Stabilized inverse Q filtering

Key takeaways

Stabilized inverse Q filtering belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Stabilized inverse Q filtering to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Stabilized inverse Q filtering from memory before moving on to harder problems.

Reference excerpt

Stabilized inverse Q filtering is a data processing technology for enhancing the resolution of reflection seismology images where the stability of the method used is considered. 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 seismic 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, Seismic inverse Q filtering (2008), 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) 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 filtering. 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:

U ( t + △ t , w ) = U ( t , w ) exp ⁡ ( | w w r | − γ | w | △ t 2 Q ( w ) ) exp ⁡ ( i | w w r | − γ w △ t ) ( 1 ) {\displaystyle U(t+\bigtriangleup t,w)=U(t,w)\exp {\bigg (}|{\frac {w}{w_{r}}}|^{-\gamma }{\frac {|w|\bigtriangleup t}{2Q(w)}}{\bigg )}\exp {\bigg (}i|{\frac {w}{w_{r}}}|^{-\gamma }w\bigtriangleup t{\bigg )}\quad (1)}

γ = ( π Q r ) − 1 {\displaystyle \gamma =(\pi Q_{r})^{-1}}

Time is denoted τ, frequency is w and i is the imaginary unit. Qr and wr 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:

U ( t + △ t , w ) = U ( t , w ) exp ⁡ ( | w | △ t 2 Q ( w ) ) exp ⁡ ( i w △ t ) ( 2 ) {\displaystyle U(t+\bigtriangleup t,w)=U(t,w)\exp {\bigg (}{\frac {|w|\bigtriangleup t}{2Q(w)}}{\bigg )}\exp {\bigg (}iw\bigtriangleup t{\bigg )}\quad (2)}

The sum of these plane waves gives the time-domain seismic signal,

U ( t + △ t ) = ∫ 0 ∞ U ( t + △ t , w ) d w . ( 2. b ) {\displaystyle U(t+\bigtriangleup t)=\int _{0}^{\infty }U(t+\bigtriangleup t,w)dw.\quad (2.b)}

On figure 1 is presented the solution of (2/2.b) 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.

… excerpt ends here. Continue reading the full article.

Illustrations

Stabilized inverse Q filtering illustration

Worked examples

Example 1 — a first encounter with Stabilized inverse Q filtering

Start with the simplest possible case. Write down what Stabilized inverse Q filtering claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Stabilized inverse Q filtering before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Stabilized inverse Q filtering ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Stabilized inverse Q filtering

In research: Stabilized inverse Q filtering appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Stabilized inverse Q filtering in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Stabilized inverse Q filtering is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geophysics, Seismology measurement, so understanding it makes those chapters shorter.
In everyday life: Look for Stabilized inverse Q filtering outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Stabilized inverse Q filtering in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Stabilized inverse Q filtering means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Stabilized inverse Q filtering out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Stabilized inverse Q filtering in simple terms?

Stabilized inverse Q filtering is a data processing technology for enhancing the resolution of reflection seismology images where the stability of the method used is considered. Q is the anelastic attenuation factor or the seismic quality factor, a measure of the energy loss as the seismic wave mov…

Why does Stabilized inverse Q filtering matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Stabilized inverse Q filtering?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Stabilized inverse Q filtering.