Multi-order analytical solving computation of rainstorm causal decomposition during typhoons using a designed key–lock quasi-Newton optimizing derivation

To precisely identify multi-dimensional spatiotemporal rain-making parameters, generate an approximate Hessian matrix, and solve the nonlinear ill-posed problem, this study uses composite logical tangent hyperbolic functions to construct the rain-generating simulation model as nonlinear algebraic equations with designed key–lock quasi-Newton optimization for deriving multi-order objective functional derivatives for rainstorm causal decomposition into advanced functional, analytical solution (lock) and Newton's conditional constraints. Specifically, the rank-two approximate structure of the Levenberg–Marquardt and Broyden–Fletcher–Goldfarb–Shanno quasi-Newton algorithms are modified as the symmetric rank-four structure to efficiently calculate a positive definite stable Hessian and solve the constrained nonlinear rain-making threshold. The model projects various rain-making factors into multi-rank loading scores, characterizing rain-generating mechanisms and causal components as associated DNAs. To accelerate/modify directional convergence, avoid local minimum, and detect global optimum, the devised vectorized limited switchable step sizes are optimized using advanced double-bracketing approaches combined with candidate parameters' correction vectors (key) and referenced step-size distributions solved by Newton's constrained analytical solution to reduce heterogeneous differences and eliminate the conventional overestimated Hessian. The identified rain-making DNAs reveal that typhoons with similar DNAs move in similar directions. Specifically, rain-making DNAs in Taipei Category 1 were correlated with wind force/direction and cloud height along PCs 1, 3, 4, and 7, and those in Category 2 were correlated with cloud-cover distribution along PCs 1, 2, and 5. The identified rain-making thresholds of typhoons with constant direction/structure showed a weaker steady state, whereas the unsteady rest produced multi-peak rainfall hydrographs. Rain evolution analysis reveals that cloudy rainbands, carried by the wind field, move along the Tamsui River valley when traveling between northeast and south-southeast of Taipei; converge with gradient and geostrophic winds when traveling between east-northeast and southwest; merge with southwest monsoon when traveling between west-southwest and northeast of Kaohsiung.