## 2 Modal Macro Strain (MMS)-Based Monitoring Strategy  
Modal macro strain within the long gauge sensor can be measured from the peak value of the power spectral density (PSD) of macro strain signals from distributed dynamic response when the pipeline is subjected to ambient excitation. The normalized modal macro strain vector can be used to identify damage efficiently. Generally, the excitation of pipeline structure is unknown, therefore, the macro strain frequency response function (FRF) extracted from the ratio between output macro strain signals and input signals is unknown. The macro strain FRF denoted by \(H_{mp}^{\overline{z}}(\omega)\) can be viewed as a ratio between the measurement from the \(m\)th sensor and the signal excitation \(p\)th degree of freedom (DOF), and can be acquired by:  
\[H_{mp}^{\overline{z}}(\omega)=\frac{\overline{\epsilon}_{\omega}(\omega)}{P_{ p}(\omega)}, \tag{1}\]where the superscript \(\overline{\epsilon}\) is macro strain, and \(P_{P}(\omega)\) is the signal excitation measured at the \(p\)th point. By studying the relationship between frequency response function and power spectral density, the power spectral density of macro strain in the dynamical response for \(m\)th long gauge sensor can be denoted as:  
\[S_{m}^{\overline{\epsilon}}(\omega)=\Big{|}H_{mp}^{\overline{\epsilon}}(\omega )\Big{|}^{2}S_{P}(\omega), \tag{2}\]  
where power spectral density of the excitation at the \(p\)th point can be denoted as \(S_{P}(\omega)\). The above equation can be represented as the following for a specific mode:  
\[S_{m}^{\overline{\epsilon}}(\omega_{r})=\Big{|}H_{mp}^{\overline{\epsilon}}( \omega_{r})\Big{|}^{2}S_{P}(\omega_{r}), \tag{3}\]  
\[\frac{\sqrt{S_{m}^{\overline{\epsilon}}(\omega_{r})}}{\sqrt{S_{P}(\omega_{r}) }}=A_{pr}|\delta_{mr}|, \tag{4}\]  
where, \(\delta_{mr}\) is the macro strain measured within the \(m\)th gauge length, and the corresponding macro strain modal constant is expressed as:  
\[A_{pr}=\frac{\phi_{pr}}{2M_{r}\xi_{r}\omega_{r}^{2}}, \tag{5}\]  
where \(\phi_{pr}\), \(M_{r}\), \(\xi_{r}\) and \(\omega_{r}\) respectively represent \(r\)th displacement mode shape at \(p\)th DOE, \(r\)th damping ratio, and \(r\)th natural frequency. Through extracting the macro strain response from the PSD of dynamic signal for every long-gauge macro strain sensor, the modal macro strain is normalized in regard to that in the supposed \(m\)th reference sensor, and it can be acquired by:  
\[\left[ \begin{array}{c}\frac{\delta_{lr}}{\delta_{mr}}\\ \frac{\delta_{2r}}{\delta_{mr}}\end{array} \right]=\left[ \begin{array}{c}\frac {\sqrt{S_{1}^{\overline{\epsilon}}(\omega_{r})}}{\sqrt{S_{m}^{\overline{ \epsilon}}(\omega_{r})}}\\ \frac{\sqrt{S_{2}^{\overline{\epsilon}}(\omega_{r})}}{\sqrt{S_{m}^{\overline{ \epsilon}}(\omega_{r})}}\\ \vdots\\ 1\\ \vdots\\ \frac{\delta_{mr}}{\delta_{mr}}\end{array} \right]=\left[ \begin{array}{c} \frac{\sqrt{S_{1}^{\overline{\epsilon}}(\omega_{r})}}{\sqrt{S_{m}^{\overline{ \epsilon}}(\omega_{r})}}\\ \frac{\sqrt{S_{2}^{\overline{\epsilon}}(\omega_{r})}}{\sqrt{S_{m}^{\overline{ \epsilon}}(\omega_{r})}}\\ \vdots\\ 1\\ \vdots\\ \frac{\sqrt{S_{2}^{\overline{\epsilon}}(\omega_{r})}}{\sqrt{S_{m}^{\overline {\epsilon}}(\omega_{r})}}\end{array} \right], \tag{6}\]  
The above equation means that the dynamic macro strain extracted from the power spectral density from all long-gauge sensors uniquely determines the structural modal macro strain. The peak values of power spectrum density of dynamic macro strain identify the natural frequencies and modal macro strain. The reference sensor is necessarily chosen for comparative study. For undamaged cases, the ratio of power spectral density at \(\omega_{i}\) of dynamic macro strain from target sensor to that of the reference sensor remains invariant. The ratio varies obviously if damage develops and occurs within the area that the target sensors cover.