A Hybrid Normal Mode‐Collocation Method for Finding the Response of Laterally Homogeneous Compressible Maxwell Viscoelastic Earth Models

Abstract

AbstractNormal mode analysis is a Laplace‐transform method for calculating the surface‐loading response of laterally homogeneous spherical Earth models with linear viscoelasticity which delivers modal decay times and amplitudes. It can locally fail owing to numerical singularities arising from the viscoelastic parameters, leading to an incomplete accounting of the surface‐loading response. Collocation methods were developed to circumvent this issue. The mixed collocation method includes least‐squares fitting to the Laplace‐transformed Earth response to determine amplitudes assuming the normal mode decay times are known, while the pure collocation method assumes a series of logarithmically regularly spaced inverse decay times for which amplitudes are determined numerically. Both collocation methods may determine amplitudes that are physically unrealistic and all three methods produce crustal motion predictions that differ significantly. The hybrid normal mode‐collocation method presented here applies the normal mode analysis, and then applies the pure collocation to the resulting residuals. This retains the modal structure, while providing an improved fit. Our implementation avoids numerical singularities that may arise from Rayleigh‐Taylor instabilities occurring at large times and can be automated. Vertical crustal motions predicted by the hybrid method for North America with the ICE‐6G_C loading model and the VM5a viscosity structure have a root mean square (RMS) of 4.49 mm/yr and RMS differences with the normal mode, pure, and mixed collocation method of 0.06, 0.23, and 0.25 mm/yr, respectively. Maximum differences reach 0.20, 0.87, and 0.63 mm/yr. The differences increase for a viscosity profile with a greater viscosity increase with depth that exhibits stronger singularity issues.