THE WAVE EQUATION. The Wave Equation in the Time Domain

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1 THE WAVE EQUATION Disturbances of materials can be described as a waveform propagating through a medium (the material itself) as particles either push or shear neighboring particles from their equilibrium (prior to time zero) position forming a traveling wave (energy). The instant of the initial motion (disturbance) of a particle is referred to as the wavefront singularity. As the main displacement energy passes the area of the particle, the particle slowly (in the seismic time scale) returns, as much as possible, to its equilibrium position (the elastic nature of the medium). The size of such displacement in the far eld is extremely small and depends on the material compressibility (for the acoustic case). Specically, a tradeo between the size of displacement of particles and the wavelength (speed of the wave) adheres to the principle of conservation of energy. Mathematically, the (real valued) amplitude of the particle displacement (wave) is represented by a scalar wave function u that depends on both space and time: u = u(x; t); () where x = (x ; x 2 ; x 3 ) = (x; y; z) (2) represents position in a three dimensional Euclidian space, and t represents time. The scalar function u commonly describes the waveeld in an acoustic medium, but well approximates the waveeld in an elastic one. In solids, it can be understood that it describes the amplitude of the particle displacement from its equilibrium position as such displacement varies with time. The Wave Equation in the Time Domain The homogeneous (in source-free regions) scalar wave equation is a hyperbolic linear second-order partial dierential equation:! 2 r 2 u(x; t) = 0: (3) c 2 2 The wave equation is derived from a combination of Newton's second law that relates force and acceleration (of the particle displacement) and Hooke's law of elasticity, that relates stress (force per area square) and strain (deformation). The wave equation is the law of the land for wave propagation and for the wave equation shown above and used often in imaging, the land is acoustic. A hyperbolic partial dierential equation of order n has a well-posed initial value problem for the rst n derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data. By a linear change of variables, any equation of the form au xx + bu xy + cu yy + ::::: = 0 ; (4)

2 The acoustic wave equation 2 Alkhalifah with and b 2 4ac > 0 ; (5) u kl = 2 u kl ; (6) can be transformed to the wave equation (regarded as hyperbolic), where "::::" represent lower-order terms, which are inessential (eects only the amplitude) for the qualitative understanding of the wave equation. This denition is analogous to the denition of a planar hyperbola. The acoustic wave equation, casted as a set of rst-order linear partial dierential equations, is given by v i p = p x i (i = ; 2; 3); = P 3 i= v i x i ; (7) where p is the stress and v (v ; v 2 ; v 3 ) is the particle velocity, is the mass density and is the bulk modulus. We recast equation 7 in terms of unknown wave variables, p and v i in matrix form: w = Aw; (8) with 0 A = B 0 x 2 x 3 x x x x C ; A w = B p v v 2 v 3 C A : (9) Thus, since the matrix A is diagonalizable and its eignvalues are real, the equation is deemed hyperbolic. The Helmholtz wave Equation As a result of its linear nature, the wave equation can be easily formulted in other domains, including the very useful frequency domain. In this case, the solution is a function of frequency, and thus, our time-domain solution is nothing but a superposition of sinusoidals of xed frequencies weighted by the frequency-domain solution ( inverse Fourier transform). If we assume a seismic wave of a xed frequency, then the time-harmonic form of the waveeld is given as: (x) = A(x)e i(x) ; (0) where is in general a complex quantity, with a separate amplitude component, A,and phase component,, dened in the Euclidian space, with x=(x; y; z or x ; x 2 ; x 3 ).

