Seismologia ja maan rakenne A Seismology and structure of the Earth

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1 Seismologia ja maan rakenne A Seismology and structure of the Earth Prac-cals 1: Seismic data analysis: poles and zeros, convolu-on and deconvolu-on Main materials: 1) New Manual of Seismological Observatory Prac-ce (NMSOP- 2): hjp:// bib.telegrafenberg.de/publizieren/vertrieb/nmsop/ hjp://geoladm.geol.queensu.ca Chapter 5. Erhardt Wielandt. Seismic sensors and their calibra-on Chapter 11. Peter Bormann, Klaus Klinge and Siegfried Wendt. Data Analysis and Seismogram Interpreta-on 2) Havskow, J., Alguacil, G Instrumenta-on in earthquake seismology FOR POLES AND ZEROS YOU CAN FIND MATERIALS IN CONTROL SYSTEMS IN INTERNET, ALSO IN YOUTUBE: hjps://

2 The basic requirements in analog and digital rou-ne observatory prac-ce i.e., to: recognize the occurrence of an earthquake in a record; iden-fy and annotate the seismic phases; determine onset -me and polarity correctly; measure the maximum ground amplitude and related period; calculate slowness and azimuth; determine source parameters such as the hypocenter, origin -me, magnitude, source mechanism, etc.. In modern digital observatory prac-ce these procedures are implemented in computer programs.

3 Seismic observables Period ranges (order of magnitudes) Sound s Earthquakes s (surface waves, body waves) Eigenmodes of the Earth 1000 s Coseismic deforma-on 1 s 1000 s Postseismic deforma-on s Seismic explora-on s Laboratory signals s s - > What are the consequences for sampling intervals, data volumes, etc.? Modern Seismology Data processing and inversion 3

4 Different factors/sub- systems (without seismic noise) which influence a seismic record (yellow boxes) and the informa-on that can be derived from record analysis (blue boxes)

5 SEISMOMETERS Output voltage is propordonal to ground velocity in certain frequency bandwidth. requirements ü Very low noise ü ResoluDon ü Broad bandwidth Weight and size MagneDc resistance Price

6 BASIC DEFINITIONS 1) Signals and systems A signal is a descrip-on of how one parameter varies with another parameter. For instance, voltage changing over -me in an electronic circuit, or brightness varying with distance in an image. A system is any process that produces an output signal in response to an input signal Con-nuous systems input and output con-nuous signals, such as in analog electronics. Discrete systems input and output discrete signals, such as computer programs that manipulate the values stored in arrays. x(t) x[n] Continuous System Discrete System y(t) y[n] Con-nuous signals are usually represented with parentheses, while discrete signals use brackets. All signals use lower case lejers, reserving the upper case for the frequency domain. Unless there is a bejer name available, the input signal is called: x(t) or x[n], while the output is called: y(t) or y[n]

7 The Laplace transformadon and transfer funcdon of a linear system A signal that has a definite beginning in -me (such as the seismic waves from an earthquake) can be decomposed into exponen-ally growing, sta-onary, or exponen-ally decaying sinusoidal signals with the Laplace integral transforma-on: st ( s) f ( t) e dt 0 F. f ( t) 1 2 j j j F( s) e st ds s=σ + jω Generally, the transfer func-on H(s) of an linear system is the complex func-on for which G( s) H ( s) F( s) with F(s) and G(s) represen-ng the Laplace transforms of the input and output signals

8 GENERAL PRESENTATION OF A TRANSFER FUNCTION OF A LINEAR SYSTEM: where B and A and are polynomials in s, M is the order of the numerator polynomial, b m is the m- th coefficient of the numerator polynomial, N is the order of the denominator polynomial, and a n is the n- th coefficient of the denominator polynomial. The zeros of the system are roots of the numerator polynomial: TF - à 0 The poles of the system are roots of the denominator polynomial: TF - à inf

9 It is oren convenient to factor the polynomials in the numerator and denominator, and to write the transfer func-on in terms of those factors: on in terms of those factors: H(s) = N(s) D(s) = K (s z 1)(s z 2 )...(s z m 1 )(s z m ) (s p 1 )(s p 2 )...(s p n 1 )(s p n ), where the numerator and denominator polynomials, N(s) and D(s), have real coefficients defined by the system s differential equation and K = b m /a n. As written in Eq. (2) the z i s are the roots of the equation N(s) =0, (3) and are defined to be the system zeros, and the p i s are the roots of the equation D(s) =0, (4) and are defined to be the system poles. In Eq. (2) the factors in the numerator and denominator are written so that when s = z i the numerator N(s) = 0 and the transfer function vanishes, that is lim H(s) =0. s z i

10 Pole- zero plot IMAGINARY REAL

11 System poles define stability of a system: system is stable when f(t)à 0 if tà inf The specifica-on of the form of components of the homogeneous response from the system pole loca-ons on the pole- zero plot: EXAMPLE: H(s)=1/(s+2) à s=- 2, if we take inverce Laplace transform, then f(t)= e - 2 H(s)=1/(s- 2) e 2: The à specification s=2, then off(t)=e the form 2 of components of the homogeneous response from the s

