## 2 Methods
### Root Mean Square Propagation Algorithm
The RMSprop algorithm uses the idea of the exponential weighted moving average (EWMA) method to calculate the exponential mean of the latest \(h\) iterations of the derivative squared of various tissues, as shown in the following:  
\[\theta_{s,k}=\alpha\theta_{s-1,k}+(1-\alpha)g_{s,k}^{2}, \tag{21}\]  
\[h=\frac{1}{1-\alpha}. \tag{22}\]  
In Equation (21), \(\theta_{s,k}\) is the exponential mean of the derivative squared of the \(k\)th tissue in the \(s\)th iteration. \(\theta_{s-1,k}\) is the exponential mean of the derivative squared of the \(k\)th tissue in the \((s-1)\)th iteration. \(g_{s,k}^{2}\) represents the derivative squared of the \(k\)th tissue in the \(s\)th iteration. \(\alpha\) represents the hyperparameter. \(h\) is the number of historical derivatives required to calculate the average value of the index. Based on the idea of the EWMA method, Equation (21) weights the derivative squared of the latest \(h\) iterations to control the contribution of the historical derivative squared to the exponential mean. The closer the derivative squared is to the current iteration, the greater the weight it multiplies. According to the weight size, the data information about derivative squared is extracted. The exponential mean obtained from Equation (21) provides reference data for the adaptive adjustment of step length. In addition, the exponential mean calculated by the RMSprop algorithm in the early iteration may be inaccurate. To solve this problem, Equation (23) is applied to correct the deviation of the exponential mean to improve the accuracy of the early exponential mean. \[\hat{\theta}_{s,k}= \begin{cases}\frac{\theta_{s,k}}{1-\alpha^{ \frac{1}{2}}},&s<h,\\ \frac{\theta_{s,k}}{1-\alpha^{\frac{1}{2}}},&s\geq h.\end{cases}  \tag{23}\]  
In Equation (23), \(\hat{\theta}_{s,k}\) is the corrected value of the exponential mean of the \(k\)th tissue in the \(s\)th iteration. The corrected value of the exponential mean of each tissue is calculated by Equation (23), which can provide the weight vector \(\mathbf{w}\) for adaptively adjusting the step length of the sound speed of each tissue. \[\mathbf{w}=\left[\frac{1}{\sqrt{\theta_{s,1}}},\ \frac{1}{\sqrt{\theta_{s,2}}}, \cdots,\ \frac{1}{\sqrt{\theta_{s,k}}},\cdots,\ \frac{1}{\sqrt{\theta_{s,k}}} \right]^{T}. \tag{24}\]  
The global step length is set to \(\lambda_{2}\), and its initial value is \(\lambda_{initial}\). To avoid the sound speed parameter producing a small range of oscillation in the later stage of iteration, the global step length slowly decays with iteration. \[\lambda_{2}=\lambda_{initial}\cdot\eta^{s}. \tag{25}\]  
where \(\eta\) is the attenuation coefficient of global step length. Along the direction of the negative gradient vector, various tissues use the corresponding step length to calculate the updated sound speed as follows:  
\[\mathbf{V}_{s+1}=\mathbf{V}_{s}-\lambda_{2}\mathbf{w}\odot\
abla\mathbf{F}(\mathbf{V}_{s}). \tag{26}\]  
where the \(\odot\) symbol is the Hadamard product, which refers to the new vector obtained by multiplying the elements at the corresponding positions of two vectors. The RMSprop algorithm can balance the updated amplitude of sound speed by adaptively adjusting the step length of the sound speed of each tissue. It alleviates the interaction between various tissues in the process of updating sound speed. If a tissue with a large proportion reaches the convergence state, the small range change of its sound speed value may still have an impact on other tissues. To solve this problem, the sound speed values of several successive iterations after this tissue convergence can be selected to create an average, and the average value is taken as the optimal solution of this tissue's sound speed. The sound speed value of this tissue is fixed as the optimal solution in the subsequent iterations to ensure the stability of the convergence process for other tissues. In summary, the proposed algorithm optimizes the inhomogeneous sound speed field iteratively through two steps.