3 The acoustic wave equation 3 Alkhalifah The waveeld in the time domain, under a homogeneous medium assumption, is given by, u(x; t) = Re n (x)e i!to ; () for a xed angular frequency,! = 2f, with f representing the wave frequency. Substituting this expression into the wave equation 3 yields the frequency form of the wave equation, also known as the Helmholtz equation: r2 + k 2 (x) = 0 (2) where k =! v = 2 (3) is the wave number, i is the imaginary unit, and (x) is the time-independent, complex-valued component of the propagating wave. Note that the propagation constant, specically the wavenumber, k, and the angular frequency,!, are linearly related to one another, a typical characteristic of seismic waves in homogeneous media. In representing the space components (x; y; z) of the waveeld by its Fourier coecients, we are eectively describing the weights of the plane wave representation (decomposition) of the waveeld, and this description provides an alternative platform for extrapolating, as well as analyzing, waveelds, with special features in describing the reection and refraction behavior of waves. Paraxial plane waves (with respect to the z-direction, vertical) For plane waves, an elementary solution to the wave equation takes the form: (x) = A(k)e jk x (4) where is the wave vector, and thus, k = fk x ; k y ; k z g (5) k = jkj = q k 2 x + k 2 y + k 2 z =! v = 2 (6) is the wavenumber, where is the wavelength. For plane waves, k for a xed frequency describes the speed of wave propagation (proportional to slowness), whereas its vector components describe the direction of the plane wave probagation. For large k z relative to k x and k y, the plane wave is traveling predominantly vertical, which is the case most of the time for reection seismology. Reformulating equation 6 as a function of the vertical wave number provides us with what is known as the dispersion relation: s! 2 k z = v 2 k2 x ky; 2 (7)

4 The acoustic wave equation 4 Alkhalifah which relates the measurable wavenumber components, k x and k y, from our recorded data at the surface, d(x; y; t), to a desired wavenumber component, k z, that can help us place reection energy in our data to its accurate reected locations inside the Earth (imaging). Since, in conventional seismic experiments, the recording surface is the Earth surface, most of the recorded waves of interest are those arriving from the subsurface taking a predominantly vertical route, especially since most of the Earth layering is nearly horizontal. Thus, a useful paraxial approximation (from near vertical propagation), is to assume that k 2 x + k 2 y k 2 z (8) or equivalently, sin (9) where is the angle between the wave vector k and the z-axis (vertical). As a result, we consider a Taylor's series expansion of the dispersion relation to the second order, in which k z = k cos k( 2 =2) (20) and thus, dene a new trial solution based on the near vertical propagation approximation as follows (r) A(k)e i(kxx+kyy) e ikz2 =2 e ikz (2) Paraxial wave equation The paraxial expansion provides an approximate trial solution to the wave equation. Thus, substituting this expression into the Helmholtz equation, we derive the paraxial wave equation as follows: r 2 T A 2ik A = 0; (22) z where r 2 T = 2 x + 2 (23) 2 y 2 is the transverse Laplacian operator, shown here in the Cartesian coordinates. Taking this wave equation to the time domain using an inverse Fourier transform in!, we obtain 2 2 A r 2 T A = 0; (24) v z as k =!. Thus, equation 2 along with the solution of equation 24 yields the v approximate paraxial solution for plane waves propagating near vertical. This wave equation is an example of many of such approximations developed mostly in the 70's by Claerbout (970) and others to simplify wave propagation simulation in complex media for imaging purposes. Higher-order approximations

5 The acoustic wave equation 5 Alkhalifah to equation 20, and specically to the cosine, yields higher-order accurate, but more complicated, wave equations. Such approximations control the angle range of validity of the wave propagation accuracy resulting in what we refer to as the 5, 45, and so fourth degree paraxial wave equations. In fact, the 5 degree wave equation is a borne of the following approximation to k z, for k 2 x + k 2 y k 2 z in the dispersion relation 7, k z = k cos = k( sin 2 ) = k( 2 kx 2 + ky 2 ): (25) 2 k 2 Multiplying both sides by the waveeld in the Fourier domain, ^u(!; k x ; k y ; k z ), and inverse Fourier transform in k x, k y, and k z yields the following paraxial wave equation: r 2 T u + 2 v 2 u z 2 2 u = 0: (26) v 2 2 The 45 degree equation is extracted from a bilinear approximation of k z as follows: k z = k cos = k resulting in the following paraxial wave equation: k 2 x +k2 y k 2 k 2 x +k2 y k 2 ; (27) 3 u 4 r2 T v u 4 r2 T z + 3 u 3 u = 0: (28) v z 2 v 2 3 These wave equations are commonly solved by extrapolating in depth. Thus, the initial condition is the data at the surface or at depth equal 0. They are used primarily to downward continue the data recorded on the surface for imaging purposes. The plane wave representation The plane wave representation (spectrum) of the waveeld is the basic foundation of the Fourier description of the waveeld. The plane wave spectrum is a continuous representation of uniform plane waves, in which there exists one plane wave component in the spectrum for every tangent point on the far-eld phase front. The amplitude of that plane wave component would be the amplitude of the waveeld at that tangent point. Of course, this is true only in the far eld of the source. The level of how ne the details of an image is governed by the bandwidth of the signal, the ner the image, a larger bandwidth is required for proper representation. A plane wave propagating parallel to the z axis has a constant value in any x y plane, and therefore is analogous to the (constant) DC component of an electrical signal that has no oscillations. Bandwidth in signals relates to the dierence between the highest and lowest frequencies present in the spectrum of the signal. The spatial frequency content (spatial bandwidth) is important in resolving features laterally. It also measures how far from the vertical axis the corresponding plane waves are tilted,

