# FUNDAMENTALS of ENGINEERING SEISMOLOGY SEISMIC SOURCES: POINT VS. FUNDAMENTALS of ENGINEERING SEISMOLOGY SEISMIC SOURCES: POINT VS. EXTENDED SOURCE; SOURCE SCALING 1 SOURCE REPRESENTATION Kinematic point source 2 Point sources Complete wave solution near-, intermediate-, far-field terms

Radiation patterns P vs. S wave amplitudes S wave spectra 3 Basic properties of seismic sources Focal mechanisms Double couple force system Brune source model

Self-similarity principle Haskell source model directivity 4 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. This is a specialized case of the Representation Theorem, using a point source and the infinite space Greens function. 5 KINEMATICS POINT SOURCE

Validity range M0 (seismic moment) r Point source approximation is allowed when the receiver is at a distance from the source larger than a few lengths of the fault. r >> L u ( x, t ) 6 KINEMATICS POINT SOURCE

Moment release Fault perimeter at different times in the rupture process. 1s 2s 3s 4s 5s 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. 7 KINEMATICS POINT SOURCE Seismic moment M 0 t A t D t M0(t)=0 before the earthquake begins. M0(t)= M0, the final seismic moment, after slip has finished everyplace on the fault. M0(t) treats this process as if it occurs at a point, and ignores the fault finiteness. 8

KINEMATICS POINT SOURCE Source time function Source time function D(t ) Dmax rise time t D (t ) 9

t KINEMATICS POINT SOURCE Simplest solution 1 N 1 u ( x, t ) A 4 4 r r/ M t d 0

r / 1 r IP 1 A 2 M0 t 2 4 r 1 r IS 1 A 2 M0 t

2 4 r 1 r FP 1 A M0 t 3 4 r 1 r FS 1

A M0 t 3 4 r true only if the medium is : Infinite Homogeneous Isotropic 3D 10 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. 11 KINEMATICS POINT SOURCE Equation terms 1 N 1 u ( x, t ) A 4

4 r r/ M 0 t d Near-field term r / 1 r IP 1 A 2 M0 t 2

4 r Intermediate-field P-wave 1 r IS 1 A 2 M0 t 2 4 r 1 r

FP 1 A M0 t 3 4 r Intermediate-field S-wave 1 r FS 1 A M0 t 3

4 r Far-field P-wave Far-field S-wave 12 KINEMATICS POINT SOURCE Radiation pattern 1 N 1 u ( x, t ) A 4 4 r

r/ M t d 0 r / 1 r IP 1 A 2 M0 t 2 4 r

1 r IS 1 A 2 M0 t 2 4 r 1 r FP 1 A M0 t 3 4

r A* is a radiation pattern. A* is a vector. A* is named after the term it is in. For example, AFS is the farfield S-wave radiation pattern 1 r FS 1 A M0 t 3 4

r 13 KINEMATICS POINT SOURCE Other constants 1 N 1 u ( x, t ) A 4 4 r r/ is material density

M t d 0 r / 1 r IP 1 A 2 M0 t

2 4 r is the P-wave velocity is the S-wave velocity. r is the source-station distance. 1 r IS 1 A 2 M0 t 2 4

r 1 r FP 1 A M0 t 3 4 r 1 r FS 1 A M0 t

3 4 r 14 KINEMATICS POINT SOURCE Temporal waveform M 0 t d r M0 t r M0 t

r M0 t r M0 t M0(t), or its first derivative, controls the shape of the radiated pulse for all of the terms. M0(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 KINEMATICS POINT SOURCE Geometrical spreading 1 N 1 u ( x, t ) A 4 4 r r/

M t d 1/r4 0 r / 1 r IP 1 A 2 M0 t 2 4 r

1 r IS 1 A 2 M0 t 2 4 r 1 r FP 1 A M0 t 3 4

r 1 r FS 1 A M0 t 3 4 r 1/r2 1/r2 1/r

1/r 16 KINEMATICS POINT SOURCE Geometrical spreading 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 M0(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. 17 KINEMATICS POINT SOURCE Temporal delays 1 N 1 u ( x, t ) A 4 4 r r/ M t d 0

Signal between the P and the S waves. r / 1 r IP 1 A 2 M0 t 2 4 r 1

r IS 1 A 2 M0 t 2 4 r 1 r FP 1 A M0 t 3 4 r

Signal for duration of faulting, delayed by P-wave speed. Signal for duration of faulting, delayed by S-wave speed. 1 r FS 1 A M0 t 3 4 r 18

