## 3 Results  
### Test of the Algorithm on the Seismic Trace Model  
A seismic trace model was built to test the efficacy of the harmonic extrapolation algorithm before applying it to real seismic data. The thickness of the model was determined from the E reservoir (of Inas Field), which is encased in between shales. Upon definition of this interval, shear wave velocity, compressional wave velocity, and the density of the reservoir were used to model a synthetic reflectivity series, and a statistical wavelet was extracted at a time interval corresponding to the reservoir zone. The vertical time thickness of the model is 200 ms (600 ms-800 ms), and its amplitude range is \(-0.22\) to \(+0.22\). The time thickness of the individual loops of the trace ranges from 11 ms to 27 ms. See Figure 11 below. A basis pursuit decomposition algorithm was applied to the modeled trace to decompose it into a superposition of layer spectra [17], and the emerging components were summed up using Equation (10) to obtain the band expanded trace shown in Figure 12 below. The result shows an improved resolution while also maintaining the characteristics of the original trace. It is observed that the seismic amplitude is reduced significantly across the time series. For instance, the maximum positive amplitude dropped from \(+0.2\) to \(+0.053\) while the maximum negative amplitude increased from \(-0.2\) to \(-0.068\). To correct for the amplitude difference, a global factor was applied by multiplying the amplitude by a constant, and the result is as shown in Figure 13 below, which is an overlay of both traces. Both traces compare very well, while the enhanced trace also maintains a very good resolution. Figure 10: Algorithmic scheme of the applied techniques. Figure 11: A plot of the original model parameter. Figure 12: A plot of the bandwidth-enhanced seismic trace model parameter showing an improved resolution after harmonic extrapolation was applied to the original spectrum. Figure 13: Comparison between the enhanced and original trace model. The time thickness resolution is sufficient, allowing very thin layers to be resolved within the modeled interval. The loop's minimum resolvable time thickness fell from 11 ms to 2 ms. The bandwidth of the original seismic model was estimated to be around 68 Hz, as illustrated in Figure 14. Using the extrapolation methods described above, its bandwidth increased from 68 Hz to 161 Hz (Figure 10), a 137 ((\((161-68)/68\)) \(\times\) 100) percent increase. The amplitude spectrum of the enhanced trace between 15 Hz and 68 Hz (bounded by the dashed red lines) mirrors that of the original normalized spectrum (Figure 14), however with a slightly different amplitude magnitude. The spectrum of the original trace is a composite of the unit impulse pairs. Due to the limitedness of the seismic impulse, this often causes a destructive interference of thin reflectors, thereby resulting in unrealistically high amplitudes. When a signal correctly resolves a layer, the impulse pairs (top and base) of the reflectors constructively interferes, retaining the correct amplitude. This suggests why the amplitude of 9a and 9b defers after signal processing. Notice that the spectrum of the enhanced signal has more notches, which indicates that more layers were resolved, whereas the spectrum of the original data has those layers lumped up in few notches. The outlined portion depicts the sampled component utilized to expand the frequency to beyond the original band limit. The extrapolation's validity is determined by how accurately the sampled component of the original data is represented in the enhanced data. As a result, data quality must be meticulously validated using the procedure described above. Bandwidth extension to 161 Hz frequency helped to resolve the reflections at about 2-11 ms time thickness and sharpened the events, as shown in Figure 12. The initial model (Figure 11) is fairly like the enhanced trace, albeit not identical as one might assume due to the added frequency.