6 The acoustic wave equation 6 Alkhalifah and so this type of bandwidth is often referred to as angular bandwidth, or fresnel zone. It takes more frequency bandwidth to produce a short pulse in a given medium, and more angular (or, spatial frequency, smaller fresnel zone) bandwidth to produce a sharp spot in an image. The plane wave spectrum arises naturally as the eigenfunction or "natural mode" solution to the homogeneous wave equation in rectangular coordinates. In the frequency domain, with an assumed time convention of e i!t, the homogeneous wave equation is known, as we saw above, as the Helmholtz equation and takes the form: r 2 + k 2 = 0 (29) where (k; x; y; z) describes the waveeld, and k = 2=! is the wavenumber of the medium. Natural mode solutions: In dierential equations, as in matrix equations, if the right-hand side of an equation is zero (i.e., the forcing function or forcing vector is zero), the equation may still admit a non-trivial solution, known in applied mathematics as an eigenfunction solution or in physics as a "natural mode" solution. Common physical examples of resonant natural modes would include the resonant vibrational modes of stringed instruments (in D). Innite homogeneous media admit the rectangular, circular and spherical harmonic solutions to the Helmholtz equation, depending on the coordinate system under consideration. There is a striking similarity between the Helmholtz equation 29, which may be written (r 2 + k 2 ) = 0; (30) and the fundemental equation for the eigenvalues/eigenvectors of a square matrix, A, (A I) x = 0; (3) particularly since both the scalar Laplacian, r 2, and the matrix, A, are linear operators on their respective function/vector spaces. It is perhaps worthwhile to note that both the eigenfunction and eigenvector solutions to these two equations respectively, often yield an orthogonal set of functions/vectors which span (i.e., form a basis set for) the function/vector spaces under consideration. We can also use other functional linear operators, which give rise to dierent kinds of orthogonal eigenfunctions such as Legendre polynomials, Chebyshev polynomials and Hermite polynomials. In the matrix case, eigenvalues may be found by setting the determinant of the matrix equal to zero, i.e. nding where the matrix has no inverse. Finite matrices have only a nite number of eigenvalues/eigenvectors, whereas linear operators can have a countably innite number of eigenvalues/eigenfunctions (in conned regions) or uncountably innite (continuous) spectra of solutions, as in unbounded regions.

7 The acoustic wave equation 7 Alkhalifah It is often the case that the elements of a matrix will be functions of frequency and wavenumber, and the matrix will be non-singular for most combinations of frequency and wavenumber,!2 6= k 2 v 2 x + ky 2 + kz, 2 but will also be singular for certain other combinations,!2 = k 2 v 2 x + ky 2 + kz. 2 By nding which combinations of frequency and wavenumber drive the determinant of the matrix to zero, the propagation characteristics of the medium may be determined. Relations of this type, between frequency and wavenumber as seen above, are known as dispersion relations and some physical systems may admit many dierent kinds of dispersion relations. Solutions to the Helmholtz equation may readily be found in rectangular coordinates via the principle of separation of variables for partial dierential equations. This principle says that in separable orthogonal coordinates, an elementary product solution to this wave equation may be constructed of the following form: u(x; y; z) = f x (x) f y (y) f z (z) (32) i.e., as the product of a function of x, times a function of y, times a function of z. If this elementary product solution is substituted into the wave equation 29, using the scalar Laplacian in rectangular coordinates: r 2 u = 2 u x u y u z 2 ; (33) then the following equation for the three individual functions is obtained f 00 x (x)f y (y)f z (z) + f x (x)f 00 y (y)f z (z) + f x (x)f y (y)f 00 z (z) + k 2 f x (x)f y (y)f z (z) = 0; (34) which is readliy rearranged into the form: fx 00 (x) f x (x) + f y 00 (y) f y (y) + f z 00 (z) f z (z) + k2 = 0: (35) Since each of the rst three terms in equation 35 is a function of a single independent variable, x, y, and z, respectively, and to insure that equation 35 is satised for all x, y, and z (everywhere in the domain), then each of these three terms must be constant. The constant for the rst term is denoted as k x?. Similarly for the y and z quotients, we end up with three ordinary dierential equations for the f x, f y and f z, along with one separation condition: d 2 dx f x(x) + kxf 2 x (x) = 0 2 (36) d 2 dy f y(y) + kyf 2 y (y) = 0 2 (37) d 2 dz f z(z) + kzf 2 z (z) = 0 2 (38) kx 2 + ky 2 + kz 2 = k 2 (39) Each of these three dierential equations has the same solution: sines, cosines or complex exponentials. We will use the complex exponential for notational simplicity,

