Group Velocity Measurement
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1 Group Velocity Measurement Distance Travelled (8.3) Group Velocity Time It Took
2 Group Velocity Measurement Distance Travelled (8.3) Group Velocity Time It Took station Earthquake t time d distance earthquake
3 Group Velocity Measurement Distance Travelled (8.3) Group Velocity Time It Took station Earthquake t = 238 s time d = 1000 km distance earthquake
4 Group Velocity Measurement Distance Travelled (8.3) Group Velocity Time It Took station Earthquake t = 238 s time d = 1000 km distance earthquake Need to know exact location and timing of an earthquake
5 Group Velocity Measurement Distance Travelled (8.3) Group Velocity Time It Took Location of stations are well known Earthquake t 1 = 238 s time d = 200 km station 1 station 2 t 2 = 285 s earthquake
6 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Filter seismograms Measure group velocity Plot group velocity vs. frequency
7 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Filter seismograms Measure group velocity Plot group velocity vs. frequency Use single, well-dispersed surface wave arrival
8 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Filter seismograms Measure group velocity Plot group velocity vs. frequency more accurate Use single, well-dispersed surface wave arrival less accurate
9 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Use single, well-dispersed surface wave arrival station d distance earthquake Earthquake: Mexico Station: CCM, Cathedral Cave, Missouri Distance: 22.4 degrees Component: Vertical
10 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Use single, well-dispersed surface wave arrival station Well-dispersed Rayleigh wave d distance earthquake Earthquake: Mexico Station: CCM, Cathedral Cave, Missouri Distance: 22.4 degrees Component: vertical
11 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Use single, well-dispersed surface wave arrival T/2 = s time = s Earthquake: Mexico Station: CCM, Cathedral Cave, Missouri Distance: 22.4 degrees Component: vertical
12 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Use single, well-dispersed surface wave arrival Period = T = 55.7 s Group Velocity (km/s) T/2 = s Period (seconds) time = s Earthquake: Mexico Station: CCM, Cathedral Cave, Missouri Distance: 22.4 degrees Component: vertical
13 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Use single, well-dispersed surface wave arrival Period = T = 42.8 s Group Velocity (km/s) T/2 = 21.4 s Period (seconds) time = s Earthquake: Mexico Station: CCM, Cathedral Cave, Missouri Distance: 22.4 degrees Component: vertical
14 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Use single, well-dispersed surface wave arrival Period = T = 38.5 s Group Velocity (km/s) T/2 = s Period (seconds) time = s Earthquake: Mexico Station: CCM, Cathedral Cave, Missouri Distance: 22.4 degrees Component: vertical
15 Group Velocity Dispersion Dispersion: dependence of wave speed on frequency Use single, well-dispersed surface wave arrival Group Velocity (km/s) Period (seconds) Earthquake: Mexico Station: CCM, Cathedral Cave, Missouri Distance: 22.4 degrees Component: vertical
16 Group vs. Phase Velocities Illustration from
17 Phase Velocity Measurement Phase Velocity: speed at which phase travels station Earthquake time d earthquake distance We need to know the initial phase of the wave generated by the earthquake single station method not reliable
18 Phase Velocity Measurement Phase Velocity: speed at which phase travels d station 2 station 1 time earthquake Cycle ambiguity
19 Phase Velocity Measurement Phase Velocity: speed at which phase travels d station 2 station 1 t time earthquake
20 Phase Velocity Dispersion Dispersion: dependence of wave speed on frequency Filter seismograms Measure phase velocity Plot phase velocity vs. frequency
21 Phase Velocity Dispersion Dispersion: dependence of wave speed on frequency Filter seismograms Measure phase velocity Plot phase velocity vs. frequency Shearer (1999)
22 Phase Velocity Dispersion Dispersion: dependence of wave speed on frequency Filter seismograms Measure phase velocity Plot phase velocity vs. frequency Shearer (1999) Rayleigh Wave Sensitivity to Depth Radius Period
23 Phase Velocity Dispersion Dispersion: dependence of wave speed on frequency Filter seismograms Measure phase velocity Plot phase velocity vs. frequency Implication for mantle velocity structure? Shearer (1999) Rayleigh Wave Sensitivity to Depth Radius Period
24 Global Surface Waves November 3, 2002 Denali, Alaska Earthquake ANTO VHZ Filtered between 3 and 8 mhz
25 Global Surface Waves November 3, 2002 Denali, Alaska Earthquake ANTO VHZ Filtered between 3 and 8 mhz
26 Global Surface Waves November 3, 2002 Denali, Alaska Earthquake ANTO VHZ Filtered between 3 and 8 mhz
27 Global Surface Waves November 3, 2002 Denali, Alaska Earthquake ANTO VHZ Filtered between 3 and 8 mhz
28 Global Surface Waves November 3, 2002 Denali, Alaska Earthquake ANTO VHZ Filtered between 3 and 8 mhz
29 Global Surface Waves November 3, 2002 Denali, Alaska Earthquake ANTO VHZ Filtered between 3 and 8 mhz
30 Global Surface Waves November 3, 2002 Denali, Alaska Earthquake ANTO VHZ ANTO VHE Filtered between 3 and 8 mhz
31
32 Standing Waves Standing Wave: Stationary wave generated by constructive/destructive interference of two waves travelling in opposite directions Standing waves or normal modes of the Earth
33 Standing Waves Standing Wave: Stationary wave generated by constructive/destructive interference of two waves travelling in opposite directions 1-D, fixed ends m = 0 m = 1 m = 2 Standing waves constitute basis functions. Sines and Cosines Fourier Transform: combination of sines and cosines describe any 1-D function Index gives the number of nodes.
