1. February 14, 2008 1John Anderson GE/CEE 479/679 Earthquake Engineering GE / CEE - 479/679 Topic 8. Fourier Spectra John G. Anderson Professor of Geophysics

2. February 14, 2008 2John Anderson GE/CEE 479/679 Last Time Response Spectra –Based on response of single-degree-of-freedom oscillators. –Five types. Measure peak displacement, peak velocity, peak acceleration, and two approximations to those.

3. February 14, 2008 3John Anderson GE/CEE 479/679 What is a Spectrum? A spectrum is, first of all, a function of frequency. Second, for our purposes, it is determined from a single time series, such as a record of the ground motion. The spectrum in general shows some frequency-dependent characteristic of the ground motion.

4. February 14, 2008 4John Anderson GE/CEE 479/679 m Earth k y0y0 F y x = y-y 0 (x is negative here) Hooke’s Law c Friction Law z(t)

5. February 14, 2008 5John Anderson GE/CEE 479/679 The differential equation for the forced oscillator is: The DuHammel integral gives a general solution. Let: The response of the oscillator to a(t) is: Notice that the solution is a function of ω 0 and h. The response spectra focus on the ω 0 dependence.

6. February 14, 2008 6John Anderson GE/CEE 479/679 Reminder: There are two different types of frequency, the angular frequency ω and the natural frequency f. They are related by: We also use the period, T, related to the frequency by:

7. February 14, 2008 7John Anderson GE/CEE 479/679 Summary: 5 types of response spectra SD = Maximum relative displacement response. SV = Maximum relative velocity response. SA = Maximum absolute acceleration response

8. February 14, 2008 8John Anderson GE/CEE 479/679 Data from Guerrero, Mexico, Anderson and Quaas (1988)

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11. February 14, 2008 11John Anderson GE/CEE 479/679 Fourier Spectrum General idea: reproduce the seismogram exactly as the sum of sine and cosine terms. The Fourier spectrum is different from the response spectrum. Depends only on the frequency (i.e. not damping). However, two numbers characterize the Fourier spectrum at each frequency, rather than just one that characterizes the response spectrum. The Fourier spectrum is a linear function of the input seismogram, while the response spectrum is a nonlinear function of the seismogram. The Fourier spectrum tells how much energy is in the seismogram at each frequency, but the response spectrum does not.

12. February 14, 2008 12John Anderson GE/CEE 479/679 Fourier Spectrum There is enough information in the Fourier spectrum to uniquely recover the original seismogram, while it is not possible to do this from the response spectrum. This characteristic makes it possible to easily carry out analyses that are very difficult otherwise. The effect of various processes on the Fourier spectrum is often a useful way to characterize a phenomenon. For example: –Response of an instrument. –Site effect. –Wave propagation.

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15. February 14, 2008 15John Anderson GE/CEE 479/679 Fourier spectrum: Definition For any time series g(t), the Fourier spectrum is: Generally expect that g(t) is limited in time.

16. February 14, 2008 16John Anderson GE/CEE 479/679 Fourier spectrum: First, note, we can carry out the integral for many different values of the angular frequency ω. Also, the value of G(ω) at every frequency is determined from the same seismogram. Considering these features, it satisfies the definition of a spectrum.

17. February 14, 2008 17John Anderson GE/CEE 479/679 Fourier spectrum: In what way are there two numbers at each frequency? From basic complex number theory: Using this, the definition can be rewritten as: Thus, the definition can be rewritten as: The two numbers at each frequency are a(ω) and b(ω) (for g(t) real).

18. February 14, 2008 18John Anderson GE/CEE 479/679 Fourier spectrum: It is also useful to define the Fourier Amplitude Spectrum:

19. February 14, 2008 19John Anderson GE/CEE 479/679 Fourier spectrum: Starting from the Fourier spectrum, it is possible to recover the original seismogram. The Inverse Fourier Transform is: It is not possible to recover the original seismogram from the Fourier Amplitude Spectrum alone.

20. February 14, 2008 20John Anderson GE/CEE 479/679 OK. So much for the math. What does it mean? Try to build an intuitive model for what the Fourier transform is telling us. Show some examples.

21. February 14, 2008 21John Anderson GE/CEE 479/679 Start with a related concept. Fourier approximation to a time series of finite duration, 0<t<D.

