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4 - Seismic Moment and Stress Aki(1966 introduced Mo Is the...

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Seismic Moment and Stress Aki (1966) introduced Mo Is the product of three factors that indicate A D μ For crustal rocks = 3.3 × dyne/cm2 μ Seismic moment ( M) is measured in N-m or dyn- cm (1 N-m = 107 dyn-cm). 11 10

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Kanamori (1977) used momemt to define the moment magnitude scale M o given here in N-m M W  = 2 / 3 * log(M o ) - 10.73
Stress Fault rupture is caused by stress in the earth. The relative slip on both sides of the dislocation surface occurs when shear stress exceeds the available friction stress. stress drop, = Δσ initial - final state of shear stress 0 1 σ σ = - Stresses are measured in bar, dyn/cm2, and N/m2 (Pa). The related conversions are: 1 bar = 106 dyn/cm2, 1 Pa = 10 dyn/cm2, and 1 Mpa = 106 Pa.

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Stress The average stress is defined as the mean value of the shear stresses acting before and after the earthquake: σ ( 29 1 1 / 2 ( / 2) o σ σ σ σ σ = + = + The total Strain energy, W, released by the fracture will be simply : . W DA av Stress slip area σ = = Some of the strain energy is lost to Friction , H. So the Radiated energy E E= W-H = f DA DA σ σ - f σ = frictional stress.
E = 1 1 ( / 2) ( ) ( ) f o f DA DA E DA σ σ σ σ σ + - = + - o= lower bound of the radiated energy = 0 ( / 2) ( / 2 ) DA M σ σ μ = 11 2 5 10 / , 50 dyne cm bar μ σ = = If ( 29 0 4 ; 2 10 o M E = 0 0 log log 4.3 E M = - Where E 0 is in ergs. Substituting M o we will get log 1.5 11.8 o w E M = + Moment and Energy

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Stress and Fault L W D Let the average slip = for a fault of length L and width W The strain According to Hook’s law , the stress drop and strain could be related as Where =Characteristic rupture dimension(either L or W) C = constant depends on the fault geometry Circular fault (radius a) strike slip fault Dip slip fault D D D or L W D ( ) D C L σ μ = L 3 7 16 o M a σ = 2 2 o M W L σ π = 2 8 3 o M W L σ π =
Rupture Model V is either P-wave or S-wave velocity The time taken by the wave to reach station from the far end of the rupture R V L V r T - = V r T T o R - =

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Rupture Model Due to the finite rupture length the radiated pulse varies in the time duration as a function azimuth. Area of the pulse is same at all azimuths, The magnitude of the source time function varies inversely with the duration. This is called Directivity effect Directivity Doppler effect. Shifts the Frequency of moving oscillator to higher frequency when oscillator moves towards the observer and lower frequency when its move away.

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Rupture Model Aki (1987) relate stress drop and slip velocity of a rupture as r D K T μ σ β = If is the time taken by the rupture to sweep a fault of length L, then r r L T V D We assume here that = shear-wave velocity. K = Scaling factor varies Between 0.5 to 1. r T R V β = r D T = slip velocity
Feb 19, 2008 11 Near Field Ground Motion If x represents the perpendicular distance from the fault surface H(t) is the Heaviside unit step function β is the shear-wave propagation velocity, the initial time function for shear stress pulse ( , ) x x t H t σ σ β = - Ground displacement close to the fault center parallel to the fault surface, u, can be obtained through integration: u x δ σ μ δ � � = � � � � 0, x u t σ β μ �� = = �� ��

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4 - Seismic Moment and Stress Aki(1966 introduced Mo Is the...

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