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2 (1) 1. 1 dx dy d x τ xx x x, stress x + dx x τ xx x+dx dyd x x τ xx x dyd y τ xx x τ xx x+dx d dx y x dy 1. dx dy d x τ xy x τ x ρdxdyd x dx dy d ρdxdyd u x t = τ xx x+dx dyd τ xx x dyd + τ xy y+dy ddx τ xy y ddx + τ x +d dxdy τ x dxdy (1) dx, dy, d ρ u x t = (τ xx x+dx τ xx x )/dx + (τ xy y+dy τ xy y )/dy + (τ x +d τ x )/d () ρ u x t = τ xx x + τ xy y + τ x (3) Hooke s law τ kl = c ijkl e ij (4)

3 3 (1) e ij (u x, u y, u ) strain e ij = 1 ( ui + u ) j, q i = x, y, (5) q j q i c ijkl summation convention: c ijkl e ij c ijkl e ij (6) i j (x, y, ) c ijkl = λδ ij δ kl + μ(δ ik δ jl + δ il δ jk ) (7) isotropic λ, μ Lamé s constants (1978) (1984) (4) (5), (7) τ xx = (λδ ij δ xx +μδ ix δ jx ) e ij (8) τ xy = μ(δ ix δ jy + δ iy δ jx ) e ij τ x = μ(δ ix δ j + δ i δ jx ) e ij (5) e ij = e ji τ xx = λ(e xx + e yy + e )+μe xx (9) τ xy = μe xy τ x = μe x. (5) ( ux τ xx = λ x + u y y + u ( uy τ xy = μ x + u ) x y ( u τ x = μ x + u ) x ) +μ u x x (10) (3) ρ u x t = λ ( ux x x + u y y + u ) + λ ( ux x x + u y y + u ) + μ u x u x x x +μ x + μ ( uy y x + u ) x + μ ( uy y y x + u ) x + μ ( u y x + u ) x + μ ( u x + u ) x = (λ + μ) ( ux x x + u y y + u ) ( ) + μ x + y + u x + λ ( ux x x + u y y + u ) + μ ( uy y x u ) x μ ( ux y u ) ( μ u x + x x x + μ u x y y + μ ) u x (11) 3

4 4 (1) y, ρ u y t = (λ + μ) ( ux y x + u y y + u ) ( ) + μ x + y + u y + λ ( ux y x + u y y + u ) + μ ( u y u ) y μ ( uy x x u ) ( x μ u y + y x x + μ u y y y + μ ) u y ρ u t = (λ + μ) ( ux x + u y y + u ) ( ) + μ x + y + u + λ ( ux x + u y y + u ) + μ ( ux x u ) μ ( u x y y u ) ( y μ u + x x + μ u y y + μ ) u (1) x y x Lamé λ, μ x, y, (11), (1) ρ u x t = (λ + μ) ( ux x x + u y y + u ( ux ρ u y t = (λ + μ) y ρ u t = (λ + μ) x + u y y + u ( ux x + u y y + u ) ( ) + μ x + y + u x ) ( ) + μ x + y + u y ) ( ) + μ x + y + u (13) = ( x, y, ) = x + y + (14) (11), (1) ρ u t =(λ + μ) ( u )+μ u + λ( u )+ μ ( u )+( μ ) u (15) λ, μ ρ u t =(λ + μ) ( u )+μ u (16) 4

5 5 (1). Eikonal (15) e iωt τ( q ) q =(x, y, ) i/ω u =exp[iω(t τ( q ))] u k ( q )( iω) k (17) ray expansion ω i/ω Ben-Menahem and Singh, 1981 k=0 (s v ) = s( v )+ s v (18) (s v ) = s( v )+ s v (s v ) = ( s) v + s v +( s ) v (17) (15) u t = ( iω) e iω(t τ) u k ( iω) k k=0 (19) ( u ) = [ ( u )+( iω)( τ( k u k )+ ( τ u k )) + ( iω) τ( τ u k )] e iω(t τ) u k ( iω) k k=0 u = [ u k +( iω)(( τ) u k +( τ ) u k )+( iω) ( τ) u k ] e iω(t τ) u k ( iω) k k=0 u = [ u k +( iω) τ u k ]e iω(t τ) u k ( iω) k k=0 u = [ u k +( iω) τ u k ]e iω(t τ) u k ( iω) k ( μ ) u = [( μ ) u k +( iω)( μ τ) u k ]e iω(t τ) u k ( iω) k (15) N( u k ) = ρ u k +(λ + μ) τ( τ u k )+μ( τ) u k (0) M( u k ) = (λ + μ)[ τ( u k )+ ( τ u k )] + μ[( τ) u k +( τ ) u k ] + λ( τ u k )+ μ ( τ u k )+( μ τ) u k L( u k ) = (λ + μ) ( u k )+μ u k + λ( u k ) k=0 k=0 + μ ( u k )+( μ ) u k e iω(t τ) u k ( iω) k [ N( u k ) M( u k 1 ) L( u k )] = 0, u u 1 0 (1) k=0 5

