Views
This case study is based on a paper by Y.H. Lee and V.P. Singh (1998) titled: Application of the Kalman filter to the Nash model. The paper is referred to throughout as L&S.
Introduction
This case study is useful as it provides an application of the straightforward linear Kalman Filter (KF) algorithm (Kalman 1960) to a rainfall-runoff prediction problem. The use of the linear KF is not commonly encountered in hydrological modelling; being mainly superseded by more complex Extended Kalman Filter (EKF) and Ensemble Kalman Filter (EnKF?) methods. However; to understand the approach and benefits afforded by these more complex approaches it is necessary to appreciate the application of the KF in its original flavour. This case study provides an excellent introduction to this.
Problem Description
A Nash model is identified (Nash 1957) and estimated from observational data to simulate the rainfall runoff relationship for the Nakdong River Basin (Korea), a catchment of ~ 473 km^2. However; L&S suggest that it is asking a little much of the initially-identified Nash model to generate accurate forecasts of discharge from rainfall throughout a storm event without allowing real time assimilation of data to condition the original model estimates in some way.
Methods
L&S identified a classic Nash model and then used a recursive KF algorithm to apply a time varying ordinate to the originally identified Instantaneous Unit Hydrograph (IUH) function i.e., as new data was collected the “height” of the original IUH could be scaled accordingly. This was performed in the following way:
-
At time period k, the observed rainfall and discharge is collected.
-
The observed discharge at k is compared with the predicted discharge made at k-1.
-
The error between these is used to calculate a new gain level for the original IUH using the KF.
-
The rainfall is applied to the original IUH, modified by this gain factor, and a Nash + Kalman forecast of discharge is made for period k+1.
-
Steps (1) through (4) are repeated.
Kalman filtering requires that the process is represented in state space form. Here, L&S used the following structure: the system model is specified as:
Equation 1.
where X is a n x 1 state vector; A is an n x n state transition matrix; B is a n x n system error matrix; and w is a n x 1 system error vector. and a measurement model:
Equation 2.
where Z is an m x 1 measurement vector; H is an m x m measurement transition matrix; and v is an m x 1 measurement error vector. These matrices and vectors are defined in Equation 3:
Equation 3.
Given the above models, a set of observations, and some initial conditions, it is now possible to proceed with a Kalman filtering exercise. The required initial conditions are:
-
an estimate of X at time zero,
-
an initial estimate of the covariance matrix for the model and measurement error vectors Q and R respectively (L&S chose zero),
-
an initial value for the state prediction error covariance matrix P (L&S chose a diagonal matrix with entries:
).
Given (1) through (3) above, the iterative Kalman filter process can be initiated and proceeds as shown in Equation 4:
Equation 4.
Outside of Equation 4, but within the same time step, the rainfall input was applied to the Nash + Kalman model in order to generate the estimated discharge (i.e., the first element of the state vector X).
These steps generate the following process:
-
the Nash model, a well accepted and widely used model, of was identified from a short period of observational data;
-
The model is used to make a prediction of discharge given a rainfall input;
-
the observed discharge is collected and, using a KF, an estimate for the discharge and ordinate adjustment factor are calculated (note: this is only optimal if all the underlying assumptions of the KF hold, most importantly Q and R should be zero mean, independent, and normally distributed; however, the performance of the KF is often good outside of these strict requirements).
Results
L&S used rainfall-runoff data from 5 storm events.
The time-invariant Nash model, with number-of-reservoirs and storage parameters estimated from the 5 storm events, was shown to perform poorly for estimating discharge for each individual event.
However, once the ordinate of the original Nash IUH was allowed to vary in time as determined by the Kalman filter's assimilation of observed discharge, the one-step ahead forecast was (prehaps not too surprisingly) very good.
L&S comment that an analysis of the time-evolution of the Kalman gain for each state, during a storm event, provides useful insight into the progress of the forecasting process: the Kalman gain for the discharge forecast remained very steady showing that the updated discharge forecast was always close to the observed discharge; the Kalman gain for the IUH ordinate height, was shown to vary considerably during the storm event - showing that the deviations between pre-updated forecasted discharge and observed discharge was considerable, this in turn results in a large error variance estimate for the ordinate state and allows the observational data to exert a large change to this state variable.
In effect, the study demonstrates that the nature of the Nash model needs to change in order to reproduce the observed rainfall-runoff relationship during different points of a storm event.
The approach used by L&S is similar to the adaptive "gain updating" step used by Romanowicz et al (2006) see Real Time Flood Forecasting case study.
References
-
Romanowicz, R. J., P. C. Young and K. J. Beven (2006). "Data assimilation and adaptive forecasting of water levels in the river Sevem catchment, United Kingdom." Water resources research 42(6)
-
Lee, Y. H. and V. P. Singh (1998). "Application of Kalman Filter to the Nash Model." Hydrological processes 12(5): 755.
-
Kalman, R. E. (1960). "A new approach to linear filtering and prediction problems." Trans. ASME Journal of Basic Engineering 82: 34-45.
-
Nash, J. E. (1957). "The form of the instantaneous unit hydrograph." IAHS Publication 147: 265-272.