Outline Formulation of the . . . Seismic Waves and the . . . Why Ricker Wavelets Are Analysis of the Problem Main Result Successful in Processing Conclusions and . . . Seismic Data: Towards a Acknowledgments Home Page Theoretical Explanation Title Page Afshin Gholamy ◭◭ ◮◮ ◭ ◮ Department of Geological Sciences University of Texas at El Paso Page 1 of 34 El Paso, TX 79968 afshingholamy@gmail.com Go Back Full Screen Close Quit
Outline 1. Outline Formulation of the . . . • Formulation of the problem Seismic Waves and the . . . Analysis of the Problem • Seismic waves and the empirical success of Ricker wavelets Main Result • Analysis of the problem Conclusions and . . . Acknowledgments • Main result Home Page • Conclusions and future work Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 34 Go Back Full Screen Close Quit
Outline 2. Outline Formulation of the . . . • Formulation of the problem Seismic Waves and the . . . Analysis of the Problem • Seismic waves and the empirical success of Ricker wavelets Main Result • Analysis of the problem Conclusions and . . . Acknowledgments • Main result Home Page • Conclusions and future work Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 34 Go Back Full Screen Close Quit
Outline 3. Seismic Data Is Very Useful Formulation of the . . . • Already ancient scientists noticed that earthquakes gen- Seismic Waves and the . . . erate waves which can be detected at large distances. Analysis of the Problem Main Result • These waves were called seismic waves , after the Greek Conclusions and . . . word “seismos” meaning an earthquake. Acknowledgments • After a while, scientists realized that from the seismic Home Page waves, we can extract: Title Page – not only important information about earthquakes, ◭◭ ◮◮ – but also information about the media through which ◭ ◮ these waves propagate. Page 4 of 34 • Different layers reflect, refract, and/or delay signals dif- ferently. Go Back Full Screen • So, by observing the coming waves, we can extract a lot of information about these layers. Close Quit
Outline 4. Seismic Data is Very Useful (cont-d) Formulation of the . . . Seismic Waves and the . . . • Since earthquakes are rare, geophysicists set up small explosions to get seismic waves. Analysis of the Problem Main Result • The resulting seismic information helps: Conclusions and . . . – geophysicists, petroleum and mining engineers, to Acknowledgments find mineral deposits; Home Page – hydrologists to find underground water reservoirs; Title Page – civil engineers to check stability of the underground ◭◭ ◮◮ layers below a future building, etc. ◭ ◮ • In particular, computational intelligence techniques are Page 5 of 34 actively used in processing seismic data. Go Back Full Screen Close Quit
Outline 5. Ricker Wavelet: Reminder Formulation of the . . . • We need to describe how the amplitude x ( t ) of a seismic Seismic Waves and the . . . signal changes with time t . Analysis of the Problem Main Result • In 1953, N. Ricker proposed to use a linear combination Conclusions and . . . of wavelets of the type Acknowledgments 1 − ( t − t 0 ) 2 − ( t − t 0 ) 2 � � � � Home Page x 0 ( t ) = · exp . σ 2 2 σ 2 Title Page • Different wavelets correspond to: ◭◭ ◮◮ – different moments of time t 0 and ◭ ◮ – different values of the parameter σ describing the Page 6 of 34 duration of this wavelet signal. Go Back • The power spectrum X 0 ( ω ) of this wavelet has the form Full Screen X 0 ( ω ) = K · ω 2 · exp( − c · ω 2 ) , where c = σ 2 . Close Quit
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Outline 6. Ricker Wavelets Are Empirically Successful Formulation of the . . . • Since the original Ricker’s paper, Ricker wavelets have Seismic Waves and the . . . been successfully used in processing seismic signals. Analysis of the Problem Main Result • In particular, Ricker wavelets are used with computa- Conclusions and . . . tional intelligence techniques, Acknowledgments • The power spectrum S ( ω ) of the seismic signal is rep- Home Page resented as a linear combination of Ricker spectra: Title Page n K i · ω 2 · exp( − c i · ω 2 ) . � S ( ω ) ≈ ◭◭ ◮◮ i =1 ◭ ◮ • This description requires 2 n parameters K i and c i . Page 8 of 34 • Often, this approximation is the most accurate of all Go Back approximations with the fixed number of parameters. Full Screen Close Quit
