Soil C measurements (Zt) may be made each year or less frequently, but measurements of R are not possible. Thus, the model has two variables that are to be estimated, but only one is observable. Furthermore, it is assumed that the SOC measurement error is normally distributed, independent in time and independent from X and R. A time series of measurements was generated using two steps to demonstrate the approach. First, a time series of true values of SOC (Xt) was computed using Equation 16.1 with the true value of the parameter R for the hypothetical field. Then, a time series of measurements was generated by randomly sampling from the distribution of Z (ez) and adding this random error to "true" SOC values at each discrete time step. Thus, Zt was generated by
Zt = measurement of SOC in year t, kg[C]/ha ez,t = error in measurement, ez,t ~ N(0,a|)
where a2 is variance of SOC measurement error. In real applications of the EnKF, actual measurements would be used.
16.3.3 The Ensemble Kalman Filter: Combining Model and Measurements
The Kalman filter is a set of mathematical equations that are used to obtain optimal estimates of the state of a system. There are two types of equations in a Kalman filter: (1) time update equations, and (2) measurement update equations (Welch and Bishop, 2002). The time update equations project forward in time the current predictions of the system state and covariance. The measurement update equations provide feedback by incorporating a new measurement to obtain an improved estimate of system state and covariance. In a discrete-time Kalman filter, a linear stochastic model is used to project the state and covariance estimates forward to the next time step. At measurement times, the model-projected state and covariance values are updated by using the measurement and its covariance characteristics. A Kalman gain matrix is computed to update estimates of system state and covariance. This process is repeated over time in a recursive fashion, projecting values for each discrete time step and updating those estimates for time steps when measurements are available.
The ensemble Kalman filter (EnKF) follows this same general approach for nonlinear models, but relies on Monte Carlo methods to project state and covariance values between measurement times (Burgers et al., 1998; Marguilis et al., 2002). The SOC model (Equation 16.1) is nonlinear due to multiplication of R and X, and both "states" of the system are estimated. The equations to update Xt and Rt at each time step in the EnKF are:
Updated Xt = Predicted Xt + KX (Zt - Predicted Xt)
Updated Rt = Predicted Rt + KR (Zt - Predicted Xt)
where KX and KR are Kalman gains for Xt and Rt, computed at each time that measurements are used to update the variables.
For this particular problem (i.e., the specific model, the variables to be estimated, and the measurements that are made), these gain factors can be computed as follows (Jones et al., 2004):
2 2 0X, t + Oz where oJ2,t is the variance of soil C predictions at time t, and oXR,t is the covariance between X and R estimates at time t. These variance and covariance values are estimated before state estimates are updated.
Note that although R is not measured, the measurement of SOC provides information for refining the estimate of R via the covariance term. Also note that these gains vary with time; they are recalculated each time a measurement is made. If measurements are not made in a particular year, model predictions provide estimates of SOC and the update step is omitted.
The Kalman gain variables are used to weight the updated estimate on the basis of error variances. Note, for example, that if measurement error variance (a|) is very small relative to model prediction variance (aX,t), then KX approaches 1.0, and the updated Xt (Equation 16.4) will be approximately the value (Zt) that was measured. In contrast, if measurement error is large relative to prediction error, KX
will be closer to 0.0, and the updated estimate will be near the predicted value. Furthermore, if the covariance term used to compute KR is small, the updated Rt will remain near its estimate from the previous step. However, if the covariance term is large, differences between measured and predicted SOC will result in adjustments to R in the update step.
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