### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model calibration in coastal modelling
- 3. Data assimilation
- 4. The sediment transport model
- 5. Morphodynamic model for Morecambe Bay
- 6. Joint state-parameter estimation
- 7. Conclusions
- Acknowledgements
- References

Data assimilation is predominantly used for state estimation, combining observational data with model predictions to produce an updated model state that most accurately approximates the true system state whilst keeping the model parameters fixed. This updated model state is then used to initiate the next model forecast. Even with perfect initial data, inaccurate model parameters will lead to the growth of prediction errors. To generate reliable forecasts, we need good estimates of both the current system state and the model parameters. This article presents research into data assimilation methods for morphodynamic model state and parameter estimation. First, we focus on state estimation and describe implementation of a three-dimensional variational (3D-Var) data assimilation scheme in a simple 2D morphodynamic model of Morecambe Bay, UK. The assimilation of observations of bathymetry derived from synthetic aperture radar (SAR) satellite imagery and a ship-borne survey is shown to significantly improve the predictive capability of the model over a 2-year run. Here, the model parameters are set by manual calibration; this is laborious and is found to produce different parameter values depending on the type and coverage of the validation dataset. The second part of this article considers the problem of model parameter estimation in more detail. We explain how, by employing the technique of state augmentation, it is possible to use data assimilation to estimate uncertain model parameters concurrently with the model state. This approach removes inefficiencies associated with manual calibration and enables more effective use of observational data. We outline the development of a novel hybrid sequential 3D-Var data assimilation algorithm for joint state-parameter estimation and demonstrate its efficacy using an idealised 1D sediment transport model. The results of this study are extremely positive and suggest that there is great potential for the use of data assimilation-based state-parameter estimation in coastal morphodynamic modelling. Copyright © 2012 Royal Meteorological Society

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Model calibration in coastal modelling
- 3. Data assimilation
- 4. The sediment transport model
- 5. Morphodynamic model for Morecambe Bay
- 6. Joint state-parameter estimation
- 7. Conclusions
- Acknowledgements
- References

An increase in extreme and hazardous weather events in recent years has led to growing concern over the effects of global climate change. Expected sea level rise combined with an increase in the frequency and intensity of storm events has profound implications for coastal regions and highlights the need for greater knowledge and understanding of how the morphology of the coastal zone is changing (Nicholls *et al.*, 2007).

Morphodynamic change can have wide-reaching environmental, human, social and economic impacts. Accurate knowledge of coastal morphology is fundamental to effective shoreline management and protection; for example, managing coastal erosion, assessing the potential impact of human use of coastal land, monitoring wildlife habitats and the mitigation of flood hazard. It is a complex and challenging subject but one which is of great practical importance.

Unfortunately, knowledge of evolving near-shore bathymetry is limited. In many regions, the topography of the sea bed can change rapidly over time. Obtaining full bathymetric surveys can be time consuming and expensive, and so it is generally not feasible to collect observational data at the spatial and temporal frequency required to track these changes effectively. Consequently, bottom topography is a large source of uncertainty in coastal inundation modelling and this can strongly influence the quality of model predictions (Brown *et al.*, 2007; Hesselink *et al.*, 2003).

Computational morphodynamic models are becoming increasingly sophisticated in an attempt to compensate for this lack of complete data (e.g. Lesser *et al.*, 2004). Modelling the continual interaction between water flow and bathymetry in the coastal zone presents a significant challenge. In addition to difficulties capturing the underlying physics, models are limited by imperfect knowledge of initial conditions and parameters. Even with a perfect model, uncertainties in initial conditions and parameters will lead to the growth of forecast error and therefore will affect the ability of a coastal morphodynamic model to accurately predict the true state of the near-shore environment. In order to generate reliable forecasts of long-term morphodynamic evolution, we need to have good estimates of both the model parameters and the current bathymetry.

Data assimilation is a mathematical technique that enables the optimal use of model and observational resources and offers the potential to generate forecasts that are more accurate than using a model alone. It is most commonly used to produce initial conditions for state estimation: estimating model variables whilst keeping the model parameters fixed. However, it is also possible to use data assimilation to provide estimates of uncertain model parameters. Data assimilation techniques have been employed in the context of atmospheric and oceanic prediction for some years but only relatively recently have begun to be explored in the context of morphodynamic modelling. In this article we present research into data assimilation methods for morphodynamic model state and parameter estimation, conducted as part of the Natural Environment Research Council (NERC)-funded Flood Risk from Extreme Events (FREE) programme. The project extends a previous feasibility study by Scott and Mason (2007) in which a 2D horizontal (2DH) decoupled morphodynamic model of Morecambe Bay, UK, was enhanced through the assimilation of partial observations in the form of waterlines derived from synthetic aperture radar (SAR) satellite images (Mason *et al.*, 2001) using a simple optimal interpolation (OI) algorithm. Despite the known deficiencies of the OI method (e.g. Lorenc, 1981), the data assimilation was shown to improve the ability of the model to predict large-scale changes in bathymetry in the bay over a three-year period.