22. February 14, 2008 22John Anderson GE/CEE 479/679 Fourier series Consider a time series g(t) defined (or measured) over a time interval of duration D, 0<t<D. The discovery is that g(t) can be written as the sum of a series of sine and cosine functions:

23. February 14, 2008 23John Anderson GE/CEE 479/679 The coefficients are found by:

24. February 14, 2008 24John Anderson GE/CEE 479/679 An example follows At the bottom, the next figure shows the original time series. Above, it shows the sum of the terms represented by n=1, n=2, n=3, … For instance, the term represented by n=2 is:

25. February 14, 2008 25John Anderson GE/CEE 479/679 Original at the bottom, with contributions of different frequencies above, residual at the top.

26. February 14, 2008 26John Anderson GE/CEE 479/679 Discussion All of the terms are shown on the same scale. Most are shifted upwards - all have a zero mean value. In the previous slide, the n=1 term is small. The n=2 term captures the main swings of this particular seismogram. The n=3 term is also fairly large.

27. February 14, 2008 27John Anderson GE/CEE 479/679 Discussion (cont) The top trace is the motion that is not explained by the first five terms of the Fourier series. This residual is much smaller in amplitude than the original seismogram. The residual is also obviously dominated by contributions at higher frequencies than the frequencies of the first five terms shown below it.

28. February 14, 2008 28John Anderson GE/CEE 479/679 Discussion (cont) Conclusion to be drawn: In this case, the first five terms of the Fourier series are able to represent the main features of the time series. The next slide reinforces this conclusion, by showing the sum of the first n terms, up to n=5.

29. February 14, 2008 29John Anderson GE/CEE 479/679 Sum of the first n terms of the Fourier series, with n increasing from bottom to top, original at the top. n=1 n=2 n=3 n=4 n=5 original

30. February 14, 2008 30John Anderson GE/CEE 479/679 Discussion (cont.) The amplitudes of the sine and cosine terms in the above example tell something about the character of the seismogram. The frequency of a sine or cosine in the n th term is f=n/D. The amplitude at each frequency tells how strongly that frequency contributes to the total seismogram. The frequency that contributes most strongly can be estimated by inspection of the seismogram.

31. February 14, 2008 31John Anderson GE/CEE 479/679 Another example The next slides show the same process for a more complicated time series. The conclusions are the same.

32. February 14, 2008 32John Anderson GE/CEE 479/679 Original at the bottom, with contributions of different frequencies above, residual at the top.

33. February 14, 2008 33John Anderson GE/CEE 479/679 Sum of the first n terms of the Fourier series, with n increasing from bottom to top, original at the top.

34. February 14, 2008 34John Anderson GE/CEE 479/679 For the first example, this slide tabulates the amplitudes of each of the terms in the Fourier series. The frequency is given by n/D. n+1f anan bnbn

35. February 14, 2008 35John Anderson GE/CEE 479/679 We can plot the amplitudes from the previous slide as a function of frequency. The result is a spectrum.

36. February 14, 2008 36John Anderson GE/CEE 479/679 Discussion Sometimes the sine contribution is larger, and sometimes the cosine term is larger. In the amplitude spectrum, the information about which is larger is lost. The sign and relative amplitudes of the sine and cosine terms are essential to reconstruct the original seismogram. This information is called the phase. However, the amplitude spectrum is more informative in telling which frequencies are present in the seismogram.

37. February 14, 2008 37John Anderson GE/CEE 479/679 Discussion (cont.) The frequency of a sine or cosine in the n th term is f=n/D. As D increases, there are more frequencies in the Fourier series. The lowest frequency is always f=1/D, or one cycle in the length of the record.

38. February 14, 2008 38John Anderson GE/CEE 479/679 Discussion (cont) As the duration of the time series increases towards infinity, the frequency step decreases towards zero, so the Fourier series approaches a continuous function of the frequency. The Fourier spectrum can be regarded as the Fourier series in this limit.

39. February 14, 2008 39John Anderson GE/CEE 479/679 Q&A What does the Fourier transform tell us? –The frequencies that contribute to the time series, and the relative and absolute amplitudes of each of those frequencies.

40. February 14, 2008 40John Anderson GE/CEE 479/679 Challenge I stated earlier that the peak frequency of the Fourier spectrum can be estimated by inspection of the record. The next figure shows noise. Can you estimate the peak frequency of the Fourier spectrum?

