Relation between the propagator matrix of geodesic deviation and the second-order derivatives of the characteristic function

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1 Journal of Electromagnetic Waves an Applications 203 Vol. 27 No Relation between the propagator matrix of geoesic eviation an the secon-orer erivatives of the characteristic function Luˇek Klimeš Faculty of Mathematics an Physics Department of Geophysics Charles University in Prague Ke Karlovu Praha 2 Czech Republic Receive 3 January 203; accepte 22 May 203) In the Finsler geometry which is a generalization of the Riemann geometry the metric tensor also epens on the irection of propagation. The basics of the Finsler geometry were formulate by William Rowan Hamilton in 832. Hamilton s formulation is base on the first-orer partial ifferential Hamilton Jacobi equations for the characteristic function which represents the istance between two points. The characteristic function an geoesics together with the geoesic eviation in the Finsler space can be calculate efficiently by Hamilton s metho. The Hamiltonian equations of geoesic eviation are consierably simpler than the Riemannian or Finslerian equations of geoesic eviation. The linear orinary ifferential equations of geoesic eviation may serve to calculate geoesic eviations amplitues of waves an the secon-orer spatial erivatives of the characteristic function or action. The propagator matrix of geoesic eviation contains all the linearly inepenent solutions of the equations of geoesic eviation. In this paper we use the Hamiltonian formulation to erive the relation between the propagator matrix of geoesic eviation an the secon-orer spatial erivatives of the characteristic function in the Finsler geometry. We assume that the Hamiltonian function is a positively homogeneous function of the secon egree with respect to the spatial graient of the characteristic function which correspons to the Riemannian or Finslerian equations of geoesics an of geoesic eviation. The erive equations which represent the main result of this paper are applicable to the Finsler geometry the Riemann geometry an their various applications such as general relativity or the high-frequency approximations of wave propagation.. Introuction In the Riemann geometry the metric tensor epens on coorinates. In the Finsler geometry which is a generalization of the Riemann geometry the metric tensor also epens on the irection of propagation. All equations applicable to the Finsler space are equally applicable to the Riemann space. Seismic electromagnetic an other waves in the high-frequency approximation often propagate accoring to the Finsler geometry. The characteristic function an the propagator matrix of geoesic eviation have foun many important applications in wave propagation see ]. The basics of the Finsler geometry were formulate by Sir William Rowan Hamilton in 832 see 2]. Hamilton s formulation is base on the first-orer partial ifferential Hamilton Jacobi equations for the characteristic function which represents the istance between two points. On the other han Finsler s 3] formulation is base on ifferential geometry. The relation between Finsler s formulation an Hamilton s formulation is well known see 4 Section 5.] but the simplicity an power of Hamilton s formulation are unerestimate too much in the Taylor & Francis