6 6 (1) ( iω) k (1) u k N( u k ) M( u k 1 ) L( u k )=0, u u 1 0 () k =0, 1,,... 0 k =0 u u 1 0 N( u 0 )= ρ u 0 +(λ + μ) τ( u 0 τ)+μ( τ) u 0 = 0 (3) (3) τ = [ ρ +(λ +μ)( τ) ]( u 0 τ) =0 (3) τ = [ ρ + μ( τ) ]( u 0 τ) = 0 (4) (4) ( τ) = α, u 0 τ = 0 (5) ( τ) = β, u 0 τ = 0 (6) α =(λ +μ)/ρ β = μ/ρ (17) τ travel time τ ray (5) 0 P α (6) 0 S β v = α or β (5), (6) ( τ) = v = s (7) s =1/v slowness 0 Eikonal eikonal τ Bruns Kravtsov and Orlov, 1990;, 1975) τ s (3) (4) Pšenčík, 197 τ =(σ 1,σ,σ 3 ), Λ = λ + μ (3) u 01 [ ρ + μ( τ) +Λσ1]+u 0 Λσ 1 σ + u 03 Λσ 1 σ 3 = 0 u 01 Λσ σ 1 + u 0 [ ρ + μ( τ) +Λσ ]+u 03Λσ σ 3 = 0 u 01 Λσ 3 σ 1 + u 0 Λσ 3 σ + u 03 [ ρ + μ( τ) +Λσ3] = 0 (8) Q = ρ + μ( τ) Q +Λσ 1 Λσ 1 σ Λσ 1 σ 3 Λσ σ 1 Q +Λσ Λσ σ 3 = 0 (9) Λσ 3 σ 1 Λσ 3 σ Q +Λσ3 6

7 7 (1) Eikonal Q (Q +Λ( τ) )= [ ρ + μ( τ) ] [ ρ +(λ +μ)( τ) ] = 0 (30) (30) P S Eikonal S SH SV (30) ρ + μ( τ) =0 (3) u 0 τ =0 ρ +(λ +μ)( τ) =0 (λ + μ) τ( u 0 τ) (λ + μ)( τ) u 0 = 0 (31) τ u 0 τ =0 7

8 8 (1) 3. Eikonal (7) 1 [ (1957) ] 1 H(q i,p i, τ)=0, p i = τ q i (3) 1 dq i H/ p i = dp i H/ q i + p i H/ τ = pi H/ p i (33) H τ 3 = dα dq i dα = H p i dp (34) i dα = H q i H Hamiltonian [ (1964) ] Eikonal (7) (34) (q 1,q,q 3 ) (1978) =( 1 1 1,, ) (35) h 1 q 1 h q h 3 q 3 Eikonal (7) 3 j=1 ( ) 1 τ = s (36) h j q j h j 1 h 1 h h 3 (x, y, ) (r, θ, ) 1 r 1 (r, θ, φ) 1 r r sin θ (36) Hamiltonian H(q i,p i )= 1 ( 3 i=1 p i /h i (q i) s (q i ) 1 ) = 0 (37) 8