KINEMATICS POINT SOURCE Solution for a Heaviside source time function D(t ) Dmax M0 Rise time = 0 r t u (r , t ) 19

KINEMATICS POINT SOURCE Solution for a Heaviside source time function D(t ) Dmax M0 Rise time = 0 r t dD(t ) dt t

20 KINEMATICS POINT SOURCE Solution for a Heaviside source time function D(t ) Dmax M0 Rise time = 0 r t u (r , t )

0 r r t 21 KINEMATICS POINT SOURCE Solution for a Heaviside source time function D(t ) Dmax

M0 Rise time = 0 r t Far field P wave u (r , t ) 0 r r

t 22 KINEMATICS POINT SOURCE Solution for a Heaviside source time function D(t ) Dmax M0 Rise time = 0 r t + Int. field P wave

u (r , t ) 0 r r t 23 KINEMATICS POINT SOURCE Solution for a Heaviside source time function D(t )

Dmax M0 Rise time = 0 r t + far field S wave u (r , t ) 0 r r

t 24 KINEMATICS POINT SOURCE Solution for a Heaviside source time function D(t ) Dmax M0 Rise time = 0 r t

+ int. field S wave u (r , t ) 0 r r t 25 KINEMATICS POINT SOURCE Solution for a Heaviside source time function

D(t ) Dmax M0 Rise time = 0 r t + near field wave u (r , t ) 0 r

r t 26 27 28 29 30 INFLUENCE OF SOURCE PARAMETERS Displacement versus acceleration (for the S-wave,

showing starting and stopping arrivals) d 2u (t ) dt 2 t du (t ) dt t u (t ) t 31 SOURCE REPRESENTATION

Kinematic point source: FAR FIELD 32 KINEMATICS POINT SOURCE Far Field 1 r FP 1 u ( x, t ) A M0 t 3 r 4

1 r FS 1 A M t 0 r 4 3

1/r geometrical spreading Signal for duration of faulting, delayed by P-wave speed. Signal for duration of faulting, delayed by S-wave speed. 33 34 Frequencies of ground-motion for engineering purposes 10 Hz --- 10 sec (usually less than about 3 sec) Resonant period of typical N story structure ~ N/10 sec Corner periods for M 5, 6, and 7 ~ 1, 3, and 9 sec

35 Horizontal motions are of most importance for earthquake engineering Seismic shaking in range of resonant frequencies of structures Shaking often strongest on horizontal component: Earthquakes radiate larger S waves than P waves Decreasing seismic velocities near Earths surface produce refraction of the incoming waves toward the vertical, so that the ground motion for S waves is primarily in the horizontal direction Buildings generally are weakest for horizontal shaking => An unfortunate coincidence of various factors 36

Radiation Patterns & Relative Amplitudes in 3D no nodal surfaces for S waves 37 Source spectra of radiated waves (far-field, point source) 38 Source spectra of radiated waves (far-field, point source) A description of the amplitude and frequency content of waves radiated from the earthquake source is the foundation on which theoretical predictions of ground

shaking are built. The specification of the source most commonly used in engineering seismology is based on the motions from a simple point source. 39 Point Source: Discussion Fault perimeter at different times in the rupture process. 1s 2s 3s 4s 5s

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. 40 Point Source: Discussion M 0 t A t D t M0(t)=0 before the earthquake begins. M0(t)= M0, the final seismic moment, after slip has finished everyplace on the fault. M0(t) treats this process as if it occurs at a point, and ignores the fault finiteness. 41

Consider: M0(t) M0 0 t This is the shape of M0(t). It is zero before the earthquake starts, and reaches a value of M0 at the end of the earthquake. This figure presents a rise time for the source time function, here labeled T. (Do not confuse this symbol with the period of a harmonic wave--- should have 42 used Tr ) Consider these relations:

M0(t) dM0(t)/dt the far-field shape is proportional to the moment rate function From M0(t), this suggests that the simplest possible shape of the far-field displacement pulse is a one-sided pulse. The simplest possible shape of M0(t) is a very smooth ramp. 43 Consider these relations: M0(t)

dM0(t)/dt d2M0(t)/dt2 d3M0(t)/dt3 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. 44 Consider these relations: M0(t) dM0(t)/dt

d2M0(t)/dt2 Far-field: displacement velocity d3M0(t)/dt3 acceleration 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 (with zero area, implying velocity = 0.0 at end of record). 45 Point Source: Discussion