This article has two main parts, each describing an independent but complementary aspect of the project. The first focuses on the state estimation problem and describes implementation of a more robust three-dimensional variational (3D-Var) data assimilation scheme for Morecambe Bay. The bathymetric component of the model describes the tidal flow and the sediment transport to predict changes in the bathymetry of the bay. The sediment transport rate is computed using a semi-empirical formula containing two parameters which must be calibrated to fit the physical characteristics of the specific study site and the particular conditions being modelled. Here, the parameters are determined by manually calibrating the model against a set of validation observations; the model is run with various parameter combinations and the fit to observations assessed using Brier skill scores.

The second part of this article addresses in more detail the problem of parameter estimation in morphodynamic modelling and explains how data assimilation can be used to estimate uncertain model parameters concurrently with the model state. In this case, the model parameters and predicted model state are updated simultaneously, rather than being treated as two individual processes, thus removing inefficiencies associated with manual calibration and making better use of the available observational data. Data assimilation techniques have the further advantage over many other parameter estimation methods in that they offer a framework for explicitly accounting for all sources of uncertainty. Supplementary material and discussions of the work presented in this section can be found in Smith *et al.* (2009a, 2009b, 2011), and Smith (2010).

We begin in section 2 with a brief summary of some of the various different approaches to model calibration in coastal modelling. In section 3 we describe the theoretical formulation of the data assimilation problem for state estimation in a general system model. We introduce the terminology and notation that we will use and outline the 3D-Var framework used in this work. Section 4 introduces the equations upon which the sediment transport models considered in this study are based. Section 5 gives an overview of the Morecambe Bay model and describes how the assimilation scheme developed by Scott and Mason (2007) was improved by replacing the original OI method with a 3D-Var algorithm in a study using data covering the period 2003 to 2005. We give details of the approach used to calibrate the parameters appearing in the sediment transport equation, discuss its limitations, and assess the results. In section 6 we explore the potential for joint state-parameter estimation in morphodynamic modelling. We outline the development of a new hybrid 3D-Var data assimilation algorithm which enables us to estimate uncertain model parameters alongside the model state variables as part of the assimilation process. Application of the technique is demonstrated using an idealised 1D sediment transport model. Conclusions and potential future developments are discussed in section 7.

### 2. Model calibration in coastal modelling

- Top of page
- Abstract
- 1. Introduction
- 2. Model calibration in coastal modelling
- 3. Data assimilation
- 4. The sediment transport model
- 5. Morphodynamic model for Morecambe Bay
- 6. Joint state-parameter estimation
- 7. Conclusions
- Acknowledgements
- References

Parameter estimation is a fundamental part of the development of a morphodynamic model. Parameters are typically used as a way of representing processes that are not completely known or understood, or where limitations to computer power constrain the model resolution and therefore the level of detail that can be described. Lack of detail in our knowledge of the various processes governing sediment transport means that sediment transport models usually contain empirical or heuristic elements derived from practical experience rather than physical laws. A consequence of this is that these models will often contain parameters that are not directly measurable and which must be ‘tuned’ in order to calibrate the model to a specific field site.

Poorly known parameters are a key source of uncertainty in sediment transport models (Soulsby, 1997). Even if the initial bathymetry is well defined, errors in the parameters will affect the accuracy of the sediment transport flux calculation, leading to the growth of forecast error, and in turn the accuracy of the predicted bed level changes. In most cases, parameter estimation is addressed as a separate issue to state estimation and model calibration is performed offline in a separate calculation. The classic approach is manual calibration or ‘tuning’ of the model against observational data, as is described in section 5. However, the increase in computational capabilities in recent years has seen the development of many new, often complex, automated parameter optimisation algorithms. A variety of schemes are presented in the coastal modelling literature. Some, such as the downhill simplex optimisation (Hill *et al.*, 2003), genetic algorithm (Knaapen and Hulscher, 2003), and hybrid genetic algorithm (Ruessink, 2005a), are based on determining a single best parameter set, whereas probabilistic approaches, such as classical Bayesian (Wüst, 2004) and Bayesian Generalised Likelihood Uncertainty Estimation (GLUE) (Beven and Binley, 1992; Ruessink, 2005b, 2006), are based on the principle that there is no single best parameter set. Instead, the parameters are treated as probabilistic variables with each parameter set being assigned a likelihood value.

