Normal Modes in a Coupled Earth Model: A New Perspective on the Compliance Method

Abstract

To understand the compliance parameters observed for atmospheric pressure waves, we develop a normal-mode approach for a coupled Earth model that includes the atmosphere and the solid Earth. Extending the method of Press and Harkrider to a coupled Earth model, we introduce two new features. First, we present a technique for computing group velocity in a coupled Earth model without relying on the variational principle. Second, we develop a framework for calculating a theoretical compliance parameter by the normal-mode approach, defined as $\eta \left(f\right)=S_Z\left(f\right)/S_P\left(f\right)$, where $f$ is frequency, $S_Z\left(f\right)$ is the power spectral density of vertical displacement, and $S_P\left(f\right)$ is that of surface pressure. Previous studies have modeled crustal deformation as a pressure-loading problem caused by atmospheric pressure. In contrast, the normal-mode approach treats this deformation as intrinsic to the eigenfunctions of modes in a coupled Earth system. We compare the normal-mode and pressure-loading approaches by computing compliance parameters and their depth sensitivity kernels within the solid Earth. The results show close agreement, provided that the phase velocity of the corresponding normal mode is used as the pressure-wave speed in the pressure-loading model. They justify the use of the pressure-loading approach for structural inversion, which offers significantly greater computational efficiency than the normal-mode approach.