## 2 Reflection Power Estimation Using Eigenbeam Spatial Correlation Model
### Methodology  
For determining the \(\gamma_{tau}(t,k|\mathbf{y}_{o})\) coefficients, we utilize the spatial correlation of higher-order spherical harmonic (eigenbeam) coefficients of the incident sound field. The estimation of the reflection power response involves two main steps:  
Step 1: Estimating spherical harmonic coefficients of the incident sound field  
In this work, since we are interested in characterizing the room response independent of the source power spectrum, we assume a sound source emitting an impulse signal and treat \(H(\mathbf{x}_{q},\mathbf{y}_{o},t,k)\) as the incident sound field on the spherical microphone array. For deducing the higher-order spherical harmonic coefficients of the incident sound field, we represent \(H(\mathbf{x}_{q},\mathbf{y}_{o},t,k)\) as the spherical harmonic decomposition of Helmholtz wave equation solution to the interior sound field problem [12] as  
\[H(\mathbf{x}_{q},\mathbf{y}_{o},t,k)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\alpha_{nm}(t,k |\mathbf{y}_{o})b_{n}(kR)Y_{nm}(\mathbf{\hat{x}}_{q}) \tag{7}\]  
where \(\alpha_{nm}(t,k|\mathbf{y}_{o})\) are the modal coefficients of the spatial sound field, \(\mathbf{\hat{x}}_{q}\) is the unit vector in the direction of the \(q\)th microphone, and  
\[b_{n}(kR)= \begin{cases}j_{n}(kR)&\text{for an open array}\\ j_{n}(kR)-\frac{j_{n}^{\prime}(kR)}{h_{n}(kR)}h_{n}(kR)&\text{for a rigid array}\end{cases}  \tag{8}\]  
with \(j_{n}(\cdot)\) and \(h_{n}(\cdot)\) denoting the spherical Bessel and Hankel functions of order \(n\), respectively, and \((\cdot)^{\prime}\) represents the first derivative operation. From (7), we can estimate \(\alpha_{nm}(t,k|\mathbf{y}_{o})\) coefficients using the orthogonal property of spherical harmonics [36] as  
\[\alpha_{nm}(t,k|\mathbf{y}_{o})=\frac{\sum_{q=1}^{Q}H(\mathbf{x}_{q},\mathbf{y}_{o},t,k)Y_ {nm}^{*}(\mathbf{\hat{x}}_{q})}{b_{n}(kR)} \tag{9}\]where \((\cdot)^{*}\) denotes the complex conjugation operation. Practically, we truncate \(\alpha_{nm}(t,k|\mathbf{y}_{o})\) to an order \(N\), such that \(N=\lceil kk\rceil\) and \(Q\geq(N+1)^{2}\), where \(\lceil\cdot\rceil\) denotes the ceiling operation, to avoid errors due to spatial aliasing and high-pass nature of higher-order Bessel functions [36]. Step 2: Estimating reflection gains using the spatial correlation model  
We can now estimate \(\gamma_{nu}(t,k|\mathbf{y}_{o})\) from the \(\alpha_{nm}(t,k|\mathbf{y}_{o})\) coefficients using the spatial correlation matrix expression [33] given by  
\[\underbrace{ \begin{bmatrix}\Lambda_{0000}\\ \Lambda_{001-1}\\ \vdots\\ \Lambda_{00 NN}\\ \Lambda_{1-100}\\ \vdots\\ \Delta_{NNNNN}\end{bmatrix} }_{\mathbf{A}(t,k|\mathbf{y}_{o})}=\underbrace{ \begin{bmatrix} \delta_{0000}&d_{00000}&\cdots&d_{0000 VV}\\ \delta_{001-1}&d_{001-100}&\cdots&d_{001-1 VV}\\ \vdots&\vdots&\vdots&\vdots\\ \delta_{0 NN}&d_{00 NN0}&\cdots&d_{00 NNVV}\\ \delta_{1-100}&d_{1-10000}&\cdots&d_{1-100 VV}\\ \vdots&\vdots&\vdots&\vdots\\ \delta_{NNNN}&d_{NNNN00}&\cdots&d_{NNNNVV}\end{bmatrix} }_{\mathbf{B}(k,\mathbf{y}_{o})} \times \begin{bmatrix}P_{\mathcal{D}}\\ \gamma_{00}\\ \gamma_{1-1}\\ \vdots\\ \gamma_{\mathcal{V}-V}\\ \vdots\\ \gamma_{\mathcal{V}V}\end{bmatrix}  \tag{10}\]  
where  
\[\Lambda_{nmur^{\prime}m^{\prime}}=\mathbb{E}\left\{\alpha_{nm}(t,k|\mathbf{y}_{o}) \alpha_{n^{\prime}m^{\prime}}^{*}(t,k|\mathbf{y}_{o})\right\} \tag{11}\]  
\[\delta_{nmur^{\prime}m^{\prime}}=16\pi^{2}i^{(n-n^{\prime})}Y_{nm}^{*}(\hat{ \mathbf{y}}_{o})Y_{n^{\prime}m^{\prime}}(\hat{\mathbf{y}}_{o}) \tag{12}\]  
\[d_{nmur^{\prime}m^{\prime}vu}=16\pi^{2}i^{(n-n^{\prime})}(-1)^{m}\sqrt{\frac{( 2v+1)(2n+1)(2n^{\prime}+1)}{4\pi}}W_{1}W_{2} \tag{13}\]  
with \(W_{1}=\left( \begin{array}{ccc}v&n&n^{\prime}\\ 0&0&0\end{array} \right)\) and \(W_{2}=\left( \begin{array}{ccc}v&n&n^{\prime}\\ u&-m&m^{\prime}\end{array} \right)\) representing the Wigner 3j symbols [37].