These results for the shape of the seismic pulses will always apply at low frequencies, for which the corresponding wavelengths are much longer than the fault dimensions--- the fault looks like a point. They will tend to break down at higher frequencies. They have important consequences for the shape of the Fourier transform of the seismic pulse. 46 Calculate the period for which the wavelength equals a given value. Assume s = 3.5 km/s. S T T S M

5.7 3.5 6.9 35 8.0 350 T 47 Calculate the period for which the wavelength

equals a given value. Assume s = 3.5 km/s. S T T S M T 5.7 3.5 1s 6.9

35 10 s 8.0 350 100 s 48 Source Time Function The Source time function describes the moment release rate of an earthquake in time For large earthquakes, source time function can be complicated For illustration, consider a simple pulse

49 Source Spectrum To explore source properties in more detail, consider the source spectrum 50 Source Spectrum To explore source properties in more detail, consider the source spectrum 51 Source Spectrum To explore source properties in more detail, consider the source spectrum

52 53 Source Spectrum Radiated energy as function of frequency Small earthquake: high frequencies (short ) Large earthquake: lower frequencies (long ) Energy release proportional to velocity spectrum Corner frequency = peak of velocity spectrum peak frequency of energy release

Displacement spectrum: flat below corner 54 Point Source: Discussion fc The Fourier transform of a onesided pulse is always flat at low frequencies, and falls off at high frequencies. The corner frequency is related to the pulse width. Commonly used equation: S 1 / [1 ( ff / 2

0 ) ] 55 Motivation for commonly used equation 56 KINEMATICS EXTENDED SOURCE Source radiation: convolution of two box functions D(t ) D (t )

Dmax Dmax t t This motivates the need to look at the frequency-domain representation of a box function t 57 Fourier spectrum of a box function: The frequency domain

representation of the point source For any time series g(t), the Fourier spectrum is: G ( ) g t expi t dt 58 Example Calculate the Fourier transform of a boxcar function. 0 b t B0

0 D t 2 D D t 2 2 D 0 t 2 B0

0 D 2 D 2 59 The answer D sin 2

G ( ) B0 D D 2 With the following behavior for low and high frequencies: G() area of pulse = B0D, 0 G() 1/, 60 Properties: The asymptotic limit for frequency -->0 is B0D. The first zero is at: D

2 D 2 f 2 1 f D 61 Corner frequency First zero Note can approximate the

spectral shape with two lines, ignoring the scalloping. The intersection of the two lines is the corner frequency, an important concept. 62 Examples of spectra for two pulses with the same area but different durations linear-linear axes log-log axes

63 Examples of spectra for two pulses with the same area but different durations. Note that the low frequency limit is the same for both pulses, but the corner frequency shifts linear-linear axes log-log axes 64 KINEMATICS EXTENDED SOURCE Source radiation: convolution of two box functions D(t )

D (t ) Dmax Dmax t t t 65 KINEMATICS EXTENDED SOURCE Omega square model Spectrum of single box function

goes as 1/f at high frequencies; spectrum of convolution of two box functions goes as 1/f2 u (t ) u~ ( f ) t w 2 d M0 corner frequency f

1 d 66 KINEMATICS POINT SOURCE Far Field 1 r FP 1 u ( x, t ) A M0 t 3 r 4

1 r FS 1 A M t 0 r 4 3 1/r geometrical spreading

Signal for duration of faulting, delayed by P-wave speed. Signal for duration of faulting, delayed by S-wave speed. 67 Static scaling before, now consider frequencydependent source excitation Changing notation, the Fourier transform of u(t) can be written: Spectrum of displacement = Source X Path X SIte Y ( M 0 , R, f ) E ( M 0 , f ) P( R, f )G ( f ) E ( M 0 , f ) CM 0 S ( M 0 , f ) C R VF 4S S3 R0

68 Simplest source model: 1 S( f ) 2 1 f f 0 This is known as the -square model. Because the acceleration source spectrum is 2 A(M 0 , f ) M 0 2f S (M 0 , f ) the scaling of the acceleration source spectrum at low frequencies goes as A(M 0 , f ) M 0 f 2 , f 0 and at high frequencies as

