Adaptive MOS methods

About every few years, NWP models undergo significant changes that make the MOS regression analysis obsolete. There are, however, techniques whereby the MOS can be updated on a regular (monthly or quarterly) basis, although this does not completely eliminate the drawback of historic inertia.

Alternatively, adaptive methods have increasingly come into use. Here the coefficients X₁ and X₂ in the error equation are constantly updated in the light of daily verification (Persson, 1991).

Figure 95 shows forecasts and observations for the location with severe systematic 2 m temperature errors depicted in Figure 94. It is not a case of  “plain bias” but of “conditional bias”, since mild forecasts are less at error than cold forecasts. A simple mean-error correction would therefore not be optimal.

Figure 95: Adaptive Kalman filtering of 2-metre temperature forecasts for Tromsö in northern Norway during winter 2011. The forecasts are too cold and over-variable, both of which are remedied by X₁ and X₂ in a 2-parameter error equation.

By a daily verification, the Kalman filter estimates the coefficients X₁ and X₂ in the error equation:

Err = X₁ + X₂·T_fc

where T_fc is the verified forecast. The coefficients are updated from a variational principle of “least effort”, whereby the equation line is translated (by modifying X₁) and rotated (by modifying X₂), so that it takes the verification into account, considering the uncertainties in the verification and the coefficients (see Figure 96).

Figure 96: A schematic illustration of the workings of an adaptable MOS by Kalman filtering. At a given time the error equation has a certain orientation (full red line) with a certain estimated uncertainty (red dashed lines). A forecast is verified and yields an error (red filled circle) that does not normally fall on the error line. Depending on the interplay between the equation uncertainty and the verification uncertainty (dashed red circle), the error equation line is translated and rotated to take the new information into account, after which this information is discarded.

Note that the system keeps information only about the error equation and its uncertainty and the last, not yet verified forecast. When the forecast is verified and the verification has affected the error equation, the verifying observation is discarded.

Figure 97 shows an ensemble forecast for the same location with severe systematic 2 m temperature errors.

Figure 97: A plume diagram for Tromsö, 12 February 2011. The forecast is too cold with 50-100% probabilities of temperatures < -15°C.

The two-dimensional error equation is able to apply corrections which are different for different forecast temperatures and thus take the flow dependence into account to some degree. The error equation is applied to all EPS members at all ranges, assuming no significant model drift (see Figure 98).

Figure 98: The same as Figure 97but after the Kalman-filtered errors equation has been applied. Mild forecasts have hardly been modified, whereas cold ones have been substantially warmed, leading to less spread and more realistic probabilities with, for example, 0% probabilities for 2 m temperature <-15°C.

A two- or multi-dimensional error equation is able not only to correct for mean errors, but also systematic over- and under-forecasting of the variability, thereby providing realistic probabilities.