2 590 L. Klimeš Finsler geometry. The characteristic function an geoesics in the Finsler space can be calculate efficiently by Hamilton s metho. The Hamiltonian formulation simplifies many equations of the Riemann geometry. In the Hamiltonian formulation the escription of geoesics an of geoesic eviation in the Finsler geometry is no more ifficult than in the Riemann geometry. The non-linear orinary ifferential equations of geoesics equations of rays) may serve to calculate geoesics the characteristic function point-to-point istance two-point travel time) with its first-orer spatial erivatives or the action istance for general initial conitions travel time for general initial conitions) with its first-orer spatial erivatives. The linear orinary ifferential equations of geoesic eviation may serve to calculate geoesic eviations amplitues of waves the secon-orer spatial erivatives of the characteristic function or the secon-orer spatial erivatives of the action. Geoesic perturbations higher-orer geoesic eviations an the perturbation erivatives an higher-orer spatial erivatives of the characteristic function or of the action can be calculate by quaratures along geoesics see 56]. In this paper we use the Hamiltonian formulation to erive the relation between the propagator matrix of geoesic eviation an the secon-orer spatial erivatives of the characteristic function in the Finsler geometry. The erive equations which represent the main result of this paper are applicable to the Finsler geometry to the Riemann geometry an to their various applications such as general relativity or the high-frequency approximations of wave propagation. We assume that the Hamiltonian function is a positively homogeneous function of the secon egree with respect to the spatial graient of the characteristic function which correspons to the Riemannian or Finslerian equations of geoesics an of geoesic eviation. The equations erive in this paper were generalize to an arbitrary Hamiltonian function by Klimeš see 7]. 2. Hamiltonian formulation of the Finsler geometry 2.. Homogeneous Hamiltonian function of the secon egree an the metric tensor We consier a smooth manifol ifferentiable manifol) an coorinates x i of its coorinate chart. At each point x i we have the tangent space containing contravariant vectors y i an the cotangent space containing covariant vectors y i such as the graients of functions. We consier Hamiltonian function Hx i y j ) which is a real-value function of coorinates x i an of covariant vector y j from the cotangent space at point x i an which is ifferentiable within its efinition omain. Hamilton 2] assume the Hamiltonian function to be a positively homogeneous function of the first egree with respect to y j but his theory is applicable to general Hamiltonian functions as well. The Riemannian or Finslerian equations of geoesics an of geoesic eviation correspon to the positively homogeneous Hamiltonian function Hx i y j ) of the secon egree with respect to y j. In this paper we thus assume that the Hamiltonian function Hx i y j ) is a positively homogeneous function of the secon egree with respect to y j especially in Section 3.2 an the whole Section 4. Note that in the Riemann geometry Hx m y n ) = 2 y i g ij x m ) y j where g ij x m ) are the contravariant components of the Riemann metric tensor. Hamilton 2] calle the subset of the unit vectors y j in the cotangent space at point x i efine by equation Hx i y j ) = 2 ) the surface of components of normal slowness. Now it is often calle the phase-slowness surface or briefly the slowness surface sometimes the inex surface. In the Finsler geometry it is referre to as the figuratrix.

3 Journal of Electromagnetic Waves an Applications 59 Contravariant vector y i corresponing to covariant vector y j is efine by relation y i = H y i x m y n ). 2) For example if covariant vector y i represents the slowness vector the corresponing contravariant vector y i represents the velocity vector. The contravariant components of the metric tensor are efine by equation g ij x m y n ) = 2 H y i y j x m y n ) 3) see 8Equation.5.4)] Action the characteristic function an the Hamilton Jacobi equations The Hamilton Jacobi equation is a partial ifferential equation of the first-orer. The Hamilton Jacobi equation for action istance for general initial conitions travel time for general initial conitions) Sx m ) reas H x i S x j x m ) ) = 2. 4) Hamilton 2] also efine the characteristic function point-to-point istance two-point travel time) Vx m x n ) 5) from the point of coorinates x n to the point of coorinates x m. The characteristic function satisfies the Hamilton Jacobi equations H x m V x n x a x b ) ) = 2 6) an H x m V x n x a x b ) ) = 2 7) see 2Equation C)]. Note that one of Equations 6) an 7) serves as the initial conitions to the other. The Hamilton Jacobi equations express the requirement that the graient of the action or of the characteristic function is unit see the efinition ) of unit covariant vectors Equations of geoesics Hamilton s equations equations of geoesics) rea τ xi = H x m y n ) y i 8) τ y i = H x i xm y n ). 9) Hamilton 2] referre to these equations as the general equations of rays. Toay they are also calle the equations of rays or the ray-tracing equations. Hamilton s equations 8) an 9) can simply be erive by ifferentiating the Hamilton Jacobi equation 4) or 6) with respect to coorinate x j. The meaning of the inepenent parameter τ along the geoesic an the sensitivity of the geoesic to the initial conitions epen on the form of the Hamiltonian function. Covariant vector y i in 8) an 9) which represents the first-orer partial erivatives of the characteristic function or action) with respect to spatial coorinates was calle the normal slowness by Hamilton see 2]. Now it is usually calle the slowness vector. Note that we may obtain analogous Hamilton s equations for initial point x j if we ifferentiate the secon Hamilton Jacobi equation 7) with respect to coorinates x j see 7Equations 9) an 0)].

