GEOPH 426/526: Signal Processing in Geophysics

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1 GEOPH 426/526: Signal Processing in Instructor: Geophysics Jeff Gu CCIS room Teaching Assistant: Jingchuan Wang Time: Place: CCIS room Tu, Th 9:30-10:50 AM CCIS L1047 Office Hours: can also for an appointment or just show up

2 Overall goal: Try to explain fundamental but somewhat boring concepts. The goal is to have a clearer understanding of what goes on in those Black Boxes that we call signal processing software, and to be able to implement simple ones of your own. Truth about myself: ---- Somewhat mathematically challenged. That s not always a bad thing. That means if I can understand it, so should you! ---- I am a person concentrating on global seismology. So naturally I will have some better examples (ideas) related to signal processing in global seismology than exploration seismology. But ultimately, the ideas are the same Last but not the least, not too proud penmanship, hence quite a bit of slides ---- I like leaking possible exam problems in class unannounced

3 Recommended Only Textbooks: 1. Statistical and Transform Methods for Geophysical Processing (*) By M. Sacchi, on-line book 2. Seismic Data Processing (Vol 1) O. Yilmaz, Society of Exploration Geophysicists ISBN: Digital Signal Processing By A. V. Openheim, and R. W Schafer. ISBN: Most useful: class notes and book 1

4 Grading (Geoph426): Homework (20%): Dropbox in CCIS for the course at specified time. Unless there is a special reason, the standard late penalty is 15% a day for each additional day. Reminder: Submit your own work (4-5 in total) Mid-term Exam (30%): in-class. Final Exam (50%) Grading (Geoph526): Homework (20%): same as above. Mid-term Exam (40%): in-class, could be the same as 426. Project Presentation (15%): a chance to showcase your newly acquired knowledge. Project Paper (25%): Start early and continue to work on it when you have time. I will make a list of topics early. Talk to me.

5 Course website: Topics: " Fourier Series and Transforms Discrete Fourier Transform and FFT Z-transform Digital signals: aliasing and Nyquist Concept Convolution and deconvolution Inverse filtering and theory Laplace transform, filtering Principle value decomposition Signal enhancement and applications

6 Yes, these images look cool, but lots of signal processing work, time and transformed domain operations. Where it all began: 1. Carl F. Gauss, 1700s, the Prince of Mathematics, principle of FFT 2. Analogue -à digital (1950s) 3. FFT --- Fast Fourier Transform (Cooley and Tukey, 1965) reasons: Speed Discreteness Microprocessors

7 Example of 2D SVD Noise Reduction Cocos Plate Nazca Plate South American Plate South American Subduction System (the subducting slab is depressing the phase boundary near the base of the upper mantle) 410 Key Signal Processing Steps: 1. Filtering 2. Averaging (stacking) 3. Depth migration 4. Next page -à (principal value Deconvolution) Courtesy of Sean Contenti 660

8 Singular Value Decomposition Largely, digital processing is about data visualization, processing, decomposition, and storage. 8

Radon Transform Transform with Inversion

9 Radon Transform Transform with Inversion Time to SS (sec) Time to SS (sec) 1 Equalize on SS 2 Distance Averaging Window Distance (deg) 3 4 Tau-p transform via inversion S400S S670S Partial stack based on distance Distance (deg) Inverse transform + resampling P (ray parm, 1/deg) Distance (deg) Gu and Sacchi, 2011

10 Spectral Domain Signals Winter? Summer? Gu & Shen (BSSA, 2012). Involved operations are: correlation, filtering and stacking. How is this done and what do these frequency signals mean? 10 Lake signal in winter???

propagation and recording of seismic waves.

uncover the internal structure of the earth.

11 What do we learn from seismic waves Seismology is a study of the generation, propagation and recording of seismic waves. It is to date the most effective means to uncover the internal structure of the earth. The theory is directly linked to theories in continuum mechanics and wave. Seismogram= source impulse*medium response*receiver electronics anomaly 31

are called "narrow" band, because they preferentially sense frequencies near 1/(15 s) and 1 hertz respectively.

