# Estimation of the Noise Source Level of a Commercial Ship Using On-Board Pressure Sensors
## 2 Method and Validation  
### Methodology  
A simplified beamforming-based source-level estimation method [31] was used in the current study. It was assumed that, at a given frequency, the noise source can be represented by a single monopole source in a quiescent fluid. With \(M\) sensors at the hull, the measured sound pressure at the \(m\)-th sensor (\(y_{m}\)) can be calculated as  
\[y_{m}=\frac{e^{-\mathrm{i}\mathbf{k}r_{m}}}{r_{m}}s+e, \tag{1}\]  
where \(r_{m}\) is the distance between the source and the receiver, \(k=2\pi f/c\) is the wavenumber, \(s\) is the source strength of the monopole, and \(e\) is unwanted noise. As Equation (1) is defined in the frequency domain, \(y_{m}\) is the pressure converted from time-series data using the Fourier transform. It should be noted that the use of the free-space Green function could lead to overprediction of the source level at low frequencies, as the measured signal would be greater due to solid boundaries around the source. For \(M\) on-board sensors, Equation (1) can be written as  
\[\mathbf{y}=s\mathbf{a}+\mathbf{e}, \tag{2}\]  
where  
\[\mathbf{y}=\left[ \begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{M}\end{array} \right],\quad\mathbf{a}=\left[ \begin{array}{c}\frac{e^{- \mathrm{i}\mathbf{k}r_{1}}}{r_{2}}\\ \frac{e^{-\mathrm{i}\mathbf{k}r_{2}}}{r_{2}}\\ \vdots\\ \frac{e^{-\mathrm{i}\mathbf{k}r_{M}}}{r_{M}}\end{array} \right],\quad\mathbf{e }=\left[ \begin{array}{c}e_{1}\\ e_{2}\\ \vdots\\ e_{M}\end{array} \right]. \tag{3}\]  
In the general beamforming method, a scan vector would be made for a virtual sound source in the region of interest to calculate the maximum beamforming power with the measurement vector \(\mathbf{y}\). In this case, however, the source strength is not given. To resolve this, an assumed source strength \(E\) was introduced; the problem is thus changed into a minimization problem and is written as  
\[\min cost=\frac{1}{M^{2}}\|\mathbf{Y}-E^{2}\mathbf{A}\|_{F}^{2}. \tag{4}\]  
Here, \(\|\cdot\|_{F}\) is the Frobenius norm, \(\mathbf{Y}=(\mathbf{y}\mathbf{y}^{\mathrm{H}})\) is the correlation matrix of the measurement vector, and \(\mathbf{A}=(\mathbf{a}\mathbf{a}^{\mathrm{H}})\) is the correlation matrix of the propagation vector. The superscript 'H' indicates the conjugate transposition. The right-hand side of Equation (4) without \(M^{2}\) can be rewritten as  
\[\|\mathbf{Y}-E^{2}\mathbf{A}\|_{F}^{2}=\mathrm{tr}(\mathbf{Y}\mathbf{Y}^{ \mathrm{H}})-E^{2}\mathrm{tr}(\mathbf{Y}^{\mathrm{H}}\mathbf{A})-E^{2} \mathrm{tr}(\mathbf{Y}\mathbf{A}^{\mathrm{H}})+E^{4}\mathrm{tr}(\mathbf{A} \mathbf{A}^{\mathrm{H}}). \tag{5}\]  
Here, the operator '\(\mathrm{tr}\)' is the trace of a matrix. The virtual source strength \(E\) can be obtained by solving Equation (6), i.e.,:  
\[E=\sqrt{\frac{\text{tr}(\mathbf{A}^{\text{H}}\mathbf{Y})}{\text{tr}(\mathbf{A}^{ \text{H}}\mathbf{A})}}. \tag{7}\]  
Substituting Equation (7) into Equation (5) yields  
\[\|\mathbf{Y}-E^{2}\mathbf{A}\|_{F}^{2}=\text{tr}(\mathbf{Y}^{\text{H}}\mathbf{ Y})-\frac{\text{tr}(\mathbf{A}^{\text{H}}\mathbf{Y})^{2}}{\text{tr}( \mathbf{A}^{\text{H}}\mathbf{A})}. \tag{8}\]  
As the first term on the right-hand side of Equation (8) depends only on the measurement, the second term should be maximized to minimize the cost function. In other words, the minimization problem can be seen as a maximization problem, which can be expressed as  
\[\max cost=\frac{\text{tr}(\mathbf{A}^{\text{H}}\mathbf{Y})^{2}}{\text{tr}( \mathbf{A}^{\text{H}}\mathbf{A})}. \tag{9}\]  
For simplicity of notation, the cost function in Equation (9) is replaced with \(\alpha\) in the rest of the paper. The meaning of Equation (8) can be explained by the following example. Assume that a single monopole source is placed at \(\mathbf{q}\) in a two-dimensional space, which is shown in Figure 1. The sound pressure is measured by an array consisting of two sensors. A virtual source region is set with two points, one of which is the true source location.