Why Ricker Wavelets Are Need for a Theoretical . . . Successful in - PowerPoint PPT Presentation

Seismic Waves: Brief . . . Seismic Data is Very . . . Ricker Wavelets: . . . Ricker Wavelets Are . . . Why Ricker Wavelets Are Need for a Theoretical . . . Successful in Processing How Each . . . What is the Joint . . . Seismic Data: Towards a Analysis of the Problem Main Result Theoretical Explanation Home Page Title Page Afshin Gholamy and Vladik Kreinovich ◭◭ ◮◮ Computational Science Program ◭ ◮ University of Texas at El Paso El Paso, TX 79968, USA Page 1 of 18 afshingholamy@gmail.com, vladik@utep.edu Go Back Full Screen Close Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 1. Seismic Waves: Brief Reminder Ricker Wavelets: . . . • Already ancient scientists noticed that earthquakes gen- Ricker Wavelets Are . . . erate waves which can be detected at large distances. Need for a Theoretical . . . How Each . . . • These waves were called seismic waves , after the Greek What is the Joint . . . word “seismos” meaning an earthquake. Analysis of the Problem • After a while, scientists realized that from the seismic Main Result waves, we can extract: Home Page – not only important information about earthquakes, Title Page – but also information about the media through which ◭◭ ◮◮ these waves propagate. ◭ ◮ • Different layers reflect, refract, and/or delay signals dif- Page 2 of 18 ferently. Go Back • So, by observing the coming waves, we can extract a Full Screen lot of information about these layers. Close Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 2. Seismic Data is Very Useful Ricker Wavelets: . . . • Since earthquakes are rare, geophysicists set up small Ricker Wavelets Are . . . explosions to get seismic waves. Need for a Theoretical . . . How Each . . . • The resulting seismic information helps: What is the Joint . . . – geophysicists, petroleum and mining engineers, to Analysis of the Problem find mineral deposits; Main Result – hydrologists to find underground water reservoirs; Home Page – civil engineers to get check stability of the under- Title Page ground layers below a future building, etc. ◭◭ ◮◮ • In particular, computational intelligence techniques are ◭ ◮ actively used in processing seismic data. Page 3 of 18 Go Back Full Screen Close Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 3. Ricker Wavelets: Reminder Ricker Wavelets: . . . • We need to describe how the amplitude x ( t ) of a seismic Ricker Wavelets Are . . . signal changes with time t . Need for a Theoretical . . . How Each . . . • In 1953, N. Ricker proposed to use a linear combination What is the Joint . . . of wavelets of the type Analysis of the Problem 1 − ( t − t 0 ) 2 − ( t − t 0 ) 2 � � � � x 0 ( t ) = · exp . Main Result σ 2 2 σ 2 Home Page • Different wavelets correspond to: Title Page ◭◭ ◮◮ – different moments of time t 0 and – different values of the parameter σ describing the ◭ ◮ duration of this wavelet signal. Page 4 of 18 • The power spectrum S ( ω ) of this wavelet has the form Go Back S ( ω ) = K · ω 2 · exp( − c · ω 2 ) , where c = σ 2 . Full Screen Close Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 4. Ricker Wavelets Are Empirically Successful Ricker Wavelets: . . . • Since the original Ricker’s paper, Ricker wavelets have Ricker Wavelets Are . . . been successfully used in processing seismic signals. Need for a Theoretical . . . How Each . . . • In particular, Ricker wavelets are used with computa- What is the Joint . . . tional intelligence techniques, Analysis of the Problem • The power spectrum S ( ω ) of the seismic signal is rep- Main Result resented as a linear combination of Ricker spectra: Home Page n K i · ω 2 · exp( − c i · ω 2 ) . Title Page � S ( ω ) ≈ ◭◭ ◮◮ i =1 ◭ ◮ • This description requires 2 n parameters K i and c i . Page 5 of 18 • Often, this approximation of the most accurate of all approximations with the fixed number of parameters. Go Back Full Screen Close Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 5. Need for a Theoretical Explanation Ricker Wavelets: . . . • Empirical fact: Ricker wavelets, in general, lead to a Ricker Wavelets Are . . . better approximation of the seismic spectra. Need