A PDE-constrained optimization method for tsunami source inversion from surface current measurements

Sudden vertical deformation of the seafloor during an earthquake is the main cause of tsunamis. Besides the generation of long waves, the perturbation of the water column induces currents that carry information about the underlying deformation. While tsunami source inversions commonly rely on seismic, GNSS, tide gauges, deep-ocean pressure sensors among other sources of data, surface currents have only recently been proposed as a complementary and potentially noise-robust data source. Existing current-based inversions, however, typically rely on restrictive assumptions such as flat bathymetry, absence of background currents or waves.

We present a PDE-constrained optimal control framework for tsunami source inversion that estimates both the initial surface elevation and the vertical seafloor deformation from time series of surface velocity fields. The governing equations are the non-linear Shallow Water Equations with spatially variable bathymetry, Coriolis forcing and optional background flows, such as tidal or wind-driven currents. The inverse problem is formulated as a regularized optimization constrained by the PDE, and can incorporate spatially and temporally variable sensor coverage, measurement errors and noise through a flexible observation operator acting on a virtual sensor array. The methodology can accommodate joint inversions to combine surface current measurements with sea-level or seafloor observations, such as tide-gauges or ocean-bottom pressure sensors.

We test the method using synthetic deformation fields over a range of bathymetric configurations, from simple idealized profiles to realistic bathymetry, and for different sensor distributions, types and sampling intervals. Background currents and different uncertainty levels are included to assess the robustness of the source inversions. Finally, the Mw 8.8 Maule 2010 event is used as a benchmark to test the methodology under a realistic coseismic deformation pattern.

Our results show that, even with sparsely distributed surface current measurements, the method can recover the main features of the tsunami source and initial surface height distribution. Within the region covered by current measurements, the spatial resolution is approximately uniform in both along-strike and along-dip directions and is mainly affected by the sensor coverage density rather than other factors, such as bathymetry, showing that even sparse, non-uniformly distributed networks may be adequate for estimating the source of tsunami events, with near homogeneous resolution above the ruptured area.  The addition of  tide-gauge or pressure sensor records significantly improves the reconstruction of complex source geometries, particularly near the shoreline and when current measurements are sparse, spatially non-uniformly distributed or strongly clustered. The inversion is robust to measurement noise and it exhibits low sensitivity to bathymetric complexity. Time-varying and incomplete sensor networks, represented in our methodology tests by random and systematic sensor dropout, degrade only moderately the solution as long as sufficient sensor coverage is maintained. Resolution is mostly homogeneous, increasing with sensor density, faster sampling rates, and the inclusion of complementary sea-level or ocean-bottom data.