# A Forward\(-\)Backward Splitting Equivalent Source Method Based on S\(-\)Difference
## 2 Forward-Backward Equivalent Source Method Based on S\(-\)Difference  
### Basic Principles of Equivalent Source Method  
According to the basic principle of equivalent source method near-field acoustic holography, the sound field arrangement is shown in Figure 1, which includes the arrangement of the microphone array, sound source, equivalent source surface, and reconstructed surface. The equivalent source surface can be either a plane or a surface surrounding the sound source. In the equivalent source method, no a priori knowledge is required to use planar equivalent source surfaces. Because the propagation direction of the sound waves on the plane equivalent source surface is known, the equivalent source method can decompose the real sound source on the plane equivalent source surface into a superposition of multiple equivalent sound sources, and the positions and sound pressures of these equivalent sources can be determined more easily. However, to adopt the spherical equivalent source surface, it is necessary to know the location of the sound source in advance and set the spherical equivalent surface at the location of the sound source. Compared with the spherical equivalent source surfaces, the use of planar equivalent source surfaces can make the simulation and modeling more flexible. In this paper, a plane equivalent source surface consisting of a rectangular grid will be used.  
Assuming that \(N\) equivalent sources are distributed on the equivalent source plane and \(M\) microphones are placed on the holographic plane, the sound pressure vector at the \(m\)-th measuring point can be expressed as:  
\[p(m)=\sum_{n=1}^{N}g(r_{m}\mid r_{n})q_{n} \tag{1}\]  
where \(g(r_{m}\mid r_{n})\) represents [PERSON]'s function in the free field. The transfer function from the NTH equivalent source to the MTH measurement point can be expressed as:  
\[g(r_{m}|r_{n})=\frac{e^{-jk\|\bar{r}_{m}\|}}{4\pi\|\bar{r}_{m}\|} \tag{2}\]  
where \(k=2\Pi f/c\) is the wave number (unit: rad/m). \(f\) is frequency, \(c\) is velocity, \(j^{2}=-1\), and \(\|\bar{r}_{m}\|\) is the distance from the equivalent source to the microphone and is the determining factor in the relationship between holographic surface sound pressure, equivalent source strength, and transfer matrix, which can be expressed in matrix form:  
\[\text{p}=\text{Gq} \tag{3}\]  
where \(\text{G}(M\times N)\) is the transfer matrix between the equivalent source plane and the holographic plane, \(\text{p}(M\times 1)\) is the complex sound pressure column vector measured by four rows of microphone array, and \(\text{q}(N\times 1)\) is the equivalent source strong column vector.  
In practical engineering applications, the number of measurement points is often limited by measurement conditions and costs, resulting in fewer measurement points than equivalent sources, that is, \(M<<N\); this disparity leads to ill\(-\)posed problems when solving Equation (3) for the inverse problem. This inverse problem is flawed because it attempts to reconstruct the evanescent wave part that is away from the sound source surface. The noise present in the measurement process can affect the accuracy of the results,  
Figure 1: Schematic diagram of the principle of the equivalent source method. (A represents the distance from the equivalent source surface to the reconstruction surface, and G represents the distance from the sound source surface to the holographic surface).  
so a regularization method was used to overcome this limitation during the solving process, which helps to improve the accuracy of sound field reconstruction.