9 9 (1) θ r y φ x. H τ (33) 3 dα = (34) τ dq i = 1 p i s h i dp i = 1 s p j h (38) j s q i s q i h 3 j=1 j τ/ q i p i (36) 3 ( ) pi = s (39) i=1 h i q =(x, y, ) h x = h y = h =1 (38) dq i dp i = p i s = 1 s s (40) q i Shooting Červený et al.(1977) Aki and Richards (1980) [ (3.) (13.5) ] 3. Eikonal (36) (37) H(q i, p i )= 3 p j h j (q i) s(q i) = 0 (41) j=1 9

10 10 (1) Hamiltonian (33) = dλ dq i = 1 p i dλ s h i dp i dλ = s p j q i s h 3 j=1 j h (4) j q i (38) s (33) 3 /s = dλ da (x, y, ) da = = dx + dy + d 3 dqj (43) j=1 da 3 (x o,y o, o ) ( x+dx, y+dy, +d ) da (x, y, ) (x, y, ) o o o 3. da (1978) da da = 3 (h j dq j ) (44) j=1 (4) 1 (41) da = dλ λ (38), (4) τ λ 10

11 11 (1) q i =(r, θ, φ) h r =1,h θ = r, h φ = r sin θ (4) (39) dr dλ = 1 s p dp r r, = s dλ r + 1 ( ) p θ s r 3 + p φ r 3 sin θ dθ dλ = 1 sr p dp θ θ, = s dλ θ + 1 cot θ s r sin θ p φ (45) dφ dλ = 1 sr sin θ p φ, dp φ dλ p r + 1 r p θ + 1 r sin θ p φ = s = s φ (4) (45) Wesson (1971) Julian and Gubbins (1977) Bending 3.3 Hamiltonian Eikonal (36) (39) 1/ H(q i, p i )= 1 ( 3 ) p i s (q i ) = 0 (46) i=1 dq i H/ p i = dp i H/ q i + p i H/ τ = pi H/ p i (47) 3 = dσ /s = dσ (38) s dq i = p i dσ dp i = s s (48) dσ q i 3.4 [ Hamilton (193) ] (34) H Hamiltonian α p i Eikonal (36) (34) (38) Hamilton-Jacobi [ (1964) ] p i (39) Hamiltonian H(q i, p i ) (Hamilton ) δ Ldα =0, L = p j q j H (49) j 11

12 1 (1) Hamiltonian Hamiltonian(37), (41) (46) δ =0, δ sdλ =0, δ s dσ = 0 (50) Fermat (50) T Hamiltonian, sdλ, s dσ (51) dt =1, dt dλ = s, dt dσ = s (5) 3.5 u 0 transport equation, P (7) v = α Červený and Ravindra (1971) p.1 k =0 (7) () (5) u 0 = u τ 0 N( u k ) τ =[ ρ +(λ +μ)( τ) ]( u k τ) 0 (53) M( u 0 ) τ = N( u 1 ) τ =0. (54) τ M(u τ 0 τ) τ =ρ( τ ) u τ 0 + u τ 0 [ρ τ +( τ )ρ] = 0 (55) (4) s/dλ = p i /dq i τ τ = p i = 1 d dq i α dλ (56) P du τ 0 dλ + uτ 0 ( α τ + 1 ρ ) dρ = 0 (57) dλ 1

13 13 (1) 4. PA PB S (6) u 0 = u 0 n + u 3 0 b n, b τ du 0 dλ 1 T u3 0 + β μ u 0 ( μ τ) = 0 (58) du 3 0 dλ + 1 T u 0 + β μ u3 0( μ τ) = 0 Červený and Ravindra, 1971 T, 1954 v 0 = u 0 +iu3 0 ( dv 0 i dλ + v 0 T + β ) μ ( μ τ) = 0 (59) 13

14 14 (1) 4. Eikonal ray equation (38) (4) (48) (39) p i q i Hamiltonian (41) ( ) d sh dq i i = s + s ( ) h j dqj h j (60) dλ dλ q i q j i dλ ( ) dq j h j = 1 (61) dλ j (60) (49) Lagrangian Lagrange ( Euler ) ( ) d L = L (6) dλ q i q i Julian and Gubbins (1977) Bending [ (7) ] Lagrange (6) Lagrangian 3.4 L = s p i = L/ q i L = s(q i ) h j q j (63) j (h x = h y = h =1) (60) ( d s dq ) i = s (64) dλ dλ q i Wesson (1971) (1) (45) ( d s dr ) = s [ ( ) dθ ( ) ] dφ dλ dλ r + s r + r sin θ dλ dλ ( d sr dθ ) = s ( ) dφ dλ dλ θ + sr sin θ cos θ dλ ( d sr sin θ dφ ) = s dλ dλ φ ( ) dr ( ) dθ ( ) dφ + r + r sin θ = 1 (65) dλ dλ dλ 14

