**Stabilizing ****inverse**** Q filtering**

**(Knut Sørsdal) Seismic inverse
Q filtering (IQF)** is a data processing technology for enhancing the
resolution of reflection seismology images. Q
is the anelastic
attenuation factor or the seismic quality factor, a measure of the energy
loss as the seismic wave moves. To obtain a solution when we make computations
with a seismic model we always have to consider the problem of instability and
try to obtain a stabilized solution for inverse Q filtering.

Basics

When a wave propagates through subsurface materials both energy dissipation and velocity dispersion takes place.
Inverse Q filtering is a method to restore the energy loss due to energy
dissipation (amplitude compensation) and to correct the time-shift of the data
due to velocity dispersion.

Wang has written an excellent book on the subject of
inverse Q filtering and discuss the subject of stabilizing the method. He writes:

“The phase-only inverse Q filter mentioned above is
unconditionally stable. However, if including the accompanying amplitude
compensation in the inverse Q filter, stability is a major issue of concern in
implementation.”

Hale (1981,1982) found that the
inverse Q filter overcompensated the amplitudes for the later events in a
seismic trace. Therefore, in order to obtain reasonable amplitude, the
amplitude spectrum of the computed filter has to be clipped at some maximum
gain to prevent undue amplitude at later times. On basis of this concept Wang
proposed a stabilized inverse Q filtering approach that was able to compensate
simultaneously for both attenuation and dispersion. The unclipped version of
Wang’s solution is presented in the wikipedia article
‘seismic inverse Q filter’. The solution is based on the theory of wavefield downward continuation. In this outline here I
will compute on a clipped version by introducing low-pass filtering. Both Hale
and Wang introduced low-passfiltering as a method for stabilization.

**Calculations**

We have the equation for seismic inverse Q filtering from
Wang:

_{} (1)

where *γ=(πQ _{r})^{-1}*

Time is denoted τ, frequency is w and i is the imaginary unit. Q_{r}
and w_{r} are reference values
representing damping and frequency for a certain frequency. To demonstrate
stability we can simply bypass using a reference frequency and get a more
simple equation:

_{} (2)

On figure 1 is presented the solution of (2) for a
seismic model for different Q-values, which clearly indicates the numerical
instability. Number on top of figure 1 corresponds with the Q number, 1=Q1, 2=Q2 etc. The
results are close to the results presented in Wang’s book (each trace is scaled
individually, so artefacts are stronger on trace 5
than on trace 4). However, Wang also considered phase compensation.
Computations here are for amplitude only inversion since the phase compensation
is unnecessary to demonstrate instability because it is always stable.

**Inverse Q filtered traces
Q1=400,Q2=200,Q3=100,Q4=50,Q5=25**

*Fig.1.For
traces with Q=400 and 200, the process restores the initial undamped
deltapulse. *

**Low-passfiltering
and inverse Q-filtering**

In practice, the artefacts
caused by numerical instability can be suppressed by a low-pass filter. Hale wrote that the unclipped IQF of a seismogram amplified the Nyquist
frequency by a factor 7x10^{6} when we had the ratio t/Q=10
and concluded that for typical seismograms with lengths longer than 1000
samples and Q value around 100, data is seldom pure enough to warrant the use
of unclipped IQF. Wang introduced
a cutoff frequency to set up a criterion for the stabilization by a
mathematical formula. However, considering Hales’ article it could be
sufficient to simply remove the Nyquist frequency.
That means to let the frequency close to Nyquist frequency
be the cutoff frequency. On fig.2 we see a seismic model giving us benchmark
data for inverse Q-filtering (red graph). We will see that IQF of this model
will amplify the Nyquist frequency by a factor little
less than 5x10^{6}.

*Fig.2. Impulseresponse
of a seismic model. Green graph is undamped and red
is damped impulse response Q=50*

*.*

Figure 3 is the amplitude-only inverse Q filtered trace of figure 2 for
Q=50 (trace 4). The result clearly indicates the numerical instability. Artefacts are seen through the whole trace.

*Fig.3. Model from fig.2. with applied invers Q-filter Q=50.*

We will try to remove the artefacts by
applying a low-pass filter on the trace of figure 3. We used MATLABS signal
processing tool and created a low-passfilter (Zero-phase
IIR-filter) on fig.4 with cutoff frequency at 120 Hz. The amplitude response of
the filter is in blue and the phase in green.

*Fig.4.Low-pass filter*

The result of filtering trace on fig.3 with the low-passfilter
of fig.4 is shown on fig.5. All artefacts are removed
and we are left with the impulse response that can be compared with the
original model on fig.2.

*Fig.5.Filtered impulserespons
where all unstable energy are removed.*

**Frequency-response**

A study of the frequency response of the trace of
figure 3 and figure 5 will give more insight into the filtering process. Figure
6 shows the magnitude of the frequency response as a function of digital
frequency before filtering. This representation gives a good picture of what
happens around the Nyquist frequency when filtering
with the low-pass filter is done. After filtering the unstable energy around
the Nyquist-frequency is completely removed, and
fig.7 give the frequency response of the impulse response of fig.5.

*Fig.6.
Frequency-response with digital frequency before filtering.*

*Fig.7.
Frequency-response with digital frequency after filtering.*