CORE MANTLE BOUNDARY TOPOGRAPHY FROM SHORT PERIOD PCP, PKP, AND PKKP DATA
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Physics of the Earth and Planetary Interiors 135 (2003) 27–46
Core mantle boundary topography from short period PcP,
PKP, and PKKP data
Edmond K.M. Sze a,∗ , Robert D. van der Hilst b,1
a Department of Earth, Atmospheric and Planetary Sciences Massachusetts Institute of Technology, E34-530A,
77 Massachusetts Avenue, Cambridge, MA 02139, USA
b Department of Earth, Atmospheric and Planetary Sciences Massachusetts Institute of Technology,
54-514, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Received 25 October 2001; accepted 27 September 2002
Abstract
We use a total of 839,369 PcP, PKPab, PKPbc, PKPdf, PKKPab, and PKKPbc residual travel times from [Bull. Seism.
Soc. Am. 88 (1998) 722] grouped in 29,837 summary rays to constrain lateral variation in the depth to the core–mantle
boundary (CMB). We assumed a homogeneous outer core, and the data were corrected for mantle structure and inner-core
anisotropy. Inversions of separate data sets yield amplitude variations of up to 5 km for PcP, PKPab, PKPbc, and PKKP
and 13 km for PKPdf. This is larger than the CMB undulations inferred in geodetic studies and, moreover, the PcP re-
sults are not readily consistent with the inferences from PKP and PKKP. Although the source–receiver ambiguity for
the core-refracted phases can explain some of it, this discrepancy suggest that the travel-time residuals cannot be ex-
plained by topography alone. The wavespeed perturbations in the tomographic model used for the mantle corrections
might be too small to fully account for the trade off between volumetric heterogeneity and CMB topography. In a sec-
ond experiment we therefore re-applied corrections for mantle structure outside a basal 290 km-thick layer and inverted
all data jointly for both CMB topography and volumetric heterogeneity within this layer. The resultant CMB model can
explain PcP, PKP, and PKKP residuals and has approximately 0.2 km excess core ellipticity, which is in good agreement
with inferences from free core nutation observations. Joint inversion yields a peak-to-peak amplitude of CMB topogra-
phy of about 3 km, and the inversion yields velocity variations of ±5% in the basal layer. The latter suggests a strong
trade-off between topography and volumetric heterogeneity, but uncertainty analyses suggest that the variation in core ra-
dius can be resolved. The spherical averages of all inverted topographic models suggest that the data are best fit if the
actual CMB radius is 1.5 km less than in the Earth reference model used (i.e. the average outer core radius would be
3478 km).
© 2002 Published by Elsevier Science B.V.
Keywords: Core–mantle boundary topography; PcP; PKP; PKKP; Excess ellipticity; Core radius
1. Introduction
∗
The core–mantle boundary (CMB) marks the
Corresponding author. Tel.: +1-617-642-4355.
E-mail addresses: edsze@erl.mit.edu (E.K.M. Sze), hilst@mit.edu
strongest change in density and viscosity within the
(R.D. van der Hilst). Earth. It separates the solid silicate mantle from the
1 Tel.: +1-617-253-6977. liquid iron–nickel outer core. CMB topography is
0031-9201/02/$ – see front matter © 2002 Published by Elsevier Science B.V.
PII: S 0 0 3 1 - 9 2 0 1 ( 0 2 ) 0 0 2 0 4 - 2
28 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 generally believed to be caused by dynamical pro- 2. Data selection and processing cesses within the mantle (e.g. hot, buoyant regions may pull the CMB upward, and cold, dense regions We extracted travel-time residuals of PcP, PKPab, may depress it). Thus, knowledge of CMB topog- PKPbc, PKPdf, PKKPab, and PKKPbc phases from raphy can put direct constraints on the modeling of the data set of Engdahl et al. (1998) for the years thermochemical mantle convection (Forte et al., 1991; 1964–1999. This data set contains nearly 100,000 Kellogg et al., 1999; Forte and Mitrovica, 2001), geo- relocated events that are well-constrained teleseismi- dynamo (Bloxham, 1993), core–mantle interactions cally by travel-time data reported to ISC and the US (Bloxham and Gubbins, 1987; Stevenson, 1987), and Geological Survey’s National Earthquake Information possibly the genesis of hotspots (Yuen and Peltier, Center (NEIC). The radially stratified ak135 earth 1980). model (Kennett et al., 1995) was used as the reference With data provided by the International Seismolog- for computing ray paths and travel time residuals. ical Center (ISC), Creager and Jordan (1986) used Along with the signal that originates other than at or PKPbc and PKPdf and Morelli and Dziewonski (1987) near the CMB (e.g. upper and lower mantle hetero- used PcP and PKPbc travel time residuals to construct geneity, and inner core structure), the high noise level maps of long-wavelength CMB topography. These in- in the travel-time data is one of the major obstacles to versions produced consistent results but later studies retrieve signal pertinent to CMB undulations. There did not agree with them. Rodgers and Wahr (1993) are different types of noise present in the data, includ- used ISC PcP, PKPab, PKPbc, and PKPdf in separate ing random noise, human errors (misreading of arrival inversions and pointed out that models of CMB topog- times and phase misidentification), and hypocenter raphy inferred from different phases