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42. February 14, 2008 42John Anderson GE/CEE 479/679 Answer This seismogram shows vibration at a rate of about 8 cycles/minute. So the frequency in the units we usually use is f=8/60=1/7.5=0.13 Hz. –(1 Hz = 1 cycle/second) The period is T=1/f=7.5 s.

43. February 14, 2008 43John Anderson GE/CEE 479/679 By the way… This is a seismogram recorded in Nevada. It was recorded when a strong storm was off the coast of California. This kind of noise is typically generated in the ocean during strong storms. Before satellites, seismologists were developing a system to use this kind of noise to track hurricanes. Satellites made that kind of a system obsolete.

44. February 14, 2008 44John Anderson GE/CEE 479/679 Example We will calculate the Fourier transform of a “boxcar” function. 0 B0B0 t

45. February 14, 2008 45John Anderson GE/CEE 479/679 Just substitute in the definition

46. February 14, 2008 46John Anderson GE/CEE 479/679 Continuing one way

47. February 14, 2008 47John Anderson GE/CEE 479/679 Continuing …

48. February 14, 2008 48John Anderson GE/CEE 479/679 Taking a different route… Because the sin function is odd and b(t) is even, and the integral of an odd function times an even function is zero.

49. February 14, 2008 49John Anderson GE/CEE 479/679 Continuing … Same as before.

50. February 14, 2008 50John Anderson GE/CEE 479/679 Next, a plot This uses D=1.0 and B 0 =1.0. The assymptotic limit for frequency -->0 is B 0 D. The first zero is at:

51. February 14, 2008 51John Anderson GE/CEE 479/679 Corner frequency First zero

52. February 14, 2008 52John Anderson GE/CEE 479/679 Discussion The fact that the low frequency limit and the first zero occur in the right place is a check that my program calculated the spectrum correctly, since it was calculated numerically. The spectrum is flat at low frequencies, then starts to decrease at a corner frequency. Above the corner frequency, the spectrum falls off as f -1, with some fine structure superimposed.

53. February 14, 2008 53John Anderson GE/CEE 479/679 More examples…

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55. February 14, 2008 55John Anderson GE/CEE 479/679 Comments on triangle function It has the same width (at the base) as the boxcar. It’s corner frequency and it’s first zero are at the same frequency. It falls off faster, proportional to f -2, above the corner frequency.

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57. February 14, 2008 57John Anderson GE/CEE 479/679 Impulse response. The following are the seismograms and spectra of DuHammel’s integral for an impulse at t=0. All oscillators have f 0 =1.0 Hz. Three dampings, h=0.02, h=0.05, and h=0.8. Question: does the peak occur at the expected frequency?

58. February 14, 2008 58John Anderson GE/CEE 479/679 So, the differential equation for the forced oscillator is: After dividing by m, as previously, this equation becomes: This is the differential equation that we use to characterize both seismic instruments and as a simple approximation for some structures, leading to the response spectrum.

59. February 14, 2008 59John Anderson GE/CEE 479/679 Sinusoidal Input It is informative to consider first the response to a sinusoidal driving function: It can be shown by substitution that a solution is: Where:

60. February 14, 2008 60John Anderson GE/CEE 479/679 Sinusoidal Input (cont.) The complex ratio of response to input can be simplified by determining the amplitude and the phase. They are:

61. February 14, 2008 61John Anderson GE/CEE 479/679 h=0.02

62. February 14, 2008 62John Anderson GE/CEE 479/679 h=0.05

63. February 14, 2008 63John Anderson GE/CEE 479/679 h=0.8

64. February 14, 2008 64John Anderson GE/CEE 479/679 h=0.05 Response oscillates Spectrum is peaked h=0.02 Response oscillates a long time Spectrum is highly peaked

65. February 14, 2008 65John Anderson GE/CEE 479/679 h=0.8 Very little overshoot Spectrum is smooth h=0.05 Response oscillates Spectrum is peaked

66. February 14, 2008 66John Anderson GE/CEE 479/679 Discussion Fourier spectrum tells you what frequencies are present in the seismogram. Response spectrum may also, but it is modulated by the peak in the instrument response, so it is not quite as simple to interpret.

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