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J ournal o f A pp http://dx.doi.or g Copyright © 2 0 N Propa g ABSTRA C A numerical- “Earth-Atmo s dynamic equ a linearized Na v with respect t o of the reduce d Keywords: S e M 1. Introdu c In mathemati c elastic mediu m b orders on v a fied on a fr e seismic wav e and the gene r waves in th e b oundary are i In the last investigations tion between mosphere. Pa p mic induction Papers [2,3] d p rocesses at t and an isoth e p apers, prop e modified La m In the pres e algorithm to s seismic and a mogeneous “ A the algorith m with a finite- d A similar a p lied Mathemat i g /10.4236/jamp . 0 13 SciRes. umeric g ation i T h Siber i C T analytical sol u s phere” model a tions of elast i v ie r -Stokes e q o time, the fi n d problem. e ismic Waves ; M etho d c tion c al simulation m , it is typic a a cuum, and b o e e surface. S p e s are assume d r ation of aco u e atmosphere i gnored. decade, some have shown t the waves i n p er [1] descri b of an acousti c d eal with theo r t he boundary b e rmal homog e e rties of the s m b waves are s t e nt paper we c o s imulate and i a coustic-gravi t A tmosphere- E m is a combi n d ifference met h a pproach to so i cs and Ph y sics , .2013.14003 P u c al Sim u i n a He t with W B. G. Mik h h e Institute of C i an Branch of t h u tion for seis m . Seismic wa v i city theory. P q uations with t n ite integral F o ; Acous t ic-Gr a of seismic w a lly assumed t h o undary cond i p ecifically, at d to be abso l u stic-gravity w and their in t theoretical a n t hat there is a s n the lithosph b es the effect c wave produc r etical investi g b etween an e l e neous atmos p s urface Stone l t udied. o nsider an eff i i nvestigate th e t y waves in a E arth” model. A n ation of inte h od. lving the pro b , 2013, 1, 12-1 7 u blished Online O u lation t eroge n W ind i n h ailenko, A. A C omputational M h e RAS, pr. Ak a Email Rece m ic and aco u v e propagatio n P ropagation o f t he wind. The o urie r transfor m a vity Waves; N ave fields in a h at the mediu m i tions are spe c the boundar y l utely reflecte w aves by elas t t eraction at t h n d experimen t s triking correl ere and the a of acoustose i ed by a vib r at o g ations of wa v l astic half-spa c p here. In the s l ey-Scholte a n i cient numeric e propagation o a spatially inh o A peculiarity o gral transfor m b lem for a ver t 7 O ctobe r 2013 ( h of Aco u n eous E n the A t A . Mikhailo v M athematics and a d. Lavrentieva, : mikh@sscc.r u ive d July 2013 u stic-gravity w n in an elastic f acoustic-gra v algorithm pr o m along the s p N avie r -Stokes a n m c i- r y, d, t ic h e t al a- a t- i s- o r. v e c e s e n d al o f o - o f m s t i- cally i n coordi n dered i n is writ t terms o Cartesi n are ass u the me d coordi n The al g form w thod c a spectra instead that is integra l trast to the ini t which t of the e thod f o was fi r for pr o [9]. T h of this guerre Fourie r h ttp://www.scir p u stic-G E arth- A t mosph v , G. V. Res h Mathematical G 6, Novosibirsk u w aves propag a half-space is v ity waves in o posed is base d p atial coordin a Equations; L a n homogeneou s n ates with no n [4]. In the p t en down as o f the velocit y n system of c u med to be fu n d ium is assu m n ate. This pro b g orithm is ba s w ith respect t o a n be conside r l method bas e of the freq u the degree o l Laguerre tra n the Fourier tr a t ial problem t t he parameter e quations and o r solving dy n r st considered o blems of vis c h e above-men t method and transform ov r transform wi t p .org/journal/ ja m ravity W A tmosp h ere h etova G eophysics, 630090, Russia a tion is appli e described by a the atmosphe d on the integ r a te with the fi n a guerre Transf o s model in a wind in the a p roblem state m a firs t -order y vector and o ordinates. T h n ctions of onl y m ed to be ho m b lem statemen t s ed on the in t o the tempora l r ed to be an a n e d on the Fo u u ency , we o f the Lague n sform with r e a nsform) mak e o solving a s y is present onl y has a recurre n n amic proble m in papers [5, 6 c oelasticity [7 t ioned papers the advantag e er the differ e t h respect to ti m m p ) W aves h ere M o e d to a heter o a system of fi r re is describe d r al Laguerre t r n ite difference o rm; Finite Di f cylindrical s y a tmos p here w a m ent, the initi a hyperbolic s y stress tensor h e medium pa r y two coordin a m ogeneous in t t is called a 2. 