# Acoustic Shooting and Bounce Ray Method for Calculating Echoes of Complex Underwater Targets
## 4 Dimension Reduction by Time Domain Integration  
In general, many plane or surface integrals are required to obtain the time-domain results of the scattered field, which has high computational complexity. Thus, it needs to be accelerated by integration. Taking the single-frequency signal as an example, the expression of the incident acoustic wave without considering the initial phase is \(P_{i}=\exp\left(i2\pi f_{0}t\right)\). Considering the motion of the target, without calculating the micro-Doppler generated by the projection relation, etc., the form of the target echo is \(P_{r}=\exp\left(i2\pi f_{0}t\right)\), where:  
\[ \begin{array}{l}f_{d}=f_{0}\sigma\\ \sigma=\frac{1+V_{m}/C}{1-V_{m}/C}\end{array}  \tag{13}\]  
\(f_{d}\) represents the Doppler center frequency of the target echo signal; \(f_{0}\) represents the center frequency of the incident acoustic signal; \(\sigma\) stands for the Doppler factor; \(V_{m}\) represents the relative projection velocity of the target; and \(C\) is the speed of sound in water. Then, the surface integral part in Equation (12) can be expressed as follows:  
\[ \begin{array}{l}\iint\limits_{S_{m}}\left(P_{i}(\mathbf{r}_{0},t-\frac{r-2z^{ \prime}}{c}-\Delta t)-P_{i}(\mathbf{r}_{0},t-\frac{r-2z^{\prime}}{c}+\Delta t) \right)cos\phi_{m}dS\\ =(e^{i2\pi f_{0}(t-\frac{r}{c}-\Delta t)}-e^{i2\pi f_{0}(t-\frac{r}{c}+\Delta t )})cos^{2}\phi_{m}\iint\limits_{S_{m}}e^{i2\pi f_{0}(\frac{2z^{\prime}}{c})} dS\end{array}  \tag{14}\]As shown in Figure 1, considering that \(z^{\prime}\) is the modulus value of the vector projected by \(\mathbf{r^{\prime}}\) to \(\mathbf{r}\), and according to the definition of vector \(\mathbf{r_{0}}\) and \(\mathbf{x}\), \(z^{\prime}\) can be expressed as:  
\[z^{\prime}=(\mathbf{r_{0}}+\mathbf{x})\cdot\frac{\mathbf{r}}{|\mathbf{r}|}=(\mathbf{r_{0}}+\mathbf{x}) \cdot\frac{\mathbf{r}}{r} \tag{15}\]  
Substitute Equation (14) into Equation (13), where the surface integral part is rewritten as:  
\[\iint\limits_{S_{m}}e^{2\pi f_{0}(\frac{2\cdot d^{\prime}}{c})}dS=e^{i4\pi \frac{f_{0}}{c}(\mathbf{r_{0}}\cdot\frac{\mathbf{r}}{r})}\iint\limits_{S_{m}}e^{i4\pi \frac{f_{0}}{c}(x\cdot\frac{\mathbf{r}}{r})}dS \tag{16}\]  
Let \(k=2\pi f_{0}/c\) and \(K=2\ k\), and let \(\mathbf{u}\) be the projection vector of vector \(\mathbf{r}/r\) on \(S_{m}\), namely: \(\mathbf{u}=\mathbf{r}/\mathbf{r}-(\mathbf{r}/\mathbf{r}\mathbf{\cdot}\mathbf{n})\mathbf{n}\). According to the vector relationship, we can know: \(\mathbf{x}\cdot\mathbf{u}=\mathbf{x}\cdot[\mathbf{r}/r-(\mathbf{r}/r\cdot\mathbf{n})\mathbf{n}]=\mathbf{x} \cdot\mathbf{r}/\mathbf{r}\). After \(\mathbf{x}\cdot\mathbf{r}/\mathbf{r}\) in Equation (16) is replaced by \(\mathbf{x}\cdot\mathbf{u}\), the [PERSON] formula condition is satisfied, then the final dimensionality reduction form of Equation (12) is:  
\[ \begin{split} P_{ss}(\mathbf{r},t)&=-\frac{1}{4\pi rc \Delta t}\sum\limits_{m=1}^{M}\Gamma_{m}(\phi_{m})\iint\limits_{S_{m}}(P_{i}( \mathbf{r}_{0},t-\frac{r-2z^{\prime}}{c}-\Delta t)-P_{i}(\mathbf{r}_{0},t-\frac{r-2z ^{\prime}}{c}+\Delta t))cos\phi_{m}dS\\ &=-\frac{1}{4\pi rc\Delta t}\sum\limits_{m=1}^{M}\Gamma_{m}( \phi_{m})(e^{i2\pi f_{0}(t-\frac{r}{c}-\Delta t)}-e^{i2\pi f_{0}(t-\frac{r} {c}+\Delta t)})cos^{2}\phi_{m}e^{iK(\mathbf{r}_{0}\cdot\frac{\mathbf{r}}{r})}\\ &\cdot\frac{i}{Ku^{2}}\sum\limits_{n=1}^{N}(\mathbf{u}^{*}\cdot \Delta a_{n})sinc(Ku\cdot\Delta a_{n}/2)exp[iKu\cdot(\mathbf{a}_{n+1}+\mathbf{a}_{n})/2 ]\end{split}  \tag{17}\]  
For the specific derivation of Equation (16), please refer to [PERSON]'s integral derivation process [9]. When Equation (16) is used to solve the target time-domain scattered sound field, only the discrete sum calculation is required instead of the two-dimensional integration, which greatly reduces the computational complexity. For the LFM signal:  
\[P_{i}=\exp(i\pi(ft+Kt^{2}/2) \tag{18}\]  
where \(K\) is the frequency modulation slope.