1. March 11, 2008 1John Anderson: GE/CEE 479/679 Lecure 15 Earthquake Engineering GE / CEE - 479/679 Topic 15. Character of Strong Motion on Rock and Ground Motion Prediction Equations John G. Anderson Professor of Geophysics

2. March 11, 2008 2John Anderson: GE/CEE 479/679 Lecure 15

3. March 11, 2008 3John Anderson: GE/CEE 479/679 Lecure 15 Corner Frequency (approx) Scaling of strong motion in Guerrero, Mexico

4. March 11, 2008 4John Anderson: GE/CEE 479/679 Lecure 15

5. March 11, 2008 5John Anderson: GE/CEE 479/679 Lecure 15 Begin the study of strong motion with an examination of the character of strong motion on rock. A subsequent step will be to consider the perturbation to the rock motions caused by surface geology.

6. March 11, 2008 6John Anderson: GE/CEE 479/679 Lecure 15 Point Source Much can be learned from the equation giving the motion in an infinite medium resulting from a small (mathematically, a point) seismic source.

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9. March 11, 2008 9John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion Both u and x are vectors. u gives the three components of displacement at the location x. The time scale t is arbitrary, but it is most convenient to assume that the radiation from the earthquake source begins at time t=0. This assumes the source is at location x=0. The equations use r to represent the distance from the source to x.

10. March 11, 2008 10John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion Near-field term Intermediate- field P-wave. Intermediate- field S-wave. Far-field P- wave. Far-field S- wave.

11. March 11, 2008 11John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion A * is a radiation pattern. A * is a vector. A * is named after the term it is in. For example, A FS is the “far- field S-wave radiation pattern”

12. March 11, 2008 12John Anderson: GE/CEE 479/679 Lecure 15 Radiation Pattern Terms

13. March 11, 2008 13John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion ρ is material density α is the P-wave velocity β is the S-wave velocity. r is the source- station distance.

14. March 11, 2008 14John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion M 0 (t), or it’s first derivative, controls the shape of the radiated pulse for all of the terms. M 0 (t) is introduced here for the first time. Closely related to the seismic moment, M 0. Represents the cumulative deformation on the fault in the course of the earthquake.

15. March 11, 2008 15John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion Imagine an earthquake source which is growing with time. At each instant in time, one could define the moment that has been accumulated so far. That would involve the area A(t) and the average slip D(t) at each point in time. Fault perimeter at different times in the rupture process. 5 s 4 s 3 s 2 s 1 s

16. March 11, 2008 16John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion M 0 (t)=0 before the earthquake begins. M 0 (t)= M 0, the final seismic moment, after slip has finished everyplace on the fault. M 0 (t) treats this process as if it occurs at a point, and ignores the fault finiteness.

17. March 11, 2008 17John Anderson: GE/CEE 479/679 Lecure 15 M0(t)M0(t) Consider: This is the shape of M 0 (t). It is zero before the earthquake starts, and reaches a value of M 0 at the end of the earthquake. This figure presents a “rise time” for the source time function, here labeled T. Do not confuse with the period of a harmonic wave. M0M0 0 t

18. March 11, 2008 18John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion 1/r 4 1/r 2 1/r

19. March 11, 2008 19John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion The far field terms decrease as r -1. Thus, they have the geometrical spreading that carries energy into the far field. The intermediate-field terms decrease as r -2. Thus, they decrease in amplitude rapidly, and do not carry energy to the far field. However, being proportional to M 0 (t), these terms carry a static offset into the region near the fault. The near-field term decreases as r -4. Except for the faster decrease in amplitude, it is like the intermediate-field terms in carrying static offset into the region near the fault.

20. March 11, 2008 20John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion Signal between the P and the S waves. Signal for duration of faulting, delayed by P-wave speed. Signal for duration of faulting, delayed by S-wave speed.

21. March 11, 2008 21John Anderson: GE/CEE 479/679 Lecure 15 M0(t)M0(t) Consider these relations: The simplest possible shape of M 0 (t) is a very smooth ramp. Thus the simplest intermediate-field term is a smooth ramp. From M 0 (t), this suggests that the simplest possible shape of the far-field displacement pulse is a one-sided pulse.

22. March 11, 2008 22John Anderson: GE/CEE 479/679 Lecure 15 M0(t)M0(t) Consider these relations: Differentiating again, the simplest possible shape of the far-field velocity pulse is a two-sided pulse. Likewise, the simplest possible shape of the far-field acceleration pulse is a three-sided pulse.

23. March 11, 2008 23John Anderson: GE/CEE 479/679 Lecure 15 M0(t)M0(t) Consider these relations: If the simplest possible far-field displacement pulse is a one-sided pulse, the simplest velocity pulse is two-sided, and the simplest acceleration pulse is three sided. velocitydisplacementacceleration Far-field:

24. March 11, 2008 24John Anderson: GE/CEE 479/679 Lecure 15 Simple S-pulse Simple P-pulse

25. March 11, 2008 25John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion These results for the shape of the seismic pulses will always apply at “low” frequencies. They will tend to break down at higher frequencies. They have important consequences for the shape of the Fourier transform of the seismic pulse.