did not agree spa- mislocation. Engdahl et al. (1998) corrected for the tially. Obayashi and Fukao (1997) inverted ISC PcP ellipticity and applied a patch station correction that data alone and obtained images consistent with the PcP compensated for uppermost crust and upper mantle models of earlier studies. From reprocessed ISC data, heterogeneity. They also re-identified phases and re- Boschi and Dziewonski (2000) also reported on the duced hypocenter mislocations. Fig. 1a and b show disagreement between the PcP and PKP models and the geographic distribution of the locations of 64,993 offered an explanation in terms of mantle anisotropy events and 3500 stations that produced the travel-time and outer-core heterogeneity. The magnitude of the residual data used in our study. CMB undulation inferred from these studies ranges Fig. 2a shows the ray paths of the core-reflected PcP from 4 to 12 km, which exceeds the inferences from and core-transmitted PKP phases. The bottomside re- geodetic studies. Garcia and Souriau (2000) used PcP, flected PKKP phases are shown in Fig. 2b. The spe- PKPbc, and PKKPbc phases selected from the data cific structure of the core produces different PKP and set by Engdahl et al. (1998) and concluded that CMB PKKP branches (e.g. ab, bc and df). The travel time topography cannot be resolved, but they used a data curves are shown in Fig. 3a and b. set that is significantly smaller than we used in our For each seismic phase we selected the data based study. on epicentral distance range and cut-off time. The se- Here, we explore CMB models inferred from dif- lection on epicentral distance range was used to avoid ferent seismic phases using the dataset by Engdahl contamination by other arrivals. The cut-off time pre- et al. (1998) which is updated to the year 2000, and it vents outliers from biasing the inversion results. In has 10% more data than the original dataset. In addi- this paper, travel-time residuals outside a ±4 s window tion to the core reflected PcP, core refracted PKPbc, around the median residual are excluded. and CMB underside reflected PKKPbc phases, we Fig. 4a shows the plot of PcP residuals versus epi- have also chosen to use PKPbc, PKPdf, and PKKPab central distance. The two solid lines show the ±4 s phases to invert for CMB topography and to inves- window around the median residual as a function of tigate the trade-off with heterogeneity just above the epicentral distance. The residuals plotted in black are CMB. We also perform uncertainty analyses and com- the data that we selected for inversion, and those plot- pare results with previous seismological and geodetic ted in gray depict the ones that we discarded. There studies. are two distance ranges that have substantial scatter
E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 29 Fig. 1. (a) Geographic distribution of the 64,993 events during the period 1964–1998; (b) Distribution of the 3500 stations that reported the core phases used in this study. due to phase misidentification: PcP travel-times co- heterogeneous D layer we did not observe greater incide with PP at around 40◦ –45◦ , and at distances scatter in the PKPab times. PKPbc runs from 143 larger than 70◦ it is difficult to separate PcP from di- to 155◦ but we only used PKPbc residuals between rect P since the travel-time of these two phases merge 146 to 154◦ (Fig. 4d). The quality of PKPbc in the at large distances (Fig. 3a). We use the PcP in the epi- data set was improved by Engdahl et al. (1998) by central distance ranges 25◦ –40◦ and 45◦ –70◦ to avoid re-identifying PKPbc arrivals that were misidentified such contamination. The final PcP data set has 57,122 by ISC as PKPdf. PKPab and PKPbc both have turning travel-time residuals. points in the liquid outer-core. PKPdf corresponds to a The caustic near 143◦ epicentral distance (Fig. 3a) phase that refracts through the inner-core; it can be ob- complicates identification of different branches of served between 110◦ and 180◦ but we used it only for PKP, and we did not select data at distance range the distance ranges 115◦ –135◦ and 152◦ –180◦ . Fig. 4d 140◦ –145◦ . PKPab can be observed between 143 shows the scatter caused by diffraction near the PKP and 180◦ (Fig. 4b) but we only used PKPab residu- caustic at 143◦ ; phases near 143◦ are more likely to be als between 155 and 180◦ . The waveform of PKPab misidentified and we, therefore, discarded PKPdf data is the Hilbert transform of that of PKPdf but the in the distance range 135◦ –152◦ . Our selection re- high-frequency (1 Hz) character of the signals renders sulted in 69,767 PKPab, 365,379 PKPbc, and 335,530 the error of mispicking due to pulse distortion to be PKPdf residuals to be included in our final data set. relatively small. In spite of the waveform distortion PKKP phases refract through the outer core and and the fact that PKPab has longer ray-paths in the reflect once at the underside of the CMB. Like PKP
30 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46
more difficult to be picked correctly at distance than
120◦ as it approaches the b caustic. PKKPbc can
be observed in the distance range 82◦ –120◦ but we
discarded data between 88 and 92◦ to avoid inter-
ference with SS. Fig. 4b and c show residual plot
of PKKPab and PKKPbc, respectively. Our final
dataset consists of 2575 PKKPab and 8996 PKKPbc
residuals.
