From Seafloor Deformation to Sea-Surface Displacement: Tsunami Generation Over Variable Bathymetry

Tsunami source models commonly convert seafloor deformation into an initial sea-surface displacement with the classical Kajiura filter, which is exact for an ocean of uniform depth. Yet earthquake source regions often span trenches and continental slopes where water depth varies on the same scales as the deformation. We derive the corresponding transfer for arbitrary smooth, submerged, single-valued three-dimensional bathymetry. Seafloor motion enters through the bottom-normal forcing $q=U_z+ abla h\cdot\Uh$, and the exact rapid-source response is $\eta=\opC q$ with $\opC=\opB^{-1}$, a nonlocal operator determined by the actual bathymetry. This is not obtained by simply replacing $H$ with $h(\xh)$ in $\sech(KH)$: slope enters the exact bottom boundary condition, and spatially varying bathymetry couples the response globally. Exact wedge and three-dimensional seamount tests reveal a clear hierarchy. Local water depth controls the dominant attenuation and broadening, gentle slopes add a mainly phase-like correction, and surrounding geometry produces a smaller but genuinely nonlocal redistribution that local formulas cannot always improve. A 2011 Tohoku-Oki application shows the same hierarchy for realistic bathymetry and demonstrates that the full solution restores global volume consistency. The operator also appears in $\eta_{tt}+g\opG\eta=\opC q_{tt}$, making generalized Kajiura transfer the rapid-source limit of a single time-dependent generation framework rather than an independent initialization rule.