4 592 L. Klimeš For the positively homogeneous Hamiltonian function H of the secon egree Hamilton s equations 8) an 9) are equivalent to the Riemannian or Finslerian equations τ xi = g ij y j 0) τ y i = Ŵ j ik y j g kl y l ) of geoesics see 8Equations.5.8) an 2.2.5)]. Here Ŵ j ik represents the Christoffel symbols of the first kin see 8Equations 2.2.3) an 2.2.7)]. For a general Hamiltonian function Equations 8) 9) an 0) ) iffer just by the inepenent parameter τ along the geoesic an by the sensitivity of the geoesic to the initial conitions Equations of geoesic eviation We efine vectors an X i = xi γ Y i = y i γ representing the geoesic eviation corresponing to some parameter γ parametrizing the initial conitions for the geoesics. Since the geoesics are parametrize by inepenent parameters τ an γ erivatives τ an γ commute. The equations for X i an Y i are then obtaine by ifferentiating Hamilton s equations 8) an 9) with respect to γ. The resulting Hamiltonian equations of geoesic eviation paraxial ray equationsynamic ray tracing equations) erive by Červený 9] rea τ X i = H X j + H j Y j 4) τ Y i = H j X j H Y j 5) where 2) 3) H j = 2 H x i x j xm y n ) 6) H = 2 H y i x j xm y n ) 7) H j = 2 H y i y j x m y n ). 8) For the positively homogeneous Hamiltonian function H of the secon egree the Hamiltonian equations 4) an 5) of geoesic eviation are equivalent to the Riemannian or Finslerian equations τ X i + Ŵ i js ys X j = g ij Ỹ j 9) τ Ỹi Ŵ j is ys Ỹ j = R irsj y r y s X j 20) of geoesic eviation see 8 Equation 4.4.6)]. Here R isrj represents any one of the relevant curvature tensors see 8Equations 4.2.2) or 4.2.5)] an Ỹ i = Y i Ŵ s ij y s X j. 2) For a general Hamiltonian function Equations 4) 5) an 9) 20) iffer.

5 Journal of Electromagnetic Waves an Applications Propagator matrix of geoesic eviation The propagator matrix of geoesic eviation from point x b to point x a is efine by equation x i x i x a x b ) = x j ỹ j y i y i 22) x j ỹ j where the erivatives are taken at fixe parameter τ along geoesics. The propagator matrix of geoesic eviation is symplectic ] an obeys the chain rule x a x c ) = x a x b ) x c x ) 23) where x x c an x a are the coorinates of three points along a geoesic. The propagator matrix of geoesic eviation contains all linearly inepenent solutions of the equations of geoesic eviation an may thus be use to calculate the geoesic eviation for any initial conitions. In the high-frequency approximations of wave propagation the submatrix xi ỹ j of the propagator matrix of geoesic eviation etermines the amplitue of the Green function of the corresponing wave fiel. The amplitue of the Green function is proportional to expression V 2 et x i ) 2 ỹ j. The whole propagator matrix of geoesic eviation may be use to calculate the amplitue for any initial conitions. The relation between the propagator matrix of geoesic eviation an the secon-orer spatial erivatives of the characteristic function is propose in Section 3.2. The Hamiltonian equations 4) an 5) of geoesic eviation for the propagator matrix rea τ xa x b ) = H H j H j H ) x a x b ) 24) with unit initial conitions. Note that it is also possible to efine the covariant version of the propagator matrix of geoesic eviation but we o not nee it here. 3. Derivatives of the characteristic function 3.. First-orer spatial erivatives of the characteristic function The first-orer spatial erivatives V x i = y i 25) V x i = ỹ i 26) of the characteristic function result from the solution of Hamilton s equations 8) an 9) see 2] Relation between the propagator matrix of geoesic eviation an the secon-orer spatial erivatives of the characteristic function The linear orinary ifferential Hamiltonian equations 24) of geoesic eviation can be use to calculate the propagator matrix 22) of geoesic eviation. The secon-orer spatial erivatives of the characteristic function can be obtaine from the propagator matrix 22) of geoesic eviation.