5 t egral Laguer r l coordinate. T n alog to a we l u rier transfor m have a para m rre polynomi a e spect to time e s it p ossible t y stem of equ a y in the righ t - h n ce relation. T m s of elasticit y 6 ] and then d e ,8] and poro u consider pec u e s of the inte e nce methods m e. JAMP o del o geneous rst order d by the r ansform solution f ference y stem of a s consi- a l system y stem in in a 3D r ameters a tes, and t he third 5 D one. r e trans- T his m e- l l-known m , where, m eter p a ls. The (in con- t o reduce a tions in h and side T his m e- y theory e veloped u s media u liarities gral La- and the B. G. MIKHAILENKO ET AL. Copyright © 2013 SciRes. JAMP 13 2. Problem Statement The system of equations for the propagation of acous- tic-gravity waves in an inhomogeneous non-ionized iso- thermal atmosphere in the Cartesian system of coordi- nates (,,) x yz with the wind directed along the horizon- tal axis x and vertical stratification along the axis z has the following form: 0 1 x xx xz uu v P vu tx xz , (1) 0 1 yy x uu P v tx y , (2) 00 1 zz x uu Pg v tx z , (3) 200 0xxzz P PP vc vuu tx txzz , (4) 0 0(, ,,) y x z x z u uu v tx xyz uFxyzt z (5) Here g is the acceleration of gravity, 0()z is the reference atmosphere density, 0()cz is the sound speed, () x vz is the wind velocity along the axis x , (,,) x yz uuuu is the velocity vector of displacement of the air particles, P and are the pressure and the density perturbations, respectively, due to a wave propa- gating from a source of mass 0 (, ,,)()() F xyztrr ft , where () f t is a given time signal in the source. As- sume that the axis z is directed upwards. Zero sub- scripts for the medium physical parameters show their values for the reference atmosphere. The atmospheric pressure 0 P and the density 0 for the reference at- mosphere in a homogeneous gravitational field are: 0 0 P g z , 01 ( )exp(/)zzH , where H is the height of the isothermal homogeneous atmosphere, and 1 is the density of the atmosphere at the Earth’s surface, that is, at 0z. The seismic waves propagation in an elastic medium is described by the well-known system of first order equa- tions of elasticity theory as the following relation be- tween the displacement velocity vector components and the stress vector components: 0 1() iik i k u F ft tx , (6) ikk i ik ik uu div txx u. (7) Here ij is the Kronecker symbol, 123 (, , ) x xx and 12 3 (, , ) x xx are the elastic parameters of the medium, 012 3 (, , ) x xx is the density, 123 (, ,)uuuu is the dis- placement velocity vector, and ij are the stress vector components. The equality 123 (, ,) x yz x yz FFFFeee describes the distribution of a source located in space, and () f t is a given time signal in the source. The combined system of equations for the propagation of seismic and acoustic-gravity waves in the Cartesian system of coordinates 123 (,,)(,,) x yzx xx can be writ- ten down as 0103 1() iiki x ix zzx k uuv g F ftKve ue txx x , (8) 0 1 ik ikk i ikik xz ik uu divK vgu txx x u (9) 0 0xz Kv divu tx z u. (10) Here ij is the Kronecker symbol, 0(,) x z is the density, (,) x z and (,) x z are the elastic parameters of the medium, 123 (, , )uuuu is the displacement ve- locity vector, and ij are the stress tensor components; 123 (, ,) x yz x yz FFF F eee describes the distribution of a source located in space, and () f t is a given time signal in the source. The medium is assumed to be ho- mogeneous along the axis y . System (1)-(5) for the atmosphere is obtained from system (8)-(10) at 1 атм K , 121323 0 , 1122 33 P , 2 00 c , 0 . Set 0 атм K in system (9)-(10), and obtain the sys- tem of Equations (6)-(7) for the propagation seismic waves in an elastic medium. In the problem in question, the atmosphere-elastic half-space interface is assumed to be the plane 30zx . In this case, the condition of contact of the two media at 0z is written as 0 00 00 ,zz zz zz z zz zz uu gu tt , 000 xz yz zz . (11) The problem is solved at the following zero initial da- ta: 00 000,1, 2, 3.1, 2,3. iij tt tt uPij (12) All the functions of the wave field components are as- sumed to be sufficiently smooth so that the transforma- B. G. MIKHAILENKO ET AL. Copyright © 2013 SciRes. JAMP 14 tions presented below be valid. 