**1. Introduction**

The torrential rainfall is becoming one of the most significant treats in worldwide human society year by year. It often brings severe disaster in social infrastructure or human lives. In Eastern Asia, the disasters caused by heavy rainfalls are repeated as every year. In Japan, many rainfall events have brought severe disasters especially in summer season. Among them, in September 2015, active rainband maintained in eastern part of Japan for several days, and severe flood happened. Eventually, it caused 20 fatalities and more than 7000 house damages [1]. In July 2018, a record-breaking and catastrophic torrential rainfall occurred over a wide area of western Japan. According to a report by Fire and Disaster Management Agency, Japan, it caused 263 fatalities, 484 injuries, and more than 30,000 home damages [2]. In another case, the monsoon

front so-called Baiu front (in Japanese) or Meiyu front (in Chinese) remained stagnant for long time in July 2020, large amounts of water vapor tended to flow into the Japan islands. This caused record-breaking rainfall over a wide area of Japan. Especially in Kumamoto prefecture located southwest part of Japan, floods of unprecedented magnitude occurred [3–6]. Besides these events, sudden local torrential rainfalls also often happened in summer season. On September 11, 2014, an isolated cumulonimbus suddenly developed within 10 minutes, and over 50 mm h−1 intense rainfall was observed, even though it was sunny until then. Recently, even in Europe, where climatological annual rainfall amount is relatively low, less than 1000 mm per year, the occurrence of torrential rains is not unusual. For example, the ECMWF reported extreme rain in Germany and Belgium in July 2021 [7].

To reduce disasters caused by severe rainfall events, a development of an accurate numerical weather prediction (NWP) system is essential. To improve the forecast accuracy, more accurate initial conditions are needed. Accurate initial values can be obtained through advanced initial value creation methods that maximize the use of information from observations, and data assimilation plays a central part in it. In modern NWP system, variational method and Kalman Filter, especially Ensemble Kalman filter [8], are widely applied. Although both schemes have their advantages and disadvantages, Ensemble Kalman Filter is generally superior to implementation and maintenance [9]. The reason why Ensemble Kalman filter rather than the traditional Kalman filter is applied in the NWP is the computational cost. It is quite difficult to apply the traditional Kalman filter technique due to the huge computational cost mainly in the estimation of error covariance matrix. The NWP models has very large degree of freedom, and it usually consists over 107 grid points and over 10 variables. On the other hand, in Ensemble Kalman Filter, since error covariance matrix is estimated from the ensembles of the order of 10–100, it is expected to significantly reduce computational costs. In this chapter, the author reports recent studies of improving the torrential rainfall forecasts. In Section 2, an impact of dense surface data assimilation on the forecast of band-shaped rainfall zone is presented. In Sections 3 and 4, the importance of rapid-update data assimilation for a local torrential rainfall having the order of 10 km or less and a practical application for the forecast are presented. In Section 5, the conclusion is described.
