Expert Knowledge Is . . . How Geophysical . . . How De-Noising under . . . Tikhonov . . . How Geophysicists’ Intuition Fuzzy-Motivated Solution Helps Seismic Data What If Signals Are . . . Limitation of a . . . Processing Solution: Using Local . . . Expert Knowledge: . . . Afshin Gholamy 1 and Vladik Kreinovich 2 Home Page Title Page 1 Department of Geological Sciences 1 Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ 500 W. University El Paso, Texas 79968, USA Page 1 of 14 afshingholamy@gmail.com vladik@utep.edu Go Back Full Screen Close Quit
Expert Knowledge Is . . . How Geophysical . . . 1. Expert Knowledge Is Needed In Geophysics How De-Noising under . . . • In geophysics, signals come with noise; this noise af- Tikhonov . . . fects the resulting images and maps. Fuzzy-Motivated Solution What If Signals Are . . . • It is therefore desirable to minimize the effect of this Limitation of a . . . noise. Solution: Using Local . . . • Sometimes, we know the probabilities of different val- Expert Knowledge: . . . ues of signal and noise. Home Page • In such situations, we can use statistical filtering tech- Title Page niques to find optimal de-noising. ◭◭ ◮◮ • Since the original Wiener filter, many filtering statisti- ◭ ◮ cal techniques have been invented. Page 2 of 14 • In geophysics, it is often impossible to directly measure the physical characteristics of the rocks at large depths. Go Back • Thus, we do not know the actual probabilities. Full Screen • So, we have to rely on expert knowledge. Close Quit
Expert Knowledge Is . . . How Geophysical . . . 2. How Geophysical Expert Knowledge Is Used How De-Noising under . . . Now Tikhonov . . . • Now, the geophysical expert knowledge is mainly used Fuzzy-Motivated Solution to select the most physically reasonable Earth model. What If Signals Are . . . Limitation of a . . . • This trial-and-error approach has led to many success- Solution: Using Local . . . ful applications, such as finding oil. Expert Knowledge: . . . • However, this search-in-the-dark processes is very time- Home Page consuming. Title Page • We need faster ways to translate expert knowledge into ◭◭ ◮◮ data processing techniques. ◭ ◮ • We show that fuzzy techniques can help with such Page 3 of 14 translation. Go Back • We show that the results are in good accordance with the empirically successful semi-heuristic methods. Full Screen Close Quit
Expert Knowledge Is . . . How Geophysical . . . 3. How De-Noising under Uncertainty Is Done How De-Noising under . . . Now: Case of Smooth Signals Tikhonov . . . • In many practical situations, we know that the actual Fuzzy-Motivated Solution signal x ( t ) is smooth. What If Signals Are . . . Limitation of a . . . • In contrast, the observed signal � x ( t ) contains non- smooth noise and is, thus, non-smooth. Solution: Using Local . . . Expert Knowledge: . . . • Often, we know how smooth is the actual signal: � Home Page x ( t )) 2 dt ≤ b 2 for some b . ( ˙ Title Page • Among all such smooth signals, we find X ( t ) which is ◭◭ ◮◮ the closest to � x ( t ): � ◭ ◮ x ( t )) 2 dt → min . ( X ( t ) − � Page 4 of 14 � x ( t )) 2 dt under the con- • We want to minimize ( X ( t ) − � � Go Back x ( t )) 2 dt ≤ b 2 . straint ( ˙ Full Screen • According to Lagrange multiplier technique, this is � � Close x ( t )) 2 dt + λ · C ( t )) 2 dt → min . ( ˙ equivalent to ( X ( t ) − � Quit
Expert Knowledge Is . . . How Geophysical . . . 4. Tikhonov Regularization: Problems How De-Noising under . . . • In Fourier transform, the resulting Tikhonov regular- Tikhonov . . . ˆ x ( ω ) � Fuzzy-Motivated Solution ization has the form ˆ X ( ω ) = 1 + λ · | ω | 2 . What If Signals Are . . . • This works well when we know how smooth is the orig- Limitation of a . . . inal signal, i.e., when we know λ . Solution: Using Local . . . Expert Knowledge: . . . • Often, we do not know smoothness. Home Page • So, after applying this idea, we realize that an addi- Title Page tional smoothing ∆ λ is needed. ◭◭ ◮◮ • Alas, applying this additional smoothing to ˆ X ( ω ), we get a result different from smoothing w/ λ ′ = λ + ∆ λ : ◭ ◮ ˆ ˆ Page 5 of 14 � x ( ω ) � x ( ω ) (1 + λ · | ω | 2 ) · (1 + ∆ λ · | ω | 2 ) � = 1 + λ ′ · | ω | 2 . Go Back Full Screen • Also, in geosciences, the signal is often discontinuous, since there are abrupt transitions (like Moho). Close Quit