In terms of data requirements, set-up and computational costs, adaptability, ease of implementation, etc., each of these methodologies has different strengths and weaknesses. Often it will be the chosen application that will make certain methods more appropriate than others; the suitability of a particular approach will depend on factors such as model complexity, availability of observational data, computational resources and user expertise. In section 6, we present a novel approach for model parameter estimation using a hybrid 3D-Var data assimilation scheme.

### 3. Data assimilation

- Top of page
- Abstract
- 1. Introduction
- 2. Model calibration in coastal modelling
- 3. Data assimilation
- 4. The sediment transport model
- 5. Morphodynamic model for Morecambe Bay
- 6. Joint state-parameter estimation
- 7. Conclusions
- Acknowledgements
- References

There are many different types of data assimilation algorithm, each varying in formulation, complexity, optimality and suitability for practical application. A useful overview of some of the most common data assimilation methods used in meteorology and oceanography are given in the review articles by Ghil and Melanotte-Rizzoli (1991), Ehrendorfer (2007) and Nichols (2009). More detailed mathematical formulations can be found in texts such as Daley (1991), Kalnay (2003) and Lewis *et al.* (2006).

One of the main objectives of this project was to demonstrate the utility of data assimilation for morphodynamic prediction, rather than to develop an operational assimilation-forecast system. It was important to use a stable and well-established method. 3D-Var (Lewis *et al.*, 2006) is a popular choice for state estimation in large problems; the approach has many advantages, such as ease of implementation (no model adjoints required), numerical robustness and computational efficiency. Although standard 3D-Var is designed to produce an analysis at a single time, by applying the method sequentially using a cyclic assimilation-forecast approach we can utilise time series of observational data. This is the primary technique used in the work presented here.

Since the 3D-Var method is applicable to a wide range of contexts, we formulate the assimilation problem for state estimation in a general system model. The model equations used in this study are introduced in section 4 with further details given in sections 5 and 6. The extension of the method to combined state-parameter estimation is given in section 6.1. Our notation broadly follows that of Ide *et al.*, 1997.

For sequential assimilation, we start at a given initial time *t*_{k} with an *a** priori* or model background state , with error . This should be a best-guess estimate of the current true dynamical system state and is typically taken from a previous model forecast.

Observation errors originate from instrumental error, errors in the operator **h**_{k} and representativeness errors (observing scales that cannot be represented in the model; Daley, 1991). It is common practice to assume that the errors for each observation type are statistically stationary and temporally and spatially uncorrelated. **R** is then taken to be a diagonal matrix with the observation-error variances along the main diagonal.

Prescription of the matrix **B**_{k} offers a significant challenge (Bannister, 2008a,b). The standard approach in 3D-Var is to assume that the errors in the background state are approximately statistically stationary. The background-error covariances can then be approximated by a fixed matrix (i.e. **B**_{k} = **B**, for all *k*). This assumption makes 3D-Var a particularly attractive option for large-scale systems. We can write **B** as the product of the background-error variances and correlations, i.e. **B** = *σρ σ*, where *σ* is a diagonal matrix of estimated background-error standard deviations and *ρ* is a symmetric matrix of background-error correlations. The data assimilation problem can be further simplified by assuming that these error correlations are homogeneous and isotropic. The matrix *ρ* can then be modelled using an analytic correlation function (e.g. Daley, 1991, gives further discussion on this).

The 3D-Var method solves the nonlinear optimization problem (3) numerically, using a gradient-based descent algorithm to iterate to the minimising solution. For the work described here, the scheme is applied cyclically as part of an assimilation-forecast algorithm. The model is evolved one step at a time, assimilating the observations in the order in which they become available. At each new assimilation time, the current observational data are combined with the current model forecast (background) state and the analysis is found through minimisation of (3). This updated model state estimate is then integrated forward using the model (1) to become the background state at the next assimilation time and the minimisation process is repeated.

### 4. The sediment transport model

- Top of page
- Abstract
- 1. Introduction
- 2. Model calibration in coastal modelling
- 3. Data assimilation
- 4. The sediment transport model
- 5. Morphodynamic model for Morecambe Bay
- 6. Joint state-parameter estimation
- 7. Conclusions
- Acknowledgements
- References

The bathymetric component of a morphodynamic model describes changes in bathymetry due to the transport of sediment by the water flow regime under consideration. The sediment transport rate is a complex function of the water properties plus characteristics of the sediment such as density and grain size (Long *et al.*, 2008). There is no universally agreed method for computing the sediment transport; numerous different formulae have been proposed, many of which are presented in van Rijn (1993) and Soulsby (1997). The work described in this article uses the power law equation

- (4)

where *q* is the sediment transport rate in the direction of the depth-averaged current, *u* = (*u*_{x}*,u*_{y}), and *A* and *n* are parameters whose values need to be set. This equation is one of the most basic sediment transport flux formulae and approximates the combined bed-load and suspended load sediment transport rate. It is based on a simplification of an equation formulated by van Rijn (1993) and has been chosen for its suitability to our joint state-parameter estimation work. More sophisticated models for the Morecambe Bay study site are investigated by Mason and Garg (2001) and Thornhill *et al.* (2012).

