Accurate prediction of tsunami waves requires complete boundary and initial condition
data, coupled with the appropriate mathematical model. However, necessary
data is often missing or inaccurate, and may not have sufficient resolution
to capture the dynamics of such nonlinear waves accurately. In this thesis we
demonstrate that variational data assimilation for the continuous shallow water
equations (SWE) is a feasible approach for recovering both initial conditions and
bathymetry data from sparse observations. Using a Sadourny finite-difference finite
volume discretisation for our numerical implementation, we show that convergence
to true initial conditions can be achieved for sparse observations arranged in multiple
configurations, for both isotropic and anisotropic initial conditions, and with
realistic bathymetry data in two dimensions. We demonstrate that for the 1-D
SWE, convergence to exact bathymetry is improved by including a low-pass filter
in the data assimilation algorithm designed to remove scale-scale noise, and with
a larger number of observations. A necessary condition for a relative L2 error less
than 10% in bathymetry reconstruction is that the amplitude of the initial conditions
be less than 1% of the bathymetry height. We perform Second Order Adjoint
Sensitivity Analysis and Global Sensitivity Analysis to comprehensively assess the
sensitivity of the surface wave to errors in the bathymetry and perturbations in
the observations. By demonstrating low sensitivity of the surface wave to the reconstruction
error, we found that reconstructing the bathymetry with a relative
error of about 10% is sufficiently accurate for surface wave modelling in most cases.
These idealised results with simplified 2-D and 1-D geometry are intended to be
a first step towards more physically realistic settings, and can be used in tsunami
modelling to (i) maximise accuracy of tsunami prediction through sufficiently accurate
reconstruction of the necessary data, (ii) attain a priori knowledge of how
different bathymetry and initial conditions can affect the surface wave error, and
(iii) provide insight on how these can be mitigated through optimal configuration
of the observations. / Thesis / Candidate in Philosophy
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/25964 |
| Date | January 2020 |
| Creators | Khan, Ramsha |
| Contributors | Kevlahan, Nicholas, Computational Engineering and Science |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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