A(M 0 , f ) M 0 f 02 , f 69 Discussion The displacement spectrum is flat at low frequencies, then starts to decrease at a corner frequency. Above the corner frequency, the spectrum falls off as f-2 (for two box functions), with some fine structure superimposed. The corner frequency is inversely related to the (apparent) duration of slip on the fault. 70 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/vr. We thus expect the pulse width (D before, but T now) is T~L/vr with some modification for direction. If we measure T, we can estimate the fault dimension. The uncertainty may be a factor of 2 or so. 71 Point Source: Discussion For a circular fault with radius rb, Brune (1970, 1971) proposed the relationship ( is shear-wave velocity, f0 is corner frequency): 2.34 rb 2 f 0

This is widely used in studies of small earthquakes. Uncertainties in rb due to the approximate nature of Brunes model are probably a factor of two or so. 72 Introducing the stress drop (also known as the stress parameter) 1 2 1 2 1 2 73

For a circular crack: There is a theoretical relation between the static stress drop (), the average slip over the crack surface (U), and the radius of the crack (rb): 7 U 16 rb Note that for a constant radius, an increasing slip gives increasing stress drop 74 For a circular crack: This can be converted into an equation in terms of seismic moment: 7 M0

3 16 rb Although developed for a simple source (a circular crack), this equation is the basis for the simulation of ground motions of engineering interest, as improbable as that seems. 75 Using the relation between source radius, corner frequency and stress drop leads to this important equation 6 f 0 4.9 10 M 0 13

where f0 is in Hz, in km/s, in bars, and Mo in dyne-cm 76 Stress Drop Static versus dynamic stress Variability over rupture area Estimation = difficult 77 Typical Stress Drop Values Typical values: 0.1 bars 500 bars 0.01 MPa 50 MPa Units: force/area (bars = cgs) Atmospheric pressure ~ 1 bar Absolute stress in earth = high, very difficult to measure

78 Example f0 = _____ r = 2.34 /(2f0) = ? meters If Mo = 79 Example R = 50 m If Mo = 1012Nm, stress drop = ____ If Mo = 1010Nm, stress drop = ____ 80

Source Scaling Recall that A(M 0 , f ) M 0 f 02 , f Using the equation relating f 0 , , and M 0 : we have f 0 4.9 106 M 0 13 A M 0 , f M 01 3 2 3. f This is an important equation, because it relates the high-frequency spectral level to a few parameters. The different dependence of the low- and high-frequency spectra on M 0 is also important in the dependence of ground motion on moment magnitude. This dependence is often known as source scaling.

81 Self Similarity and Scaling at High Frequencies U/rb = constant for self similarity AHF M01/3 (2/3) M0 1/3 constant stress parameter (drop) scaling (a common assumption) 82 INFLUENCE OF SOURCE PARAMETERS Magnitude u~ ( f ) Scaling if is

constant ulf1 M 01 2 2 ulf M0 w 2 u 1 hf uhf2

1 0 2 0 M M f 1 c f 13

2 c This is an important figure, as it indicates that the magnitude scaling of ground motion will be a function of frequency, with stronger scaling for low frequencies than high frequencies. One consequence is that the spectral shape of ground motion will be magnitude dependent, with large earthquakes having relatively more low-frequency energy than small earthquakes f

83 (From J. Anderson) 84 (From J. Anderson) 85 Scaling of high-frequency ground motions: Typical scaling of spectra observed for earthquakes with M<7 : 2 displacement spectral falloff and constant stress drop with respect to seismic moment 1/ 3 0 uhf ( f ) M

E( f ) A 86 INFLUENCE OF SOURCE PARAMETERS Stress drop If the moment is fixed, an increase of stress drop means an increase of the corner frequency value u~ ( f ) f c2 L1 2 3 1

fc L2 1 u~ 1 ~ u 2 1 hf 2 hf f 1

c f 2 c f 2 3 87 Scaling difference: Low frequency A M0, but log M0 1.5M, so A 101.5M.

This is a factor of 32 for a unit increase in M High frequency A M0(1/3), but log M0 1.5M, so A 100.5M. This is a factor of 3 for a unit increase in M Ground motion at frequencies of engineering interest does not increase by 10x for each unit increase in M 88 Time: 14:49:29

M = 7.5 Date: 2003-09-15; 1000 100 File: C:\metu_03\rec_proc_strong_motion\FAS_XCA.draw; Equal M implies the same spectra at low frequencies Fourier Acceleration Spectrum (cm/s)

10000 M = 4.5 10 decay at high f due to source or site (I prefer the latter) AB95 H96

Fea96 (no site amp) BC92 J97 1 0.1 0.01 0.1 1 Frequency (Hz) 10 100 89

is a KEY parameter for ground-motion at frequencies of engineering interest Units: bars, MPa, where 1 MPa= 10 bars Also, M0 in dyne-cm or N-m, where 1 N-m=10^7 dynecm (log M0=1.5M+16.05 for M0 in dyne-cm). 90 Why Stress Drop Matters Increase stress drop more high frequency motion Structural response depends on amplitude of shaking and frequency content Frequencies of Engineering concern 10 Hz --- 10 sec (usually less than about 3 sec) Resonant period of typical N story structure ~ N/10 sec Resonance period of 20 storey structure?

