The dislocation theory: a consistent way for including the gravity effect in (visco)elastic plane‐earth models

SUMMARY A homogeneous or stratified viscoelastic half‐space has often been used as a simple earth model to calculate the deformation induced by an earthquake or a regional deglaciation. Recent publications have shown that, when taking into account the coupling between the earth's gravity and the dilatational motion, the response of the model exhibits singularities for wavenumbers below a critical value ( k g ). It is shown here that the boundary‐value problems based on such earth models have no solution at all for this low‐wavenumber region ( k ≤ k g ) . In fact, previous authors tried unintentionally to solve an unsolvable system by analytical or numerical extrapolation from the valid wavenumber range ( k > k g ) to the invalid range ( k ≤ k g ) . Therefore, non‐physical singularities appear in their numerical results. In particular, the previous elastic‐gravitational dislocation theory includes an incorrect treatment of these singularities, which may lead to significant errors in the deformation models. This paper analyses the mathematical and physical causes for the unsolvability and proposes a consistent way to solve the difficulty. In particular, numerical tests show that an elastic‐gravitational deformation model would become even less reliable than the corresponding elastic model, if the coupling between gravity and deformation were considered in the incorrect way published previously.