The geographic distributions of the bounce points
of PcP, and the entry and exit points of PKPab, PKPbc,
and PKPdf are shown in Fig. 5. PcP coverage of the
CMB is nonuniform. It only has extensive coverage
beneath Eurasia, the west Pacific, Alaska, and Cen-
tral America. There are large areas with no adequate
coverage, notably beneath the Pacific Ocean and the
Southern Hemisphere. PKP phases have good cover-
age beneath the Pacific rim, the Middle East, and Eu-
rope. The distributions of the reflection, entry, and exit
points of PKKP phases (Fig. 6) are also non-uniform,
especially for PKKPab. In addition, the PKKP data
are far less abundant than any other core phases. Nev-
ertheless, including PKKP data helps us isolate the
CMB topography from the overlying volumetric het-
erogeneity.
There are several differences in data selection crite-
ria between this study and that of Garcia and Souriau
(2000), hereafter GS2000. First, we used a more re-
cent revision of the EHB dataset with event origin
times through December 2000; the EHB data used by
GS2000 was complete through August 1998. Second,
GS2000 cut the PcP dataset in three parts, namely,
PcP1, PcP2, and PcP3. PcP1 are residuals with rays of
nearly vertical incidences, PcP2 are residuals of epi-
central distances between 29◦ and 40◦ , and PcP3 are
residuals with higher incidence angles. GS2000 se-
lected only PcP2, and discarded PcP1 and PcP3 based
Fig. 2. (a) Ray paths of PcP and PKP phases; (b) Ray paths of
on their higher noise levels. However, we do not feel
PKKP phases. that the differences in noise level a warrant to discard
PcP1, PcP2, or PcP3, given the significant increase of
data coverage. Third, in contrast to GS2000, we also
phases, different branches of PKKP turn at different included PKPab, PKPdf, and PKKPab phases. PKPab
depths in the core (Fig. 2b). PKKPab and PKKPbc and PKKPab were rejected by GS2000 because of
have turning points in the outer-core and PKKPdf their sensitivity on D heterogeneities. However, this
turns in the inner-core. The amplitude of PKKPdf greater sensitivity helps us to reduce trade-offs be-
is weak due to inner-core attenuation and the small tween effects of D heterogeneities and CMB topog-
near-vertical reflection coefficient at the CMB and raphy. Including PcP, PKPab, PKPdf, and PKKPab
is, therefore, rarely observed. We selected PKKPab phases gave us a large dataset of travel time residuals
in the distance range 105◦ –120◦ . PKKPab becomes with ray geometry that had various sensitivities bothE.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 31
Fig. 3. (a) Travel time curves of PcP and PKP; (b) Travel time curves of PKKP.
on CMB topography and D heterogeneities. Tables 1
and 2 summarizes the type and number of data used
in our study and by GS2000. Table 1
Following Morelli and Dziewonski (1987), we Variance reduction achieved by inversion of the data
constructed summary rays by partitioning the Earth’s Phase Separate Joint PKP All (%) All + basal
surface into 5◦ × 5◦ bins and averaging the residuals inversion (%) + PKKP (%) layer (%)
associated with paths that originated from the same PcP 7.2 −13.8 −0.5 10.5
source cell and arrived at the same receiver cell. PKKP 12.2 9.3 7.6 19.2
The bundle of rays having the same pair of cells are PKPab 12.2 5.9 2.5 29.7
treated as one ray. We did not impose a maximum on PKPbc 6.2 7.5 6.2 20.2
PKPdf 13.4 8.3 9.6 21.4
the number of individual rays constituting a summary All 8.3 7.2 20.1
ray. The main purposes of forming summary rays are32 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 Fig. 4. Travel time residual vs. epicentral distance plots for (a) PcP, (b) PKKPab, (c) PKKPbc, (d) PKPab, (e) PKPbc, and (f) PKPdf. Residuals plotted in black were selected for inversion and those plotted in gray were discarded.
E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 33
Fig. 5. Distribution of hit points on the CMB for (a) PcP, (b) PKPab, (c) PKPbc, and (d) PKPdf.