8 The acoustic wave equation 8 Alkhalifah compatibility with usual FT notation, and the fact that a two-sided integral of complex exponentials picks up both the sine and cosine contributions. As a result, the elementary product solution for u is: u(x; y; z) = e i(kxx+kyy+kzz) = e i(kxx+kyy) e ikzz = e i(kxx+kyy) e izp k 2 k 2 x k 2 y ; (40) which represents a propagating or exponentially decaying uniform plane wave solution to the homogeneous wave equation. The minus sign is used for a wave propagating/decaying in the +z direction and the plus sign is used for a wave propagating/decaying in the z direction. This eld represents a propagating plane wave when the quantity under the radical is positive, and an exponentially decaying wave when it is negative (in passive media, the root with a non-positive imaginary part must always be chosen, to represent uniform propagation or decay, but not amplication). The complete solution: the superposition integral A general solution to the homogeneous wave equation in rectangular coordinates may be formed as a weighted superposition of all possible elementary plane wave solutions as: Z Z (k; x; y; z) = u(k; k x ; k y ) e j(kxx+kyy) e izp k 2 k 2 x k 2 y dk x dk y : (4) This plane wave spectrum representation of the waveeld is the basic foundation of Fourier imaging (this point cannot be emphasized strongly enough), because when z=0, the equation above simply becomes a Fourier transform (FT) relationship between the recorded waveeld at the surface (z = 0) and its plane wave content. All spatial dependence of the individual plane wave components is described explicitly via the exponential functions. The coecients of the exponentials are only functions of spatial wavenumber k x, k y, just as in ordinary Fourier analysis and Fourier transforms. Evanescent waves The waveeld solution in equation 4 is a superposition of plane waves described by their wavenumber vectors. However, if k 2 x + k 2 y > k 2 ; (42) which implies that the considered components of the horizontal wavenumbers has an amplitude bigger than the presumed amplitude governed by the medium property (speed, k =! ), then the plane waves are evanescent (decaying in the z-direction, in v this case). This happens when a plane wave encounters a horizontal interface with

9 The acoustic wave equation 9 Alkhalifah medium properties (i.e. large velocity) that reduces k, and thus, for certain plane waves with large enough k x and k y values (dipping, large angles from vertical) it turns evanescent in the new medium and it decays exponentially as a function of z. The concept of angular bandwidth Equation 40 may be evaluated asymptotically in the far eld (using the stationary phase method) to show that the eld at the point (x; y; z) is indeed due solely to the plane wave component (k x ; k y ; k z ) which propagates parallel to the vector (x; y; z), and whose plane is tangent to the phase front at (x; y; z). The result of performing a stationary phase integration on the expression 40, (r; ; ) = 2i (k cos ) e jkr u(k sin cos ; k sin sin ); (43) r which clearly indicates that the waveeld at (x; y; z) is directly proportional to the spectral component in the direction of (x; y; z), where, and x = r sin cos (44) y = r sin sin (45) z = r cos (46) k x = k sin cos (47) k y = k sin sin (48) k z = k cos (49) Stated another way, the radiation pattern of any planar eld distribution is the Fourier transform of that source distribution (known as the Huygens-Fresnel principle). Note that this is NOT a plane wave. The e jkr radial dependence is a spherical wave - both r in magnitude and phase - whose local amplitude is the Fourier transform of the source plane wave distribution at that far eld angle. Equation 43 is critical in making the connection between spatial bandwidth and angular bandwidth, in the far eld. The wavenumber domain and plane waves The Fourier transformation of the physical space domain to its wavenumber component allows the proper description of plane waves. The separation condition, k 2 x + k 2 y + k 2 z = k 2 (50) which is identical to the equation for the Euclidian metric in 3 dimensional conguration space, suggests the notion of a k-vector in 3 dimensional "k-space," dened (for propagating plane waves) in rectangular coordinates as: k = k x^x + k y^y + k z^z (5)

