12.3.2 Bayesian approach  12.3 Recently Developed Analysis  12.3 Recently Developed Analysis

12.3.1 Adaptive filtering

The Kalman filter is an adaptive filter in that the statistics that govern it evolve in time. Introduced to the meteorological community by Daniel Petersen (1968), the Kalman filter is a generalization of optimal interpolation in the following ways:

• A linear prediction model is included. In optimal interpolation, the prediction model supplies the background fields, but does not appear in the mathematical formulation.
• The interpolation of parameter values from grid points to observation location is included explicitly.
• The forecast error covariances are updated after each analysis. In optimal interpolation, the covariances are specified once and for all.
• The Kalman filter can be used as a vehicle for essentially continuous assimilation. Optimal interpolation is used for intermittent assimilation.
• The Kalman filter is applied globally (all observations are used to analyze all gridpoints simultaneously), whereas optimal interpolation is usually applied locally (the analysis at each gridpoint depends only upon a selection of nearby observations).

For more extensive information on the Kalman filter, consult, for example, Gelb (1974, pp. 102-119). Here, only the framework is discussed with the help of a few definitions. Let x_tbe the true state vector at time t. x_t gives the true values of all variables described by the system at all gridpoints in the domain. The state vector evolves in time according to the model

x_t = M_t-1x_t-1 + w_t-1  (12.37)

 

where M_t-1 is the transition matrix describing the (linear) evolution of the state from one time to the next. x_t-1 is the state vector at time t-1, and w_t-1 is the random error vector associated with the prediction model. w_t is assumed to follow a multivariate normal distribution with <w_t>=0 and Q_t=<w_tw_t^T>. Q_t is the equivalent of the background error covariance matrix B in optimal interpolation.

Let the vector of observations at time t be y_t.  In the Kalman filter technique, the state x_t and the observations of the state y_t are assumed to be linearly related by

y_t = L_tx_t + v_t.  (12.38)

 

In the simplest case, L_t is the matrix that interpolates the gridpoint value to the observation point. In the more general case, L_t also converts between the state variables and the observed variables, should they be different. For example, the state variables might be temperature and relative humidity, but the observed variables are radiances measured by satellite in different wavelength intervals. L_t would convert profiles of temperature and humidity, contained within x_t, into channel radiances. In this case, L_t would not be linear. v_t is a random error vector that includes both errors in measurement and errors in representativeness (the observations may be affected by features smaller than the grid can resolve). Like w_t, v_t is assumed to follow a multivariate normal distribution with <v_t>=0 and O_t=<v_tv_t^T>. O_t is the equivalent of the observation error covariance matrix O in optimal interpolation.

The analysis/prediction cycle for the Kalman filter begins with initial estimates of the state vector $\hat x_0$ (an analysis) and the state error covariance

$$\hat A_0 = <(\hat x_0-x_0)(\hat x_0-x_0)^T>.$$ (12.39)

 

The hat ( $\>\hat { }\>$) denotes an estimated quantity. $\hat A_0$ is the equivalent of the analysis error covariance E discussed in connection with optimal interpolation. At each subsequent analysis time, the analysis is produced in two steps:

Step 1: Time extrapolation ( $t-1 \rightarrow t$)
The state evolves from time t-1 to time t by means of the linear prediction equation.

$$\hat x_t(-) = M_{t-1}\hat x_{t-1}(+)$$ (12.40)

 

(-) refers to an estimate before observations are assimilated.
(+) refers to an estimate after observations are assimilated.

The state error covariance evolves as well according to

$$\hat A_t(-) = M_{t-1}\hat A_{t-1}(+)M_{t-1}^T + Q_{t-1},$$ (12.41)

 

where, as mentioned before, Q_t-1 allows for model error growth.

Step 2: State update.
First, one computes what is called the Kalman gain matrix,

$$G_t = \hat A_t(-)L_t^T[L_t\hat A_t(-)L_t^T+O_t].$$ (12.42)

 

G_t is analogous to the matrix of weights in optimal interpolation. Next, the observations are assimilated by means of

$$\hat x_t(+) = \hat x_t(-) + G_t[y_t - L_t\hat x_t(-)].$$ (12.43)

 

This corresponds to the analysis equation in optimum interpolation. In particular, one can recognize within the square brackets the residual between the observed and "interpolated" background values. Finally, the error covariance is updated through

$$\hat A_t(+) = [I - G_tL_t]\hat A_t(-)$$ (12.44)

 

where I is an identity matrix having the same dimension as the state vector x. This operation has no equivalent in optimal interpolation. It has the effect of reducing the error covariance of the system upon each addition of new observations.

The Kalman filter has the same advantages and disadvantages as optimal interpolation, but, because of its large dimensionality, it requires far more computer time, too much to allow for operational implementation at numerical prediction centers at this time.

   
 12.3.2 Bayesian approach  12.3 Recently Developed Analysis  12.3 Recently Developed Analysis