Data assimilation and Morphodynamic Modelling Of Morecambe Bay Gillian Thornhill 1, Tania Scott 1, Sarah Dance 2, David Mason 1, Nancy Nichols 2, Michael Baines 2, Kevin Horsburgh 3, Peter Sweby 2 1 Environmental Systems Science Centre and 2 Mathematics Department, University of Reading, 3 Proudman Oceanographic Laboratory The use of the 3D-Var data assimilation scheme in combination with a simple morphodynamic model has produced a large improvement in the predictive ability of the model. The model was calibrated with respect to the sediment transport equation, and the definition of the errors for the assimilation system. The use of observations from disparate data sets has been successfully demonstrated, both partial bathymetry from waterlines derived from SAR data, and more complete coverage from a ship-borne survey. The improvement due to the additional observations covering the larger part of the model domain was investigated and characterised. Summary Coastal Morphodynamics Morphodynamic modelling of coastal areas is used to predict the effects of storm surges and assess the impacts of coastal flooding. The knowledge of the near-shore bathymetry is essential for the correct prediction of areas at risk, but the predictive ability of the models is often inadequate due to inaccurate representation of the sediment transport processes and imperfect knowledge of initial conditions. Frequent surveys of areas with dynamically changing bathymetry are prohibitively expensive, resulting in a lack of timely data. Data assimilation is a technique that combines limited observations and model results in a mathematically rigorous way, and can produce substantial improvements in the predictive power compared to using a model alone (Scott and Mason 2007). Study site: Morecambe Bay has a very dynamic sediment transport regime, and channels can migrate by several kilometres in just a few years. The changes in channel position and morphology can have a dramatic effect on wildlife habitats and storm-induced flood risk for the coastal towns on the bay. Figure 1: Morecambe Bay Schematic Map Method The morphodynamic model combines a hydrodynamic component which predicts the water flow due to the tides (wave action is not modelled), and a sediment transport model based on the Soulsby-van Rijn equations. The initial bathymetry for the model was produced by taking bathymetry from (Mason et al 1999) and assimilating waterlines from January 2003 to January 2004 to evolve the out-of-date bathymetry to a state closer to that which existed in January The model is then run from January 2004 to November 2005 using the remaining waterlines and the swath data. The model was calibrated for both the length scale used in the error covariance matrix for the 3D-Var scheme, and for the sediment transport equation. This was done by comparing the assimilation systems predicted bathymetry against the swath bathymetry immediately before these data are assimilated. The run is then continued to November 2005 for comparison with the LiDAR data survey taken at that time. Mason, D. C., Amin, M., Davenport, I. J., Flather, R.A., Robinson, G.J., Smith, J. A., Measurement of recent intertidal sediment transport in Morecambe Bay using the waterline method. Estuarine, Coastal and Shelf Science 49, Scott, T.R., Mason, D.C., Data assimilation for a coastal area morphodynamic model: Morecambe Bay. Coastal Engineering 54, Issue 2, References The data that was used for the assimilation method consists of waterlines (quasi-contours of bathymetry derived from SAR satellite data) over the period , and swath bathymetry from a survey carried out in February The waterlines are not available at regular intervals, and only cover a small part of the model domain, so the assimilation process accommodates the spatial and temporal variability of these observations. The swath data covers a larger proportion of the domain, but represents a single point in time. There is a also a full LiDAR survey of the Bay, which is used for validation purposes, and thus is not used in the assimilation. Water elevation ( bathymetry) Figure 4: SAR data showing water extent Figure 5: Waterline contour derived from SAR image Data for assimilation process Figure 3: Swath data coverage Figure 6: LiDAR validation data from November 2005 Results Figure 7 and figure 8 below show the differences between the LiDAR data and the bathymetry predicted by the model in November 2005 for model runs with and without assimilation. In the case where assimilation is not used, the differences are more pronounced, particularly in areas where the channels have moved between the state of the initial bathymetry and that in the LiDAR data. The effect of including the swath bathymetry in the assimilation was tested by running the model using only the waterline data, and comparing this to the model run with the swath data included. In figure 9 the differences between the bathymetry produced by these two model runs is shown for the end of the run, in November The swath data updates the bathymetry in the vicinity of the observations, and the extent of the influence of the data on the model is controlled by the choice of length scale in the definition of the error covariance matrix (B in eqn. [1]) in the assimilation scheme. Removing the swath bathymetry resulted in greater differences between the LiDAR data and the model output, as can be seen by comparing figure 8 with figure 10. However, there is an interval of nine months between the swath data and the final LiDAR survey, and any evolution of the bathymetry in that time will result in differences appearing in the comparison of the model output with the LiDAR data. The assimilation of two waterlines in this interval updates the model to partially account for these changes, but as these observations cover a small part of the domain, they may not fully compensate for bathymetry changes in areas not covered by the data. Thus there will not be an exact correspondence between the changes in the model output and the differences seen between the two model runs when compared with the LiDAR data. Fig 7: Differences between model output and LiDAR data without assimilation Fig. 8: Differences between model output and LiDAR data with assimilation Fig. 10: Differences between model output and LiDAR data excluding the swath bathymetry Fig. 9 : Differences in model bathymetry with and without swath data Data assimilation Model forecast Partial observations of the true bathymetry Model bathymetry at the observation time Assimilation New model bathymetry Data assimilation is a technique for combining observations with model predictions, including a representation of the errors in both the observations and the model predictions. This in effect produces a weighted mean of the observations and the model at a given time. The observational data is incorporated into the model to produce an adjusted result (called the analysis) which maintains a more accurate representation of the true bathymetry than using the model alone. Data from waterlines (providing partial bathymetry) and echo-sounding from a survey has been assimilated into the simple morphodynamic model using a 3D variational data assimilation scheme. The basic mathematical formulation of 3D variational analysis requires the minimisation of a cost function : where z b is the model state vector z is the analysis state vector y is the observation vector h is the observation operator B and R are the covariance matrices of the background and observation errors J(z) = (z - z b ) T B -1 (z - z b ) + (y - h(z)) T R -1 (y – h(z)) ……….(1) Figure 2: Schematic diagram of data assimilation process