3. Solution algorithm At the first step, we use the finite cosine-sine Fourier transform with respect to the spatial coordinate y, where the medium is assumed to be homogeneous. For each component of the system, we introduce the correspond- ing cosine or sine transform: 0 cos( ) (,,,)(, ,,)() sin( ) a n n ky x zntxyztd y ky WW , 0,1, 2,...,nN; (13) with the corresponding inversion formula 1 12 (, ,,)(,0,,)(,,,)cos() N n n x yztxztxnztk y WW W (14) or 1 2 (, ,,)(,,,)sin() N n n x yztxnzt ky WW , (15) where n n ka . At rather a large distance a , consider a wave field up to the time tT, where T is a minimum propaga- tion time of a pressure wave to the boundary ra . As a result of this transformation, we obtain 1N indepen- dent 2D unsteady problems. At the second step, we apply to the thus obtained 1N independent problems the integral Laguerre transform with respect to time 2 0 (,,)(,,,)( )( )( ), 0,1,2,... pp x n zxn zthtlht dht p WW (16) with the inversion formula 2 0 ! (,,,)()(,,)( ) ()! pp p p x nzthtxnzl ht p WW, (17) where () p lht are the orthogonal Laguerre functions. The Laguerre functions () p lht can be expressed in terms of the classical standard Laguerre polynomials () p Lht (see paper [10]). Here we select an integer pa- rameter 1 to satisfy the initial data and introduce the shift parameter h>0. Then we have the following representation: 2 () ()exp(2)() pp lhththt Lht . We take the finite cosine-sine Fourier transform with respect to the coordinate x, similar to the previous trans- form with respect to the coordinate y with the corres- ponding inversion formulas: 1 1 (,, )(0,, ) π 2(,, )cos() π pi pi M pim m xnz nz mnzk x WW W (18) or 1 2 (,, ,)(,, ,)sin() π M iim m x nz pmnzpkx WW , (19) where m m kb . It should be noted that the medium in this direction is inhomogeneous. The finite difference approximation for the system of linear algebraic equations with respect to z using the staggered grid method was applied (see paper [11]) pro- viding second order accuracy approximation. This scheme is used for FD approximation within the compu- tation domains in the atmosphere and in the elastic half-space, the fitting conditions at the interface being exactly satisfied. As a result of the above transformations, we obtain 1N systems of linear algebraic equations, where N is the number of harmonics in the Fourier transform with respect to the coordinate y . The sought for solution vector W is represented as follows: 01 ( )((),( ),...,())T K ppp pWVVV, [(0,...,; ),(0,...,; ),... iixxi mMzmMz V ...(0,...,;),( 0,...,;)]T iz i PmMzu mMz. Then for every n-th harmonic (0,...,nN) the sys- tem of linear algebraic equations can be written down in the vector form: ()()(1) 2 h AE pp WF . (20) Note that only the right-hand side of the obtained sys- tem of algebraic equations includes the parameter p (the degree of the Laguerre polynomials) and has a re- current p dependence. The matrix A is thus inde- pendent of p . A sequence of wave field components in the solution vector V is chosen to minimize the number of diagonals in the matrix A. The main diagonal of the matrix has the components of this system multiplied by the parameter h (the Laguerre transform parameter). By changing the parameter h, the condition number of the matrix can be considerably improved. Solving the system of linear algebraic equations (20) determines spectral values for all the wave field components (,,,) i mnzpW. Then, using the inversion formulas for the Fourier trans- form (14), (15), (18), (19), and the Laguerre transform (17), we obtain a solution to the initial problem (8)-(12). In the analytical Fourier and Laguerre transforms, when B. G. MIKHAILENKO ET AL. Copyright © 2013 SciRes. JAMP 15 determining functions by their spectra, inversion formu- las in the form of infinite sums are used. A necessary