for a Theoretical . . . How Each . . . • Problem: What is the Joint . . . – there are many possible families of approximating Analysis of the Problem functions, and Main Result – only few of these families were actually tested. Home Page • Natural question: Title Page ◭◭ ◮◮ – are Ricker wavelets indeed the best or – they are just a good approximation to some even ◭ ◮ better family of approximating functions? Page 6 of 18 • What we show: Ricker wavelets are the best. Go Back Full Screen Close Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 6. How Each Propagation Layer Affects the Seis- Ricker Wavelets: . . . mic Signal Ricker Wavelets Are . . . • Layers are not homogeneous; as a result: Need for a Theoretical . . . How Each . . . – the same seismic signal, when passing through dif- What is the Joint . . . ferent locations on the same layer, Analysis of the Problem – can experience different time delays. Main Result • Thus, a unit pulse signal at moment 0 is transformed Home Page into a signal m ( t ) which is distributed in time. Title Page • A signal x ( t ) can be represented as a linear combina- ◭◭ ◮◮ tion of pulses of amplitudes x ( s i ) at moments s i . ◭ ◮ • Each such pulse is transformed into m ( t − s i ) · x ( s i ). Page 7 of 18 • So, each layer transforms the original signal x ( t ) into Go Back � the new signal m ( t − s ) · x ( s ) ds. Full Screen Close Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 7. What is the Joint Effect of Propagating the Sig- Ricker Wavelets: . . . nal Through Several Layers? Ricker Wavelets Are . . . • A signal x 0 ( t ) passes through the first layer, and is thus Need for a Theoretical . . . � transformed into x 1 ( t ) = m 1 ( t − s ) · x 0 ( s ) ds. How Each . . . What is the Joint . . . • The signal x 1 ( t ) passes through the second layer, and � Analysis of the Problem is, thus, transformed into y ( t ) = m 2 ( t − s ) · x 1 ( s ) ds. Main Result • Substituting the expression for x 1 ( s ), we conclude that Home Page � y ( t ) = m ( t − u ) · x 0 ( u ) du, where Title Page � m ( t ) = m 1 ( s ) · m 2 ( t − s ) ds. ◭◭ ◮◮ ◭ ◮ • The formula is known as the convolution of two func- Page 8 of 18 tions m 1 ( t ) and m 2 ( s ) corresponding to the two layers. Go Back • In general, the joint effect of several layers is a convo- Full Screen lution of several functions m i ( t ). Close Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 8. How to Describe Convolutions of Several Func- Ricker Wavelets: . . . tions? Ricker Wavelets Are . . . • A similar problem appears in probability theory. Need for a Theoretical . . . How Each . . . • If independent random variables x i have pdf’s ρ i ( x i ), then the pdf ρ ( x ) of x = � x i is a convolution: What is the Joint . . . Analysis of the Problem � ρ ( x ) = ρ 1 ( x 1 ) · ρ 2 ( x − x 1 ) dx 1 . Main Result Home Page • According to the Central Limit Theorem: Title Page – if we have a large number of small independent ran- ◭◭ ◮◮ dom variables, ◭ ◮ – then the distribution for their sum is close to Gaus- sian (normal). Page 9 of 18 • Different layers are independent. Go Back • Thus, the joint effect of several layers is described by Full Screen − t 2 � � the Gaussian formula m ( t ) = C · exp . Close 2 σ 2 Quit

Seismic Waves: Brief . . . Seismic Data is Very . . . 9. Fourier Transform Helps Ricker Wavelets: . . . � • We know that y ( t ) = m ( t − s ) · x ( s ) ds. Ricker Wavelets Are . . . • If we know n values of m ( t ) and x ( t ), we need n 2 com- Need for a Theoretical . . . How Each . . . putations to follow this formula. What is the Joint . . . • FFT computes Fourier transform Analysis of the Problem � Main Result ˆ def exp( − i · ω · x ) · f ( x ) dx in time O ( n · ln n ) ≪ n 2 . f ( ω ) = Home Page Title Page • In terms of Fourier transforms, ˆ y ( ω ) = ˆ m ( ω ) · ˆ x ( ω ). ◭◭ ◮◮ • For Gaussian m ( t ), its Fourier transform is also Gaus- 2 · σ 2 · ω 2 � − 1 � ◭ ◮ sian, so ˆ y ( ω ) = const · exp · ˆ x ( ω ) . Page 10 of 18 def x ( ω ) | 2 • We are interested in the power spectra X ( ω ) = | ˆ def Go Back y ( ω ) | 2 , so and Y ( ω ) = | ˆ Full Screen def Y ( ω ) = const · exp( − α · ω 2 ) · X ( ω ) , where α = σ 2 . Close Quit