## 2 Materials and Methods
### Signal Attributes  
Signal attributes as used in seismic and GPR data are enhancement mathematical computations utilized to improve the anomaly of targets within the subsurface on the data plots. They are found in the instantaneous parameters of the signals. Principles of instantaneous parameters correlate with geometry and the physical property changes of the material by which the GPR signal travels [31]. The basic purpose of the attributes is to reveal subtle features by identifying the same trend or pattern by quantifying particular characteristics. Thus, sought targets are enhanced by removing or hiding some portion of the signal. Most signal attributes are derived through filters, statistics, and transforms [32]. The commonly used attributes in GPR are the instantaneous variation of different parameters, which are computed through Hilbert transform [33]. Instantaneous amplitudes, frequency, and phase are mostly computed by the transformation of the real data by the Hilbert function. What the Hilbert transform does is a filtering function that goes through the amplitude of the spectral components uninterrupted but alters the phases' spectral components by ninety degrees [28]. Basic relations between instantaneous frequency and seismic signal attenuation have been given in the work of [33]. The author considered that geophysical signal (e.g., seismic or GPR time traces) \(s(t)\) could be represented by its envelope \(a(t)\) and phase \(\vartheta(t)\), and thus  
\[s(t)=a(t)\text{-}cos(\vartheta(t)) \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline
**S/N** & **Field Measurement Input Parameters** & **Title 3** \\ \hline
1 & Antenna frequency & 800 MHz \\
2 & Trace interval/station spacing & 0.010 m \\
3 & Sampling frequency & 8000 MHz \\
4 & Number of samples & 500 \\
5 & Stacking times & 16 \\
6 & Time window & 60 nS \\
7 & Antenna spacing & 0.14 m \\ \hline \end{tabular}
\end{table} 
Table 1: Field data acquisition parameters. and also the quadrature (imaginary) trace of the same signal may be represented as  
\[s^{*}(t)=a(t)\cdot sin(\vartheta(t)) \tag{2}\]  
and the complex (analytical) trace \(z(t)\) may then be given by the following equation:  
\[z(t)=s(t)+i\cdot s^{*}(t)=a(t)\cdot ei\cdot\vartheta(t) \tag{3}\]  
The work concluded that the quadrature (imaginary) trace is the Hilbert transform of the real trace. It is obtained by the phase shifting of the recorded trace by 90 degrees. The complex trace comprises the real trace (recorded) and the imaginary trace [34]. Once the quadrature trace is found, the instantaneous amplitude \(a(t)\) and the instantaneous phase \(\vartheta(t)\) can be expressed as  
\[a(t)=[s(t)2+s^{*}(t)2]1/2 \tag{4}\]  
and, respectively,  
\[\vartheta\left(t\right)=tan-1[(s^{*}(t))/(s(t))] \tag{5}\]  
Further details of other derivative attributes such as frequency are found in [33; 35]. Computed instantaneous attributes have different applications depending on the sought targets. For instance, instantaneous amplitude outputs the envelope of the selected data at the same location, which enhances lateral variations within events [36]. The attribute considered for the interpretation of the field data in this study is the instantaneous phase. It calculates phases at the same location, which emphasizes the spatial continuity/discontinuity by providing a way for weak and strong events to appear with equal strength. Essentially, it responds mostly to dielectric changes [24], which can be used to distinguish subsurface material properties. The instantaneous phase has no amplitude information but relates to the phase component of the wave propagation [37]. Hence amplitude suppression due to earth convolution and attenuation may not directly influence phase attributes. It is on this premise that the phase attribute was computed on the field data to delineate the changes that accompany the soil horizons as water infiltrates through its matrix, which can be a good pointer to its potential status. Mathematically, the instantaneous phase is expressed as shown in Equation (5).