34 Basis Functions 1-D Sines and Cosines Fourier Transform: combination of sines and cosines describe any 1-D function 2-D (Spherical Surface) What are the standing waves or basis functions we can use to describe any 2-D functions on a sphere?
35 Basis Functions 2-D (Spherical Surface) Longitude: Sines/Cosines Index: angular order m Latitude: Legendre Functions Index: angular degree l and order m m = 0 m = 0 l = 0 m = 1 Rule: -l m l m = 1 l = 1 m = 2 m = 2 l = 2 m = 3 m = Longitude (degrees) l = 3 Latitude (degrees)
36 Basis Functions 2-D (Spherical Surface) Longitude: Sines/Cosines Index: angular order m Latitude: Legendre Functions Index: angular degree l and order m m = 0 m = 0 l = 0 m = 1 Rule: -l m l m = 1 l = 1 m = 2 m = 2 l = 2 m = 3 m = Longitude (degrees) l = 3 Latitude (degrees)
37 Spherical Harmonics m = 0 m = 1 m = 0 l = 1 l = 0 m = 2 l = 2 m = 3 l = 3 m = 4 l = 4
38 Basis Functions 3-D (Sphere) Spherical Surface: Spherical Harmonics Indices: angular degree l and order m Radius: Bessel Functions (homogeneous sphere) Index: number of zero crossings n Radius n = 0 n = 1 n = 2 n = 3
39 Basis Functions 1-D Sines and Cosines Fourier Transform: combination of sines and cosines describe any 1-D function 2-D (Spherical Surface) Longitude: Sines and Cosines Index: angular order m Latitude: Legendre Polynomials Index: angular degree l and order m Spherical Harmonic Transform: combination of spherical harmonics describe any function on a spherical surface Rule: -l m l 3-D (Sphere) Spherical Surface: Spherical Harmonics Indices: angular degree l and order m Radius: Bessel Functions (homogenous sphere) Index: number of zero crossings n Normal Modes: combination of the Earth s normal modes describe any motion
40 Normal Mode Nomenclature Earth: Sphere Spherical Harmonics and radial function to describe standing waves Need three indices: n = radius; l = latitude; m = longitude Type of Motion Toroidal pure shear, denote by T SH waves, Love waves Spheroidal Combination of shear and change in shape denote by S SV waves, Rayleigh waves
41 Characteristic Frequency Each mode has its characteristic frequency and decay constant. Degeneracy If Earth is Spherically symmetric Isotropic Non-rotating Laterally homogeneous then i.e., modes with same overtone number n, and angular degree l same characteristic frequency regardless of angular order m Mode names are often denoted and
42 Attenuation amount of energy dissipated quality factor Q initial energy e -wt/2q amplitude time
43 Seismograms Combination of normal modes can describe ANY motion on spherical Earth Seismogram observed at a station from a given earthquake Constants determined by earthquake source mechanism, and station/hypocentral locations. Fourier Transform
44 Earth s Free Oscillations (Spheroidal Mode)
45 Earth s Free Oscillations (Toroidal Mode)
46 Normal-Mode Central Frequency
47 Normal-Mode Central Frequency
48 Synthetic Seismograms Combination of normal modes can describe ANY motion on spherical Earth Generate synthetic seismograms Seismogram observed at a station from a given earthquake Constants determined by earthquake source mechanism, and station/hypocentral locations.
49 Synthetic Seismograms Combination of normal modes can describe ANY motion on spherical Earth Generate synthetic seismograms November 3, 2002 Denali, Alaska Earthquake ANTO VHZ observed ANTO VHZ synthetic Filtered between 3 and 8 mhz
50 Synthetic Seismograms Combination of normal modes can describe ANY motion on spherical Earth Generate synthetic seismograms November 3, 2002 Denali, Alaska Earthquake observed ANTO VHZ observed synthetic ANTO VHZ synthetic Filtered between 3 and 8 mhz
51 Normal-Mode Splitting 0 S 2
52 Earth s Free Oscillations ( 0 T 2 Receiver Strips) Frequency (mhz)
53 Earth s Free Oscillations (Spheroidal-Toroidal Coupling)
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