12 Signal Processing is inherent Filters & Convolution (in this case, instrument response function) Broadband vs. low/high freq seismometers The broad band instrument senses most frequencies equally well; the long-period and short period instruments are called "narrow" band, because they preferentially sense frequencies near 1/(15 s) and 1 hertz respectively. The yellow region is the low end of the frequency range audible to most humans (we can hear waves around 20 hertz to 20,000 hertz). Bottom line: Signals are naturally filtered, regardless

13 Fourier Synthesis, in preparation Signal = something that convey information on the state or behavior of a physical system, a function of one or more variables. Representations: To develop an intuitive understanding of abstract concepts, it is often useful to have the same idea expressed from different view points. For example, Fourier Transforms: Example: N-dimensional Spaces 1D 2D 3D Geometrical + analytical N-Dimension? Example 2 (ancient Greek problem): Can a circle be filled by a polygon? (the quadrature problem)

14 Why bother with this class, if I can use matlab? (1) The purpose of computing is insight, not numbers (2) Fourier analysis is a natural tool for describing physical phenomenon that are periodic. For instance, tides, sunspots, electrical oscillations (AC) ---> see next page (3) You need to be able to perform operations beyond simple commands (4) Without knowing the details, your interpretation of the outcomes will be limited Math Review: Y Description of vector quantities, components Geometric θ R X Algebraic Polar Form: (R, θ) R = magnitude θ = direction Cartesian Form: (X, Y) X = component #1 Y = component #2 14

15 Vector Addition! Given vectors A = ( A, ) and x Ay A! + B! = A + B, A + Scalar Multiplication! A = A x, A ) ( x x y By! B = i.e. to add vectors you add their components. What about subtraction (A-B)?? Treat it as adding a negative B to A. ) ( B x, By Given a vector and a scalar, scalar multiplication is DEFINED as Example! F =! ma! s A = s ( sa x, say When we write we really mean: F, F ) = ( ma, ma ) Mul:plica:on ( y ) ) ( x y x y (a) Dot Product (or scalar product )! A B! = A! B! cosθ = a1* b1+ a2 * b Math Review: That is F F x y = = ma ma 15 x y

16 Ideas of complex numbers So, there are some benefits of expressing geometrical relations algebraically, or vise versa. Key ---> orthogonal (perpendicular) vectors are separate and independent of each other ---> components add or multiplied separately. Lets emphasize this point further. X = Y= A A A B B B A + B A + B A + B A Y Y Geometric A θ A X X Algebraic A A = A 2 2 X + A Y = modulus of A So the square root of this product is magnitude, or modulus 16

17 What if I multiply the Y component of A by some special quantity i We can kind of write A = A X + ia Y where A X and A Y don t mix since they are cakes and pizzas A = A X + ia Y = A cosθ + i A sinθ = A (cosθ + isinθ) Euler said: e iθ = cosθ + isinθ A = A e iθ We know now a good choice of A = A e iθ i = ± 1 Amplitude and Phase amplitude phase e iθ e iφ = e i(θ+φ ) (prove this) 17

18 Useful Trig Identities: cos(a + b) = cos(a)cos(b) - sin(a)sin(b) sin(a + b) = sin(a)cos(b) + cos(a)sin(b) cos(2a) = cos 2 a - sin 2 a sin(2a) = 2(cos a) (sin a) Math Review: Based on the above, we can derive the following relations (prove): 18

19 Derivatives (and anti-derivative---integrals) (1) Polynomial (2) Product rule: f (x) = x n df n 1 + c (c = constant) = nx df dx = dg dx dx h + g dh dx f (x) = g(x)h(x) ( f, g, h are func. of x) (3) Chain rule: i.e., f (x) = (x n + c) m (c = constant, m,n = integer) df dx = (m)(x n + c) m 1 (n)x n 1 = mnx n 1 (x n + c) m 1 df (g(x)) = f '(g(x))g'(x) dx (4) Trigonometric functions: d(sin x) dx = cos x (5) log/exponential functions: d(cos x) dx d(ln(x)) dx = 1 x = sin x d(e x ) dx = ex 19

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.

Partial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions. Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,