SPRD6 (Ishii & Tromp, 1999, 2001)

Using a large collection of free-oscillation data and additional constraints imposed by the free-air gravity anomaly, lateral variations in shear velocity, compressional velocity, and density within the mantle, dynamic topography on the free surface, and topography on the 660-km discontinuity and the core-mantle boundary are determined. The velocity models are consistent with existing models based upon travel-time and waveform inversions. In the lowermost mantle, near the core-mantle boundary, denser than average material is found beneath regions of upwellings centered on the Pacific Ocean and Africa characterized by slow shear velocities. These anomalies suggest the existence of compositional heterogeneity near the core-mantle boundary.

SPRD6 Model Coefficients

SPRD6 contains 3D models of shear-wave velocity, compressional-wave velocity, and density within the mantle and 2D models of topography on the free-surface (dynamic), the 660-km discontinuity, and the core-mantle boundary. The 3D models are parameterised in terms of spherical harmonics laterally (up to degree 6) and Chebyshev polynomials radially (up to order 13).
There are two spherical harmonic conventions used. One is the fully normalised "complex" spherical harmonics (Edmonds, 1960) and the other is the "real" spherical harmonics in which most Harvard 3D model coefficients are given (e.g., Su & Dziewonski, 1997). Model coefficients are available in both conventions.
The model coefficients are given in relative perturbations (delta x/x).
For convenience, there is also a tar file (plot_sprd6.tar) which contains files of model coefficients, FORTRAN programs and a README file.

The files (except for those containing real spherical harmonic coefficients of 3-D models) contain

k l m model (real/cosine) model (imaginary/sine)
where k is the order of the Chebyshev polynomial, l is the spherical angular degree and m is the angular order. For 2-D models, the k values are not included.

FORTRAN Programs

There are 5 FORTRAN source codes (which have to be compiled once they are transferred to your machine).

  • plot.f
    This program reads a model (either 3D or 2D) and outputs a file (with specified name) which contains longitude, latitude and value of the model. For 3D models, the values are in percentage, and for 2D models, the values are in kilometres. The increment of latitude/longitude can be changed by changing the value of parameter "block_size".
    In order to run this program, one needs all the following subroutines and also 3D models (in real spherical harmonics - sprd6_s, sprd6_p, and sprd6_r) and 2D models (in complex spherical harmonics - sprd6_fs.XCS, sprd6_d660.XCS, and sprd6_dcmb.XCS).
  • sub_chebyshev.f
    This subroutine calculates the values of Chebyshev polynomials at a given depth (within the mantle) with appropriate normalisation as discussed in Su (1992).
  • sub_kernel.f
    This subroutine builds a matrix that contains values of (real) spherical harmonics at each latitude and longitude.
  • sub_readModel.f
    This subroutine reads a file containing model coefficients and converts them (if necessary) into appropriate spherical harmonics convention to be consistent with spherical harmonic convention used to create the matrix by the subprogram sub_kernel.f.
  • sub_switchLatLon.f
    This subroutine switches latitude and longitude of the data vector so that the output file (from plot.f) will contain values of longitude, latitude and model.
  • Makefile
    If one prefers, download this file and type
    make plot
    to compile plot.f with its subroutines (all the FORTRAN codes must be in the same directory). Then, typing
    will run the program plot.f.

Supplementary Material for Ishii & Tromp (1999, 2001)

  • Edmonds, A.R., 1960.
    Angular Momentum in Quantum Mechanics, Princeton Univ. Press, Princeton, N.J.
  • Ishii, M., and Tromp, J., 1999.
    Normal-mode and free-air gravity constraints on lateral variations in velocity and density of Earth's mantle. Science 285, 1231-1236.
  • Ishii, M., and Tromp, J., 2001.
    Even-degree lateral variations in the Earth's mantle constrained by free oscillations and the free-air gravity anomaly. Geophys. J. Int., 145, 77-96.
  • Su, W.-J., 1992.
    The Three-Dimensional Shear-Wave Velocity Structure of the Earth's Mantle. Ph.D. Thesis, Harvard Univ.
  • Su, W.-J., and Dziewoński, A.M., 1997.
    Simultaneous inversion for 3-D variations in shear and bulk velocity in the mantle. Phys. Earth Plnet. Inter. 100, 135-156.

Back to Downloads (Three-Dimensional Mantle Models)

Department of Earth and Planetary Sciences / Harvard University / 20 Oxford Street / Cambridge / MA 02138 / U.S.A. / Telephone: +1 617 495 2350 / Fax: +1 617 496 1907 / Email: reilly@eps.hartvard.edu