12 e 2: The specification of the form of components of the homogeneous response from the s 1. A real pole p i = σ in the left-half of the s-plane defines an exponentially decaying component, Ce σt, in the homogeneous response. The rate of the decay is determined by the pole location; poles far from the origin in the left-half plane correspond to components that decay rapidly, while poles near the origin correspond to slowly decaying components. 2. A pole at the origin p i = 0 defines a component that is constant in amplitude and defined by the initial conditions. 3. A real pole in the right-half plane corresponds to an exponentially increasing component Ce σt in the homogeneous response; thus defining the system to be unstable. 4. A complex conjugate pole pair σ ± jω in the left-half of the s-plane combine to generate a response component that is a decaying sinusoid of the form Ae σt sin (ωt + φ) where A and φ are determined by the initial conditions. The rate of decay is specified by σ; thefrequency of oscillation is determined by ω. 5. An imaginary pole pair, that is a pole pair lying on the imaginary axis, ±jω generates an oscillatory component with a constant amplitude determined by the initial conditions. 6. A complex pole pair in the right half plane generates an exponentially increasing component.

13 Example A linear system is described by the differential equation Find the system poles and zeros. Solution: d 2 y dt 2 +5dy dt +6y =2du dt +1. From the differential equation the transfer function is H(s) = which may be written in factored form 2s +1 s 2 +5s +6. (5) H(s) = 1 s +1/2 2 (s + 3)(s + 2) = 1 s ( 1/2) 2 (s ( 3))(s ( 2)). (6) The system therefore has a single real zero at s = 1/2, and a pair of real poles at s = 3 and s = 2.

14 Seismic instruments: The standard iner-al seismometer with damping Spring force Damping force It is convenient to define a resonant angular frequency and a dumping parameter as follows: These subs-tu-ons give: This equa-on shows that the Earth accelera-on may be recovered by measuring the displacement of the mass and its -me deriva-ves.

15 EXERCICE&1:&write&the&expression&for&transfer&func8on& Example A linear system is described by the differential equation Find the system poles and zeros. Solution: d 2 y dt 2 +5dy dt +6y =2du dt +1. From the differential equation the transfer function is H(s) = which may be written in factored form 2s +1 s 2 +5s +6. (5) H(s) = 1 s +1/2 2 (s + 3)(s + 2) = 1 s ( 1/2) 2 (s ( 3))(s ( 2)). (6) The system therefore has a single real zero at s = 1/2, and a pair of real poles at s = 3 and s = 2.

16 Exercice 1: A second- order system has a pair of complex conjugate poles a s = 2±j3 and a single zero at the origin of the s- plane. Find the transfer func-on and use the pole- zero plot

17 Solution: From the problem description H(s) = s K (s ( 2+j3))(s ( 2 j3)) = s K s 2 +4s + 13 The pole-zero plot is shown in Fig. 6. From the figure the transfer function is

18 Poles and zeros must either be real or symmetric to the real axis, as men-oned above. When the numerator polynomial is s m, then s = 0 is an m- fold zero of the transfer func-on, and the system is a high- pass filter of order m. (Zeros at nonzero frequency do normally not appear in the transfer func-on of broadband seismographs because, if they occur mathema-cally, their effect must prac-cally be cancelled by nearby poles; otherwise the response would not be called broadband.) Adding a zero at s = 0 produces the transfer func-on Hz(s) = sh(s), and the step response of this system is purely the deriva-ve of the step response of the original system. Depending on the order n of the denominator and accordingly on the number of poles, the response may be flat at high frequencies (n = m), or the system may act as a low- pass filter there (n > m). The case n < m can occur only as an approxima-on in a limited bandwidth because no prac-cal system can have an unlimited gain at high frequencies.

19 Seismic instruments: The standard iner-al seismometer We obtained: and: When h=1, the system is said to be cri-cally damped. Seismometers generally perform op-mally at values of damping close to cri-cal. A polarity reversal at high frequencies.

20 EXERSICE 2: Plot poles and zeros for broadband seismometers Seismograph Zeros Poles STS2 (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) STS1(GRF) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) STS1(VBB)) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) (-3.674E-2, E-3) (-3.674E-2, 3.675E-3) ( , ) ( , ) ( , 0.0) ( , 4.574) ( , ) (-7.006, ) (-7.006, ) ( , ) ( , ) ( E-02, E-02) ( E-02, E-02)

21 on in terms of those factors: H(s) = N(s) D(s) = K (s z 1)(s z 2 )...(s z m 1 )(s z m ) (s p 1 )(s p 2 )...(s p n 1 )(s p n ), The transfer function may be evaluated for any value of s = σ + jω, and in general, when s is complex the function H(s) itself is complex. It is common to express the complex value of the transfer function in polar form as a magnitude and an angle: with a magnitude H(s) and an angle φ(s) given by R {} I {} H(s) = H(s) e jφ(s), (17) H(s) = R {H(s)} 2 + I {H(s)} 2, (18) ( ) I {H(s)} φ(s) = tan 1 (19) R {H(s)}

