With the accumulation of meteorological big data, data-driven models for short-term precipitation forecasting have shown increasing promise. We focus on Koopman operator analysis, which is a data-driven scheme to discover governing laws in observed data. We propose a method to apply this scheme to phenomena accompanying advection currents such as precipitation. The proposed method decomposes time evolutions of the phenomena between advection currents under a velocity field and changes in physical quantities under Lagrangian coordinates. The advection currents are estimated by kinematic analysis, and the changes in physical quantities are estimated by Koopman operator analysis. The proposed method is applied to actual precipitation distribution data, and the results show that the development and decay of precipitation are properly captured relative to conventional methods and that stable predictions over long periods are possible.

1. [1] J. R. Holton and G. J. Hakim, “An Introduction to Dynamic Meteorology,” 5th Edition, Academic Press, 2013.
2. [2] H. R. Pruppacher and J. D. Klett, “Microphysics of Clouds and Precipitation,” 2nd Edition, Springer, 1996.
3. [3] R. Kato, K. Shimose, and S. Shimizu, “Predictability of Precipitation Caused by Linear Precipitation Systems During the July 2017 Northern Kyushu Heavy Rainfall Event Using a Cloud-Resolving Numerical Weather Prediction Model,” J. Disaster Res., Vol.13, No.5, pp. 846-859, doi: 10.20965/jdr.2018.p0846, 2018.
4. [4] A. P. Khain et al., “Representation of Microphysical Processes in Cloud-Resolving Models: Spectral (Bin) Microphysics Versus Bulk Parameterization,” Reviews of Geophysics, Vol.53, No.2, pp. 247-322, 2015.
5. [5] B. Zhao and B. Zhang, “Assessing Hourly Precipitation Forecast Skill with the Fractions Skill Score,” J. of Meteorological Research, Vol.32, No.1, pp. 135-145, 2018.
6. [6] T. Miyoshi et al., “‘Big Data Assimilation’ Revolutionizing Severe Weather Prediction,” Bulletin of the American Meteorological Society, Vol.97, No.8, pp. 1347-1354, 2016.
7. [7] X. Shi et al., “Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting,” Proc. of the 28th Int. Conf. on Neural Information Processing Systems (NIPS’15), Vol.1, pp. 802-810, 2015.
8. [8] X. Shi et al., “Deep Learning for Precipitation Nowcasting: A Benchmark and a New Model,” Proc. of the 31st Int. Conf. on Neural Information Processing Systems (NIPS’17), pp. 5622-5632, 2017.
9. [9] S. Agrawal et al., “Machine Learning for Precipitation Nowcasting from Radar Images,” arXiv: 1912.12132, 2019.
10. [10] S. Rifai, Y. N. Dauphin, P. Vincent, Y. Bengio, and X. Muller, “The Manifold Tangent Classifier,” Proc. of the 24th Int. Conf. on Neural Information Processing Systems (NIPS’11), pp. 2294-2302, 2011.
11. [11] Y. Bengio, A. Courville, and P. Vincent, “Representation Learning: A Review and New Perspectives,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol.35, No.8, pp. 1798-1828, 2013.
12. [12] S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems,” Proc. of the National Academy of Sciences of the United States of America, Vol.113, No.15, pp. 3932-3937, 2016.
13. [13] S. H. Rudy, S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Data-Driven Discovery of Partial Differential Equations,” Science Advances, Vol.3, No.4, Article No.e1602614, 2017.
14. [14] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Machine Learning of Linear Differential Equations Using Gaussian Processes,” J. of Computational Physics, Vol.348, pp. 683-693, 2017.
15. [15] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations,” SIAM J. on Scientific Computing, Vol.40, No.1, pp. A172-A198, 2018.
16. [16] J. Berg and K. Nyström, “Data-Driven Discovery of PDEs in Complex Datasets,” J. of Computational Physics, Vol.384, pp. 239-252, 2019.
17. [17] I. Mezić, “Spectral Properties of Dynamical Systems, Model Reduction and Decompositions,” Nonlinear Dynamics, Vol.41, Nos.1-3, pp. 309-325, 2005.
18. [18] C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, “Spectral Analysis of Nonlinear Flows,” J. of Fluid Mechanics, Vol.641, pp. 115-127, 2009.
19. [19] I. Mezić, “Analysis of Fluid Flows via Spectral Properties of the Koopman Operator,” Annual Review of Fluid Mechanics, Vol.45, pp. 357-378, 2013.
20. [20] J. N. Kutz, J. L. Proctor, and S. L. Brunton, “Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems,” Complexity, Vol.2018, Article No.6010634, 2018.
21. [21] B. O. Koopman, “Hamiltonian Systems and Transformation in Hilbert Space,” Proc. of the National Academy of Sciences of the United States of America, Vol.17, No.5, pp. 315-318, 1931.
22. [22] J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz, “On Dynamic Mode Decomposition: Theory and Applications,” J. of Computational Dynamics, Vol.1, No.2, pp. 391-421, 2014.
23. [23] N. Takeishi, Y. Kawahara, and T. Yairi, “Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition,” Proc. of the 31st Int. Conf. on Neural Information Processing System (NIPS’17), pp. 1130-1140, 2017.
24. [24] J. Sesterhenn and A. Shahirpour, “A Characteristic Dynamic Mode Decomposition,” Theoretical and Computational Fluid Dynamics, Vol.33, Nos.3-4, pp. 281-305, 2019.
25. [25] N. Takahashi et al., “Development of Multi-Parameter Phased Array Weather Radar (MP-PAWR) and Early Detection of Torrential Rainfall and Tornado Risk,” J. Disaster Res., Vol.14, No.2, pp. 235-247, doi: 10.20965/jdr.2019.p0235, 2019,
26. [26] Geospatial Information Authority of Japan, GSI Tiles List, https://maps.gsi.go.jp/development/ichiran.html (in Japanese) [accessed April 13, 2022]