6 594 L. Klimeš The relations between the secon-orer spatial erivatives of the characteristic function an the propagator matrix of geoesic eviation can be erive in the following way: The relations between the secon-orer spatial erivatives of the characteristic function an the propagator matrix of geoesic eviation in ray-centre coorinates in isotropic meia given by Arnau 0] an Červený Klimeš & Pšenˇcík ] can simply be generalize to the propagator matrix 2 Equation 59)] of geoesic eviation in ray-centre coorinates in anisotropic meia i.e. in the Finsler geometry. These relations can then be transforme to general coorinates using the transformation equations by Klimeš see 2Equations 64) an 65)] an Červený an Klimeš see 3Equation 36)]. The Finslerian equations of geoesic eviation correspon to a positively homogeneous Hamiltonian function of the secon egree. For the positively homogeneous Hamiltonian function of the secon egree with respect to the spatial graient of the characteristic function the erive unique relations between the secon-orer spatial erivatives of characteristic function 5) an the propagator matrix 22) of geoesic eviation rea 2 V x i x j + ) V V x j V x i x j = y i 27) ỹ k ỹ k 2 V x i x j + ) V V x j V x i x j = δi k ỹ 28) k x i ỹ j 2 V x j x k + V ) V V x j x k = xi x k 29) where Kronecker elta δ k i represents the components of the ientity matrix. These relations which represent the main result of this paper are prove in Section 4. Only three submatrices of the propagator matrix 22) of geoesic eviation are use in Equations 27) 29). Note that the fourth submatrix y i / x k of matrix 22) carries no aitional information; it can be calculate from the three submatrices use in Equations 27) 29) using the symplectic property of the propagator matrix 22) of geoesic eviation. 4. Proof of the relation between the propagator matrix of geoesic eviation an the secon-orer spatial erivatives of the characteristic function We first calculate the limits of Equations 27) 29) at the initial point of a geoesic i.e. for point x a approaching the initial point x a of the geoesic an show that Equations 27) 29) hol at the initial point. We then assume that Equations 27) 29) are satisfie at point x a of the geoesic calculate the erivatives of Equations 27) 29) along the geoesic at point x a an show that these erivatives hol. In this way we prove that Equations 27) 29) hol along the whole geoesic. 4.. Limit of the relations at the initial point of a geoesic For small istances between x a an x b the characteristic function can be approximate as 8 Equation..4)] Vx b x a ) = F x m x n x n ) + OV 2 ) 30)

7 Journal of Electromagnetic Waves an Applications 595 where Fx i y j ) is the funamental function which specifies the length of the infinitesimally small vectorial istance y j an is thus positively homogeneous of the first egree with respect to y j. The approximate velocity vector along the geoesic from x a to x a reas y i = The slowness vector corresponing to this velocity vector is x i x i F x m x n x n ). 3) y i x m y n ) = F 2 2 y i xm y n ) 32) see 8Equation.5.6)] an the corresponing covariant metric tensor is efine by relation g ij x m y n ) = 2 F 2 2 y i y j xm y n ) 33) see 8Equation.3.)]. Inserting 3) into the right-han sie of 33) an consiering the positive homogeneity of the funamental function Fx i y j ) an of its erivatives with respect to y j we obtain g ij x m y n ) = 2 F 2 2 y i y j xm x n x n ). 34) Since 34) contains the square of the funamental function F we express approximation 30) as We now ifferentiate approximation 35) once V x i xb x a ) = 2 F 2 F 2 y i V x i xb x a ) = an twice 2 V x i x j xb x a ) = 2 V x i x j xb x a ) = 2 V x i x j xb x a ) = 2 V x i x j xb x a ) = Vx b x a ) = F 2 x m x n x n ) + OV 2 ). 35) 2 F 2 F 2 y i 2 2 F 2 F 2 y i y j 4 F 2 ] F 2 F 2 y i y j 4 F 2 ] F 2 F 2 y i y j Vx b x a ) 4 F 2 ] 3 ] x m x n x n ) + OV ) 36) ] x m x n x n ) + OV ) 37) ] F 2 F 2 y i y j x m x n x n ) + OV 0 ) 38) ] F 2 F 2 y i y j ] F 2 F 2 y i y j Inserting relations 34) 35) 36) an 37) into 38) 40) we arrive at g ij x m y n ) V 2 V x i x j xb x a ) = 2 V x i x j xb x a ) = Vx b x a ) Vx b x a ) g ij x m y n ) V x m x n x n ) +OV 0 ) 39) x m x n x n ) + OV 0 ). 40) x i xb x a ) V ] x j x x c ) + OV 0 ) 4) x i xb x a ) V ] x j x x c ) + OV 0 ) 42) ] + OV 0 ). 43) g ij x m y n ) V x i xb x a ) V x j x x c )

