Adjoint-based sensitivity analysis for a numerical storm surge model

Highlights

• We calibrate an unstructured-mesh finite element storm surge model of the North Sea

• We use an adjoint approach to perform sensitivity analysis for modelled skew surges

• We compare spatial and temporal patterns of sensitivity with respect to three inputs

• We demonstrate the physical insight available through numerical adjoint methods.

Abstract

Numerical storm surge models are essential to forecasting coastal flood hazard and informing the design of coastal defences. However, such models rely on a variety of inputs, many of which carry uncertainty. An awareness and understanding of the sensitivity of model outputs with respect to those uncertain inputs is therefore essential when interpreting model results. Here, we use an unstructured-mesh numerical coastal ocean model, Thetis, and its adjoint, to perform a sensitivity analysis for a hindcast of the 5th/6th December 2013 North Sea surge event, with respect to the bottom friction coefficient, bathymetry and wind stress forcing. The results reveal spatial and temporal patterns of sensitivity, providing physical insight into the mechanisms of surge generation and propagation. For example, the sensitivity of the skew surge to the bathymetry reveals the protective effect of a sand bank off the UK east coast. The results can also be used to propagate uncertainties through the numerical model; based on estimates of model input uncertainties, we estimate that modelled skew surges carry uncertainties of around 5 cm and 15 cm due to bathymetry and bottom friction, respectively. While these uncertainties are small compared with the typical spread in an ensemble storm surge forecast due to uncertain meteorological inputs, the adjoint-derived model sensitivities can nevertheless be used to inform future model calibration and data acquisition efforts in order to reduce uncertainty. Our results demonstrate the power of adjoint methods to gain insight into a storm surge model, providing information complementary to traditional ensemble uncertainty quantification methods.

Introduction

Storm surge poses a significant hazard for coastal communities worldwide. Allowing for investment in adaptation measures (e.g. rising flood defences), global flood losses in 136 of the world’s largest coastal cities have recently been estimated to rise from US$6 bn per year in 2005 to US$60–63 bn per year in 2050 (Hallegatte et al., 2013). Globally, the increase in extreme sea levels (Stocker et al., 2013) will result in critical flood defence thresholds being reached more frequently and therefore the risk of flooding will increase. The UK is vulnerable to storm surges, particularly along its North Sea coast; a large number of severe storms have impacted the UK in the last century (Haigh et al., 2016), with the two most severe of those events occurring in the North Sea in 1953 and 2013. The approximate economic impacts of the coastal flooding resulting from these events (for year 2014) were £1.2 bn and £0.25 bn respectively; the impact of the latter event was reduced through mitigation action taken after the 1953 event (Wadey et al., 2015). With continued development of the coastal zone in flood risk areas (ASC, 2014), the role of storm surge modelling remains vital.

Essential to the intelligent application of any storm surge model is an understanding of the model’s sensitivity to its uncertain inputs. In a forecast scenario, the greatest model uncertainty arises from the meteorological forcing, namely the surface stress due to wind, and the atmospheric pressure gradient. For this reason, it is common to employ ensemble methods for uncertainty quantification, whereby the surge model is run multiple times, with each run using a different sample from the uncertain distribution of meteorological inputs (Flowerdew et al., 2010). While such ensemble methods provide a practical approach to uncertainty quantification within an operational forecast framework, they provide little insight into the patterns (in space and/or time) of the underlying model sensitivity, and they depend on the choice of meteorological ensemble.

An alternative sensitivity analysis approach is provided by adjoint methods. In the context of numerical modelling, adjoint methods are used to efficiently compute gradients of model outputs with respect to model inputs, which can in principle vary in both space and time. Such methods have been used within a meteorological context since the 1980s (e.g. Hall et al., 1982), and have a variety of applications within the field of coastal ocean modelling. Adjoint-derived sensitivities to model inputs can be used for gaining physical insight into a modelled system (e.g. Losch and Heimbach, 2007, Massmann, 2010, Verdy et al., 2014, Nowak, 2015, Villaret et al., 2016), or can be used within frameworks for model calibration, data assimilation and parameter estimation (e.g. Lardner et al., 1993, Canizares et al., 1998, Heemink et al., 2002, Lu and Zhang, 2006, Zhang et al., 2011, Li et al., 2013, Chen et al., 2014). Adjoint methods have previously been applied to the analysis of storm surge model sensitivity to wind stress (Wilson et al., 2013, Warder et al., 2019), and this paper represents an extension to these works.

Here, we apply a numerical coastal ocean model, Thetis, and its adjoint, to perform a storm surge sensitivity analysis with respect to multiple model inputs, namely the bottom friction coefficient, bathymetry and wind stress. We use the resulting sensitivities to gain physical insight into surge generation and propagation in the North Sea, and to estimate and compare the uncertainty in surge model outputs arising from each of these inputs, and at different locations in the model domain. We first introduce the numerical model in Section 2, and perform a brief model calibration in Section 3. The adjoint approach to sensitivity analysis is described in Section 4, and sensitivity analysis results are presented in Section 5, using the extreme December 2013 storm surge event as a case study. The results of the sensitivity analysis are discussed in Section 6, and conclusions are made in Section 7.