LECTURE 6 EARTHQUAKES AS TSUNAMI SOURCES

Transcription

1 LECTURE 6 EARTHQUAKES AS TSUNAMI SOURCES Northwestern University, 2007

2 TSUNAMI GENERATION by EARTHQUAKE SOURCES CLASSICAL APPROA CH 1. Consider model of EarthquakeRupture 2. Compute Static Deformation of Ocean Bottom 3. Interpret as Initial Condition for Vertical Surface Displacement with ZeroInitial Velocity

3 CLASSICAL (a) APPROA CH (b) t = 0 (d) (c) t = 0 + (e) TSUNAMI t = 0 ++ t =+

4 GENERIC EARTHQUAKE DISLOCATION Involves MANY parameters Earthquake moment M 0 Earthquake geometry φ, δ, λ Earthquake depth h Water depth H Epicentral distance to shore L Beach slope β M 0 :{ Fault Length L F Fault width W Slip on Fault u

5 FIRST STEP Position a point force F in an infinite homogeneous elastic medium F = {F j } u = {u i } r = r{γ k } Obtain the Dynamic displacement field of the deformation [Aki and Richards, 1980; p. 73, Eqn. (4.23)] The STATIC displacement is simply obtained by putting t. [This expression is known as the Somigliana Tensor]

6 SECOND STEP Replace Single Force by Double-Couple Simply use Somigliana s tensor as a Green s function and take appropriate derivatives. Note that these are the P and S waves of the near [and far] field[s]. NEAR FIELD NEAR FIELD NEAR FIELD [Far Field] [Aki and Richards, 1980; p. 79; Eqn. (4.29)]

7 THIRD STEP Include effect of free surface Integrate over finite area of faulting (Combine with "reflection" of equivalent P and S waves) Reflected P Incident P Reflected S The problem has an analytical solution [Stein and Wysession, 2002] TWO equivalent algorithms Mansinha and Smylie [1971] Okada [1985] Only difference: Okada allows for tensile crack (non-double-couple solution).

8 STATIC DEFORMATION OF OCEAN BOTTOM Straightforward, if somewhat arcane analytical formulæ [Mansinha and Smylie, 1971; Okada, 1985]

9 STATIC DEFORMATION OF OCEAN BOTTOM EXAMPLE: VALPARAISO, CHILE 17 AUGUST 1906 [Okal, 2005] 250 km M 0 = dyn-cm ZAPALLAR km PICHILEMU -150 LLICO φ f = 3 ; δ = 15 ; λ = L F = 200 km; W = 75km; u = 5. 3 m AMPLITUDE (cm) Maximum Uplift: 1.83 m Maximum Subsidence: 0.68 m (in this geometry, occurs under continent; does not contribute to tsunami source)

10 Use this static deformation field (limited to its oceanic portion) as the initial condition (t = 0 + )ofthe evolution of the tsunami. Justification: The seismic source is generally MUCH FASTER than any tsunami process, hence it can be taken as instantaneous (even in the case of SLOW, so-called "Tsunami" earhtquakes) This source can be used for numerical simulations [[ see Chapter EIGHT ]]

11 NORMAL MODE FORMALISM: A different approach [Ward, 1980] Atvery long periods (typically 15 to 54 minutes), the Earth, because of its finite size, can ring likeabell. Such FREE OSCILLATIONS are equivalent to the superposition of two progressive wavestravelling in opposite directions along the surface of the Earth. T=54 minutes T=21.5 minutes "FOOTBALL Mode" [After Lay and Wallace, 1995] "BREATHING Mode" Ward [1980] has shown that Tsunamis come naturally as a special branch of the normal modes of the Earth, provided it is bounded by an ocean, and gravity is included in the formulation of its vibrations.

