## 2 Problem Formulation  
According to the compound-Gaussian model, vector \(\mathbf{c}\in\mathbb{C}^{N\times 1}\) represents the clutter in the cell under test (CUT), which can be denoted as  
\[\mathbf{c}=\sqrt{\tau}\mathbf{x} \tag{1}\]  
where \(\tau\) represents the slowly changing texture component, which is generally a non-negative random variable and \(\mathbf{x}\) represents the rapidly changing speckle component, following a circular symmetric complex Gaussian distribution with conditional covariance matrix \(\text{E}\big{[}\mathbf{x}\mathbf{x}^{H}|\tau\big{]}=\tau\mathbf{R}\), where \(\mathbf{R}\in\mathbb{C}^{N\times N}\) represents the unknown clutter covariance matrix. The training samples can be expressed as \(\mathbf{c}_{l}=\sqrt{\tau_{l}}\mathbf{x}_{l}\), \(l=1,2,\cdots,L\), whose conditional covariance matrix is \(\text{E}\big{[}\mathbf{x}_{l}\mathbf{x}_{l}^{H}|\tau_{l}\big{]}=\tau_{l} \mathbf{R}\), \(l=1,2,\cdots,L\).  
Assume that the radar system possesses \(N\) channels. If a signal exists, it can be expressed as \(N\times 1\) vector \(\mathbf{s}\). For subspace signals, assuming \(\mathbf{s}\) is located in the subspace \(\mathbf{H}\), it can be represented as \(\mathbf{s}=\mathbf{H}\xi\). Therefore, in the power heterogeneous clutter environment, the detection problem can be represented by the following binary hypothesis test  
\[\begin{split}\text{H}_{0}:\left\{\begin{array}{l}\mathbf{z} \sim\mathcal{CN}_{N}(0,\mathbf{R})\\ \mathbf{z}_{l}\sim\mathcal{CN}_{N}(0,\gamma_{l}\mathbf{R}),\ l=1,2,\cdots,L\\ \text{H}_{1}:\left\{\begin{array}{l}\mathbf{z}\sim\mathcal{CN}_{N}(\mathbf{ H}\xi,\mathbf{R})\\ \mathbf{z}_{l}\sim\mathcal{CN}_{N}(0,\gamma_{l}\mathbf{R}),\ l=1,2,\cdots,L \end{array}\right.\end{split}\right.\end{split} \tag{2}\]  
where \(\mathbf{z}\in\mathbb{C}^{N\times 1}\) is the data to be detected, \(\mathbf{z}_{l}\in\mathbb{C}^{N\times 1}\) represents the training samples, \(\mathbf{H}\in\mathbb{C}^{N\times p}\) is a known column-full-rank signal matrix, \(\xi\in\mathbb{C}^{p\times 1}\) denotes an unknown signal coordinate vector, \(\mathbf{R}\in\mathbb{C}^{N\times N}\) represents the unknown clutter covariance matrix, and \(\gamma_{1},\gamma_{2},\cdots,\gamma_{l}>0\) are unknown parameters that denote the different power mismatch amounts between each training sample and the data to be detected.  
As mentioned above, the detection model in (2) can be transformed from the compound-Gaussian model. Specifically, dividing all the clutter data by \(\sqrt{\tau}\) yields the clutter in the CUT as \(\mathbf{c}^{\prime}=\mathbf{x}\) with conditional covariance matrix \(\mathbf{R}\), where the value of \(\tau\) can be estimated by [24]. Meanwhile, the training samples can be rewritten as \(\mathbf{c}_{l}^{\prime}=\mathbf{c}_{l}/\sqrt{\tau}\). According to the definition of \(\mathbf{c}_{l}\) above, we set \(\gamma_{l}=\sqrt{\tau_{l}}/\sqrt{\tau}\), \(l=1,2,\cdots,L\), so the training samples can be written as \(\mathbf{c}_{l}^{\prime}=\gamma_{l}\mathbf{x}_{l}\), \(l=1,2,\cdots,L\) with conditional covariance matrix being \(\gamma_{l}\mathbf{R}\), \(l=1,2,\cdots,L\). In addition, this paper regards \(\gamma_{l}\), \(l=1,2,\cdots,L\) as definite unknown parameters, which can increase the applicability of this model; that is, it is not limited to a particular compound-Gaussian clutter with a predetermined texture distribution.