**4. The importance of rapid-update Kalman filtering for the local torrential rainfall**

This section provides an explanation of the importance of rapid-update Kalman Filtering for the local torrential rainfall. Here, the "local" refers to a special scale of the order of 10 km or less.

#### *On the Use of the Ensemble Kalman Filter for Torrential Rainfall Forecasts DOI: http://dx.doi.org/10.5772/intechopen.107916*

In the first place. The weather system has strong nonlinearity, and the error grows exponentially as the time progresses. Moreover, the weather system has multiscale structure. Large-scale phenomena such as jet stream and have slower error growth, whereas the smaller special scale phenomena have the faster error growth. In torrential rainfalls, cumulus convection plays a main part, and it is the key issue that how to accurately estimate the state of cumulus convection. A relation between special scale and predictive skill is well described in Forecast User Guide by ECMWF [27]. According to **Figure 7(b1)** in the document by ECMWF [27], cumulus convection has 1-km order special scale, and it develops within 10 minutes; thus, fine temporal resolution is required to capture the developing process of the cumulus convection. Recently, sensing technologies have been exploring, and they also have been applied to weather observation instruments, such as radars or satellites. An example of the observation of the cumulus convection has already shown in **Figure 6**. Here, let us see the rapid growth of a convective system in **Figure 6**, carefully. At 0805 JST, only weak radar echo is found. It is a signal of the initiation of rainfall system (**Figure 6a**). At this time, the rainfall intensity is about only 1 mm h−1. The minute-by-minute observation appears to be continuously changing (**Figure 6a**–**d**). However, this system rapidly evolved within 10 minutes, an active convection having the intense precipitation was formed by 0815 JST (**Figure 6f**).

As mentioned in Section 1, the data assimilation has become an important technique, and Kalman filter is widely applied to convective-scale weather forecasts. The important issue in Ensemble Kalman filter is how to set the length of the data assimilation window. In the first place, Kalman filter assumes linear theory. To apply it to the nonlinear system such as NWP model, we assume that a linear approximation holds within one data assimilation window. So, setting too long assimilation window will destroy the assumption of a linear approximation, resulting in a significant reduction in estimation accuracy. It is repeatedly mentioned that an active convection system, which brings a torrential rainfall, rapidly develops within 10 minutes or so [27] (**Figure 6**). Thus, short-time data assimilation window should be taken in case of applying the ensemble Kalman Filter to torrential rainfall forecast. Here, how long should the data assimilation window be short? Now, let us see **Figure 6** again. Even though the convective cloud is rapid growth system, it can be considered linear growth within a few minutes range. Of course, the shorter the time progress, the more linearity is ensured. From the perspective of Ensemble Kalman Filter, the shorter data assimilation window is taken, the higher accuracy is expected, because the Kalman Filter assumes linear theory. On the other hand, the computational cost will explosively increase by the very short window; however, this issue has been overcome with the innovation of supercomputers. In the previous sections, the author reported a 30-second-update data assimilation experiments for a torrential rainfall event by the previous Japanese flagship supercomputer "K" based on the contents of a published paper. Recently, more powerful supercomputers, such as the current Japanese flagship supercomputer "Fugaku," are operated all over the world. With the development of supercomputers, more accurate NWP system will be performed in operation.
