  The abundance of observed data in recent years has increased the number of
statistical augmentations to complex models across science and engineering. By
augmentation we mean coherent statistical methods that incorporate measurements
upon arrival and adjust the model accordingly. However, in this research area
methodological developments tend to be central, with important assessments of
model fidelity often taking second place. Recently, the statistical finite
element method (statFEM) has been posited as a potential solution to the
problem of model misspecification when the data are believed to be generated
from an underlying partial differential equation system. Bayes nonlinear
filtering permits data driven finite element discretised solutions that are
updated to give a posterior distribution which quantifies the uncertainty over
model solutions. The statFEM has shown great promise in systems subject to mild
misspecification but its ability to handle scenarios of severe model
misspecification has not yet been presented. In this paper we fill this gap,
studying statFEM in the context of shallow water equations chosen for their
oceanographic relevance. By deliberately misspecifying the governing equations,
via linearisation, viscosity, and bathymetry, we systematically analyse
misspecification through studying how the resultant approximate posterior
distribution is affected, under additional regimes of decreasing spatiotemporal
observational frequency. Results show that statFEM performs well with
reasonable accuracy, as measured by theoretically sound proper scoring rules.