condition in the numerical implementation is to deter- mine the number of terms of the summable series to con- struct a solution with a given accuracy. For instance, the number of harmonics in the inversion formulas of the Fourier transform (14), (15), (18), (19) depends on a mi- nimal spatial wavelength in the medium and on the size of the spatial calculation domain of the field given by the finite limits of the integral transform. In addition, the convergence rate of the summable series depends on smoothness of functions of the wave field. The number of the Laguerre harmonics for determining functions by formula (17) depends on a signal given in the source () f t, the parameter h, and the time interval of the wave field. Papers [5-8] consider in detail how one can determine the required number of harmonics and choose an optimal value of the parameter h. 4. Numerical Results Figures 1-3 show the results of numerical calculations of a wave field as snapshots at a fixed time for the horizon- tal component of the displacement velocity (, ,) x uxyz . Figure 1 presents a snapshot of the wave field for (, ,) x uxyz in the plane XZ at the time t=15 sec. This model of the medium consists of a homogeneous elastic layer and an atmospheric layer separated by a plane boundary. The physical characteristics of the layers are as follows: the atmosphere: sound speed 340 p cm·sec–1. Den- sity versus coordinate z was calculated by the formula 01 ( )exp(/)zzH , where 3 11.22510 g·cm–3, 6700Hm; the elastic layer: pressure wave velocity 300 p c m·s e c–1, shear wave velocity 200 s c m·sec–1, den- sity 01.2 g·cm–3. A bounded domain, (,, ) x yz=(15km,15km, 10 km), was used for the calculations. A wave field from a point source (a pressure center) located in the elastic medium at a depth of ¼ of the length of a pressure wave with coordinates 000 (,,) x yz=(6 km,7.5km,0.075 km) was simulated. The figure shows the wave fields for the ho- rizontal component x u of the displacement velocity in the plane XZ at 07.5 kmyy : without wind (top), with the wind speed in the atmosphere of 50 m·se c –1 (bottom). The elastic medium-atmosphere interface is shown by the solid line. This figure demonstrates that in the elastic medium, in addition to the spherical P-pressure wave and the conic S-shear wave, there also propagates a “non-ray” spherical wave S*, and then there follows a surface Stoneley-Scholte wave. An acoustic-gravity wave refracted at the Earth-atmosphere boundary propa- gates in the atmosphere. At the boundary, this wave ge- nerates the corresponding pressure and shear waves in Figure 1. A snapshot at t = 15 sec for the velocity component ux in the plane (XZ) without wind (top), with wind (bottom) (wind speed 50 m·sec –1). the elastic medium. Figures 2 and 3 present snapshots of the wave field when the seismic waves velocity in the elastic medium is greater than the sound speed in the atmosphere. In this model, the physical characteristics of the elastic medium and the atmosphere are as follows: the atmosphere: sound speed 340 p cm·sec–1. Den- sity versus coordinate z was calculated by the formula 01 ( )exp(/)zzH , where 3 11.22510 g·cm–3, 6700H m; the elastic layer: pressure wave velocity 800 p c m·s e c–1, shear wave velocity 500 s c m·sec–1, den- sity 01.5 g·cm–3. A bounded domain(,,)(20 km, 16 km, 14 km)xyz, was used for the calculations. A wave field from a point source (the pressure center) located in the elastic medium at a depth of ¼ of the length of a pressure wave with the coordinates 000 (,,)(10 km,8 km,0.2 km)xyz was B. G. MIKHAILENKO ET AL. Copyright © 2013 SciRes. JAMP 16 Figure 2. A snapshot at t = 12 sec for the velocity component ux in the plane (XZ) without wind (top), with wind (bottom) (wind speed 50 m·sec–1). simulated. Figure 2 shows the wave fields for the horizontal component x u, of the displacement velocity in the plane XZ at 08yy km: without wind (top), with wind speed in the atmosphere of 50 m·sec–1 (bottom). The elastic mediumatmosphere interface is shown by the solid line. This figure shows that in the atmosphere, in addition to the conical P- pressure wave and the conical S-shear wave, there also propagates