91 Why Stress Drop Matters Ground motion prediction methods: stress drop = input parameter Intraplate earthquakes (longer recurrence) higher stress drop 92 Use of mb/Mw in the Search for High Stress-Parameter Earthquakes in Regions of Tectonic Extension Jim Dewey and Dave Boore 93 We have 21,179 events, h(PDE) or h(GCMT) < 50 km, 1976 Sept 2007, for

which mb(PDE) and Mw(GCMT) are both available Conventional wisdom: high mb with respect to Mw implies high stress parameter Assumptions for theoretical curves random-vibration source with -squared source-spectrum mb measured on WWSSN SP seismograph same raypath attenuation for all source-station pairs 94 SOURCE EFFECTS Complex source phenomena

95 SOURCE EFFECTS ON STRONG GROUND MOTION Influence of source phenomena Directivity and rupture velocity Super shear velocity Rupture in surface Hanging wall/foot wall

Stopping phases Concept of asperities and barriers Self similar slip distribution 96 60 min Haskell source model: Simple description of a moving source. 97 Directivity: Ground motion pulse duration will be shortened in duration in the direction in which wave front is advancing, as waves radiating from near-end of fault pile up on top of waves radiating from the far end. This directivity effect increases wave amplitudes in the rupture propagation direction. 98

Example of observed directivity effects in the Landers earthquake ground motions near the fault. Directivity was a key factor in causing large ground motions in Kobe, Japan, and a major damage factor. It probably also played a role in the recent San Simeon, CA, earthquake 99 COMPLEX SOURCE PHENOMENA Directivity formulation r t0 c

L r L cos tL vr c L d t L t0 vr vr 1 cos c

1 1 90 fc fc d vr 1 cos c 100 COMPLEX SOURCE PHENOMENA Directivity coefficient

For an unilateral fault : vr / 1 Cd vr 1 cos 0 180 .8

5 0.9 .9 10 0.83 101 COMPLEX SOURCE PHENOMENA Directivity effect on radiation Hirasawa (1965) 102

COMPLEX SOURCE PHENOMENA Directivity effect on acceleration spectrum ~ u ( f ) Cd 2 w2 Cd f c f

For very low frequencies, the wavelengths are much longer than the fault length, and directivity has no impact on the motion, which is controlled by the seismic moment; this is why the two spectra are the same at low 103 frequencies in this cartoon. COMPLEX SOURCE PHENOMENA Directivity effect on displacement spectrum ~ u ( f ) Haskell (1964) Frankell (1991) Non directive Cd f

104 Directivity Directivity is a consequence of a moving source Waves from far-end of fault will pile up with waves arriving from near-end of fault, if you are forward of the rupture This causes increased amplitudes in direction of rupture propagation, and decreased duration. Directivity is useful in distinguishing earthquake fault plane from its auxiliary plane because it destroys the symmetry of the radiation pattern. 105 SOURCE REPRESENTATION Kinematics extended source 106

Fault kinematics Distribution of fault slip as a function of space and time Often parameterized by velocity of rupture front, and rise time and total slip at each point of the fault 107 KINEMATICS EXTENDED SOURCE An extended source is a sum of point sources surface 108 Slip on an earthquake fault START

Surface of the earth Depth Into the earth 100 km (60 miles) Distance along the fault plane 109 Slip on an earthquake fault Second 2.0 110 Slip on an earthquake fault

Second 4.0 111 Slip on an earthquake fault Second 6.0 112 Slip on an earthquake fault Second 8.0 113 Slip on an earthquake fault Second 10.0 114

Slip on an earthquake fault Second 12.0 115 Slip on an earthquake fault Second 14.0 116 Slip on an earthquake fault Second 16.0 117 Slip on an earthquake fault Second 18.0

118 Slip on an earthquake fault Second 20.0 119 Slip on an earthquake fault Second 22.0 120 Slip on an earthquake fault Second 24.0 121

Rupture on a Fault Total Slip in the M7.3 Landers Earthquake 122 End 123