 89, 6003±6015.
Hager, B.H. & Clayton, R.W., 1989. Constraints on the structure
of mantle convection using seismic observations, ¯ow models,
and the geoid, in Mantle Convection: Plate Tectonics and Global
Dynamics, pp. 657±763, ed. Peltier, W.R., Gordon and Breach,
New York.
Hager, B.H. & O'Connell, R.J., 1978. Subduction zone dips and ¯ow
driven by the plates, Tectonophysics, 50, 111±134.
Hager, B.H. & O'Connell, R.J., 1981. A simple global model of
plate dynamics and mantle convection, J. geophys. Res., 86,
4843±4867.
Hager, B.H. & Richards, M.A., 1989. Long-wavelength variations in
the Earth's geoid: physical models and dynamical implications, Phil.
Trans. R. Soc. Lond., A328, 309±327.
Inoue, H.Y., Fukao, K. & Tanabe, & Ogata, Y., 1990. Whole
mantle P-wave travel time tomography, Phys. Earth planet. Inter., 59,
294±328.
Ishii, M. & Tromp, J., 1999. Normal-mode and free-air gravity
constraints on lateral variations in velocity and density of the Earth's
mantle, Science, 285, 1231±1236.
Jeanloz, R., 1989. High pressure chemistry of the Earth's mantle and
core, in Mantle Convection: Plate Tectonics and Global Dynamics,
pp. 203±259, ed. Peltier, W.R., Gordon and Breach, New York.
Jordan, T., 1988. Structure and formation of the continental
tectosphere, J. Petrol., Special lithosphere issue, 11±37.
Karason, H. & van der Hilst, R., 1999. Improving global tomography
models of P-wavespeed I: incorporation of differential times for
refracted and diffracted core phases (PKP, Pdiff), J. geophys. Res.,
submitted.
Karato, S.-I., 1989. Plasticity-crystal structures systematics in
dense oxides and its implications for the creep strength of the
Earth's interior: a preliminary result, Phys. Earth planet. Inter., 55,
234±240.
Karato, S.-I., 1993. Importance of anelasticity in the interpretation of
seismic tomography, Geophys. Res. Lett., 20, 1623±1626.
King, S.D., 1995. Radial models of mantle viscosity: results from a
genetic algorithm, Geophys. J. Int., 122, 725±734.
Lerch, F.J., Klosko, S.M. & Patch, G.B., 1983. A re®ned gravity model
from LAGEOS (GEM-L2), NASA Technical Memorandum, 84986.
Li, X.D. & Romanowicz, B., 1996. Global mantle shear-velocity model
developed using nonlinear asymptotic coupling theory, J. geophys.
Res., 101, 22 245±22 272.
Liu, X.-F., Su, W.-J. & Dziewonski, A.M., 1994. Improved resolution
of the lowermost mantle shear wave velocity structure obtained using
SKS-S data, EOS, Trans. Am. geophys. Un., 75, 232±233.
Masters, G., Johnson, S., Laske, G. & Bolton, H., 1996. A shear
velocity model of the mantle, Phil. Trans. R. Soc. Lond., A354,
1385±1414.
Mitrovica, J.X. & Forte, A.M., 1997. Radial pro®le of mantle viscosity:
results from the joint inversion of convection and post-glacial
rebound observables, J. geophys. Res., 102, 2751±2769.
Nakiboglu, S.M., 1982. Hydrostatic theory of the Earth and
its mechanical implications, Phys. Earth planet. Inter., 28,
302±311.
O'Connell, R.J., Gable, C.W. & Hager, B.H., 1991. Toroidal-poloidal
partitioning of lithospheric plate motion, in Glacial Isostasy, SeaLevel and Mantle Rheology, pp. 535±551, eds Sabadini, R.,
Lambeck, K. & Boschi, E., Kluwer, Boston.
Olson, P. & Bercovici, D., 1991. On the equipartitioning
of kinetic energy in plate tectonics, Geophys. Res. Lett., 18,
1751±1754.
Panasyuk, S.V., 1998. The effect of compressibility, phase transformations, and assumed density structure on mantle viscosity inferred
from Earth's gravity ®eld, PhD Thesis, MIT, Cambridge, MA.
Panasyuk, S.V. & Hager, B.H., 1998. A model of transformational superplasticity in the upper mantle, Geophys. J. Int.,
133, 714±755.