a “non-ray” spherical wave P *, and then there follows a surface Stoneley- Scholte wave. Figure 3 presents snapshots of a 3D wave field at 10t sec for the velocity component x u with the wind speed of 50 m·se c –1 in the atmosphere. The numerical simulation results have revealed some new peculiarities of the wave propagation with wind in the atmosphere. Specifically, the influence of the wind on the propagation velocity of the surface Stoneley Figure 3. A snapshot of a wave field for the horizontal ve- locity component ux(x,y,z), at t = 10 sec with wind in the atmosphere (wind speed 50 m·sec–1). waves in an elastic medium has been demonstrated. The numerical results have also shown that the velocity of these waves increases downwind and, hence, it decreases upwind by a quantity equal to the wind speed. The same influence of wind is on a non-ray spherical exchange acoustic-gravity wave propagating in the atmosphere from a source located in a solid medium. Another fact of the wind influence that has been established is that the surface wave changes in the amplitude along its front. This manifests itself as an increase in the amplitude in that part of the wave front that propagates downwind and a decrease in the wave front propagating upwind but with conservation of the total wave energy. 5. Conclusion The approach proposed to the statement and solution of the problem makes it possible to simulate the effects of the wave field propagation in a unified mathematical earth-atmosphere model and to study the exchange waves at their boundary. The numerical simulation of these processes makes it possible to investigate the peculiari- ties of the wind effects on the propagation of the acous- tic-gravity atmospheric waves and surface Stoneley waves. 6. Acknowledgements This work was supported by the Russian Foundation for Basic Research (projects no. 11-05-00937, 13-05-00076 and 13-05-12051). REFERENCES [1] A. S. Alekseev, B. M. Glinsky, S. I. Dryakhlov, et al., “The Effect of Acoustic-Seismic Induction in Vibroseis- B. G. MIKHAILENKO ET AL. Copyright © 2013 SciRes. JAMP 17 mic Sounding,” Dokl. RAN [in Russian], Vol. 346, No. 5, 1996, pp. 664-667. [2] L. A. Gasilova and Yu. V. Petukhov, “On the Theory of Surface Wave Propagation along Different Interfaces in the Atmosphere,” Izv. RAN. Fizika Atmosfery i Okean [in Russian], Vol. 35, No. 1, 1999. pp. 14-23. [3] A. V. Razin, “Propagation of a Spherical Acoustic Delta Wavelet along the Gas-Solid Interface,” Izv. RAN. Fizika zemli [in Russian], No. 2, 1993, pp. 73-77. [4] B. G. Mikhailenko and G. V. Reshetova, “Mathematical Simulation of Propagation of Seismic and Acoustic-Gravity Waves for an Inhomogeneous Earth-Atmosphere Model,” Geologiya i geofizika [in Russian], Vol. 47, No. 5, 2006, pp. 547-556. [5] B. G. Mikhailenko, “Spectral Laguerre Method for the Approximate Solution of Time Dependent Problems,” Applied Mathematics Letters, Vol. 12, No. 4, 1999, pp. 105-110. http://dx.doi.org/10.1016/S0893-9659(99)00043-9 [6] G. V. Konyukh, B. G. Mikhailenko and A. A. Mikhailov, “Application of the Integral Laguerre Transforms for Forward Seismic Modeling,” Journal of Computational Acoustics, Vol. 9, No. 4, 2001, pp. 1523-1541. [7] B. G. Mikhailenko, A. A. Mikhailov and G. V. Reshetova, “Numerical Modeling of Transient Seismic Fields in Viscoelastic Media Based on the Laguerre Spectral Me- thod,” Journal Pure and Applied Geophysics, Vol. 160, No. 7, 2003, pp. 1207-1224. http://dx.doi.org/10.1007/s000240300002 [8] B. G. Mikhailenko, A. A. Mikhailov and G. V. Reshetova, “Numerical Viscoelastic Modeling by the Spectral La- guerre Method,” Geophysical Prospecting, Vol. 51, No. 1, 2003, pp. 37-48. http://dx.doi.org/10.1046/j.1365-2478.2003.00352.x [9] Kh. Kh. Imomnazarov and A. A. Mikhailov, “Use of the Spectral Laguerre Method to Solve a Linear 2D Dynamic Problem for Porous Media,” Sib. Zh. Industr. Matem [in Russian], Vol. 11, No. 2, 2008, pp. 86-95. [10] P. K. Suetin, “Classical Orthogonal Polynomials,” Nauka, Moscow, 1974. [11] J. Virieux, “P-SV Wave Propagation in Heterogeneous Media: Velocity-Stress Finite-Difference Method,” Geo- physics, Vol. 51, No. 4, 1986, pp. 889-901. |