# Numerical Analysis of Ultrasonic Multiple Scattering for Fine Dust Number Density Estimation
## 2 Theoretical Background on Multiple Scattering of Acoustic Waves
Considering that Equation (1) is implicit, the exact closed-form solution for \(\mathrm{G}(\omega,\ \mathbf{r},\mathbf{r}^{\prime})\) is not available. When the distance (\(|\mathbf{r}-\mathbf{r}^{\prime}|\)) is much shorter than \(l_{S}\), \(\mathrm{G}(\omega,\ \mathbf{r}_{1},\mathbf{r}^{\prime})\) in Equation (1) can be approximated as \(G_{0}(\omega,\mathbf{r}_{1}-\mathbf{r}^{\prime})\), which is called the Born approximation. However, as the distance (\(|\mathbf{r}-\mathbf{r}^{\prime}|\)) increases, the Born approximate solution diverges from the actual solution. To obtain an accurate solution that considers the multiple scattering effects, the integral equation (Equation (1)) needs to be expanded to nearly infinite orders with respect to \(G_{0}\), which is computationally intractable. Instead of seeking an exact solution for Equation (1), a statistical solution that is computationally tractable can be sought. Taking ensemble average responses, the mean Green's function \(\langle\mathrm{G}(\omega,\ \mathbf{r}-\mathbf{r}^{\prime})\rangle\) can be obtained as follows:  
\[\big{\langle}\mathrm{G}\big{(}\omega,\ \mathbf{r}-\mathbf{r}^{\prime}\big{)}\big{\rangle} =G^{0}\big{(}\omega,\mathbf{r}-\mathbf{r}^{\prime}\big{)}+\int\ G_{0}( \omega,\mathbf{r}-\mathbf{r}_{1})\Sigma(\omega,\mathbf{r}_{1}-\mathbf{r}_{1})\big{\langle} \mathrm{G}\big{(}\omega,\ \mathbf{r}_{2}-\mathbf{r}^{\prime}\big{)}\big{\rangle}\,d\mathbf{r}_{1}d\mathbf{r}_{2}, \tag{3}\]  
where the notation \(<\) \(>\) represents ensemble averaging and \(\Sigma\) is the self-energy that contains information about multiple scattering of acoustic waves. Equation (3) is known for the Dyson equation [17]. Note that the mean Green's function \(<\)G\(>\) shown in Equation (3) is a nonlocal (translation-invariant) operator, similar to \(\mathrm{G}_{0}\), unlike the Green's function \(\mathrm{G}\) shown in Equation (1). Then, the mean Green's function in the \(\mathbf{k}\) domain is given by  
\[\langle\mathrm{G}(\omega,\mathbf{k})\rangle\,=\frac{1}{{k_{0}}^{2}(\omega)-\Sigma (\omega,\mathbf{k})-k^{2}}. \tag{4}\]  
A comparison between Equations (2) and (4) reveals that the self-energy modifies the dispersion relation resulting in a modified wave speed and attenuation in the mean wavefield. Numerous multiple scattering theories have been proposed to model the self-energy term and hence to compute the effective wavenumber (\(k_{e}\)) [18; 19; 20; 21; 22]. In this study, we incorporate the independent scattering approximation (ISA) method [22] to estimate the number density of fine dust in the air from the measured acoustic wave responses. In the ISA method, each scatterer is assumed to be uncorrelated. The self-energy \(\Sigma\) is then expressed by the product of the scattering contribution of one scatterer and the number density (\(n\)) of scatterers  
\[\Sigma(\omega,\mathbf{k})=n\mathbf{T}_{\mathbf{k},\mathbf{k}}, \tag{5}\]  
where \(\mathbf{T}_{\mathbf{k},\mathbf{k}}\) is the T-matrix that describes the contribution of a single scatterer [23]. Within a certain range of frequencies where the wavelength is not much larger than the scatterers, the self-energy becomes independent of \(\mathbf{k}\). In this regime, the random medium can be considered a quasi-homogeneous medium (effective medium) with an effective wavenumber \(k_{e}\) given by  
\[{k_{e}}^{2}(\omega)={k_{0}}^{2}(\omega)-\Sigma(\omega). \tag{6}\]  
Note that \(k_{e}\) is a complex number whose real part is related to the phase velocity of the effective medium and imaginary part the attenuation. An important characteristic distance for coherent wave propagation is the mean free path \(l_{S}\) given by  
\[l_{s}=\frac{1}{2\text{Im}(k_{e})}=\frac{1}{\alpha(\omega)}, \tag{7}\]  
where \(\alpha\) is the attenuation of the coherent wave. Within a few mean free paths, the coherent wave propagates with the speed of \(\omega/\text{Re}(k_{e})\) and decays exponentially with attenuation \(\alpha\). Here, the coherent wave attenuation is caused by scattering.