15 15 (1) Jacob (1970) (7) s ds/dλ Jacob (1970) (9) ds dλ = s dr r dλ + 1 s dθ r θ dλ + 1 s dφ r sin θ φ dλ (66) 1 ds/dλ, 3 r 1 ds dλ = s dr r dλ + s dθ θ dλ + s dφ φ dλ (67) (65) 1 4 (67) ( s d r dλ r s ) [( ) dθ ( ) ] dφ r + rs +sin θ + s dr dθ dλ dλ θ dλ dλ + s dr dφ φ dλ dλ = 0 (68) Julian and Gubbins (1977) (A.3) (67) (66) 4. (38) (4) (39) (39) p i (39) p x + p y + p = s (69) p x = s sin i cos j p y = s sin i sin j p = s cos i (70) τ = / q i p i (70) i, j 5 (70) (38) Červený et al.(1977) (3.5) Julian and Gubbins (1977) (A1.3) dx dy d di dj = v sin i cos j = v sin i sin j = v cos i ( v = cos i x 1 = sin i ) v cos j + y sin j + v sin i ( ) v v sin j x y cos j 15 (71)

16 16 (1) ray i p j y x 5. i, j y x j =0, v/ y =0 dx d dθ = v sin θ = v cos θ i θ = v v cos θ + sin θ (7) x p r + 1 r p θ + 1 r sin θ p φ = s (73) p r = s cos i p θ = sr sin i cos j p φ = sr sin θ sin i sin j (74) i r ( ) j (74) (38) Aki and Richards (1980) (13.8) i π i i j j π j j p r = s cos i p θ = sr sin i cos j p φ =+sr sin θ sin i sin j (75) 16

17 17 (1) Jacob, 1970: Julian and Gubbins, 1977 (75) (38) dr = v cos i dθ = v sin i cos j r dφ =+ v sin i sin j r sin θ di ( v = r + v r dj = 1 ( v θ sin i r sin j ) sin i cos i ( v θ r cos j + v ) φ r sin θ sin j v ) φ r sin θ cos j + v sin i sin j cot θ r (76) v =1/s v r,v θ,v φ v Julian and Gubbins (1977) (A1.4) 4, 5 5/6 p i i j (70) (75) τ λ (70) (75) (4) (45) 17

18 18 (1) Aki, K. and P. G. Richards, 1980, Quantitative Seismology, Vol. II, W. H. Freeman and Company, San Francisco. Ben-Menahem, A. and S. J. Singh, 1981, Seismic Waves and Sources, Springer-Verlag, 1108pp. Červený, V., I. A. MolotkovandI. Pšenčík, 1977, Ray Method in Seismology, Univerita Karlova, Praha, 14 pp. Červený, V. and R. Ravindra, 1971, Theory of Seismic Head Waves, Univ. Tronto Press, Toronto. Hamilton, W. R., 193. Mathematical Papers, Cambridge Univ. Press, London., 1957,,, 366pp. Jacob, K. H., 1970, Three-dimensional seismic ray tracing in a laterally heterogeneous spherical Earth, J. Geophys. Res., 75, Julian, B. R. and D. Gubbins, 1977, Three-dimensional seismic ray tracing, J. Geophys., 43, , 1975, 4, I,, 496pp. Kravtsov, Yu. A. and Yu. I. Orlov, Geometrical Optics of inhomogeneous Media, Springer-Verlag, Berlin, 31pp., L.D., E. M., 1964,,, 48pp., 1978,,, 454pp., 1954,,, 7pp., 1984,,, 310pp. Wesson, R. L., 1971, Travel-time inversion for laterally inhomogeneous crustal velocity models, Bull.Seism.Soc.Am., 61,

k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x

k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m A f i x i B e e e e 0 e* e e (2.1) e (b) A e = 0 B = 0 (c) (2.1) (d) e