Expert Knowledge Is . . . How Geophysical . . . 5. Fuzzy-Motivated Solution How De-Noising under . . . • We know that X ( t ) ≈ � x ( t ) . Tikhonov . . . • Smoothness means that if t and t ′ are close, then Fuzzy-Motivated Solution X ( t ) ≈ X ( t ′ ) and thus, X ( t ) ≈ � x ( t ′ ). What If Signals Are . . . Limitation of a . . . • Let µ ( t, t ′ ) describe closeness; then, for each t , we get Solution: Using Local . . . x ( t ′ ) with degree µ ( t, t ′ ). X ( t ) = � Expert Knowledge: . . . • The usual centroid defuzzification leads to Home Page � µ ( t, t ′ ) · � x ( t ) dt ′ � X ( t ) = Title Page . µ ( t, t ′ ) dt ′ ◭◭ ◮◮ • Under reasonable assumptions, closeness is described ◭ ◮ by a Gaussian membership function � � Page 6 of 14 − ( t − t ′ ) 2 µ ( t, t ′ ) = exp . 2 σ 2 Go Back � � − ( t − t ′ ) 2 � Full Screen x ( t ′ ) · exp dt ′ . • Thus, we get X ( t ) = C · � 2 σ 2 Close Quit
Expert Knowledge Is . . . How Geophysical . . . 6. Fuzzy-Motivated Solution (cont-d) How De-Noising under . . . � � − ( t − t ′ ) 2 � Tikhonov . . . x ( t ′ ) · exp dt ′ . • We get X ( t ) = C · � 2 σ 2 Fuzzy-Motivated Solution � 1 � What If Signals Are . . . 2 · σ 2 · ω 2 X ( ω ) = ˆ • In Fourier transform, ˆ x ( ω ) · exp � . Limitation of a . . . Solution: Using Local . . . • In this case, two consecutive smoothings are equivalent Expert Knowledge: . . . to a single smoothing of this type: Home Page � 1 � � 1 � 2 · ( σ ′ ) 2 · ω 2 2 · ( σ ′′ ) 2 · ω 2 = ˆ ˆ X ( ω ) · exp x ( ω ) · exp Title Page � , ◭◭ ◮◮ = σ 2 + ( σ ′ ) 2 . where ( σ ′′ ) 2 def ◭ ◮ • Moreover, one can prove that this is the only smoothing Page 7 of 14 with this property. Go Back • The resulting shape regularization has indeed been suc- Full Screen cessfully used in geophysics (Sergey Fomel et al.). Close Quit
Expert Knowledge Is . . . How Geophysical . . . 7. What If Signals Are Sometimes Not Smooth? How De-Noising under . . . • Many geophysical structures have abrupt boundaries. Tikhonov . . . Fuzzy-Motivated Solution • Thus, the signals are not smooth: they have abrupt What If Signals Are . . . transitions corresponding to these boundaries. Limitation of a . . . • To describe such signals, it is thus natural to use piece- Solution: Using Local . . . wise smooth models ( splines ). Expert Knowledge: . . . Home Page • Splines have indeed been efficiently used in data pro- cessing, in particular, in seismic data processing. Title Page • However, splines do not explain what type of a transi- ◭◭ ◮◮ tion this is. ◭ ◮ • A more adequate description should take into account Page 8 of 14 the specifics of the corresponding transitions. Go Back • In general, specifics means that we have a family of Full Screen models m ( t,� c ), with parameters � c = ( c 1 , . . . , c k ). Close Quit
Expert Knowledge Is . . . How Geophysical . . . 8. Families of Models How De-Noising under . . . • A smooth dependence can be locally well described by Tikhonov . . . a linear model Fuzzy-Motivated Solution What If Signals Are . . . x ( t ) ≈ m ( t ) = c 1 + c 2 · t. Limitation of a . . . • A wave can be locally described by a sinusoid Solution: Using Local . . . x ( t ) ≈ m ( t ) = c 1 · cos( c 2 · t + c 3 ) . Expert Knowledge: . . . Home Page • There are also finite-parametric models for such non- Title Page smooth phenomena as phase transitions. ◭◭ ◮◮ • In all the above cases, a model with a finite number of ◭ ◮ parameters is only an approximation: Page 9 of 14 – the longer the period of time that we need to cover, Go Back – the less accurate the model becomes. Full Screen • Thus, a reasonable idea is to use each such model only locally. Close Quit
Expert Knowledge Is . . . How Geophysical . . . 9. Limitation of a Traditional Piece-Wise Smooth How De-Noising under . . . (Spline) Approach: Example Tikhonov . . . • A smooth function x ( t ), in the vicinity of each point Fuzzy-Motivated Solution t 0 , can be well approximated by a linear function: What If Signals Are . . . Limitation of a . . . x ( t ) ≈ m ( t ) = x ( t 0 ) + ˙ x ( t 0 ) · ( t − t 0 ) . Solution: Using Local . . . • The further t from t 0 , the less accurate the correspond- Expert Knowledge: . . . ing approximation. Home Page Title Page • To make approximation accurate, we need to use differ- ent linear approximations on different time intervals: ◭◭ ◮◮ • first m ( t ) = x ( t 0 ) + ˙ x ( t 0 ) · ( t − t 0 ), ◭ ◮ • then m ( t ) = x ( t 1 ) + ˙ x ( t 1 ) · ( t − t 1 ), etc. Page 10 of 14 • At the border between two intervals, the derivative Go Back m ( t ) changes from ˙ ˙ x ( t 0 ) to ˙ x ( t 1 ). Full Screen • However, the original signal x ( t ) was smooth! Close Quit
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