Slope at downstream boundary

The effect of changing the parameter controlling the bathymetry slope at the down- stream boundary can be seen in the following simulations. In each case, a symmetrical domain with a central channel was initially filled with water to a depth of approximately 4 m as shown in cross section in figure 3.6. The domain has a slope of 0.09% which caused the water to flow downhill under the influence of gravity and leave the domain at the

downstream end. No extra water was added to the domain during the simulations, which ran for 1500 seconds. Some water remained in the domain at the end of the simulations. For one case the downstream slope was extrapolated into the ghost cells, and for the other case the ‘no slope’ condition was used, where the elevation in each ghost cells is set to be the same as the domain cell next to it on the inside of the boundary.

Figure 3.6: Cross section of the domain, showing the channel filled with water at the start of the simulations. The green line shows the elevation of the domain and the blue points show the water depth.

At the end of the simulations, the water profile is different for the two cases as shown in figure 3.7. For the extrapolated boundary slope condition, the water is able to leave the domain cleanly; the flow over the boundary is the same as elsewhere in the domain. For the no slope condition, water cannot leave the domain as fast and therefore builds up at the boundary.

Chapter 3: Hydrodynamic model

(a) Extrapolated slope; water levels relative to topography in 3D

(b) No slope; water levels relative to topography in 3D

(c) Extrapolated slope; water levels relative to topography in plan view

(d) No slope; water levels relative to topography in plan view

Figure 3.7: Water profiles for the slope and no slope boundary conditions after 1500s.

3.3

Chapter summary

In this chapter we have described the inundation model used for the simulations in chapters 4, 5 and 6 of this thesis. We have described the boundary condition options in Clawpack; our description of the treatment of source terms is presented as part of our paper Cooper et al. [2018b], reproduced in this thesis as chapter 4 (see section 4.8 for treatment of source terms). In this chapter we have also presented results of studies investigating the sensitivity of modelled water levels to the domain friction parameter and the downstream topography slope.

Chapter 4:

Effect of channel friction

estimation on observation impact

In this chapter we address the first research question outlined in chapter 1; How does estimation of the channel friction parameter affect observation impact in data assimilation for inundation forecasting? In particular we wish to find out:

• Can the ETKF retrieve the correct channel friction parameter in synthetic experi- ments using SAR-like observations, and does that improve the forecast?

• Is error in the channel friction parameter distinguishable from error in inflows? • Does it matter if we assume zero momentum when restarting after assimilation?

We describe a series of synthetic experiments. The remainder of this chapter, except for the chapter summary (section 4.9), has been published and is reproduced from Cooper et al. [2018b].

4.1

Abstract

Accurate inundation forecasting provides vital information about the behaviour of fluvial flood water. Using data assimilation with an Ensemble Transform Kalman Filter we combine forecasts from a numerical hydrodynamic model with synthetic observations of water levels. We show that reinitialising the model with corrected water levels can cause an initialization shock and demonstrate a simple novel solution. In agreement with others, we find that although assimilation can accurately correct water levels at observation times, the corrected forecast quickly relaxes to the open loop forecast. Our

Chapter 4: Effect of channel friction estimation on observation impact

new work shows that the time taken for the forecast to relax to the open loop case depends on domain length; observation impact is longer-lived in a longer domain. We demonstrate that jointly correcting the channel friction parameter as well as water levels greatly improves the forecast. We also show that updating the value of the channel friction parameter can compensate for bias in inflow.

Keywords Data assimilation, inundation forecasting, fluvial flooding, observation impact, joint state-parameter estimation, ensemble Kalman filter.

Highlights

• Data assimilation is applied to simulated flood forecasts and SAR-like observations • Reinitialisation shock due to water level correction is removed using a novel method • Observation impact is linked to domain length when updating only water levels • Updating the channel friction parameter leads to marked improvement in forecast

skill

• Updating the channel friction parameter can compensate for biased inflow

Software Availability

The inundation simulations in this work were generated using Clawpack 5.2.2, a col- lection of FORTRAN and python code available from http://www.clawpack.org/. Details of the amended Clawpack source code as used in this work are freely available on request from the corresponding author, as is the python code used to perform data assimilation on the inundation simulation output. Please contact [email protected] for details.