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2 precipitation). Let us point out that LP includes some growth and decay. For example, if there is some persistent dissipation at the rear of a system this will result in a motion pattern that is confluent. If the dissipation at the rear continues into the future the advection will capture the narrowing of the pattern. predictable, and it was predicted. The decrease in intensity within this area, however, was not predictable by LP. The example above illustrates that the method of determining the velocity field and pattern advection are critical to the method s skill. Another physical limitation of LP is the evolution of the motion field. Figure 3 shows that around 20% of loss of skill (defined here as the time to decorrelation to 1/e) is due to time evolution of motion field. Figure 3: Decorrelation between nowcasted and observed reflectivity patterns using stationary motion filed and a posteriori observed motion field during the 60 h of nowcast. Figure 2.- Precipitation patterns 2:30 hours apart. The differential motion along the trajectories at the initial time leads to a prediction of the observed narrowing of the pattern. An example is given in Fig. 2, where at the initial time it can be seen within the circled area that the motion vectors at the rear are longer that at the front. At the later time the pattern has narrowed. This was Predictability of meteorological fields is scale dependent and the fundamental limit to predictability due to non-linearity of the processes governing precipitation is a difficult issue beyond the scope of this discussion. Here, the term predictability is used denote the limit of skill of the method of forecast used. In relation to LP we note that small cells evolve rapidly and therefore are less predictable that large scale systems. Haar wavelet scale decomposition was used to compare nowcasts of radar precipitation patterns and radar observed patterns at the nowcast time for

3 different scales. An analysis of four days of data is summarized in Figure 4. that the ensemble represents the probabilities as determined above. In addition, each member of the ensemble is a possible realization of the precipitation field that respects the statistical properties (such as mean, variance, decorrelation time and distance) of the observations made in the immediate past. Figure 4.- Life-time (as defined by the time to decorrelation to 1/e) for four days of data as a function of filtered out scales. Since the loss of predictability is faster the smaller the scale of the precipitation pattern, a scale filtering, increasing with leadtime, is needed to avoid attempts at predicting the unpredictable. An evaluation of the skill in precipitation detection of a particular algorithm of LP nowcast (MAPLE, McGill Algorithm for Precipitation nowcasting by Lagrangian Extrapolation), the filtered nowcast (O- MAPLE, Optimized MAPLE * ) as well as numerical model outputs is shown in Fig. 5. The small scales can be only predicted by a probabilistic method. One approach is nowcasting of fields of probability by LP. The fields of probability are determined in the immediate past and advected by the motion field. Another, more complete, possibility is the generation of ensembles of nowcasts so * The algorithm is described in three papers: Germann and Zawadzki (2002), Germann and Zawadzki (2004), Turner et al (2004). Figure 5.- skill scores of nowcasting by LP (unfiltered and filtered) compared to numerical model outputs. In relation to this there is another issue that requires a statistical approach. Nowcasting of precipitation intensity based on radar data relies on the transformation from reflectivity Z to rain rate R. The Z-R relationship has a strong stochastic component that is related to the physical processes responsible for the formation of different raindrop size distributions. These processes are not totally random (in the Gaussian sense). On the contrary, they are quite structured and correlated in space and time. Thus the differences in the fields of Z and

4 R are also structured with spatial and temporal scales of correlation. Figure 6.- A rain rate field obtained from Z to R transformation by a climatological Z-R relationship and four stochastic realizations of the R-filed that have the same mean, variance and correlation structure of the variability in the Z-R relationship around the climatological one. Figure 6 shows four plausible fields of rain rate that correspond to a single reflectivity filed. These four fields are members of an ensemble that incorporates the uncertainty in the Z-R relationship. To produce these realizations we analyze the fluctuations around a mean Z-R relationship as given by disdrometric observations. In particular we determine the correlation between Z and R, the variance of the fluctuations around a mean regression and the time structure function of these fluctuations. A self-afine cascade is then generated that respects the observed stochastic properties and the result is added to the deterministic R-field obtained by the transformation from Z to R using the mean Z-R regression. Each realization of the self-affine cascade produces a member of the ensemble. The average of the ensemble is the deterministic R-field. The procedure is similar to the one used by Bowler et al. (2004) to downscale model outputs in order to generate ensemble nowcasts. In the same manner one can generate an ensemble that contains as well other errors in radar measurements (notably extrapolation

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