The parameter *A* is a dimensional constant whose value depends on various properties of the sediment and water, such as flow depth and velocity range, sediment grain size and kinematic viscosity (Soulsby, 1997). The derivation of the parameter *n* is less clear, but it generally takes a value in the range 1 ≤ *n* ≤ 4. Typically, *A* and *n* are specified by determining theoretical values, or by calibrating the model against observations. The optimal values can vary from site to site and calibration of the model is generally carried out by running the model with different combinations of the parameters and then comparing the results with available observations to find the set that yield the best results. In regions where the maximum depth-averaged current is around 1.0m s^{−1}, the sediment transport rate is not strongly dependent on *n*. In the Morecambe Bay model, this occurs on the tidal flats. In general, adjusting *A* will affect the accretion and erosion over the whole model domain, whereas adjusting *n* will affect the volumes in the tidal channels only. The effect of increasing *n* for higher values of *A* can be seen in Figure 1, where the value of the sediment transport rate is plotted for different combinations of *A* and *n* using a value of *u* = 2.0m s^{−1}, representing the maximum velocity in the tidal channels.

The bathymetry is updated by solving the sediment transport equation

- (5)

where *z* is the bathymetry, *t* is time, *q*_{x} and *q*_{y} are the *x* and *y* components of the sediment transport rate, and ε is the sediment porosity (Soulsby, 1997). The sediment porosity is a non-dimensional value, expressed as a fraction between 0 and 1, which depends on the degree of sediment sorting and compaction. Its value can be obtained using the measurement techniques described in Soulsby (1997). In the absence of any information on sediment characteristics, a default value of ε = 0.4 is recommended; this corresponds to a natural sand bed with average sorting and packing.

The sediment transport models employed in sections 5 and 6 below each take a different approach to solving (5). A full description of the numerical model implemented for the Morecambe Bay study site is given in Scott and Mason (2007). Details of the idealised 1D model used in the development of the joint state-parameter data assimilation scheme are given in section 6.3.

### 6. Joint state-parameter estimation

- Top of page
- Abstract
- 1. Introduction
- 2. Model calibration in coastal modelling
- 3. Data assimilation
- 4. The sediment transport model
- 5. Morphodynamic model for Morecambe Bay
- 6. Joint state-parameter estimation
- 7. Conclusions
- Acknowledgements
- References

In section 5.4, we saw how the manual calibration of the Morecambe Bay model produced different parameter values depending on the type, coverage and survey time of the validation dataset. In this section we consider an alternative approach to morphodynamic parameter estimation using sequential data assimilation. Although data assimilation is predominantly used for state estimation, it is also possible to use the technique to estimate uncertain model parameters concurrently with the model state. We do this using the method of state augmentation. This approach has the advantage that it enables the parameters to be estimated and updated ‘online’ as new data become available rather than calibrating against historical data. It also has the potential of being able to estimate parameters that are expected to vary over time. Here, we describe the development of a novel hybrid sequential 3D-Var data assimilation algorithm for joint state-parameter estimation and demonstrate its application using an idealised 1D sediment transport model.

#### 6.1. State augmentation

State augmentation is a conceptually simple technique which can in theory be applied to any standard data assimilation scheme; the parameters we wish to estimate are appended to the model state vector, the model state forecast equations are combined with the equations describing the evolution of the parameters, and the chosen assimilation algorithm is simply applied to this new augmented system in the usual way (Jazwinski, 1970). This framework enables us to estimate the model parameters and update the predicted model state simultaneously (rather than being treated as two individual processes), thereby saving on calibration time, facilitating better use of the available data and potentially delivering more accurate model forecasts. The technique has previously been successfully used in the treatment of systematic model error or forecast bias in atmosphere and ocean modelling using sequential and four-dimensional variational (4D-Var) assimilation methods (e.g. Martin *et al.*, 1999; Griffit and Nichols, 2000; Bell *et al.*, 2004; Dee, 2005). The review article by Navon (1997) discusses its application to parameter estimation in the context of meteorology and oceanography using 4D-Var, and more recently the approach has been employed for parameter estimation in simplified numerical models using variants of the extended and ensemble Kalman filters (Hansen and Penland, 2007; Trudinger *et al.*, 2008).

In this study, we assume that the required parameters are constants, that is, they are not altered by the forecast model from one time step to the next. The parameter estimates will only change when they are updated by the data assimilation at each new analysis time. Note that this is not a necessary assumption, but one which is appropriate to the model we consider here. In this case, the evolution model for the parameters is given by the simple equation

- (7)

The parameter model (7) together with the model forecast equation (1) constitute an augmented state system model

- (8)

where

- (9)

is the augmented state vector and

- (10)

with .

