## 2 Materials and Methods
### Reflectivity Spectrum Extrapolation Method
\[Q(t)=r_{1}\delta\left(t-t_{1}\right)+r_{2}\;\delta\left(t-t_{1}-n.dt\right) \tag{5}\]  
where \(Q(t)\) is the time domain impulse response or [PERSON]'s function, \(r_{1}\) and \(r_{2}\) are the reflectivity at the top and bottom reflectors of the layer, \(t\) is time from zero to the corresponding times of \(r_{1}\) or \(r_{2}\), \(t_{1}\) is a time sample at the top reflector, and \(n.dt=t_{1}-t_{2}\) is the two-way travel-time thickness of the thin bed; \(\delta(t)=\) the Dirac \(\delta\) function and is considered for our purpose as a unit impulse at time zero. Figure 5: Schematic illustration of the convolutional model. The analysis point is usually located at the center of the layer because a phase shift occurs if the analysis window is not centered on the layer [37]. Therefore, Equation (3) can be re-written as follows (See Figure 7):  
\[Q(t,n)\ =\ r_{1}\delta\ (t\ -\ \frac{1}{2n.dt})\ +\ r_{2}\ \delta\ (t\ +\ \frac{1}{2n.dt}) \tag{6}\]  
Spacing between spectral peaks and notches in the frequency domain is precisely the inverse of the layer thickness in the time domain [24; 26]. Since layer thickness can be determined robustly from a narrow band of frequencies with a high signal-to-noise ratio, we obtained the frequency spectrum of the impulse pair through a Fourier transform of the time domain impulse pair (Equation (6)), and when simplified by trigonometric identities this is given as Equation (7):  
\[Q(f,n)=2r_{e}cos(\pi f\ n.dt)+i2r_{o}sin(\pi.f.n.dt) \tag{7}\]  
\[q(f,n)=2r_{e}cos(\pi f.n.dt) \tag{8}\]  
Figure 6: A two-layer reflectivity model. Figure 7: Locating the analysis point at the center of the layer (red horizontal line). \[R_{c}=\frac{A_{i2}-A_{i1}}{A_{i2}+A_{i1}} \tag{4}\]0  
The real part (Equation (8)) of the equation corresponds to the even component (\(r_{e}\)), whereas the imaginary part (Equation (9)) corresponds to the odd component (\(r_{o}\)) [34]. The authors of [37] affirmed that the even component constructively interferes as layer thickness approaches zero whereas the odd components interfere destructively as thickness approaches zero. Thus, the even component is more robust against noise as thickness tends towards zero and can be easily used to extend the conventional seismic beyond its original bandwidth [37]. The analysis of peak amplitudes versus bed thickness for each of these components (odd and even) shows that the odd component behaves similarly to the Widess model [34; 45]. The maximum amplitude is reached at tuning thickness (\(\lambda/4\)) and then interferes destructively and cancels itself as thickness decreases towards zero (Figure 1). In contrast, the even component becomes more important below \(\lambda/8\) and behaves just the opposite way [2]. To demonstrate what happens in a general multilayer reflector, the authors of [19] created a sparse reflectivity sequence consisting of three distinct components: a largely even impulse pair, a predominantly odd impulse pair, and a single impulse. The predominantly even reflector pair had a real spectrum with a larger amplitude than the imaginary spectrum and vice versa for the primarily odd pair, whereas the single reflector's real and imaginary spectrums were of the same amplitude, but they were one-quarter of the spectral period apart. This suggested that the individual components respond differently to reflectors, and their spectra can be linearly summed up to obtain a composite reflectivity spectrum (See Figure 8). Using dipole decomposition, the reflector pair can be further decomposed into a unit even pair (\(\frac{1}{2n.dt}\)) and a unit odd pair (\(\frac{1}{2n.dt}\)) weighted by coefficients \(r_{e}\) and \(r_{o}\), respectively. The two reflection coefficients for the impulse pair can thus be represented as (\(r_{e}+r_{o}\)) and \(r_{e}-r_{o}\), either of which ranges from \(-1\) to \(1\) and could be zero denoting a single impulse. The frequency spectrum of the impulse pairs can be calculated from Equations (8) and (9).