Panasyuk, S.V. & Hager, B.H., 2000. Models of isostatic and dynamic
topography, geoid anomalies, and their uncertainties, J. geophys.
Res., in press.
Panasyuk, S.V., Hager, B.H. & Forte, A.M., 1996. Understanding the
effects of mantle compressibility on geoid kernels, Geophys. J. Int.,
124, 121±133.
Quinn, K.J. & McNutt, M.K., 1998. Inversion of topography and
geoid for mantle viscosity beneath the Paci®c plate using genetic
algorithms, J. geophys. Res., submitted.
Ranalli, G., 1991. The microphysical approach to mantle rheology, in
Glacial Isostasy, Sea-Level and Mantle Rheology, pp. 343±378, eds
Sabadini, R., Lambeck, K. & Boschi, E., Kluwer, Boston.
Ribe, N.M., 1992. The dynamics of thin shells with variable viscosity
and the origin of toroidal ¯ow in the mantle, Geophys. J. Int., 110,
537±552.
Ricard, Y. & Vigny, C., 1989. Mantle dynamics with induced plate
tectonics, J. geophys. Res., 94, 17 543±17 559.
Ricard, Y., Fleitout, L. & Froidevaux, C., 1984. Geoid heights
and lithospheric stresses for a dynamic Earth, Ann. Geophys., 2,
267±286.
Ricard, Y., Vigny, C. & Froidevaux, C., 1989. Mantle heterogeneities,
geoid, and plate motion: a Monte Carlo inversion, J. geophys. Res.,
94, 13 739±13 754.
Ricard, Y., Richards, M.A., Lithgow-Bertelloni, C. & LeStunff, Y.,
1993. A geodynamic model of mantle heterogeneity, J. geophys. Res.,
98, 21 895±21 909.
Richards, M.A. & Hager, B.H., 1984. Geoid anomalies in a dynamic
Earth, J. geophys. Res., 89, 5987±6002.
Richards, M.A. & Hager, B.H., 1989. Effects of lateral viscosity
variations on long-wavelength geoid anomalies and topography,
J. geophys. Res., 94, 10 299±10 313.
Sammis, C.G. & Dein, J.L., 1974. On the possibility of transformational superplasticity in the Earth's mantle, J. geophys. Res.,
79, 2961±2965.
Sammis, C.G., Smith, J.C., Schubert, G. & Yuen, D.A., 1977.
Viscosity-depth pro®le in the Earth's mantle: effects of polymorphic
phase transitions, J. geophys. Res., 85, 3747±3761.
Simons, M. & Hager, B.H., 1997. Localization of the gravity ®eld and
the signature of glacial rebound, Nature, 390, 500±504.
Su, W.-J., Woodward, R.L. & Dziewonski, A.M., 1994. Degree 12
model of shear velocity heterogeneity in the mantle, J. geophys. Res.,
99, 6945±6980.
Tarantola, A., 1997. Inverse Problem Theory, Elsevier, Amsterdam.
van der Hilst, R.D., Widiyantoro, S. & Engdahl, E.R., 1997. Evidence
for deep mantle circulation from global tomography, Nature, 386,
578±584.
#
2000 RAS, GJI 143, 821±836
Mantle viscosity inversion using geoid and topography
Zhang, S. & Christensen, U.R., 1993. Some effects of lateral viscosity
variations on geoid and surface velocities induced by density
anomalies in the mantle, Geophys. J. Int., 114, 531±547.
Note that A can now be represented as a sum of four matrix
expressions instead of three terms as in Panasyuk et al.