8 596 L. Klimeš From the Hamiltonian equations 24) of geoesic eviation we see that x i x i x j ỹ j y i y i = δ i j + 2 H y i x j V 2 H V y i y j x j 2 H ỹ j x i x j V δ j i 2 H + OV 2 ). 44) x i V y j Inserting 3) into 44) we express x i ỹ j = Vg ij + OV 2 ). 45) We multiply 4) 43) by 45) an obtain approximations 2 V x i x j + ) V V x j V x i x j = δi k + OV ) 46) ỹ k 2 V x i x j + ) V V x j V x i x j = δi k + OV ) 47) ỹ k 2 V x i x j + ) V V x k V x i x j = δi k + OV ). 48) ỹ j From 44) an 46) 48) we see that Equations 27) 29) are satisfie for small istances between x a an x a with the accuracy of OV ) an are thus satisfie for x a x a Differentiating the Hamilton Jacobi equations Hereinafter a subscript following a comma enotes the partial erivative with respect to coorinate x i e.g.h = H or V x i = V. A subscript with a tile following a comma enotes the partial x i erivative with respect to initial coorinate x a e.g.v ã = V x a. A superscript following a comma enotes the partial erivative with respect to slowness vector component y i e.g.h = y H i. We ifferentiate the Hamilton Jacobi equation 6) with respect to x i an obtain equation. H + H k V ki = 0 49) see 2Equations Q) I) K)]. We ifferentiate the Hamilton Jacobi equation 6) with respect to x a an obtain equation H k V kã = 0 50) see 2Equations U) I)]. We ifferentiate Equation 49) with respect to x j an obtain equation H j + H m V mj + V m H m + V m H mn V nj + H k V kij = 0. 5) We ifferentiate Equation 49) with respect to x a or Equation 50) with respect to x j an obtain equation V ãm H m + V ãm H mn V nj + H k V kãj = 0. 52) We ifferentiate Equation 50) with respect to x b an obtain equation V ãm H mn V n b + Hk V kã b = 0. 53)

9 Journal of Electromagnetic Waves an Applications Derivatives along geoesics Equation 8) yiels We apply Equation 54) to V an obtain τ = Hk x k. 54) τ V = Hk V k. 55) Because of the positive homogeneity of Hamiltonian function H Equation 55) reas Equation 49) with 54) yiels Equation 50) with 54) yiels Equation 5) with 54) yiels the Riccati equation τ V =. 56) τ V = H. 57) τ V ã = 0. 58) τ V j = H j H m V mj V m H m V m H mn V nj 59) for V j. Equation 52) with 54) yiels equation τ V ãj = V ãm H m V ãm H mn V nj 60) which represents for given V j the linear orinary ifferential equation for V ãj. Equation 53) with 54) yiels expression τ V ã b = V ãm H mn V n b 6) for the erivative of V ã b along the geoesic in terms of V ãj. Equations 24) rea x i x k = H τ x c k x c + y Hk k x c 62) x i = H x k k + H ky k τ 63) y i τ x c = H x k y k k Hk x c x c 64) y i x k = H k H k y k. τ 65) 4.4. Derivative of 27) We express the erivative of Equation 27) along the geoesic using Equations 56) 57) 59) 63) an 65) Hj H m V mj V m H m V m H mn V nj V H V + V H ) V 2 ] V V x j ) + V j + V V V ) H k xk + H k y k x ỹ c = H k ỹ c k H k y k ỹ c. ỹ c 66)

