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Normal-Mode and Free-Air Gravity Constraints on Lateral Variations In R ESEARCH A RTICLES ficient quality. There are density models based on current and past plate motions in which the Normal-Mode and Free-Air sources of heterogeneity are cold subducted slabs (5), and an upper mantle density model, Gravity Constraints on Lateral containing strong high-degree components, has been determined with surface wave data (6). Variations in Velocity and A density model of the mantle is im- portant to geophysics because mantle-flow calculations cannot be performed without Density of Earth’s Mantle it (7, 8). However, a whole-mantle den- Miaki Ishii* and Jeroen Tromp sity model is usually obtained by scaling an S velocity model, which is justifiable With the use of a large collection of free-oscillation data and additional con- if the heterogeneity has a single cause straints imposed by the free-air gravity anomaly, lateral variations in shear that affects both, for example, variations in velocity, compressional velocity, and density within the mantle; dynamic to- temperature. pography on the free surface; and topography on the 660-km discontinuity and In recent years, mainly as a result of the 9 the core-mantle boundary were determined. The velocity models are consistent June 1994 Bolivia earthquake, the quantity and with existing models based on travel-time and waveform inversions. In the quality of normal-mode, or free-oscillation, lowermost mantle, near the core-mantle boundary, denser than average ma- data have improved substantially (9), allowing terial is found beneath regions of upwellings centered on the Pacific Ocean and us to constrain the long-wavelength structure of Africa that are characterized by slow shear velocities. These anomalies suggest Earth. We used this data set, combined with the existence of compositional heterogeneity near the core-mantle boundary. further constraints imposed by Earth’s gravity Seismologists have produced numerous shear develop a whole-mantle density model because Department of Earth and Planetary Sciences, Harvard (S) and compressional (P) velocity models of short-period body waves are not directly sensi- University, Cambridge, Massachusetts 02138, USA. the mantle based mainly on body-wave data tive to density and long-period normal-mode *To whom correspondence should be addressed. E- (1–4). Unfortunately, they have been unable to observations have been too scarce and of insuf- mail: [email protected] A 0 Fig. 1. Comparison of the BC S part of SPRD6 and SKS12WM13. (A) Relative perturbations in S velocity at (from the top) 100 (Ϯ 4.5%) 600 (Ϯ 1.0%), 1300 (Ϯ 1.0%), 1800 (Ϯ 0.8%), 2300 (Ϯ 1.0%), 1000 and 2850 (Ϯ 3.0%) km, where the values in paren- theses indicate the scale for the maps at each depth. The S part of mod- Depth (km) el SPRD6 is shown on the left, and model 2000 SKS12WM13 (1) is shown on the right. Blue colors indicate regions of higher than average velocities, and red colors indicate re- gions of slower than aver- age velocities. Coastlines 3000 have been superimposed 0.0 1.0 2.0 0.0 0.5 1.0 for reference. Because the RMS amplitude (%) Correlation maximum harmonic de- gree in SPRD6 is 6, we truncated degree 12 model SKS12WM13 at the same degree. (B) Root-mean- square (RMS) amplitudes of the S part of SPRD6 and SKS12WM13 as a function of depth. The red line is SPRD6, and the black line is SKS12WM13. (C) Correlation between the S model of SPRD6 and SKS12WM13 as a function of depth. For this number of model parameters, two maps are correlated at the 95% confidence level if the correlation is greater than 0.24. www.sciencemag.org SCIENCE VOL 285 20 AUGUST 1999 1231 R ESEARCH A RTICLES ␦␤ ␤ t ␦␣ ␣ t field, to determine lateral variations in S veloc- velocity ( / )s, P velocity ( / )s, density mantle flow, which depend on lateral variations ␦␳ ␳ t ity, P velocity, and density within the mantle, in ( / )s, and topography on discontinuities in density and the relative viscosity of the layers ␦ t addition to dynamic topography on the free ( h/h)s by (12) above and below the discontinuity. Frequently, surface and topography on the 660-km discon- the objective of mantle-flow calculations is to a tinuity (660) and the core-mantle boundary t t t find a viscosity profile and velocity-to-density c ϭ ͵ ͓͑␦␤/␤͒ K␤ ϩ ͑␦␣/␣͒ K␣ (CMB). s s s scaling that reconcile a given velocity model b with the observed gravity anomaly. Here, Theory ϩ ͑␦␳ ␳͒t ϩ ͸͑␦ ͒t we determined lateral variations in density / s K␳]dr h/h s Kd (2) The sensitivity of a normal mode to lateral d and boundary topography simultaneously; heterogeneity