Fig. 6. (a) Distribution of transmission hit points of PKKPab, (b) underside reflection hit points of PKKPab, (c) transmission hit points of
PKKPbc, and (d) underside reflection hit points of PKKPbc on the CMB.34 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46
Table 2 side of each histogram (Fig. 7) shows the number
Number of summary rays of the data of travel-time residuals (N) in the data set, standard
Phase No. of No. of Spherical No. of deviation (σ ) and the interquartile range (ρ), all after
residuals summary average of summary correction for 3-D mantle structures. The change of
rays residuals (s) rays (GS200) standard deviation of the data (σ = σ − σ0 ) and
PKPab 69,767 2453 −0.22 the change of interquartile range (ρ = ρ − ρ0 ) are
PKPbc 365,379 7632 −0.10 989
also shown, where σ 0 and ρ 0 are the standard devi-
PKPdf 335,530 12,760 −0.12
PKKPab 2575 3572 0.36 ation and the interquartile range of the data before
PKKPbc 8996 3572 0.36 265 mantle correction, respectively. We can see that the
PcP 57,122 3420 0.11 452 histograms show (slightly) sharper peaks and nar-
rower spreads after 3-D mantle corrections, indicated
by negative values for σ and ρ.
to enhance the signal to noise ratio by averaging out PKPdf waves pass through the inner-core, which has
the random component of the data, to reduce the pre- been suggested to be anisotropic (Morelli et al., 1986;
dominance of regions with dense data coverage, and Shearer et al., 1988; Creager, 1992; Su and Dziewon-
to reduce the size of the inverse problem. ski, 1995). Travel times are several seconds faster for
polar than for equatorial paths (Poupinet et al., 1983).
We corrected the PKPdf travel-time residual data for
3. Correction for mantle structure and inner core inner-core anisotropy based on the 3-D inner-core
model of Su and Dziewonski (1995), but some skew-
One of the problems of inferring CMB topogra- ness remains in the histogram of PKPdf data.
phy is to remove other sources of the observed travel
time residuals. It is commonly assumed that the liq-
uid outer core is homogeneous (Stevenson, 1987), 4. Inversion method
which leaves mantle and—for PKPdf—inner core
structure to be accounted for. In order to suppress Model parameterization in seismic tomography can
the contribution of mantle heterogeneity and isolate be done in different ways, for example, by represent-
the signal due to CMB topography, we used the re- ing the perturbation by spherical harmonics or by dis-
cent three-dimensional mantle P-wavespeed model of cretizing the model and solving for the perturbation in
Kárason and van der Hilst (2001), hereon KH2001, cells or on nodes. We chose to partition the CMB into
to correct the travel-time residuals. Their model was 2592 cells, each of dimension 5◦ × 5◦ . With appropri-
inferred from nearly eight million direct P and pP ate regularization, a relatively fine parameterization by
travel-time data, along with PKP and Pdiff which im- means of local basis functions can help reduce spuri-
proved the data coverage of lower-mantle structure. ous anomalies in sparsely sampled areas of the CMB
We realize that the lowermost mantle part of this model (Pulliam and Stark, 1993).
could be influenced by the ignored CMB topogra- The linearized relationship between CMB topog-
phy, but we traced rays through the 3-D P-wavespeed raphy (δr) and travel-time residuals (δt) is given by
model to calculate the travel-time perturbations due to δt = −2δrη+ for PcP, δt = δr{η− − η+ } for PKP,
mantle structure. This signal was then subtracted from and δt = −δr{η+ + η− } for PKKP, where
the residual time data set to obtain the “corrected”
residual time data set. Fig. 7 shows the histograms of 2 1/2
p 2 v+
1
the average travel-time residuals binned in 0.1 s bins η+ = 1− 2 ,
v+ r0
after 3-D mantle correction. We quantify the spread
of travel-time residual by the interquartile range, ρ, 2 1/2
p 2 v−
1
which is computed as the difference between the 75th η− = 1− 2 ,
v− r0
percentile and the 25th percentile. This measure is
robust against outliers when the residual distribution v+/− is the P-wave velocity above/below the CMB,
is non-Gaussian. The upper corner on the right hand p is the ray parameter, and r0 is the CMB radiusE.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 35 Fig. 7. Histograms of travel-time residuals after mantle correction: (a) PcP, (b) PKKPab, (c) PKKPbc, (d) PKPab, (e) PKPbc, and (f) PKPdf.
36 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46
(3479.5 km as in ak135). These expressions form a jointly with PKKPbc because the number of PKKPab
system of linear equations that can be expressed as arrivals is small. The spherical average and variance
Ar mr = d, where Ar is the matrix of the sensitiv- reduction associated with each inversion is shown in
ity kernels relating topography to travel-time residu- Table 1.
als, mr the vector of topography perturbations, and d Several long-wavelength features can be observed
is the vector of observed travel-time residuals. in the topography map inferred from PcP. For example,
Travel time residuals can also arise from volumetric there is a negative anomaly beneath Southeast Asia,
heterogeneity above the CMB. In the second experi- the Philippine Sea, and Indonesia, positive anoma-
ment, therefore, we inverted data for both topography lies beneath northern high latitudes (>45◦ ), Eurasia,
and volumetric heterogeneity. The sensitivity kernel and Alaska, and a weak negative anomaly under Cen-
relating travel-time perturbation and rays that pass tral America. Our PcP map is consistent with that of
through a thin layer with velocity perturbation, δv, is Rodgers and Wahr (1993) and Obayashi and Fukao
rl (1997).