10 598 L. Klimeš We consier the secon-egree positive homogeneity of Hamiltonian function H Hj H m V mj V m H m V m H mn V nj V H V + V H ) V 2 ] V V x j + V j H k + 2V V H k ) xk + V ỹ c j H k + V V H k ) y k x = H k ỹ c k H k y k ỹ c. ỹ c 67) We collect the terms containing x j H m an the terms containing y l V mj V m H mn V nj V H V + V V H V 2 V V ] x j + V m H mn + V V H n + H n ] yn = 0. 68) We insert 27) for y l H m V mj V m H ml V lj V H V + V V H V 2 V V V m H mn + V V H n + H n )V nj + V V n V ) ] x j = 0. 69) We consier the secon-egree positive homogeneity of Hamiltonian function H H m V mj V m H mn V nj V H V + V V H V 2 V V + V m H mn V nj + V V H m V mj + H m V mj + V V m H m V + V 2 V V + 2V H V ] x j = 0. 70) We now execute the summation in 70) an arrive at V H V + V V H + V V H m V mj + V V m H m ] V x j = 0. 7) This equation hols in consequence of Equation 49) Derivative of 28) We express the erivative of Equation 28) along the geoesic using Equations 56) 57) 58) 60) an 63) Vãm H m V ãm H mn V nj V V ã H V 2 ) V ã V x j + V ãj + V ) V ã V H x l l + H l y ) l ỹ c = 0. 72) ỹ c We consier the secon-egree positive homogeneity of Hamiltonian function H Vãm H m V ãm H mn V nj V V ã H V 2 ) V ã V x j + V ãj H l + 2V ) V ã H x l l + V ỹ c ãj H l + V V ã H l) y l = 0. 73) ỹ c We collect the terms containing x j an the terms containing y l Vãm H mn V nj + V V ã H V 2 V ã V ) x j + V ãm H mn + V V ã H n) y n = 0. 74)

11 Journal of Electromagnetic Waves an Applications 599 We insert 27) for y n Vãm H mn V nj + V V ã H V 2 V ã V + V ãm H mn + V V ã H n )V nj + V V n V ) ] x j = 0. 75) We consier the secon-egree positive homogeneity of Hamiltonian function H Vãm H mn V nj + V V ã H V 2 V ã V + V ãm H mn V nj + V V ã H n V nj + V V ãm H m V + V 2 V ã V ) x j = 0. 76) We now execute the summation in 76) an arrive at V V ã H + V V ã H n V nj + V V ãm H m ) V x j = 0. 77) This equation hols in consequence of Equations 49) an 50) Derivative of 29) We express the erivative of Equation 29) along the geoesic using Equations 56) 58) 6) 62) an 63) x i ỹã Vãm H mn V n b V 2 ) V ã V b + H m xm + H m y ) m ỹã ỹã Vã b + V V ã V b) = H m xm + x b Hm y m. 78) x b Consiering Equation 29) the terms containing H m cancel out x i ỹã Vãm H mn V n b V 2 V ã V b ) + H m y m ỹã Vã b + V V ã V b ) = H m y m. 79) x b Consiering 50) an the secon-egree positive homogeneity of H we may express Equation 79) as xi ỹã Vãm + V ) V ã V m H mn V n b + V ) V n V b + H m y m ỹã Vã b + V ) V ã V b = H m y m. 80) x b We multiply Equation 80) from the right-han sie by matrix xk ỹ b xi ỹã Vãm + V ) V ã V m H mn V n b + V V n V b + H m y m ỹã Vã b + V V ã V x k b) = H m y m ỹ b x b ) x k ỹ b x k. 8) ỹ b Equation 28) implies ) x i V ỹã ãj + V V ã V = δ i j. 82) We insert 29) an 82) into Equation 8) an obtain H k + H m y m x k ỹã xã Equation 83) hols thanks to the consequence of the symplectic property of matrix 22). = Hm y m x b x k. 83) ỹ b y i x k x c y i x k x c = δi k 84)