Outline 7. Need for a Theoretical Explanation Formulation of the . . . • Empirical fact: Ricker wavelets, in general, lead to a Seismic Waves and the . . . better approximation of the seismic spectra. Analysis of the Problem Main Result • Problem: Conclusions and . . . – there are many possible families of approximating Acknowledgments functions, and Home Page – only few of these families were actually tested. Title Page • Natural question: ◭◭ ◮◮ – are Ricker wavelets indeed the best or ◭ ◮ – they are just a good approximation to some even Page 9 of 34 better family of approximating functions? Go Back • What we show: Ricker wavelets are the best. Full Screen Close Quit
Outline 8. Outline Formulation of the . . . • Formulation of the problem Seismic Waves and the . . . Analysis of the Problem • Seismic waves and the empirical success of Ricker Main Result wavelets Conclusions and . . . • Analysis of the problem Acknowledgments • Main result Home Page Title Page • Conclusions and future work ◭◭ ◮◮ ◭ ◮ Page 10 of 34 Go Back Full Screen Close Quit
Outline 9. Wavelets as a Way to Describe Seismic Waves Formulation of the . . . • Waves are ubiquitous in nature, from the ocean waves Seismic Waves and the . . . to electromagnetic waves. Analysis of the Problem Main Result • In most physical examples, waves last many periods. Conclusions and . . . • Such waves are well described by the traditional sine Acknowledgments and cosine functions – or by their linear combinations. Home Page • In contrast, earthquakes usually have a reasonably short Title Page duration. ◭◭ ◮◮ • Thus, the related seismic waves also have a short du- ◭ ◮ ration. Page 11 of 34 • Such short-duration waves also happen in other phys- Go Back ical situations. Full Screen • To describe such “small waves” (“wavelets”), researchers use special functions called wavelets . Close Quit
Outline 10. Geophysical Applications of Ricker Wavelets: Formulation of the . . . classification Seismic Waves and the . . . • Ricker wavelets x 0 ( t ) are used in all aspects of seismic Analysis of the Problem data processing. Main Result Conclusions and . . . 1) Ricker wavelets x 0 ( t ) are used to simulate : Acknowledgments – the original earthquakes, Home Page – the explosions set up by the geoscientists, and Title Page – the seismic waves generated by these earthquakes ◭◭ ◮◮ and explosions. ◭ ◮ 2) Functions x 0 ( t ) are used to process the seismic data Page 12 of 34 x ( t ), by representing x ( t ) as a linear comb. of x 0 ( t ). Go Back 3) Ricker wavelets x 0 ( t ) are also used to process other geophysical signals. Full Screen Close Quit
Outline 11. Ricker Wavelets are Useful in Simulating the Formulation of the . . . Original Earthquakes Seismic Waves and the . . . • Often, Ricker wavelets x 0 ( t ) provide the best descrip- Analysis of the Problem tion of the near-fault ground motion. Main Result Conclusions and . . . • In Apostolou et al. 2007 and Gerolymos et al. 2005, Acknowledgments x 0 ( t ) simulate the effect caused by earthquakes: Home Page – on “block-size” structures, in which the height is Title Page commeasurable with width, and ◭◭ ◮◮ – on tall structures – like high-rise building and tall bridge piers. ◭ ◮ • Zania et al. 2008 uses Ricker wavelet to simulate the Page 13 of 34 effect of earthquakes on waste landfills. Go Back • This is important: e.g., 1994 Northridge and 1995 Kobe Full Screen earthquakes caused serious damage to landfills. Close Quit
Outline 12. Ricker Wavelets Are Useful in Simulating Source Formulation of the . . . Signals Seismic Waves and the . . . • Ricker wavelets are also often the best in simulating the Analysis of the Problem source signal , i.e., the explosion set up by geoscientists. Main Result Conclusions and . . . • In Chakraborty et al. 1995 and Zhang et al. 2002, such Acknowledgments a simulation is used: Home Page – to select the best method Title Page – for processing seismic data. ◭◭ ◮◮ ◭ ◮ Page 14 of 34 Go Back Full Screen Close Quit
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