10 The acoustic wave equation 0 Alkhalifah and in the spherical coordinate system as k x = k sin cos (52) k y = k sin sin (53) k z = k cos (54) and The two-dimensional Fourier transform pairs: F (k x ; k y ) = with normalizing factor of 4 2 Z Z f(x; y)e j(kxx+kyy) dxdy; (55) f(x; y) = Z Z F (k x ; k y )e j(kxx+kyy) dk x dk y ; (56) 4 2 included as a response to using the angular frequency. The Earth system The Earth reectivity can be represented by a system operating on plane waves in which the system consists of an input plane wave, and output plane wave, and a set of components that transforms the function f formed at the input into a dierent function g formed at the output. The output function is related to the input function by convolving the input function with an impulse response, h (known as we will see later as the Green's function). The impulse response uniquely denes the input-output behavior of the Earth response system (the convolution operation). By convention here, considering that the acquisition surface of the seismic experiment to be normal to the z-axis, the two functions and the impulse response are all functions of the transverse coordinates, x and y (extrapolation in depth). The impulse response of this Earth system is the output plane waveeld, which is produced when the input is a point source. In practice, it is not necessary to have an ideal point source in order to determine an eective impulse response. This is because any source bandwidth which lies outside the bandwidth of the system won't matter anyway (since it cannot even be captured), so therefore its not necessary in determining the impulse response (as we will see later). In imaging, the input plane wave function (extracted from the recorded data) is dened as the locus of all points such that z = 0. The input function, f, is therefore, f(x; y) = U(x; y; z)j z=0 : (57) The output plane wave function can be dened as the locus of all points such that z = d, a depth of interest. The output function, g, is therefore, g(x; y) = U(x; y; z)j z=d : (58)

11 The acoustic wave equation Alkhalifah The 2D convolution of input function with the impulse response function, h, is i.e., g(x; y) = Z g(x; y) = h(x; y) f(x; y); (59) Z h(x x 0 ; y y 0 ) f(x 0 ; y 0 ) dx 0 dy 0 : (60) In this formulation, h, is the operator capable of downward continuing the wave- eld from depth zero to depth d. It has an amplitude component and a phase shift component that can be readily observed in its Fourier domain representaton. The convolution operation The extension of the convolution operation to two space dimensions is trivial, except for the fact that causality exists in the time domain, but not in the spatial domain. Causality means that the impulse response h(t t 0 ) of an Earth response system, due to an impulse applied at time t 0, must of necessity be zero for all times t such that t t 0 < 0. Obtaining the convolution representation of the response requires representing the input signal as a weighted superposition over a train of impulse functions by using the sifting property of Dirac delta generalized functions. f(t) = Z (t t 0 )f(t 0 )dt 0 : (6) Of course, the system under consideration is linear, that is to say that the output of the system due to two dierent inputs (possibly at two dierent times) is the sum of the individual outputs of the system to the two inputs, when introduced individually. Thus, the Earth system is approximated to contain no nonlinear behavior (i.e. no multiples). The output of the system, for a single delta function input is dened as the impulse response of the system, h(t t 0 ). By our linearity assumption (i.e., that the output of system to a pulse train input is the sum of the outputs due to each individual pulse), we can now say that the general input function f(t) produces the output: Output(t) = Z h(t t 0 )f(t 0 )dt 0 (62) where h(t t 0 ) is the (impulse) response of this linear system to the delta function input (t t 0 ), applied at time t 0. The convolution equation is useful because it is often much easier to nd the response of a system to a delta function input - and then perform the convolution above to nd the response to an arbitrary input - than it is to try and nd the response to the arbitrary input directly. Also, the impulse response (in either time or frequency domains) usually yields insights into relevant information on the system. The same logic is used in connection with the Huygens-Fresnel principle, wherein the "impulse response" is referred to as the Green's function of the system. So the