(1996):
A(r)~A0 zs(r)A1 z
APPENDIX A: CONTINUOUS
VARIATIONS OF VISCOSITY HANDLED
BY THE MATRIXANT APPROACH
The general physical assumptions and their mathematical
representation are based on the theory presented in Panasyuk
et al. (1996); therefore, we omit a detailed description here. The
equations used to derive the matrix differential equation,
L0 ul ~Al ul zbl ,
(A1)
are true both for constant and continuously varying viscosity
g(r). The exponent matrix Al has ®ve non-diagonal terms containing the normalized viscosity. Since mantle viscosity could
vary by several orders of magnitude, the Al matrix could
change signi®cantly within the layer, prohibiting the use of the
matrixant approach. We resolve this complication by introducing a new set of poloidal variables (see also Panasyuk 1998;
the notation is similar to that used in Panasyuk et al. 1996):
u(r)~[ y1 g
y2 g "
( y3 zo0 y5 )r
y4 r"
y5 ro"
T,
y6 r2 o]
(A2)
where g*=g0(r)/gÅ is the viscosity as a function of radius,
normalized by the reference value. In analogy with the
compressibility factor,
s(r)~
r Lo0 (r)
,
o0 (r) Lr
(A3)
we introduce a viscosity factor,
f(r)~
r Lg0 (r)
,
g0 (r) Lr
(A4)
2
{(2zs{f)
6
6
{"
6
6
6
6 (12z4s)
6
6
6
6
6 {(6z2s)"
6
6
6
0
6
4
0
#
"
0
0
0
1zf
0
1
0
{6"
1
"
so0
"o
2(2"2 {1)
{" {2
0
0
2000 RAS, GJI 143, 821±836
0
0
0
0
0
1
"
0
r1
o (r1 )
o (r1 ){o0 (r2 )
Prr12 ~ exp A0 ln zA1 ln 0
zA2 0
r2
o0 (r2 )
o
g (r1 )
:
zA3 ln 0
g0 (r2 )
(A6)
(A7)
To be able to apply the propagator technique, we ought
to make sure that the argument of the exponential does not
vary signi®cantly within the layer. Analysis of the variations
due to the density change was performed in Panasyuk et al.
(1996). The variations due to a non-constant viscosity are
expressed by the last term. It is only when the viscosity changes
exponentially between r1 and r2 that the A3 term is constant
within the layer. Where the slope of the exponent changes or
the viscosity change is discontinuous, it is necessary to stop to
tie up the propagators. Similarly, when the viscosity changes
dramatically within/across a thin layer such as a phase change
region, it is reliable to treat it as a boundary and couple
the solutions across it. We approximate the u vector across the
density/viscosity discontinuity and derive the boundary conditions for the new system. In the limit of a thin phase change
region, there is a jump condition on u. The ®rst two terms are
related to the ¯ow velocity change as
gz{ ozz zz
u
gzz oz{ 1
and
uz{
2 ~
gz{ zz gz{ *z zz
u z u :
gzz 2
gz z 4
(A8)
3
7
07
7
7
7
07
7
7
7:
7
07
7
7
7
"7
5
0
s(r)o0 (r)
A2 zf(r)A3 :
o
Here A0 is equivalent to the matrix for incompressible ¯ow
(Hager & Clayton 1989). The matrix A1 gives the effect of
compressibility on ¯ow, A2 is related to a speci®c combination
of stress and potential into one variable (Panasyuk et al. 1996),
and the matrix A3 accounts for the continuously changing
viscosity (Panasyuk 1998). Solving the matrix differential
equation, we write the propagator between the upper (ocean±
mantle boundary) and lower boundaries (core±mantle interface)
as a product of subpropagators (similar to the handling of
compressible ¯ow). Each subpropagator now has one more
additional term,
uz{
1 ~
so that the matrix Al becomes
835
The third and fourth u components are approximated across
the phase change as
(A5)
uz{
3 &4
*oz gz zz
*oz zz
zz
u
u
zu
z
1
3
oz{ gzz
"o 5
uz{
4 &{2"
*oz gz zz
u zuzz
4 :
oz{ gzz 1
and
(A9)
836
S. V. Panasyuk and B. H. Hager
The gravity-related ®fth and six components are continuous
since we assume that the phase change or the thermal boundary
is located at a constant radius.
We combine the internal and external boundary conditions
with the matrixant approach to obtain a system of equations
similar to eq. (42) in Panasyuk et al. (1996) and solve it with
respect to the ocean±mantle boundary de¯ection, da, and the
potential anomaly, V1e (r) at the ocean surface (r=e). The geoid
kernels displayed in the ®gures are normalized by the geoid
de¯ection due to a mass at the surface,
Gl (r)~
(2lz1) e
V1 (r) :
4nce
(A10)
Note that the normalized geoid kernels are identical to the
potential kernels normalized by the potential due to a mass
at the surface. In the main text we omit the superscript e,
assuming that the gravity ®eld is considered at the ocean
surface.
#
2000 RAS, GJI 143, 821±836