  We mathematically derive the sensitivity of data assimilation results for
tsunami modelling, to perturbations in the observation operator. We consider
results of variational data assimilation schemes on the one dimensional shallow
water equations for (i) initial condition reconstruction, and (ii) bathymetry
detection as presented in Kevlahan et al. (2019, 2020). We use variational
methods to derive the Hessian of a cost function J representing the error
between forecast solutions and observations. Using this Hessian representation
and methods outlined by Shutyaev et al. (2017, 2018), we mathematically derive
the sensitivity of arbitrary response functions to perturbations in
observations for case (i) and (ii) respectively. Such analyses potentially
substantiate results from earlier work, on sufficient conditions for
convergence, and sensitivity of the propagating surface wave to errors in
bathymetry reconstruction. Such sensitivity analyses would illustrate whether
particular elements of the observation network are more critical than others,
and help minimise extraneous costs for observation collection, and efficiency
of predictive models.