δtv = The CMB topography inferred from PKPab, PKPbc,
v+
2η
−
PKPdf, and joint PKKPab and PKKPbc is in good
where rl is the thickness of the thin layer. This linear agreement with the results of Creager and Jordan
forward problem that can be expressed as Av mv = d, (1986) from PKPab and PKPdf, and Rodgers and Wahr
where Av is the sensitivity kernel matrix relating ve- (1993) from separate PKP branches. The topography
locity and travel-time residuals and mv is the vector models inferred from PKPab, PKPbc, and PKPdf are
of velocity perturbations. The joint problem can then correlated with one another, except in poorly sampled
be formulated as areas (see the correlation coefficients in Table 3). All
three PKP maps show negative anomalies for high
mr
d = [ Ar Av ] . latitudes under Northern Hemisphere, and positive
mv
anomalies under West Pacific, East Pacific and South
To focus on long-wavelength variations, we mini- America, and the North Atlantic. PKPbc and PKPdf
mized the roughness of CMB topography and vol- maps both show positive anomalies under South-
umetric heterogeneity by incorporating Tikhonov east Asia, and a negative anomaly east of Australia.
second-difference regularization matrices into our We also notice the agreement between the PKP and
inversion and also apply norm damping. To be con- PKKP maps, but the latter are smoother due to sparser
sistent, we use the same Tikhonov regularization and sampling and the enhanced effect of regularization.
norm damping for seismic phases and all calculations The observation that the PKPab, PKPbc, PKPdf, and
of which results are shown here. The spherical aver- PKKP maps are consistent suggests that the anoma-
age of each data set is calculated and subtracted from lies are due to real structure in the Earth and justifies
the data set prior to inversion. The systems of equa- a joint inversion for topography (see Fig. 8f). The re-
tions are solved using the LSQR algorithm (Paige and sulting model explains about 8.3% of the variance of
Saunders, 1982), and the data variance reduction, de- the combined data set. Pronounced negative anomalies
fined as (σ )2 = 1 − ((d − Am)T (d − Am)/d T d) × remain beneath Northern Eurasia, North America, and
100%, is calculated for all inversions.
Table 3
5. Results Correlation coefficients between models inverted by different core
phases
5.1. CMB topography from separate data sets PKPab PKPbc PKPdf PKKP PcP
PKPab 1 −0.1137 0.4441 −0.4089 −0.0889
Fig. 8a–e show the maps of the CMB topography in- PKPbc 1 0.1487 0.6312 −0.5732
ferred from PcP, joint PKKPab and PKKPbc, PKPab, PKPdf 1 −0.0510 −0.2819
PKPbc, and PKPdf, respectively. A cell containing PKKP 1 −0.2762
PcP 1
no bounce points are left blank. PKKPab is invertedE.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 37
Fig. 8. The topographic models of CMB, inferred from independent inversions of (a) PcP, (b) PKKPab and PKKPbc, (c) PKPab, (d)
PKPbc, (e) PKPdf. Part (f) is a CMB topography model inferred jointly from PKPab, PKPbc, PKPdf, PKKPab, and PKKPbc. KH2001
(Kárason and van der Hilst, 2001) mantle model, derived from P, PKP and Pdiff, was used to correct for 3-D mantle structure. PKPdf
travel-times were also corrected for inner-core anisotropy (Su and Dziewonski, 1995). Blue indicates a decrease in core radius and red
indicates an increase in core radius. Scale is measured in km. Cells with no hit points on the CMB are blanked.
South Pacific. Positive anomalies are detected beneath model. We randomly discarded 10% of the data and
South America, Western Europe, and Southeast Asia. then performed the inversion. After repeating this pro-
cess 100 times we calculated the standard deviation
5.2. Uncertainty calculations at each cell. This gives us an estimate of errors asso-
ciated with sampling geometry and random noise. We
Assuming that the errors in the data are indepen- performed the bootstrap error uncertainty calculation
dent of each other we used a bootstrap method (Efron for all models inferred by different phases (Fig. 9).
and Tibshirani, 1986) to test the uncertainty of the One of the pitfalls of applying bootstrap method to38 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 Fig. 9. Bootstrap error uncertainty, shows the standard deviation of the 100 models at each point for (a) PcP, (b) PKPab, (c) PKPbc, (d) PKPdf, and (e) PKKP. under-determined problems such as ours is that areas Pacific, especially southeast of New Zealand, and for with no data generally have small standard devia- PKPab, west of South America. These areas usually tions. In these regions the standard deviations thus correspond to regions where data coverage are sparse. give misleading sense of a satisfactory result, whereas the solution in these areas is, in fact, controlled by 5.3. Differences between PcP and the regularization imposed on the inversions. There- PKP/PKKP results fore, the CMB cells that have no bounce points are shown in light green in the map. PcP in Fig. 9a There are few common anomalies in PKP/PKKP shows small uncertainty in general. PKKP shows and PcP maps. Moreover, the model produced by the larger uncertainties in East Asia and South Pacific. combined PKP and PKKP data actually increases the The PKP maps have larger uncertainties in the South variance of PcP by almost 21% (Table 1). The poor
E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 39 correlation between maps inferred from core reflected 5.4. CMB topography versus D velocity and refracted data was first noted by Rodgers and heterogeneity Wahr (1993), observed by Boschi and Dziewonski (2000), and is reminiscent of the discrepancy between The differences between PcP and PKP/PKKP as core refracted PKP and diffracted Pdiff data noted by well as the large amplitude of inferred CMB topogra- KH2001. There are several potential explanations of phy suggest that travel-time anomalies originate not the problem, including (1) data noise, in particular PcP only from topography. They can be a combined effect data; (2) differences in data sampling at the CMB; of CMB topography, volumetric heterogeneity and (3) source–receiver ambiguity for PKP and PKKP; (4) anisotropy within the D and overlying mantle, and strong, short wave length heterogeneity within the D outer core heterogeneity (Boschi and Dziewonski, region; (5) mantle anisotropy, and (6) outer-core struc- 2000). There is no convincing evidence from seis- ture. mology for outer-core heterogeneity, and geodynamic The noise level in the PcP data is high (Fig. 4a), and magnetohydrodynamic modeling argues strongly even after reprocessing by Engdahl et al. (1998), against the presence of detectable seismic hetero- which certainly complicates the interpretation through geneity in the vigorously convecting outer-core (e.g. inversion. As mentioned above, we reduced the sys- Stevenson, 1987). Measurable seismic anisotropy is tematic errors in the data (i.e. misidentification) by believed to be present in the mantle, especially in the windowing out crossovers and imposing a cut-off in upper mantle and in the D layer (Kendall and Silver, residual times. Furthermore, we formed summary rays 1996; Wysession et al., 1998; Matzel et al., 1996), to reduce random errors in the data. However, these but here we did not account for anisotropy in the D data processing techniques are far from perfect to region. Inner core heterogeneity and anisotropy can suppress data noise. Morelli and Dziewonski (1987), only effect CMB maps inferred from the PKPdf data. Rodgers and Wahr (1993), and Garcia and Souriau The use of core diffracted and refracted data in the (2000) performed stochastic data analyses and ob- construction of model KH2001 has enhanced struc- tained larger random component of the data variance tures near the base of the mantle compared to mod- than the coherent component, by a factor between els constructed without core phases, but the P-wave two and seven. To reassure that the data variance velocity perturbations remain fairly small (±0.8%). reduction after inversion was not a result of fitting Studies of the lowermost mantle suggest a highly het- random data set, we perform a bootstrap analysis by erogeneous D layer with local wavespeed variations randomly assigning travel-time residuals to ray paths, in the lowermost mantle that far exceed the detec- then inverted them. We repeated this process for 100 tion level of global tomography (e.g. Garnero, 2000). times. These inversions of random data sets yielded Thus, KH2001 does probably not account fully for the 3.7–5.2% of variance reduction, which is significantly volumetric heterogeneity in the lowermost mantle and lower than the reductions obtained for the real data we, therefore, add extra free parameters to account for (Table 1) suggesting that we can extract structural the remaining trade-off between volumetric structures signal. and topography. For this set of calculations, we cor- Core-refracted phases (PKP and PKKP) pierce the rected for contributions of 3-D mantle structure out- CMB twice, and one has to estimate two variables side a 290 km thick basal layer and inverted the PcP, (topography at two locations) from one travel-time PKP, and PKKP data both for CMB topography and residual. With no additional information, least-squares for lateral wavespeed variations within the basal layer. minimization will just yield the average of the con- Combining all data in a joint inversion gives addi- tribution from the travel-time residual and assign tional constraints to sort out the relative contributions them equally to the two locations on the CMB. This of volumetric heterogeneities within the D and the source–receiver ambiguity becomes a problem when CMB topography. The combined data almost achieves data are few and coverage is uneven. This may ex- global coverage. plain some differences in CMB maps but inverting all A major concern regarding the mapping of CMB the PcP, PKP and PKKP data jointly for CMB topog- topography is whether or not we can resolve trade-offs raphy alone did not fully remove the discrepancies. between CMB topography and D heterogeneity using
40 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 Fig. 10. Checkerboard test with an input model of ±2.5 km CMB topography inside cells of unit area of 30◦ × 30◦ : (a) results for CMB topography, (b) results for D heterogeneities. Checkerboard test with an input model of ±5% D heterogeneities inside cells of unit area of 30◦ × 30◦ : (c) results for CMB topography, (d) results for D heterogeneities. In the second test, the presence of CMB topography signal is still smaller in most areas than that in the first test, despite the strong perturbation (±5%) on the D heterogeneities.
E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 41 the currently available dataset. Through checkerboard topography and D heterogeneities. The results of tests GS2000 demonstrated that they could not resolve these inversions are shown in Fig. 10. Although we the trade-offs between them, using a dataset consisted can see that there is substantial mutual contamination of PcP, PKPbc, and PKKPbc. Here, we perform simi- between the signal of CMB topography and the signal lar checkerboard tests to see if with our larger dataset of D heterogeneity, there are certain areas that we can and additional phases we can get a more encourag- resolve their relative contributions, i.e. Central Asia, ing result. In our first test, we assigned 30◦ × 30◦ Middle East, Indonesia, Australia, Central and South topographic checkerboard anomalies of ±2.5 km at America. The spurious CMB topography signal due to the CMB, and in another test we used 30◦ × 30◦ incorrect mapping of velocity heterogeneity (Fig. 10e) velocity checkerboard heterogeneities of ±2.5% at is smaller than the signal due to CMB topography the D basal layer in the second test. Synthetic travel itself (Fig. 10b). We applied stronger regularization times were calculated for all the relevant phases and and stronger damping to the D velocity component ray paths. Random noise was added so that the syn- than that of CMB topography. The dependence on thetic data have the same standard deviations as the damping of rms amplitudes of CMB topography and real dataset. Then, they are inverted for both CMB D heterogeneity is shown in Fig. 11. The inverted Fig. 11. Damping coefficient of the basal layer velocity vs. rms amplitude CMB topography and rms amplitude of heterogeneities in the basal layer. The amplitude of the degree 2 zonal harmonic component topography is shown by the dash line. The arrow indicates the value of damping coefficient used to obtain our final model, and it gives a value of excess ellipticity that is consistent with the study of Mathews et al. (2002).
42 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 Fig. 12. Joint tomography ( PcP, PKPab, PKPbc, PKPdf, PKKPab and PKKPbc) model of CMB: (a) CMB undulation, (b) velocity perturbation of the basal layer, and (c) bootstrap error of the CMB undulation model, (d) bootstrap error of the basal layer heterogeneities. C20 component is stable and has values between 0.31 the signal originated from CMB topography and D and 0.55 km. For a very large damping coefficient (i.e. heterogeneities can be resolved. 1000), we obtained a 0.84 km rms CMB topography, Bootstrap errors of the joint inversion are generally zero rms heterogeneity, and a 0.34 km C20 component. small both for the CMB topography and D velocity Fig. 12a and b show the results of inversion of the heterogeneities (Fig. 12c and d), but relatively large EHB PcP, PKP, PKKP data corrected for 3-D mantle errors occurred under South Pacific, Indian Ocean, and structures and inner core anisotropy. Notice that we North Atlantic where there are still relatively few data modified the color scale compared to Fig. 8 to account points. for the reduced magnitude of CMB topography yielded by this joint inversion. The spherical averages of the travel time residuals, which correspond to a decrease 6. Discussion of core radius that between 0.85 and 3.44 km, are again subtracted from the data. We infer from Fig. 12a that The amplitudes of the topography derived from the core radii are larger than average under Australia, the separate data sets are all relatively large com- Indonesia, and Southern Africa. Relatively small core pared to independent, geophysical results. Hager radii are inferred beneath Central and North Amer- et al. (1985) spherical harmonics for degrees two ica, and Central Eurasia. The checkerboard tests sug- to three model of CMB dynamic topography has a gest that in these areas the relative contributions of 1.5 km peak-to-peak amplitude. Inversions based on
E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 43 Fig. 13. (a) Our CMB topography model derived from joint tomography, expressed by the spherical harmonics up to degree 4; (b) A model of the CMB topography inferred from seismic tomography incorporating velocity heterogeneity in the D layer above the CMB (Gudmundsson, 1990), expressed also up to spherical harmonic degrees 4.
