## 1 Introduction
Figure 1: Synthetic seismic response of a theoretical wedge model, (**a**) wedge and seismic traces, (**b**) amplitude extracted from top/base seismic marker (red) and measured through to peak time distance (green). Several techniques have been particularly employed to boost the spectrum of the seismic data by arbitrarily increasing the data frequency. The authors of [18] proposed the loop reconvolution technique, in which the SEG-Y data is oversampled to avoid aliasing, and a new sparse spike reflectivity series weighted by the interpolated amplitudes at all the maxima and minima locations is generated. Then, the resulting reflectivity series is convolved with a suitably analytical high-frequency wavelet that is randomly calculated. The optimum frequency band for that dataset is determined by visual preference, without any mathematical control, and filters off all unwanted frequencies. The problem with this technique is that the spike does not represent the true reflectivity of the data. This method produces output that is correlated with the original peaks and troughs, which can lead to a misleadingly high correlation to synthetic data if the original well tie is also good [19]. Although it brings out better continuity of events in the fault zone, the author of [20] further confirmed that this technique also suffers from the drawback of boosting some noise. Another enhancement technique as described by [19] is phase acceleration, which uses the instantaneous frequency of original data as a fundamental frequency and increases the seismic frequency by increasing the rate of change of the instantaneous phase. According to complex trace analysis, the bandlimited seismic signal can be viewed as the real component of a complex number representation (called the analytic trace), which provides a definitive separation for amplitude of the trace envelope and local phase content [21]. This representation permits explicit calculation and modification of the instantaneous frequency, which is the first derivative of the instantaneous phase in the time domain. The analytical signal \(S(t)\), of which the seismogram \(s(t)\) is the real part, is given by Equation (1) below. \[S(t)=s(t)+is\bot(t)=A(t)e^{\chi(t)} \tag{1}\]  
where \(\dot{\mathbb{V}}(t)\) is the instantaneous phase, \(A(t)\) is the instantaneous amplitude, and \(s\bot(t)\) is the quadrature series (Hilbert transform) of the real seismic trace. The author of [22] demonstrated that the dominant frequency of the data is the instantaneous frequency at the peak of the envelope. As a result, by multiplying the phase term by any desired amount, the instantaneous frequency and dominant frequency of the original trace can be directly changed. Equation (1) can then be used to calculate the frequency extrapolated trace \(SE(t)\). \[SE(t)=A(t)\sum_{i=1}^{k}(cos\ m_{i}(t)) \tag{2}\]  
where \(m_{i}\) can be varied to sum an arbitrary number of invented bands to the original data. The drawback of this method is that it randomly invents frequency by varying the \(m_{i}\) which is an integer (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) parameter, and this can alter the original seismic signals. In addition to seismic resolution enhancement, spectral decomposition and coherency have been employed to image stratigraphic and structural details, respectively [23]. Layer thicknesses [24,25], stratigraphic geometries [26], direct hydrocarbon indicators [27, 28, 29, 30], enhancement of seismic bandwidth [23], and depositional models [31] have all been determined using spectral decomposition. Furthermore, spectral decomposition divides the seismic signal into discrete frequency sub-bands or horizons [25]. This is accomplished by converting seismic data from the time domain to the frequency domain [32]. However, the limitedness of the seismic bandwidth limits the geologic details that can be seen using this technique.