# Multiple-Penalty-Weighted Regularization Inversion for Dynamic Light Scattering
## 2 Dynamic Light Scattering and Multiple-Penalty-Weighted Regularization  
In DLS measurement, the scattering light intensity ACF at the scattering angle \(\theta\) can be described as follows:  
\[G_{\theta}^{(2)}\big{(}\tau_{j}\big{)}=\lim_{M\to\infty}\sum_{k=1}^{M}i_{ \theta}(\tau_{k})\cdot i_{\theta}\big{(}\tau_{k+j}\big{)}/M \tag{1}\]  
where \(G_{\theta}^{(2)}(\tau_{j})\) is the discrete ACF of the scattered light intensity, \(i_{\theta}(\tau_{k})\) the scattered light intensity at the delay time \(\tau_{k}\), \(\tau_{j}\) the discrete delay time, and \(M\) the total number of samples acquired by the correlator. The light intensity ACF is related to the normalized electric field ACF, \(g_{\theta}^{(1)}\big{(}\tau_{j}\big{)}\), by the Siegert relationship [33]  
\[G_{\theta}^{(2)}\big{(}\tau_{j}\big{)}=B\bigg{(}1+\beta(\theta\ )\Big{|}g_{ \theta}^{(1)}\big{(}\tau_{j}\big{)}\Big{|}^{2}\bigg{)} \tag{2}\]  
where \(B\) is the measured baseline of \(G_{\theta}^{(2)}\big{(}\tau_{j}\big{)}\), \(\beta\) (\(\leq\)1) is the instrumental coherence constant, and \(j\) (\(1\leq j\leq\) M) is the channel number of the correlator. The normalized electric field ACF has the form [8]  
\[g_{\theta}^{(1)}\big{(}\tau_{j}\big{)}=\sum_{i=1}^{N}\exp\Bigg{(}-\frac{16\pi k _{\ B}T\eta_{m}^{2}\sin\ ^{2}(\theta/2)}{3\eta\ d_{i}\lambda_{0}^{2}}\cdot\tau_{j}\Bigg{)}f(d_{i}) \tag{3}\]  
where \(d\), \(k_{B}\), \(T\), \(\lambda_{0}\), \(n_{m}\), \(\eta\), and \(f(d_{i})\) are the diameter of spherical particle, the Boltzmann constant, the absolute temperature of the medium, the wavelength of the incident light, the refractive index of the nonabsorbing suspending medium, the viscosity of the dispersing medium, and the discrete PSD, respectively. Equation (3) can be simplified as  
\[\mathbf{g}_{\theta}^{(1)}=\mathbf{A}_{0}\mathbf{f} \tag{4}\]where \(\mathbf{g}_{\theta}^{(1)}\) is a vector with elements \(g_{\theta}^{(1)}(\tau_{j}),f\) a vector with elements \(f(d_{i})\), and \(A_{\theta}\) a kernel matrix corresponding to the measured ACF at the scattering angle \(\theta\). The elements of \(A_{\theta}\) are given by  
\[A_{\theta}(j,i)=\exp\left(-\frac{16\pi k_{\ B}Tn_{m}^{2}\sin\ ^{2}(\theta/2)}{3 \eta\ d_{i}\lambda_{0}^{2}}\cdot\tau_{j}\right) \tag{5}\]  
Tikhonov regularization method for DLS inversion needs to minimize the objective function  
\[\mathbf{M}^{\alpha}(\mathbf{f},\mathbf{g}_{\theta}^{(1)})=\|\mathbf{A}_{ \theta}\mathbf{f}-\mathbf{g}_{\theta}^{(1)}\|^{2}+\alpha\|\mathbf{L}\mathbf{f }\|^{2} \tag{6}\]  
where \(\alpha\) is the regularization parameter, \(L\) the regularization matrix, \(f\) the estimated PSD, \(\|\cdot\|\)the Euclidean norm, and \(\|\mathbf{L}\mathbf{f}\|^{2}\) the penalty functional factor. Generally, the regularization parameter \(\alpha\) is determined by L-curve criterion [34; 35] to control the accuracy and stability of the solution, and the identity matrix is chosen as the regularization matrix. The weighted regularization inversion form [31; 32] of Equation (6) is  
\[\mathbf{M}_{\theta}^{\alpha}(\mathbf{f},\mathbf{g}_{W}^{(1)})=\|\mathbf{A}_{W }\mathbf{W}_{\theta}\mathbf{f}-\mathbf{g}_{W}^{(1)}\|^{2}+\alpha\|\mathbf{L} \mathbf{f}\|^{2} \tag{7}\]  
where \(\mathbf{W}_{\theta}\) is the weighted matrix, and  
\[\mathbf{g}_{W}^{(1)}=\left[ \begin{array}{c}\mathbf{W}_{\theta_{1}}\mathbf{g }_{\theta_{1}}^{(1)}\\ \mathbf{W}_{\theta_{2}}\mathbf{g}_{\theta_{2}}^{(1)}\\ \mathbf{W}_{\theta_{m}}\mathbf{g}_{\theta_{m}}^{(1)}\end{array} \right]= \left[ \begin{array}{c}k_{1}\mathbf{W}_{\theta_{1}}A_{\theta_{1}}\\ k_{2}\mathbf{W}_{\theta_{2}}A_{\theta_{2}}\\ k_{m}\mathbf{W}_{\theta_{m}}A_{\theta_{m}}\end{array}