12 In the normal mode formalism, the solution of the vertical displacement (both in the water and solid Earth) is sought as u z (x; t) = u z (r, θ, φ; t) = y 1 (r) Y m l (θ, φ) exp(i ω t) = y 1 (r) P m l (θ, φ) e imφ exp(i ω t) where Y m l is a spherical harmonic of order l and degree m; P m l is the Legendre polynomial of order l and degree m;and {r, θ, φ}isasystem of spherical polar coordinates. This allows for the separation of the variables {r, θ, φ}. The problem is complemented by similar expressions for the overpressure p = y 2 in the tsunami wave, the horizontal displacement u x = l y 3,and the change in the gravity potential y 5. Under the linear approximation, the equations of hydrodynamics transform into a system of linear differential equations of the first order. For any given l, i.e., wavenumber k = (l + 1/2) (a radius of the Earth), the system has non trivial solutions for only one value of ω. The relationship between l and ω is the Dispersion Relation of the Tsunami.

13 SPHEROIDAL MODE HAS 6 COMPONENT EIGENFUNCTION SATISFYING: dy 1 dr 2λ (λ + 2µ) r 1 (λ + 2µ) L 2 λ (λ + 2µ) r y 1 dy 2 dr ω 2 4µ(3λ + 2µ) ρ + (λ + 2µ) r 2 4ρg r 4 µ (λ + 2µ) r L 2 ρ g r 2µ(3λ + 2µ) (λ + 2µ) r 2 L 2 r 0 ρ y 2 dy 3 dr dy 4 dr ρ g r 1 r 2µ(3λ + 2µ) (λ + 2µ) r 2 λ (λ + 2µ) r 0 1 r ω 2 ρ + 4µ L2 (λ + µ) (λ + 2µ) r 2 2µ r 2 1 µ 3 r 0 0 y 3 = ρ r 0 y 4 dy 5 dr 4π G ρ y 5 dy 6 dr 0 0 4π L 2 G ρ r 0 L 2 r 2 2 r y 6 y 1 :Vertical displacement y 3 :Horizontal displacement y 2 :Normal stress y 4 :Tangential stress y 5 :Gravity potential y 6 :Auxiliary gravity EASILYSOLVED WITH APPROPRIATE BOUNDARYCONDITIONS

14 EIGENFUNCTIONS of SPHEROIDAL MODES Rayleigh Mode l=200; T =52s Tsunami Mode l=200; T =908 s 0 y 1 Vertical Displacement y 3 Horizontal Displacement y 2 Pressure 5km y 1 ; y in solid!! 200 km TSUNAMI EIGENFUNCTION is CONTINUED (SMALL) into SOLID EARTH

15 EXCITATION OF TSUNAMI in NORMAL MODE FORMALISM Gilbert [1970] has shown that the response of the Earth to a point source consisting of a single force f can be expressed as a summation overall of its normal modes u(r, t) = Σ s n (r) N s* n(r s ) f(r s ) 1 cos ω nt exp( ω n t/2q n ) ω 2 n, the EXCITATION of each mode being proportional to the scalar product of the force f by the eigen-displacement s at location r s. Now, an EARTHQUAKE is represented by a system of forces called a double couple: Normal to Fault Plane Direction of Slip The response of the Earth to an earthquake isthus u(r, t) = Σ s n (r) N ε n(r * s ): M (r s ) 1 cos ω nt exp( ω n t/2q n ) ω 2 n where the EXCITATION is the scalar product of the earthquake s MOMENT M with the local eigenstrain ε at the source r s. This formula is directly applicable to the case of a tsunami represented by normal modes of the Earth.

16 ADVANTAGES of NORMAL MODE FORMALISM Handles anyocean-solid Earth Coupling Including Sedimentary Layers Works well at Higher Frequencies No need to assume Shallow-Water Approximation IMMEDIATE RESULTS Eigenfunction very small in Solid Requires HUGE Earthquake 0 y 3 Tsunami Mode l=200; T =908 s y 1 Vertical Displacement y 2 Pressure Horizontal Displacement Eigenfunction decays slowly in Solid Depth has minimal influence on tsunami excitation (h 70 km ) 5km y 3 present in solid. All geometries, including strike slip excite tsunamis. DRAWBACKS of NORMAL MODE FORMALISM y 1 ; y in solid!! Must assume Laterally Homogeneous Structure Linear Theory -- Does not allowfor Large Amplitudes 200 km

