## 2 Materials and Methods
### Reflectivity Spectrum Extrapolation Method  
The reflectivity spectrum extrapolation method proposed in this study is aimed at recovering the original earth reflectivity spectrum recorded in the seismic data, which is usually zeroed out by the narrowband wavelets, rather than randomly inventing frequency as in the methods described in Section 2.1 above. The processed seismic data is assumed to be expressed by a simple convolutional model [41] in which the seismic trace, T (s), is the convolution of a constant seismic wavelet (pulse), W, and a reflectivity series, R, plus random noise, n.  
\[\text{T}\left(\text{s}\right)=\text{W}^{*}\text{R}+\text{n} \tag{3}\]  
Figure 5 is an illustration of the convolutional model, in which the geologic section represents the true geology of the formation, and the acoustic impedance log is a product of the density of the geologic elements through which the seismic waves are propagated and the velocity at which it travels through the formations. Although there could be some intra-layer contrasts, acoustic impedance for a single layer is usually assumed to be homogeneous and isotropic. The reflection coefficient is a parameter that describes how much of a wave is reflected at the impedance boundaries and can be expressed as Equation (4):  
\[R_{c}=\frac{A_{i2}-A_{i1}}{A_{i2}+A_{i1}} \tag{4}\]  
This model presumes that the subsurface geology is laterally layered [42], with continual rock properties inside those layers, and that reflections are engendered at interface boundaries between those adjacent layers [43], ignoring many waves propagation effects. These are assumed to be fixed during seismic processing. The convolutional model is usually used to create synthetic traces as well as to invert seismic traces into band-limited reflectivity [44]. The bandlimited seismic wavelet causes the problem of geologic reflector resolution by shaping and/or cutting off the higher resolution reflectivity band of the seismic data and returning the spectrum outside the band to zero. In most cases, the wavelet removes both high and low frequencies from the reflected spectrum. Figure 4: Seismic base map showing the well locations. To recover and enhance the true seismic frequencies, the seismic wavelet must be removed by deconvolution processes [14]. It is possible to improve the bandwidth of the seismic data spectrum beyond the original band limit proposed by [5]. This is plausible because seismic data is a transient signal, and all transient signals have an unbounded frequency response that can be predicted to some extent if enough of the spectrum is sampled [4]. To do this, these transient signals must meet two requirements: (1) The spectrum must be simple and distinct, and (2) the signal must have clear frequency periodicities with low noise. Since the reflectivity spectrum is a superposition of sinusoidal layer frequency responses, a blocky earth structure provides a physical basis for valid bandwidth extension [37]. Therefore, any layer with a single reflection coefficient at the top and bottom, as demonstrated by [24] and investigated in detail by [26], can be depicted as an impulse pair reflectivity series. Figure 6 illustrates a two-layer reflectivity model with reflection interfaces at the top and base. The expression (Equation (5)) for an impulse pair in the time domain can be derived from the parameters shown in the figure. This reflection-coefficient pair can be decomposed into even and odd elements, with even elements having an equal magnitude and sign, while the odd components have an equal magnitude but opposite sign [44]. Various layers can be modelled using the weighted sums of the even and odd impulse pair responses.