12 600 L. Klimeš 5. Conclusions The propagator matrix 22) of geoesic eviation contains all the linearly inepenent solutions of the equations of geoesic eviation. It can be calculate using the Hamiltonian equations 4) an 5) of geoesic eviation which are consierably simpler than the Riemannian or Finslerian equations 9) an 20) of geoesic eviation. In the high-frequency approximations of wave propagation the propagator matrix 22) of geoesic eviation may be use to calculate the wave amplitue for any initial conitions. The amplitue of the Green function of the corresponing wave fiel is proportional to expression V 2 et x i ) 2 ỹ j = V et V x i x j + V V x i V x j ) 2. The main result of this paper relations 27) 29) between the propagator matrix 22) of geoesic eviation an the secon-orer spatial erivatives of the characteristic function are applicable to the Finsler geometry to the Riemann geometry an to their various applications such as general relativity or the high-frequency approximations of wave propagation see 4]. The equations erive in this paper for the Hamiltonian function positively homogeneous of the secon egree with respect to the spatial graient of the characteristic function were generalize to an arbitrary Hamiltonian function by Klimeš see 7]. Acknowlegements This paper has been inspire by the iscussions with Bartolomé Coll an José Maria Pozo in Salamanca in January I am grateful to Ivan Pšenˇcík whose suggestions mae it possible for me to improve the paper. The research has been supporte by the GrantAgency of the Czech Republic uner contracts 205/07/0032 an P20/0/0736 by the Ministry of Eucation of the Czech Republic within research project MSM an by the members of the consortium Seismic Waves in Complex 3-D Structures see References ] Červený V. Seismic ray theory. Cambrige: Cambrige University Press; ] Hamilton WR. Thir supplement to an essay on the theory of systems of rays. Trans. Roy. Irish Aca. 837;7: 44. 3] Finsler P. Über Kurven un Flächen in Allgemeinen Räumen PhD Thesis]. Göttingen; 98 Reeition Basel: Birkhäuser; 95]. 4] Synge JL. Geometrical mechanics an e Broglie waves. Cambrige: Cambrige University Press; ] Klimeš L. Secon-orer an higher-orer perturbations of travel time in isotropic an anisotropic meia. Stu. Geophys. Geo. 2002;46: ] Klimeš L. Transformation of spatial an perturbation erivatives of travel time at a general interface between two general meia. In: Seismic Waves in Complex 3-D Structures Report 20. Prague: Department of Geophysics Charles University; 200. p Available from: 7] Klimeš L. Relation between the propagator matrix of geoesic eviation an the secon-orer erivatives of the characteristic function for a general Hamiltonian function. In: Seismic Waves in Complex 3-D Structures Report 23. Prague: Department of Geophysics Charles University in Prague; 203. p Availabile from: 8] Run H. The ifferential geometry of Finsler spaces. Berlin: Springer; ] Červený V. Seismic rays an ray intensities in inhomogeneous anisotropic meia. Geophys. J. R. Astr. Soc. 972;29: 3. 0] Arnau JA. Moe coupling in first-orer optics. J. Opt. Soc. Am. 97;6: ] ČervenýV Klimeš LPšenˇcík I. Paraxial ray approximations in the computation of seismic wavefiels in inhomogeneous meia. Geophys. J. R. Astr. Soc. 984;79:89 04.

13 Journal of Electromagnetic Waves an Applications 60 2] Klimeš L. Transformations for ynamic ray tracing in anisotropic meia. Wave Motion. 994;20: ] Červený V Klimeš L. Transformation relations for secon-orer erivatives of travel time in anisotropic meia. Stu. Geophys. Geo. 200;54: ] Červený V Iversen E Pšenˇcík I. Two-point paraxial traveltimes in an inhomogeneous anisotropic meium. Geophys. J. Int. 202;89: