## 3 Ga-Gwo Hybrid Algorithm
### GA-GWO Hybrid Algorithm  
The GWO algorithm offers robust search capabilities; however, its convergence properties mean that the GA algorithm can effectively address the issue of local optima through genetic variations and selection operations. Additionally, the competitive behaviour among GWO's grey wolves facilitates a local search mechanism. Therefore, the two algorithms are mixed, through which the hybrid algorithm can better balance the global search and local search and take into account the solution needs of the global optimum and local optimum while optimising the hybrid algorithm:  
1. Piecewise chaotic mapping is used to initialise the total grey wolf population, and the expression of piecewise mapping is as follows: \[x(t+1)=\left\{ \begin{array}{c}\frac{x(t)}{p},0\leq x(t)<p\\ \frac{x(t)-p}{0.5-p},p\leq x(t)<0.5\\ \frac{1-p-x(t)}{0.5-p},0.5\leq x(t)<1-p\\ \frac{1-x(t)}{p},1-p\leq x(t)<1\end{array} \right.\] (21) \(P\in[0,1],\ p=0.4;x(1)=rand\). 2. Convergence factor and adjustment strategy \[a=\left\{ \begin{array}{c}a_{final}+(a_{initial}-a_{final})\times\cos^{ \Lambda}2\Big{[}\frac{t-1}{2(T-1)}*\pi\Big{]},t\leq 0.5T\\ a_{initial}-\Big{[}a_{final}+(a_{initial}-a_{final})\times\sin^{\Lambda}2( \frac{t-1}{2(T-1)}*\pi)\Big{]},0.5T\leq t\leq T\end{array} \right.\] (22) \(a_{final}=0;a_{initial}=2\); is the current iteration number, and T is the maximum iteration number. 3. Position update formula Proportional weights based on step size and Euclidean distance 128, 256, 512, and 512, respectively. Through four rounds of downsampling, the input river remote sensing image is condensed into a 32 pixels \(\times\) 32 pixels \(\times\) 512 size output image, effectively capturing the essential features of the river in the remote sensing image. \[L_{1}=\frac{|X_{1}|}{|X_{1}|+|X_{2}|+|X_{3}|}\] (23) \[L_{2}=\frac{|X_{2}|}{|X_{1}|+|X_{2}|+|X_{3}|}\] (24) \[L_{3}=\frac{|X_{3}|}{|X_{1}|+|X_{2}|+|X_{3}|}\] (25) \[X(t+1)=\frac{L_{1}X_{1}+L_{2}X_{2}+L_{3}X_{3}}{3}{(\frac{t-1}{T-1})}^{\lambda _{1}}+{(X_{1}\frac{t}{T})}^{\lambda_{2}}\] (26)  
included among these is \(\lambda_{1}\), \(\lambda_{2}\in rand[0,1]\). 4. GA parameter design based on hormone regulation Typically, the population evolution process is divided into two phases: (1) the early phase, which features high crossover and low mutation probabilities, promoting rapid population convergence and preserving desirable traits; and (2) the late phase, characterised by low crossover and high mutation probabilities, which enhance search refinement and maintain population diversity. However, traditional GAs have fixed crossover and mutation probabilities, making parameter tuning challenging and impacting optimisation outcomes. Consequently, an improved adaptive genetic algorithm inspired by the regulatory mechanisms of endocrine hormones, which feature ascending and descending functions, is proposed. Meanwhile, population diversity is also closely related to the selection operator, so it is necessary to set an adaptive selection threshold regulated by hormones. The fundamental law of endocrine hormones was proposed by [PERSON] [32] in 2004, which reveals that hormones have monotonic and non-negative change characteristics. The ascending regularity function \(F_{up}(G)\) and the descending regularity function \(F_{down}(G)\) of hormone regulation follow [PERSON]'s function law, which is expressed in the following equation:  
\[F_{np}(G)=\frac{G^{n}}{D^{n}+G^{n}} \tag{27}\]  
\[F_{down}(G)=\frac{D^{n}}{D^{n}+G^{n}} \tag{28}\]  
G is the function independent variable; D is the threshold value; D > 0; n is the Hill coefficient; and n 1; n and D determine the slope of the upward descent curve. If hormone x is regulated by hormone y, the relationship between the secretion rate \(S_{x}\) of hormone x and the concentration \(C_{y}\) of hormone y is as follows:  
\[S_{x}=\alpha F(C_{y})+S_{x0} \tag{29}\]  
\(S_{x0}\) is the basal secretion rate of hormone x, and \(\alpha\) is a constant coefficient.