Glacial-Isostatic Adjustment—II. The Inverse Problem

Summary The principle of correspondence is invoked to determine Laplace transform domain solutions to the surface mass loading problem for radially stratified visco-elastic (Maxwell) spheroids. These Laplace transform domain solutions are expressed in terms of visco-elastic analogues of the conventional surface load Love numbers of elasticity. These visco-elastic Love numbers may be approximately transformed to the time domain using an extremal technique. Application of this technique shows that the Love number time histories may be well approximated by the linear superposition of a discrete set of purely exponential relaxations plus a constant term. Alternatively the discrete spectrum of relaxation times involved in the synthesis of each Love number time history may be found exactly by solving the associated homogeneous problem. Such solutions determine the set of eigen-decay times associated with the normal modes of viscous gravitational relaxation of the visco-elastic planetary model. The solution of the inhomogeneous problem may be expressed in the form of a normal mode expansion. This normal mode expansion is employed as the basis for the construction of a rigorous first-order perturbation theory for the inference of the viscosity of the deep interior of the planet. A variational principle is derived which determines to first order that shift in position of a free decay pole in the relaxation spectrum which is forced by the addition of a radially-distributed perturbation of viscosity to the starting model. This determines the differential kernels required for the solution of the inverse problem. The uniqueness of the state of isostatic equilibrium for the viscously incompressible Maxwell models employed in this analysis is demonstrated and this uniqueness allows us to calculate the shift in the residue which is associated with the shift in position of a free decay pole for the inhomogeneous problem. The inhomogeneous problem is thus closed to first order. A formula is derived for the differential kernels appropriate to the inhomogeneous problem. The time domain form of these kernels may be calculated analytically. The structure of the full inverse theory is sufficiently simple that it may be employed to rigorously test the compatibility of the simple Maxwell model with the observed relaxation of the Earth's shape which accompanied deglaciation at the end of the last ice age.

[1]  G. Darwin.  On the Bodily Tides of Viscous and Semi-Elastic Spheroids, and on the Ocean Tides upon a Yielding Nucleus , 1879 .

[2]  N. A. Haskell.  The Motion of a Viscous Fluid Under a Surface Load , 1935 .

[3]  N. A. Haskell.  The Motion of a Viscous Fluid Under a Surface Load. Part II , 1936 .

[4]  N. A. Haskell.  The Viscosity of the Asthenosphere , 1937 .

[5]  Z. Alterman,et al.  Propagation of Rayleigh Waves in the Earth , 1937 .

[6]  Conyers Herring,et al.  Diffusional Viscosity of a Polycrystalline Solid , 1950 .

[7]  W. Heiskanen,et al.  The Earth And Its Gravity Field , 1959 .

[8]  Walter Munk,et al.  The rotation of the earth , 1960 .

[9]  M. Crittenden.  Effective viscosity of the Earth derived from isostatic loading of Pleistocene Lake Bonneville , 1963 .

[10]  R. McConnell,et al.  Isostatic adjustment in a layered Earth , 1965 .

[11]  G. Backus,et al.  Numerical Applications of a Formalism for Geophysical Inverse Problems , 1967 .

[12]  G. Backus.  Converting Vector and Tensor Equations to Scalar Equations in Spherical Coordinates , 1967 .

[13]  R. McConnell,et al.  Viscosity of the mantle from relaxation time spectra of isostatic adjustment , 1968 .

[14]  R. Dicke.  Average acceleration of the Earth's rotation and the viscosity of the deep mantle , 1969 .

[15]  R. O’Connell.  Pleistocene Glaciation and the Viscosity of the Lower Mantle , 1971 .

[16]  A. Toomre,et al.  Some remarks on polar wandering , 1969 .

[17]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[18]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[19]  J. Weertman.  The creep strength of the Earth's mantle , 1970 .

[20]  F. Gilbert.  Excitation of the Normal Modes of the Earth by Earthquake Sources , 1971 .

[21]  W. Farrell.  Deformation of the Earth by surface loads , 1972 .

[22]  R. Walcott.  Late Quaternary vertical movements in eastern North America: Quantitative evidence of glacio-isostatic rebound , 1972 .

[23]  W. Paterson.  Laurentide Ice Sheet: Estimated volumes during Late Wisconsin , 1972 .

[24]  W. Peltier.  Penetrative convection in the planetary mantle , 1973 .

[25]  P. C. Luh.  The Normal Modes of the Rotating, Self-Gravitating Inhomogeneous Earth. , 1974 .

[26]  W. Peltier.  The impulse response of a Maxwell Earth , 1974 .

[27]  C. Brennen.  Isostatic recovery and the strain rate dependent viscosity of the Earth's mantle , 1974 .

[28]  L. Cathles,et al.  The Viscosity of the Earth's Mantle , 1975 .

[29]  F. Gilbert,et al.  An application of normal mode theory to the retrieval of structural parameters and source mechanisms from seismic spectra , 1975, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[30]  On Postglacial Sea Level , 2007 .

[31]  W. Peltier,et al.  Glacial-Isostatic Adjustment—I. The Forward Problem , 2007 .