## 3 Proposed Model  
### Detection for an 8-Way GPR Target  
In Figure 6, the output of equalizer \(z[j,k]\) is presented as follows:  
\[z[j,k]=d[j,k]+w_{\mathbf{F}}[j,k]=\sum_{n=-1}^{1}\sum_{m=-1}^{1}a[j-n,k-m]g_{n,m}+w_{\mathbf{F}}[j,k], \tag{5}\]  
where \(w_{\mathbf{F}}[j,k]\) is the color noise, which is minimized in the MMSE algorithm. Therefore, we ignore \(w_{\mathbf{F}}[j,k]\) when analyzing the estimated signal to design detection. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**GPR**SNR(dB) & **10** & **11** & **12** & **13** & **14** & **15** \\ \hline
**2-way** & 0.0955 & 0.0784 & 0.0638 & 0.0522 & 0.0424 & 0.0344 \\
**3-way** & 0.0962 & 0.082 & 0.0726 & 0.0614 & 0.0577 & 0.0504 \\
**4-way** & 0.0858 & 0.0702 & 0.0571 & 0.0465 & 0.0377 & 0.0305 \\ \(\widetilde{8}\)**-way (3)** & 0.0793 & 0.0651 & 0.053 & 0.0434 & 0.0353 & 0.0287 \\
**8-way (4)** & **0.0707** & **0.0577** & **0.0467** & **0.038** & **0.03** & **0.0248** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MMSE results according to the signal-to-noise ratio (SNR). \begin{table}
\begin{tabular}{c c c} \hline \hline
**GPR**Operator & **Multiplication and Division** & **Addition and Subtraction** \\ \hline
**2-way** & 8 & 24 \\
**3-way** & 32 & 96 \\
**4-way** & 128 & 384 \\
**8-way** & 224 & 672 \\
**8-way** & 512 & 1536 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complexity of detection for each GPR target. In this section, we use the 8-way GPR target. Therefore, matrix \(\mathbf{G}\) has the form  
\[\mathbf{G}= \begin{bmatrix}t&p&t\\ r&1&r\\ t&p&t\end{bmatrix} .\]  
We can expand (5) as follows:  
\[\mathbf{g}_{1}^{T}\mathbf{a}_{k-1}+\mathbf{g}_{0}^{T}\mathbf{a}_{k}+\mathbf{g }_{-1}^{T}\mathbf{a}_{k+1}=\sum_{i=-1}^{1}\mathbf{g}_{i}^{T}\mathbf{a}_{k-i}, \tag{6}\]  
where  
\[\mathbf{g}_{-1}= \begin{bmatrix}t&r&t\end{bmatrix} ^{T}, \tag{7}\]  
\[\mathbf{g}_{0}= \begin{bmatrix}p&1&p\end{bmatrix} ^{T}, \tag{8}\]  
\[\mathbf{g}_{1}= \begin{bmatrix}t&r&t\end{bmatrix} ^{T}, \tag{9}\]  
\[\mathbf{a}_{k-1}= \begin{bmatrix}a[j-1,k-1]&a[j,k-1]&a[j+1,k-1]\end{bmatrix} ^{T}, \tag{10}\]  
\[\mathbf{a}_{k}= \begin{bmatrix}a[j-1,k]&a[j,k]&a[j+1,k]\end{bmatrix} ^{T}, \tag{11}\]  
\[\mathbf{a}_{k+1}= \begin{bmatrix}a[j-1,k+1]&a[j,k+1]&a[j+1,k+1]\end{bmatrix} ^{T}. \tag{12}\]  
From (6), we see that the signal \(z[j,k]\) contains the ISI signal \(\mathbf{g}_{0}^{T}\mathbf{a}_{k}\) and ITI signals \(\mathbf{g}_{-1}^{T}\mathbf{a}_{k-1}\) and \(\mathbf{g}_{1}^{T}\mathbf{a}_{k+1}\) If we know the signals \(a[j,k-1]\), \(a[j,k+1]\), \(a[j-1,k-1]\), \(a[j-1,k+1]\), \(a[j+1,k-1]\), and \(a[j+1,k+1]\), we can remove the ITI signal to create a signal that only includes ISI terms:  
\[s[j,k]=\mathbf{g}_{0}^{T}\mathbf{a}_{k}+w_{\mathbf{F}}[j,k]. \tag{13}\]  
To implement the above situations, we propose an interference estimator for an 8-way GPR target, as shown in Figure 6. In this model, we first use the detection for the 2-way  
Figure 6: Proposed detection scheme using the multilayered 2D GPR target. GPR target to easily estimate the original signal \(a_{\rm{2 way}}[j,k]\). Then, to increase the accuracy, we supplied the signal \(a_{\rm{2 way}}[j,k]\) for the detection of the 4-way GPR target. After detection of the 4-way GPR target, an improved estimation of the original signal \(a_{\rm{4 way}}[j,k]\) is recovered. Then, the signal \(a_{\rm{4 way}}[j,k]\) achieves a higher accuracy than the signal \(a_{\rm{2 way}}[j,k]\). Finally, the signal \(a_{\rm{4 way}}[j,k]\) is used to detect the 8-way GPR target. Consequently, the detection of the 8-way GPR target recovered the original signal \(\bar{a}[j,k]\) with the highest accuracy. To analyze this model, we first recall (6). Generally, the value of parameter \(t\) is very small and can be ignored. Therefore, we obtain the following expression:  
\[z_{\rm{4 way}}[j,k]=r_{\rm{4 way}}a[j,k-1]+\mathbf{g}_{0}^{T}\mathbf{a}_{k}+r_ {\rm{4 way}}a[j,k+1]. \tag{14}\]  
From (14), we can see that the target \(\mathbf{G}\) has the form  
\[\mathbf{G}_{\rm{4 way}}=