The equation for the observations (2) can be re-written in terms of the augmented system as

- (11)

where , and

- (12)

#### 6.2. Error covariances

In basic state estimation the background-error covariances govern how information is spread throughout the model domain, passing information from observed to unobserved regions and smoothing data if there is a mismatch between the resolution of the model and the density of the observations. Since it is not possible to observe the parameters themselves, the parameter estimates will depend upon the observations of the state variables. Successful application of the state augmentation technique relies strongly on the relationships between the parameters and state variables being well defined and assumes that we have sufficient knowledge to reliably prescribe them.

For joint state-parameter estimation, it is the state-parameter cross-covariances, , that determine the influence of the observed data on the estimates of the unobserved parameters and therefore play a crucial role in the parameter updating. A good *a** priori* specification of these covariances is fundamental to reliable joint state and parameter estimation. Since, by the nature of the problem, the statistics of these errors are not known exactly, they have to be approximated in some manner.

In initial experiments, we applied the same principle as for the state background-error covariance matrix, and prescribed the state-parameter cross-covariances a static functional form, but this failed to produce reliable parameter estimates (Smith, 2010). Further investigation showed that we could yield accurate estimates of both the parameters and the state variables by using a flow-dependent structure for (Smith *et al.*, 2009a). Updating the full background-error covariance matrix at every time step can be computationally expensive and impractical for large-scale systems. Importantly, the results of these experiments also showed that it is not necessary to explicitly propagate the full augmented matrix (13). We were able to recover accurate estimates of both the parameters and state variables by combining a time-varying approximation of the state-parameter cross-covariance matrix with an empirical, static representation of the state background-error covariance **B**_{zz} and static parameter-error covariance matrix **B**_{pp}.

This result led to the development of a novel hybrid assimilation algorithm. The new scheme combines ideas from 3D-Var and the extended Kalman filter (EKF) to construct a hybrid background-error covariance matrix that captures the flow-dependent nature of the state-parameter errors without the computational complexities of explicitly propagating the full system covariance matrix. The method is relatively easy to implement and computationally inexpensive to run. A 3D-Var approach is adopted for the state and parameter background-error covariances; the matrices **B**_{zz} and **B**_{pp} are prescribed at the start of the assimilation and held fixed throughout as if the forecast errors were statistically stationary. The state-parameter cross-covariances are estimated using a simplified version of the EKF forecast step. The derivation of this approximation is summarised below. A comprehensive description of the formulation of this unique hybrid system is given in chapter 6 of Smith (2010).

In the EKF, the background-error covariance at time *t*_{k+1} is determined by propagating the analysis-error covariance forward in time from *t*_{k} using a linearisation of the forecast model (Jazwinski, 1970). If we assume that the state-parameter cross-covariances are initially zero and consider the form of the augmented background-error covariance matrix (13) after a single step of the EKF, we find that the state-parameter cross-covariance at time *t*_{k} can be approximated as (Smith *et al.*, 2011; Smith, 2010)

- (14)

where

- (15)

is the Jacobian of the state forecast model with respect to the parameters, and is the parameter analysis-error covariance.

A step-by-step summary of the procedure for updating the hybrid augmented matrix **B** in the context of cycled 3D-Var can be found in section 6.4 of Smith (2010).

Practical implementation of this approach involves computing an approximation to the matrix **N**_{k} at each new analysis time. Explicitly calculating the Jacobian of complex functions can be a difficult task, requiring complicated derivatives if done analytically or being computationally costly if done numerically. For the model presented in section 6.3, the Jacobian is computed using a local finite-difference approximation as described in chapter 8 of Smith (2010) and in Smith *et al.* (2011). Since the number of parameters to be estimated is typically quite small, this computation does not add significantly to the overall cost of the assimilation scheme.

Key advantages of this new hybrid scheme are that the background-error covariance matrix needs to be updated only at each new analysis time rather than at every model time step, and it does not require the previous cross-covariance matrices to be stored. It also avoids many of the issues associated with the implementation of fully flow-dependent algorithms such as the extended and ensemble Kalman filters which include filter divergence, dynamic imbalances and rank deficiency (e.g. Houtekamer and Mitchell, 1998, 2005; Ehrendorfer, 2007).

#### 6.3. The model

The potential applicability of the hybrid technique to morphodynamic modelling was investigated using an idealised 1D version of the sediment transport model employed in the Morecambe Bay model. The purpose of this study was to gain insight into some of the issues associated with practical implementation of the state augmentation framework in order to help guide the development of a data assimilation-based state-parameter estimation system for the full model. We chose to use a relatively simple, low-order system as a first step as this allowed us to concentrate on developing, testing and evaluating ideas rather than dealing with modelling complexities.