17 NOTE: Energy scales as L 4, i.e., as M 4/3 0.

18 ENERGY of a TSUNAMI -- STATIC THEORY [Kajiura, 1981] E = 1 2 ρ w g µ 2 α 2/3 F(δ, λ, h, R) M 4/3 0 = 1 ρ w g 2/3 2 4/3 ε µ 4/3 max F M 0 4/3 * α = invariant ratio of M 0 to S 3/2 * F : dimensionless factor expressing geometry of faulting, and aspect ratio R of fault rupture area. NOTE: Energy of Tsunami grows faster than Seismic Moment Energy released by rupture, proportional to M 0 : ε, grows likemoment. Hence, Fraction of EarthquakeEnergy transferred to Tsunami Grows with EarthquakeSize Fortunately,itremains VERYSMALL (max. 1.3% for Chile, 1960)

19 TSUNAMI ENERGY COMPUTED from NORMAL MODE THEORY [Okal, 2003] Compute Kinetic Energy of water in Normal Mode Formalism Note that most energy is carried by HORIZONTAL FLOW Weigh by excitation function for each mode for given seismic moment M 0. (averaged over focal geometry) Sum over individual modes (equivalent to integrating over frequency) Account for source spectrum (according to seismic scaling laws) Account for Finite extent of source depth. Essentially Equivalent to Kajiura s. E = ρ w g 2/3 ε µ 4/3 max M 0 4/3 E grows as M 4/3 0 Sumatra 2004: E erg (100 times Hiroshima)

20 WHAT ABOUT THE ATMOSPHERE? If the tsunami eigenfunction is prolonged into the Solid Earth which is not totally rigid, It should be possible to prolong it into the atmosphere, which is not a perfect vacuum. (The sea surface is not a totally "free" boundary) This idea, hinted at by Yuen et al. [1970], was proposed by Peltier [1976]. <<<<<< STAY TUNED >>>>>>

21 M TSU Use high seas tsunami waveforms recorded by DART system Consider tsunami as free oscillation branch of Earth s normal modes [Ward, 1980] Recall Magnitude M m for seismic mantle waves; Define M TSU = log 10 X(ω ) + C D + C S + C 0 Then, log 10 M 0 = M TSU + 20 IT WORKS!! [Okal and Titov, 2006]

22 Case Study: KURIL ISLANDS, 04-OCT-1994 Single DARTStation: WC AK59 AK M 0 = dyn cm WC61 WC First Large Event Recorded by DART AK 59 AK 60 M TSU = ± Published (CMT): 8.48 Equivalent wave height at surface (cm) Time (hours) in Julian Day 277 Time (hours) in Julian Day 277 WC 61 WC 62 WC62 Time (hours) in Julian Day 277 Time (hours) in Julian Day 277 ALL 4 DARTs

23 SUCCESSFUL OPERATIONAL USE 17 NOV 2003 This is a smaller earthquake which was not recorded at the Alaskan and West Coast DART gauges. However, a new station, D-171, is only 900 km from the epicenter, and clearly recorded the tsunami, although at a very coarse sampling (1 minute). Despite this limitation, the event can be successfully processed M 0 = dyn-cm (CMT) D Published (CMT): M TSU = ± Rayleigh (aliased) TSUNAMI This estimate was used in real-time to call off an alert for Hawaii.

24 APPLICATION of M TSU to JASON SATELLITE TRACE DETECTION by SATELLITE ALTIMETRY gives first definitive measurement of MAJOR tsunami on HIGH SEAS (previous detection by Okal et al. [1999] during 1992 Nicaragua tsunami -- 8 cm -- at the limit of noise) :02 QUESTION: Can we quantify the JASON trace, i.e., recoverfrom it the source of the tsunami? PROBLEM : JASON is neither a time series nor a space series. SOLUTION : Rebuild an approximate times series from the JASON trace, then process through M TSU :59 02:52 Equivalent Time Series Original Jason Trace measures 70 cm across Bay of Bengal cm Satellite at the right place at the right time! CONCLUSION: IT WORKS!!

25 M TSU: CONCLUSION The algorithm succesfully retrieves the seismic moment of the parent earthquake. The examples tested suggest that the precision is sufficient to avoid false alarms and failures to warn.