44 E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 Table 4 The spherical harmonic expansion of the final CMB topography model up to degree 4 (standard geophysical normalization) l/m −4 −3 −2 −1 0 1 2 3 4 0 0.1804 1 −0.0870 −0.0197 −0.0722 2 −0.0657 −0.0648 −0.1165 0.0719 −0.0648 3 0.0302 0.0332 −0.0270 0.1774 0.0862 0.01097 0.0873 4 0.0440 0.0050 −0.0094 0.0450 −0.0318 −0.0162 0.0811 0.0400 0.0361 geodetic measurements (Bowin, 1986) constrain the damping (Fig. 11), but the FCN result is close to the peak-to-peak CMB topography to be less than 3 km. inverted C20 component for a wide range of damping This suggests that large signal in the observed travel parameters we explored. time anomalies is due not only to topography but Expansion into spherical harmonics (Table 4) shows also to volumetric heterogeneities in the D region. that the long wavelength pattern (l ≤ 4) of our model The large amplitude inferred from PKPdf may be due (Fig. 13a) agrees well with the result of Gudmundsson to insufficient correction of inner-core structure (e.g. (1990) (Fig. 13b), with a correlation coefficient of anisotropy). 0.64. The 3 km peak-to-peak CMB topography is also We compared our results with the independent con- consistent with the stochastic analysis performed by straint on CMB topography provided by geodetic ob- Garcia and Souriau (2000), in which they conclude servations. The free-core nutation (FCN) is caused by that 95% of the CMB topogarphy is smaller than a misalignment of the rotation axis of the fluid core ±4 km at wavelengths larger than 5◦ . Bowin (1986) and of the mantle. It has retrograde motion and a long calculated that CMB topography must be less than period (∼432 days) in the celestial frame. Very long 3 km if geoid anomalies arise solely from CMB topog- baseline interferometry (VLBI) gives the most accu- raphy. From analysis of PcP travel-time residuals Neu- rate measurements and data collected by this method berg and Wahr (1981) concluded that at wavelengths of have been used to infer the dynamical ellipticity of the 50–400 km, which is well below the length scale that CMB (Herring et al., 1986; Gwinn et al., 1986). Gwinn can be resolved by our data, CMB topography is less et al. (1986) explained the deviation of the VLBI mea- than 3 km. The velocity variations obtained from our surement of the FCN from its theoretical predicted pe- joint inversion (Fig. 12b) are on the order of ±4.5%. riod (Wahr, 1981) by suggesting that the CMB has a Effectively, the velocity perturbation in the basal layer second zonal harmonic deviation from the hydrostatic serves as a “dustbin” for the inversion to absorb signal equilibrium ellipticity, with peak-to-peak amplitude of due to volumetric wavespeed variations. Nevertheless, 490 ± 110 m. Mathews et al. (2002) accounts for mag- the resultant map shows features that are similar to netic coupling at the CMB and the outer and inner the P-wave mantle velocity model of KH2001, which core boundary, and their new results have the flatten- suggests that we can resolve volumetric heterogene- ing of the CMB of 380 m larger than the hydrostatic ity and CMB topography separately in certain regions value. Initially we did not use this as a constraint but that have dense data coverage of both core-reflected spectral analysis of the results of our joint inversion PcP and core-refracted phases. for topography and volumetric wavespeed yielded a value of approximately 0.2 km of the degree 2 zonal component. After finding that our initial inversion re- 7. Conclusions sult agrees well with these independent constraints, we used Mathews et al. (2002) result to adjust the value We used 57,122 PcP, 69,767 PKPab, 365,379 of the damping coefficient on the basal layer veloc- PKPbc, 335,530 PKPdf, 2575 PKKPab and 8996 ity to obtain the final CMB topography model. The PKKPbc travel time residuals to invert for long- magnitude of the degree two zonal harmonic compo- wavelength CMB topography. Inversion of the sepa- nent as inferred from our inversion is dependent on rate PcP, PKPab, PKPbc, PKPdf, and PKKP data sets
E.K.M. Sze, R.D. van der Hilst / Physics of the Earth and Planetary Interiors 135 (2003) 27–46 45
yielded large peak-to-peak values for topography as topography model of Gudmundsson (1990) (corre-
well as inconsistencies between PcP and PKP mod- lation coefficient = 0.64). Smaller CMB radii were
els. This concurs with earlier studies (Rodgers and inferred beneath the Central and North America and
Wahr, 1993; Obayashi and Fukao, 1997; Boschi and Central Asia, and larger CMB radii below Indonesia,
Dziewonski, 2000). From uncertainty estimates of the Australia, and Southern Africa.
inferred models we conclude that it is unlikely that
the systematic differences between the core-reflected
and core-refracted phases can all be explained by data Acknowledgements
noise. Uneven data coverage on the CMB remains an
issue, however, as well as the source–receiver ambi- We thank Hrafnkell Kárason for running ray-tracing
guity for PKP and PKKP phases. The improved data program through his P-wave tomographic model and
coverage by the combined set of PKP, PKKP, and PcP helpful discussions, and two anonymous reviewers
reduces the source–receiver ambiguity, but inverting for their constructive comments which have helped
the data jointly for CMB topography did not remove to improve the paper. This work was supported
the discrepancies. This suggests that the data cannot be by the National Science Foundation under grant
explained by CMB topography alone. There may well NSF-EAR9909492.
be other reasons for the inferred discrepancy, but to ac-
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