is usually visualized in the form The integration over radius r is from the CMB this circumvents the need for a viscosity of a splitting function (10), which represents a with radius b (3480 km) to Earth’s surface with model when fitting the gravity anomaly. local radial average of Earth’s three-dimension- radius a (6371 km), and the summation is over The spherical harmonic coefficients of the t al (3D) heterogeneity. Each mode has its own all discontinuities d. The sensitivity kernels K␤, free-air gravity anomaly, f s (15), are lin- unique splitting function, which at high degrees K␣, K␳, and Kd are given in terms of the eigen- early related to those describing lateral ␦␳ ␳ t corresponds to a surface-wave phase velocity functions of the modes that define the splitting variations in density, ( / ) s, and boundary ␦ t map. An isolated mode of degree l “sees” even- function (12). The number of unknown param- topography, ( h/h) s,by(16) degree structure up to degree 2l, and some eters can be reduced by assuming that lateral clusters of modes have sensitivity to Earth’s variations in P velocity and density are propor- a t ϭ ͑␦␳ ␳͒t Ј ϩ ͸͑␦ ͒t Ј f ͵ / K␳dr h/h K (3) odd-degree heterogeneity (11). A normal-mode tional to lateral variations in S velocity. In the s s s d d splitting function ␴ depends on colatitude ␪ and past, this has been the common approach (12, b ␾ ␦ ␣ ␦ Ј Ј longitude and may be expanded in spherical 13), with typical scaling values of ln / ln The kernels K␳ and Kd describe the sensitiv- t ␤ϭ ␦ ␳ ␦ ␤ϭ harmonics Ys of degree s and order t as (12) 0.55 and ln / ln 0.2 to 0.4 (8, 14). ity of the gravity anomaly to 3D density and s Lateral variations in density, combined with boundary topography, respectively. ␴͑␪ ␾͒ ϭ ͸ ͸ t t ͑␪ ␾͒ , csY s , (1) flow-induced topography on discontinuities, s ϭ 0 tϭϪs determine Earth’s gravity anomaly. Undula- Velocity and Density Models t The splitting-function coefficients cs are lin- tions on boundaries are determined by the den- The splitting functions were corrected for the early related to 3D relative variations in S sity contrast and radial stresses imposed by effects of Earth’s crust with the recent crustal A 0 Fig. 2. Comparison of the P part of SPRD6 and BC P16B30. (A) Relative perturbations in P ve- locity at six discrete depths throughout the mantle, as in Fig. 1. The scale for the maps at 1000 each depth is (from the top) Ϯ2.5, Ϯ0.8, Ϯ0.5, Ϯ0.5, Ϯ0.5, and Ϯ0.7%. The P part of model SPRD6 Depth (km) is shown on the left, and model P16B30 2000 (3), truncated at de- gree 6, is shown on the right. The color scheme is the same as in Fig. 1. (B) RMS am- plitudes of the P part of SPRD6 and P16B30 3000 as a function of depth. 0.0 0.5 1.0 1.5 0.0 0.5 1.0 The red line is SPRD6, RMS amplitude (%) Correlation and the black line is P16B30. (C) Correlation between the P model of SPRD6 and P16B30 as a function of depth. For this number of model parameters, two maps are correlated at the 95% confidence level if the correlation is great- er than 0.24. 1232 20 AUGUST 1999 VOL 285 SCIENCE www.sciencemag.org R ESEARCH A RTICLES model Crust5.1 (17). We assumed an isostati- ic free-surface topography, and topography on normal-mode and gravity data, explains 92% of cally compensated crust in the calculation of the 660 and the CMB (18). Initially, we invert- the variance in the mode data and 96% of the the gravity field. Our “dynamic topography,” ed for topography on the 410-km discontinuity variance in the gravity data, which correspond ␹2 ϭ ␹2 ϭ ϭ therefore, is the nonisostatic topography of (410) as well but found that undulations on this to /N 2.0 and /Nf 0.07, where Nf 27 the free surface that is produced by density boundary are poorly constrained by the data is the number of free-air gravity coefficients. anomalies within the entire mantle, including (19). Therefore, topographic variations on 410 The fit to the free-air gravity anomaly is excel- the lithosphere. are those determined in a recent travel-time lent; however, it is well known that the gravity In the inversion, we allowed for lateral vari- study (20). The starting model for the inversion anomaly can be relatively easily fit because of ations in S velocity, P velocity, density, dynam- included S velocity model SKS12WM13 (1) its highly nonunique dependence on density and P velocity model P16B30 (3). The density and boundary topography. Therefore, we did component of the starting model was obtained not allow free-air gravity to change models of by scaling velocity model SKS12WM13 by a density and boundary topography substantially factor of 0.2.
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