Inverse torsional eigenvalue problems

Technical Report · Wed Jun 01 04:00:00 EDT 1983

OSTI ID:5518640

Willis, C

We undertake a numerical and theoretical investigation of the inverse problem for the reconstruction of the density rho and S-wave velocity ..beta.. of the Earth from its torsional oscillations. We assume a spherically symmetric, non-rotating Earth which consists of a perfect elastic, isotropic material and transform the differential equation governing the torsional oscillations to a Sturm-Liouville problem. We present a numerical method for determining rho and ..beta.. in the upper mantle when rho and ..beta.. are smooth functions of radius. The method, based on the Rayleigh-Ritz method, solves iteratively for the coefficients of a generalized Fourier series for the potential. We reconstruct several earth models to 2% accuracy. However, the method is sensitive to error in the data. This is not true of the inversion for the density alone and suggests that the simultaneous inversion for the density and velocity from free oscillation data may be unstable. The smoothness assumption is a serious limitation of our numerical method, since most earth models have a discontinuity at the crust and many have gradients with discontinuities in the upper mantle. We study the associated discontinuous Sturm-Liouville problem and prove that if the eigenfunctions have two discontinuities and if the potential is known in half the interval then the potential in the whole interval is uniquely determined from one spectrum. We apply this theorem to the discontinuous earth model to prove that given rho in the lower mantle and ..beta.. in the mantle and crust, then the torsional spectra of one angular order uniquely determine rho in the upper mantle. In addition, if ..beta.. is known only in the lower mantle, then two torsional spectra uniquely determine both rho and ..beta.. in the upper mantle.

• Research Organization: Lawrence Berkeley Lab., CA (USA)
• DOE Contract Number: AC03-76SF00098
• OSTI ID: 5518640
• Report Number(s): LBL-16692; ON: DE84002568
• Country of Publication: United States
• Language: English

Related Subjects

58 GEOSCIENCES
580201* -- Geophysics-- Seismology & Tectonics-- (1980-1989)
DATA
DENSITY
DIFFERENTIAL EQUATIONS
EARTH CRUST
EARTH MANTLE
EARTH PLANET
EIGENFUNCTIONS
EIGENVALUES
EQUATIONS
FUNCTIONS
INFORMATION
MATHEMATICAL MODELS
NUMERICAL DATA
NUMERICAL SOLUTION
OSCILLATIONS
PHYSICAL PROPERTIES
PLANETS
SEISMIC P WAVES
SEISMIC S WAVES
SEISMIC WAVES
STURM-LIOUVILLE EQUATION
THEORETICAL DATA
TORSION
VELOCITY
WAVE PROPAGATION