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27. March 11, 2008 27John Anderson: GE/CEE 479/679 Lecure 15 Fourier spectrum: Definition For any time series g(t), the Fourier spectrum is:

28. March 11, 2008 28John Anderson: GE/CEE 479/679 Lecure 15 Parseval’s Theorem

29. March 11, 2008 29John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion The Fourier transform of a one-sided pulse is always flat at low frequencies, and falls off at high frequencies. The corner frequency is related to the pulse width. fcfc

30. March 11, 2008 30John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion A high corner frequency corresponds to a short pulse duration. A low corner frequency corresponds to a long pulse duration.

31. March 11, 2008 31John Anderson: GE/CEE 479/679 Lecure 15 To get more from the spectrum We will calculate the Fourier transform of a “boxcar” function. 0 B0B0

32. March 11, 2008 32John Anderson: GE/CEE 479/679 Lecure 15 We derived …

33. March 11, 2008 33John Anderson: GE/CEE 479/679 Lecure 15 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:

34. March 11, 2008 34John Anderson: GE/CEE 479/679 Lecure 15 Corner frequency First zero

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36. March 11, 2008 36John Anderson: GE/CEE 479/679 Lecure 15 Discussion The spectrum is flat at low frequencies, then starts to decrease at a corner frequency. We will treat the corner frequency as half of the frequency of the first zero in this case, i.e. f c =1/(2D) Above the corner frequency, the spectrum falls off as f -1, with some fine structure superimposed. The corner frequency is inversely related to the duration of slip on the fault.

37. March 11, 2008 37John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion The duration of the pulse gives information about the size of the source. Expect that rupture will cross the source with a speed (v r ) that does not depend much, if at all, on magnitude. Thus, the duration of rupture is ~L/v r. We thus expect the pulse width (T in the last figure) is T~L/v r. If we measure T, we can estimate the fault dimension. The uncertainty may be a factor of 2 or so.

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39. March 11, 2008 39John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion For a circular fault with radius r b, Brune (1970, 1971) proposed the relationship: This is widely used in studies of small earthquakes. Uncertainties in r b due to the approximate nature of Brune’s model are probably a factor of two or so.

40. March 11, 2008 40John Anderson: GE/CEE 479/679 Lecure 15 Static Stress Drop In general, there is no way to measure the absolute stress in the Earth at depths of earthquakes. Seismologists do measure a static stress drop, commonly written as Δτ s. The static stress drop is estimated from the slip in the earthquake. In general, C is a dimensionless constant. W is the small dimension of the fault. This is called a “W-model”, since for constant stress drop slip is proportional to W.

41. March 11, 2008 41John Anderson: GE/CEE 479/679 Lecure 15 The constant C depends on the fault type. For a small circular rupture that does not reach the surface, replace W with a E, the radius of the fault. Then… (assume ) Rupture TypeStress Drop Circular Strike Slip Normal, thrust

42. March 11, 2008 42John Anderson: GE/CEE 479/679 Lecure 15 Point Source: Discussion Thus, seismologists can estimate the stress drop of the earthquake using the estimate of the radius. The equation is:

43. March 11, 2008 43John Anderson: GE/CEE 479/679 Lecure 15

44. March 11, 2008 44John Anderson: GE/CEE 479/679 Lecure 15 Fourier transform An important property is how the Fourier transform of a derivative of a time series is related to the Fourier transform of the time series itself.

45. March 11, 2008 45John Anderson: GE/CEE 479/679 Lecure 15

46. March 11, 2008 46John Anderson: GE/CEE 479/679 Lecure 15 Consequences: Fourier spectrum: Increases at low frequencies, Flattens at middle frequencies Need to explain roll off at high frequencies.

47. March 11, 2008 47John Anderson: GE/CEE 479/679 Lecure 15 Anderson and Hough (1984) defined a parameter κ to characterize the high-frequency slope High frequency spectral behavior. Over a fairly broad band on large events, the acceleration spectrum is flat, implying that the source displacement spectrum falls off as f -2. Above about 5 Hz, the acceleration spectrum also falls off, as seen on these plots of the same spectrum on log and semilog axes.

48. March 11, 2008 48John Anderson: GE/CEE 479/679 Lecure 15 Anderson (1986) This figure suggests that the high frequency behavior is due to attenuation at the site. Parameter kappa is larger, consistent with lower Q, for deep sediments than for rock sites, as suggested in the model below. Anderson (1986) suggested that κ results from both a site term and a path term:

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50. March 11, 2008 50John Anderson: GE/CEE 479/679 Lecure 15 Scaling law of the seismic spectrum. First described by Aki (1967). This figure based on Brune (1970), modified to include the effect of attenuation through the parameter κ. Figure is from Anderson (1986).

51. March 11, 2008 51John Anderson: GE/CEE 479/679 Lecure 15

52. March 11, 2008 52John Anderson: GE/CEE 479/679 Lecure 15 Corner Frequency (approx) Scaling of strong motion in Guerrero, Mexico

53. March 11, 2008 53John Anderson: GE/CEE 479/679 Lecure 15

54. March 11, 2008 54John Anderson: GE/CEE 479/679 Lecure 15

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58. March 11, 2008 58John Anderson: GE/CEE 479/679 Lecure 15 Representation Theorem Slip on the fault Green’s function Integral over the fault surface Convolution over time Displacement at the station at location x Elastic constants

59. March 11, 2008 59John Anderson: GE/CEE 479/679 Lecure 15 7 7.

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61. March 11, 2008 61John Anderson: GE/CEE 479/679 Lecure 15 L vrvr r1r1 r2r2 O S-wave pulse duration at O: Slip functions D(t) E F

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