## 4 Methods
### Inverse Q Filtering and Active Seismic Data Processing  
[PERSON] [36] defined that seismic waves propagating through an absorbing geological medium are subjected to a frequency-dependent attenuation and an accompanying phase distortion caused by velocity dispersion. Assuming Q is constant with frequency, [PERSON] derives the following amplitude and phase-velocity dispersion pair shown in the equations:  
\[A(\omega,t)=A(\omega,0)\exp\biggl{(}\frac{-\omega t}{2\text{Q}}\biggr{)} \tag{4}\]  
\[\frac{V(\omega)}{V(\omega_{c})}=\biggl{(}1+\frac{1}{\pi\text{Q}}\ln\biggl{(} \frac{\omega}{\omega_{c}}\biggr{)}\biggr{)},(\omega<\omega_{c}) \tag{5}\]  
\[V(\omega)=constant\ (\omega\geq\omega_{c}) \tag{6}\]  
where  
Q = the medium Q-factor. \(t\) = the travel path time. \(A(\omega,\)0) = the amplitude spectrum of the initial wavelet. \(A(\omega,\)t) = the amplitude spectrum of the wavelet at a traveltime \(t\). \(V(\omega)\) = the phase velocity at a frequency \(\omega\). Q effects can be modeled through amplitude and phase filters following the Kolsky-Futterman [37] relationship:  
\[A(\omega,t^{*})=\exp\biggl{(}\mp\frac{\omega}{2}t^{*}\biggr{)} \tag{7}\]  
\[P(\omega,t^{*})=\exp\biggl{(}\pm\frac{\omega}{\pi}t^{*}\ln\biggl{(}\frac{ \omega}{\omega_{r}}\biggr{)}\biggr{)} \tag{8}\]  
where  
\(\mp\) characterizes the forward filters (modeling Q effects),  
\(\pm\) characterizes the inverse filters (compensating for Q effects),  
\(\omega\) is the (angular) frequency,  
\(\omega_{r}\) is the reference (angular) frequency,  
\(t^{*}\) is the attenuated travel time. The attenuated travel time \(t^{*}\) is defined as the integration of the phase velocity \(V\) and Q effects along the propagation path of the wave. \[t^{*}=\int_{l}V^{-1}(l)\text{Q}^{-1}(ldl)=\frac{t}{\text{Q}_{eff}} \tag{9}\]  
where Q is interval-Q and Q\({}_{eff}\) is effective-Q: characteristics of the 'cumulative' attenuation effects undergone by the wave during propagation. Attenuation increases with both increasing frequency and increasing travel path length; phase velocity increases with frequency up to an upper cutoff frequency and is constant thereafter. In fact, there is a constant fractional loss of energy per cycle travelled (constant dB/cycle). Application of IQF can correct for:  
* Amplitude Q-related effects
* Both phase and amplitude Q-related effects Conventional windowed deconvolution can only provide a very approximate treatment for the continuously time-variant earth filter. Further errors can arise from the presence of noise and from non-whiteness of the reflection series. The deterministic inverse Q-filtering technique first proposed by [PERSON] [37] uses an approximate filter derived from a series expansion of the inverse Q-filter. The phase compensation is performed by a fast algorithm that is closely related to that used in Stolt migration see [38]. For Q independent of depth, the algorithm accurately removes dispersion-generated phase distortion. The algorithm applies phase and amplitude compensation by means of inverse filtering within short, overlapping windows of data. Function treats depth-variable Q structures by a cascaded series of constant-Q compensations. The Q model consists of a number of layers, each defined by the time of the horizon at the bottom of the layer and by the Q value for that layer. The first layer is assumed to start at zero time. The last horizon time that is specified is adjusted so that it is equal to the length of the trace to be processed. Q values may be specified either as an average, effective Q at the time of a horizon or as the interval Q value for the layer above that horizon. We can use constant-Q models, layered Q models or spatially variant Q models. In all cases, after application of this process, the intention is that the resulting data should be suitable for amplitude and phase sensitive processes, such as inversion and AVO (Amplitude versus offset) analysis, which assume the medium is elastic.