22 The frequency response may be written in terms of the system poles and zeros by substituting jω for s directly into the factored form of the transfer function: H(jω) =K (jω z 1)(jω z 2 )...(jω z m 1 )(jω z m ) (jω p 1 )(jω p 2 )...(jω p n 1 )(jω p n ). (30)

23 EXERCISE 3: Convolu-on and deconvolu-on in seismic signal processing

24 ConvoluDon W Input displacement Output displacement Amplitude Input Output Convolu-on (Kanasewich 1981) is a mathema-cal opera-on defining the change of shape of a waveform resul-ng from its passage through a filter. Thus, for example, a seismic pulse generated by an explosion is altered in shape by filtering effects, both in the ground and in the recording system, so that the seismogram (the filtered output) differs significantly from the ini-al seismic pulse (the input). Time Fig. 2.9 The principle of filtering illustrated by the perturbation of a suspended weight system. As a simple example of filtering, consider a weight suspended from the end of a ver-cal spring. If the top of the spring is perturbed by a sharp up- and- down movement (the input), the mo-on of the weight (the filtered output) is a series of damped oscilla-ons out of phase with the ini-al perturba-on.

25 The effect of a filter is described mathema-cally by a convolu-on opera-on such that, if the input signal g(t) to the filter is convolved with the impulse response f(t) of the filter, known as the convolu-on operator, the filtered output y(t) is obtained (asterisk denotes a convolu-on operator): yt ()= gt ()* f() t Spike input Output = Impulse response Filter

26 The mathema-cal implementa-on of convolu-on involves -me inversion (or folding) of one of the func-ons and its progressive sliding past the other func-on, the individual terms in the convolved output being de- rived by summa-on of the cross- mul-plica-on products over the overlapping parts of the two func-ons. In general, if g i (i = 1, 2,..., m) is an input func-on and f j ( j = 1, 2,..., n) is a convolu-on operator, then the convolu-on output func-on y k is given by y = g f k = 12,,..., m+ n-1 k m  i= 1 i k-i ( )

27 In Fig the individual steps in the convolu-on process are shown for two digital func-ons, a double spike func-on given by g i = g 1,g 2,g 3 = 2,0,1 and an impulse response func-on given by f i = f 1, f 2, f 3, f 4 = 4, 3, 2, 1, where the numbers refer to discrete amplitude values at the sampling points of the two func-ons. From Fig it can be seen that the convolved output y i = y 1, y 2, y 3, y 4, y 5, y 6 = 8, 6, 8, 5, 2, 1. Note that the convolved output is longer than the input waveforms; if the func- -ons to be convolved have lengths of m and n, the convolved output has a length of (m + n - 1) Cross-products Sum = = = = = 2 Fig A method of calculating the convolution of two digital functions = 1

28 Deconvolu-on or inverse filtering (Kanasewich 1981) is a process that counteracts a previous convolu-on (or filtering) ac-on. Consider the convolu-on opera-on given in equa-on yt gt f t ()= () () * y(t) is the filtered output derived by passing the input waveform g(t) through a filter of impulse response f(t). Knowing y(t) and f(t), the recovery of g(t) represents a de- convolu-on opera-on. In the seismic case, y(t) is the seismic record resul-ng from the passage of a seismic wave g(t) through a por-on of the Earth, which acts as a filter with an impulse response f(t).

29 Reflectivity and convolution The seismic wave is sensitive to the sequence of impedance contrasts The reflectivity series (R) We input a source wavelet (W) which is reflected at each impedance contrast The seismogram recorded at the surface (S) is the convolution of the two S = W * R

30 Convolution Reflectivity series 1 ½ ½ Reflec-vity series describes the geological medium with reflec-vity contrasts Source wavelet ½ -½ 1 This is the seismic wave propaga-ng through the medium Output 0 Recorded waveform TASK: Calculate and plot the recorded seismic waveform using convolu-on operator

31 Spiking deconvolution Recorded waveform 1-1 ¾ -½ Deconvolution operator ¼ 1 1 Output 0 1 Recovered reflectivity series TASK: recover the reflec-vity series using spiking deconvolu-on operator

32 The Fourier transformadon, spectra, amplitude and phase response of a linear system ~ F( ) f ( t) e j t dt f 1 ~ j ( ) ( ) t t F e d, 2 The signal is here assumed to have a finite energy so that the integrals converge. The condition that no signal is present at negative times can be dropped in this case. The Fourier j t e transformation decomposes the signal into purely harmonic (sinusoidal) waves. The direct and inverse Fourier transformation are also known as a harmonic analysis and synthesis. We may consider the Fourier transformation as a special version of the Laplace transformation for real frequencies, i.e. for s = jω. In fact, F (ω ) = F ( jω ), i.e. the Fourier transform for real angular frequencies ω is identical to the Laplace transform for imaginary s = jω. In the case of s=jω the function H(ω) is called the complex frequency response of the system. The absolute value of H (ω ) is called the amplitude response, and the phase of H (ω ) the phase response of the system. The distinction between H(ω) and H(s) is essential when systems are characterized by their poles and zeros. Usually, poles and zeros are given for the Laplace transform.

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