## 3 Augmented Tikhonov Regularization Method  
The augmented Tikhonov regularization is an application of the Bayesian principle in the field of inverse problems. The basic idea of the Bayesian theory is to obtain the posterior probability density function through the likelihood function and the prior probability density function, and then estimate the unknown quantity according to the posterior probability density. In the model \(\mathbf{y}=\mathbf{H}\mathbf{f}\) mentioned above, the output signal \(\mathbf{y}\) will always contain some noise \(\mathbf{\mu}\). \[\mathbf{y}=\mathbf{H}\mathbf{f}+\mathbf{\mu}. \tag{9}\]  
According to the Bayesian principle, the posterior probability density function of the unknown quantity \(\mathbf{f}\) can be obtained:  
\[p(\mathbf{f}\big{|}\mathbf{y})\propto p(\mathbf{f})p(\mathbf{y}\big{|}\mathbf{f}), \tag{10}\]  
where \(p(\mathbf{f}\big{|}\mathbf{y})\) is the posterior probability density function, while \(p(\mathbf{f})\) is the prior probability density function. The variable \(p(\mathbf{y}\big{|}\mathbf{f})\) is the likelihood function. Thus, the loads \(\mathbf{f}\) can be estimated by the maximum posterior probability density function:  
\[\hat{\mathbf{f}}_{map}=\arg\max_{\mathbf{y}}p(\mathbf{f}\big{|}\mathbf{y}). \tag{11}\]  
The likelihood function represents the probability density function of \(\mathbf{y}\) when the random variable \(\mathbf{f}\) is given. The form of the likelihood function is generally determined by the noise \(\mu\). Usually, the white Gaussian noise with a zero mean and a standard deviation of \(\sigma\) can provide a proper representation for \(\mu\). The noise likelihood function can then be represent as  
\[p(\mathbf{y}\big{|}\mathbf{f})\propto(\sigma^{2})^{-n/2}\exp(-\frac{\|\mathbf{H}\mathbf{f}- \mathbf{y}\|_{2}^{2}}{2\sigma^{2}}), \tag{12}\]  
where \(n\) is the data length of \(\mathbf{y}\). The above probability density function needs to be selected according to the empirical method. In this study, Markov random field is selected as the prior model. \[p(\mathbf{f})\propto\exp(-\sum_{i\sim j}W_{ij}\phi(\gamma(f_{i}-f_{j}))), \tag{13}\]  
where \(\gamma\) is a magnification factor, while \(\phi(\gamma(f_{i}-f_{j}))\) is a non-negative function that determines the specific form of this Markov field, and \(\mathbf{W}_{ij}\) is a non-negative weight coefficient. The function \(\phi\) is then set to the form \(\phi(t)=1/2t^{2}\), so that the Markov random field is  
\[p(\mathbf{f})\propto\lambda^{m/2}\exp(-\frac{1}{2}\lambda\mathbf{f}^{T}\mathbf{W}\mathbf{f}), \tag{14}\]  
where the variable \(\lambda\) denotes the scaling parameter, and \(m\) the dimension of vector \(\mathbf{f}\). \(\mathbf{W}\) represents the symmetric matrix composed of \(\mathbf{W}_{ij}\). The posterior probability density function can be obtained by substituting Equations (12) and (14) into Equation (10), and the following is obtained:  
\[p(\mathbf{f}\big{|}\mathbf{y})\propto(\sigma^{2})^{-n/2}\lambda^{m/2}\exp(-\frac{\| \mathbf{H}\mathbf{f}-\mathbf{y}\|_{2}^{2}}{2\sigma^{2}})\exp(-\frac{1}{2}\lambda\mathbf{f}^{T} \mathbf{W}\mathbf{f}). \tag{15}\]In the Bayesian hierarchical model, the two unknown parameters \(\lambda,\sigma\) in Equation (15) can be regarded as random variables; hence, they play the role of regularization parameters. For convenience, we set \(\tau\!=1/\sigma^{2}\). The prior distributions of \(\lambda,\tau\) are assumed to be a Gamma distribution. \[p(\lambda)=\frac{\beta_{1}^{\alpha_{1}}}{\Gamma(\alpha_{1})}\lambda^{\alpha_{1 }-1}\exp(-\beta_{1}\lambda), \tag{16}\]  
\[p(\tau)=\frac{\beta_{2}^{\alpha_{2}}}{\Gamma(\alpha_{2})}\tau^{\alpha_{2}-1} \exp(-\beta_{2}\tau), \tag{17}\]  
where \(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}\) are the parameters of the Gamma distribution. Meanwhile, due to the introduction of two new random variables, the likelihood function and the prior probability density function are different from the previous ones.