Data assimilation framework¶
The MIKE 1D model can be described in a deterministic, discrete dynamical system setting as
Equation: Discrete state-transition model for the MIKE 1D dynamic system.
where
xk is the state variables of the system at time step k in terms of h- and Q-points for each branch in the river system and concentration components in the advection-dispersion model (if used).
uk is the forcing terms of the system incorporating all boundary conditions, lateral inflow/outflow and possible wind fields.
F(×) is the model operator representing the numerical scheme of MIKE 1D (one time-step-ahead prediction operator).
It is assumed that water level, discharge and concentration measurements are available at one or more locations in the modelled river system. The measurements need not be located at grid points, the mapping of state space to measurement space is taken care of by Eq. the measurement operator mapping model state to observation space.
Equation: Measurement operator mapping model state to observation space.
where
zk is the vector of modelled values at the measurement location.
Ck is a matrix that describes the relation between measurements and state variables, i.e. a mapping of state space to measurement space.
The data assimilation process is a succession of two steps. First, the model is employed to issue a forecast, and then the observed data are meld with the forecast to provide an updated state (analysis step). The analysis step is chosen to be a linear combination of the data and the model, i.e.
Equation: Kalman filter analysis step combining model forecast with observations via gain matrix.
where
\(\overline{x}_{k}^{f}\) is the forecast state vector obtained from Eq. the discrete state-transition model for the MIKE 1D dynamic system.
\(x_k^a\) is the analysed or updated state vector.
Gk is a weighting matrix (gain matrix).
The vector \((z_k - C_k x_k^f)\) is denoted the innovation vector and quantifies the differences between the measurements and their model forecasted equivalents.
A fundamental assumption in MIKE 1D's implementation of the updating is that the measurements are independent. Because of this assumption, a sequential update procedure that processes one measurement at a time can be applied.
Equation: Sequential measurement update for data assimilation processing independent observations.
where
p is the number of measurements.
\(z_{k,j}\) is measurement No. j.
\((z_{k,j} - x_k^a(j - 1))\) is the innovation in measurement point No. j after processing j-1 measurements.
\(g_{k j},\) is a weighting vector (gain vector) corresponding to measurement No. j that describes how the innovation \((z_{k,j} - x_k^a(j - 1))\) is distributed on the state vector.
Processing one measurement at a time makes it easy to use each measurement for updating as long as it contains values - so the updating can continue even if some measurements have missing values or "runs out" of values before time of forecast.
The other fundamental assumption in MIKE 1D's implementation is that the measurements are resampled (by linear interpolation) to the model time step, and the model utilizes the interpolated measurements to update every model time step.
The shape of the gain reflects the confidence we have in the measurements weighed up against the confidence we have in the model forecast. The shape of the gain also reflects the correlation of the states between the measurement point and all other points in the model. The gain function natu-rally peaks at the measurement point and clings off with the distance to it. The gain can at maximum be one, which corresponds to 100% confidence in the measurement.
The formulation of the gain matrix in Eq. the kalman filter analysis step combining model forecast with observations via gain and the gain vectors in Eq. the sequential measurement update for data assimilation processing independent are the most essential parts of the filtering scheme, and the different schemes mainly differ from each other in the way the gains are calculated. The most comprehensive linear melding scheme is the Kalman filter where the gains are determined based on a minimisation of the expected error of the analysed state vector in terms of the errors of both model dynamics and data. In this case the gains are determined sequentially based on a dynamical evolution of the forecast covariance matrix of the state vector.
For high-dimensional systems, as considered here, the propagation of the covariance matrix is the main bottleneck, in most cases imposing an unacceptable computational burden for real-time applications. The ensemble Kalman filter (EnKF) presented by Evensen 1 approximates the error covariance modelling by an ensemble of state vectors that are propagated according to the model equation the discrete state-transition model for the MIKE 1D dynamic system. This approximation reduces significantly the computational burden.
Experiences with the EnKF shows that the computational requirements for obtaining a proper representation of the covariance matrix are of the order of 100 model runs, which may be too expensive in real-time applications. The stationary gain method developed by Madsen and Skotner 2 is based on predefined gain vectors that are assumed to be time-invariant. In this case the filtering update is only slightly more expensive than a normal model run.
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Evensen, G., 1994, Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res., 99(C5), 10,143-10,162. ↩
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Madsen, H., and Skotner, C., Adaptive state updating in real-time river flow forecasting - A combined filtering and error forecasting procedure, Journal of Hydrology, Volume 308, Issues 1-4, pp.302-312, 2005, ISSN 0022-1694, https://doi.org/10.1016/j.jhydrol.2004.10.030. ↩