We use a generic test case consisting of a smooth, initially symmetric, isolated bed-form in an open channel. The bed level changes are governed by the 1D sediment conservation equation (cf. (5)):

- (16)

where *z*(*x,t*) is the bed height, *t* is time, *q* is the total (suspended and bedload) sediment transport rate in the *x* direction, and ε is the sediment porosity.

To calculate *q* we use the same simple power law equation (4) as in the Morecambe Bay model, except that now the depth-averaged current *u* = *u*(*x,t*) is one-dimensional in the *x* direction. A primary reason for choosing this simplified scenario is that, under certain assumptions, it is possible to derive an approximate analytical solution to (16) and this is useful for model validation purposes.

We assume that the amplitude of the bed-form is sufficiently small relative to the water depth such that any variation in the elevation of the water surface can be ignored. The water height, *h*, can then be taken to be a positive constant. If we further assume that the water flux, *F*, is constant across the whole domain, we can set *u*(*h* − *z*) = *F*. This enables us to express the sediment transport rate *q* as a function of bed height *z* rather than *u*. (16) can then be rewritten in the quasi-linear advection form

- (17)

where the advection velocity of the bed *c*(*z*) is now a function of the bed height *z* only. We note that the solution derived under these assumptions is only strictly valid when the migration speed of the bedform is slow relative to the flow velocity (Hudson, 2001).

To maintain numerical stability and ensure that the solution for *z* remains smooth and physically realistic, we add a small diffusive term, *κ∂z*^{2}*/∂x*^{2}, to the right-hand side of (17). The model for the evolution of the bed-form is hence described by a nonlinear advection-diffusion equation. This is discretised using a hybrid semi-Lagrangian Crank–Nicolson algorithm based on that presented in Spiegelman and Katz (2006). Implementation of the algorithm is described in Smith (2010).

#### 6.4. Experiments and results

The scheme was tested via a series of twin experiments using pseudo-observations with a range of spatial and temporal frequencies. For the purpose of these experiments, we assume that the values of *h*, *F* and ε are known and constant, but that the true values of the parameters *A* and *n* are uncertain, as in the full Morecambe Bay model. The water height and flux are specified as *h* = 10.0m, *F* = 7.0m and the sediment porosity ε = 0.4.

We generate a reference or ‘true’ solution by running the model on the domain *x* ∈ [0,500] with grid spacing Δ*x* = 1.0*m*, time step Δ*t* = 30 min, diffusion coefficient *κ* = 0.001, and parameter values *A* = 0.0018*ms*^{−1} and *n* = 3.4. These are the parameter values that gave the best Brier skill score in the initial calibration of the Morecambe Bay model against the swath data. The initial profile for the true bathymetry is specified using a Gaussian function. The evolution of the bed-form over a 72 h period is illustrated in Figure 5.

The *a priori* bathymetry for the assimilation is given by the same Gaussian function, but with different scaling factors so that it has a different height and width and is in a slightly different starting position relative to the truth. The state background-error covariance matrix **B**_{zz} = **b**_{ij} is modelled using the isotropic correlation function (Rodgers, 2000)

- (18)

with *ρ* = *exp*(−Δ*x/ℓ*) where *ℓ* is a correlation length-scale that is defined empirically. For the experiments described here, *ℓ* is set at four times the observation spacing. A particular benefit of this model is that the matrix inverse can be calculated explicitly and has a particularly simple tri-diagonal form.

The state-parameter cross-covariance matrix is recalculated at each new analysis time by computing the Jacobian of the state forecast model with respect to the parameters using a local finite-difference approximation as described in Smith (2010) and Smith *et al.* (2011).

Observations of the bed height *z*(*x,t*) are generated by sampling the true solution on a regularly spaced grid and are assimilated sequentially at regular time intervals. The space and time frequency of the observations remains fixed for the duration of each individual assimilation experiment, but is varied between experiments as described in the discussion of our results below.

The augmented cost function is minimised iteratively using a quasi-Newton descent algorithm (Gill *et al.*, 1981). The minimisation is terminated by one of:

- i
a sufficiently small gradient ||∇*J*(**w**_{k})||_{∞} ≤ ε_{1},

- ii
a sufficiently small step length ||**w**_{k} − **w**_{k−1}||_{2} ≤ ε_{2}(ε_{2} + ||**w**_{k}||_{2}),

- iii
the number of iterations exceeding a user-specified maximum *k*_{max}, or

- iv
a zero step.

For the experiments presented here, the tolerances on these stopping criteria are set at ε_{1} = 10^{−12}, ε_{2} = 10^{−12} and *k*_{max} = 500. The addition of the parameters to the estimation problem increases the dimension of the control vector to *m* + 2; since this increase is small relative to the size of the state vector, it does not have a significant impact on the cost of the minimisation compared to basic state estimation.

At the end of each assimilation cycle the current values of *A* and *n* in the model are replaced with the new estimates, **p**^{a}, the current model state is updated with the state analysis, **z**^{a}, and the model is forecast forward to the next observation time.

##### 6.4.1. Perfect observations

At first, we assume that the observations are perfect. The initial parameter estimates for *A* and *n* are taken as *A* = 0.006*ms*^{−1} and *n* = 4.2 which are the lower and upper extremes of the calibration range used in the Morecambe Bay model. The observation- and background-error variances are set at and respectively and the parameter cross-correlation *ρ*_{An} is taken to be strongly negative.

Figures 6 and 7 show the parameter updates when observations are taken at varying temporal and spatial frequencies. Figure 6 shows the effect of reducing the temporal frequency of the observations from every 2 to every 24 h. The spatial frequency of the observations is fixed at 25*m*. The speed of convergence of the estimates decreases as the time frequency of the observations decreases, but the scheme successfully retrieves the true values of *A* and *n* to a high level of accuracy in all cases. Even when the observation frequency is further reduced to every 48 h, the estimates eventually converge to close to their exact values after around 6 days (not shown).

Decreasing the spatial frequency has a similar effect. Figure 7 shows the parameter updates produced when the grid spacing between the observations is increased from every 10Δ*x* to every 50Δ*x*. Once again, the true parameter values are successfully estimated to high accuracy. There is a large difference in the rate of convergence when the observation interval is doubled from 25Δ*x* to 50Δ*x*. If this is increased further to 75Δ*x*, the scheme fails to recover the correct parameter values. In this case, the spatial resolution of the observations is too low to be able to consistently capture key features of the bed-form, in particular the height and position of its peak. This information is crucial to the accurate tracking of the movement of the bed across the domain and in turn to the reliable estimation of *A* and *n*.

##### 6.4.2. Imperfect observations

We examined the impact of observation errors on the scheme by adding random Gaussian noise to the observations. In these examples, we reverse the direction of the parameter errors; the initial estimate for *n* is taken as *n* = 2.2 which is the lower limit of the range used in the Morecambe Bay calibration; parameter *A* is set at *A* = 0.018*ms*^{−1} which is ten times greater than the true value. Observations were taken at 25*m* intervals and assimilated every 2 h. The background and observations are now given equal weight with . The parameter error variances and cross-covariance are defined as described above for the perfect observation case.

We found that smoother, more accurate parameter estimates could be obtained by averaging over a moving time window as the assimilation is running, as illustrated in Figure 9. Averaging is started at *t* = 24 h to allow time for the scheme to settle. Here, we use a time window of 6 h. The smoothed estimates are extremely close to those obtained in the perfect observation case.

We have tested the efficacy of our hybrid method in a variety of scenarios in addition to those presented here. As we would expect, there are bounds on the success of the approach. As is the case with any data assimilation scheme, the quality of the analysis is highly dependent on the quality of the information fed into the assimilation algorithm; if observational data are too infrequent or too noisy or if the initial state and parameter background estimates are particularly poor, then we cannot expect the scheme to yield reliable results. Generally, we saw a deterioration in the parameter estimates and model state predictions as the quality and frequency of the observation decreased. The estimated error variances and the relative weighting between the background and observations were also found to be extremely important to the success of the method. Further experiments with more detailed results and discussions are given in Smith *et al.* (2009b, 2011) and Smith (2010).

### 7. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Model calibration in coastal modelling
- 3. Data assimilation
- 4. The sediment transport model
- 5. Morphodynamic model for Morecambe Bay
- 6. Joint state-parameter estimation
- 7. Conclusions
- Acknowledgements
- References

Up-to-date knowledge of near-shore coastal bathymetry is important in flood prediction and risk management. In this article we have demonstrated how the application of data assimilation techniques in morphodynamic modelling can be used to generate forecasts of bathymetry that are more accurate than using a model alone by (i) producing improved estimates of the current model bathymetry, and (ii) providing estimates of uncertain morphodynamic model parameters.

Section 5 described the implementation of a sequential 3D-Var data assimilation scheme for model state estimation in Morecambe Bay. The results of this section illustrated how data assimilation can be used to greatly improve model predictions of the evolution of the bathymetry in the bay. Even though the model used here is simple relative to state-of-the-art engineering models, the inclusion of partial observations of bathymetry taken at irregular and sometimes large intervals is sufficient to produce a reasonable match to the true state of the bathymetry after a two-year model run. This suggests that extremely complex models are not necessarily required, and that more computationally efficient, simplified models may be adequate if a cycled data assimilation scheme is employed. A key difficulty with morphodynamic models is their dependence on parameters whose values are typically uncertain and often dependent on the characteristics of the study site. In the Morecambe Bay study, the model parameters were set by running the model with different parameter combinations and examining the results to find the values which produced the closest match to the observed data. This is a standard approach, but is time consuming and can produce biased estimates if the observational data available for calibration are limited. The parameters providing the best results using our calibration dataset did not produce the best results when the modelled bathymetry was compared with validation data for the whole of the bay. This discrepancy is possibly due to the calibration dataset covering only the deeper parts of the bay and channels, thus biasing the parameter selection towards values that more closely reproduced the conditions in these areas.

In section 6 we explained how the problem of morphodynamic model parameter estimation can be addressed using sequential data assimilation. We introduced the technique of state augmentation and described how the approach can be used to estimate poorly known model parameters jointly with the model state variables. The augmented assimilation framework enables us to estimate the model state and parameters simultaneously, rather than treating state and parameter estimation as two separate problems. The approach offers an attractive alternative to traditional calibration techniques, allowing more efficient use of observational data and potentially producing improved model state forecasts. A particular advantage of sequential data assimilation is that observational data can be used as they arrive in real time. This allows for the inclusion of information from new observations which become available after earlier observations have been assimilated. We gave details of the development of a novel hybrid 3D-Var assimilation algorithm and demonstrated the efficacy of the new method using an idealised 1D version of the sediment transport model implemented for Morecambe Bay. The results were excellent; the scheme is able to successfully recover the true parameter values, even when the observational data are noisy, and this has a positive impact on the forecast model. In the experiments with noisy observations, we found averaging the parameters over a moving time window to be very effective at improving the accuracy of the estimates. The results of this study are extremely promising and suggest that there is great potential for the use of data assimilation-based joint state and parameter estimation in coastal morphodynamic modelling.

The hybrid algorithm developed in this work offers an effective and versatile approach to approximating the state-parameter cross-covariances demanded by the augmented system. The scheme is relatively simple to implement and computationally inexpensive to run. In this article we have focused on application of the approach in the context of morphodynamic modelling, but the method has also been applied in a range of simple dynamical systems models with equally positive results (Smith *et al.*, 2010; Smith, 2010). We expect that this new methodology will be easily transferable to larger more realistic models with more complex parametrisations.

The next step is to implement a joint state-parameter estimation data assimilation scheme in the Morecambe Bay model. This will offer a number of challenges but will provide a useful real-world test case for the hybrid technique. The opportunity to further explore some of the practical issues identified in this initial study will lead to an improved understanding of the process of taking our new method from theory into operational practice.

The most complex part of applying the method to the 2D model will be construction of the augmented background-error covariance matrix. The success of the state augmentation approach relies heavily on the relationship between the state and parameters (as described by the state-parameter cross-covariances) being well defined. The approximation of the state-parameter covariance matrix used in the hybrid scheme requires computation of the Jacobian of the forecast model with respect to the model parameters. For the 1D model, this was calculated using a local finite-difference approximation. Because the number of parameters was small and the dimension of the state vector relatively low, this calculation did not add significantly to the overall cost of the assimilation scheme. This calculation will be computationally intensive for the full model but does not render the approach infeasible.

The definition of the parameter background-error covariance matrix **B**_{pp} is relatively simple but its specification requires some level of *a** priori* knowledge of the parameter error statistics and an understanding of the relationship between individual parameters. Difficulties can arise when parameters exhibit strong interdependence; in this case the prescription of the parameter correlations can have a significant effect on the accuracy of the estimates obtained. In our simple model experiments, we knew the magnitude and direction of the errors in our initial parameter estimates and were able to set the parameter error variances and cross-covariances accordingly. In practical situations, where the true statistics of the errors are not known, a simple sensitivity analysis can be used to help identify the existence and degree of interdependence between parameters.

A further potential issue is the type and frequency of availability of observational data. Our 1D model experiments found that the quality of the state and parameter estimates varied with the spatial and temporal frequency of the observations. Here, we used synthetic observations that were direct, evenly spaced and assimilated at regular time intervals. The Morecambe Bay model is run on a much larger domain and over much longer time-scales. The observational data are indirect and spatially and temporally sparse. Work with this model will provide a useful insight into whether the space and time frequencies of observations available on an operational scale are sufficient for a combined state-parameter estimation scheme to be effective. It could also potentially aid the design of an optimal observation network for coastal monitoring.

Ultimately, we hope to use the experience gained from this research to help guide the application of data assimilation-based state and parameter estimation in an operational setting. The extension of the Morecambe Bay 3D-Var assimilation scheme to concurrent state-parameter estimation is an important intermediate step that will enable us to assess the feasibility of our hybrid assimilation algorithm in the context of operational-scale coastal morphodynamic modelling, and to evaluate whether the approach offers a viable, cost-effective alternative to the model calibration techniques currently in use.