12 The acoustic wave equation 2 Alkhalifah spatial domain operation of a linear system is analogous in this way to the Huygens- Fresnel principle. where If the last equation above is Fourier transformed, it becomes: Output(!) H(!) F(!) Output(!) = H(!) F (!); (63) is the spectrum of the output signal is the system transfer function is the spectrum of the input signal Likewise, equation 59 may be Fourier transformed to yield: G(k x ; k y ) = H(k x ; k y ) F (k x ; k y ) (64) This equation takes on its real meaning when G(k x ; k y ) is associated with the coecient of the plane wave whose transverse wavenumbers are (k x ; k y ). Thus, the input plane wave spectrum is transformed into the output plane wave spectrum through the multiplicative action of the system transfer function. The system transfer function for waves propagating in the Earth is extracted from the appropriate wave equation for the propagating medium, which is approximated in our case by the acoustic condition. As we saw earlier, it given by H(k x ; k y ) = e izp k 2 k 2 x k 2 y : (65) Another observation is that though theoretically h is the response to a perfect impulse, since the convolution process, necessary to take advantage of the sifting property of the delta function, is a multiplication in the Fourier domain, the output function will most likely have the bandwidth of the input function. Thus, the pulse, used to calculate the response, only needs to have a at spectrum in the bandwidth range of the input function. Solving the acoustic wave equation The acoustic Model (only compressional waves), is good enough for most of our objectives in imaging, especially the main objective: imaging the structure. In inhomogeneous media in the continuum, we consider the medium to be made up of innitesimally small locally homogenous regions. The locality of the homogeneous region is dened by the range of inuence of the second-order space derivatives. For discrete second-order nite dierence approximations, its spans over three samples, but as we take the descretization down to zero, it reduces to a continuum representation. Thus, the rst-order acoustic wave equation, shown above, now relates (x)

13 The acoustic wave equation 3 Alkhalifah (material density), (x) (bulk modulus), p(x; t) (pressure), v(x; t) (particle velocity vector), f(x; t) (force density, sound source), as follows: and v p = rp + f; (66) = r:v; (67) with the proper initial and boundary conditions. and wave speed v = q. Considering the acoustic eld potential u(x; t) = R t ds p(x; s): p = u ; v = ru: (68) An equivalent form for the set of rst order equations, the second-order wave equation for potential is given by: 2 u v 2 r Z t 2 ru = dsr with proper initial and boundary conditions.! f : (69) The Born approximation The forward mapping operator, S, the impulse response or the time history of pressure for each x s at receiver locations x r (predicted seismic data), depends on velocity eld v(x): S[v] = p(x r ; t; x s ): (70) Thus, the reection seismic inverse problem, given an observed seismic data S o, includes nding a velocity, v, so that S[v] S o : (7) Since this inverse problem is very large, governed, mainly, by a desired high resolution inverted model, and since the problem is highly nonlinear, linearization is an indispensable tool of the inversion process. Thus, we consider v = v 0 ( + r) and treat r as a small number representing the relative rst-order perturbation about v 0, resulting in a perturbation in the pressure eld p = u = 0. Assuming causality, the perturbation in the pressure eld satises the wave equation with a source guided by the background source waveeld! 2 v 0 2 r2 u = 2r 2 u v 2 0 : (72) 2

14 The acoustic wave equation 4 Alkhalifah Thus, we dene the linearized forward map F by F [v 0 ] contains the scattered waveeld. As a result, the linearization error is given by and such errors are F [v 0 ]r = fp(x r ; t; x s )g (73) S[v 0 ( + r)] (S[v 0 ] + F [v 0 ]r); (74) small when v 0 smooth, r rough or oscillatory on wavelength scale - well-separated scales large when v 0 not smooth and/or r not oscillatory - poorly separated scales The general inversion: given S o, we nd v 0 and r so that Otherwise, via Born, given S o and v, we nd r so that S[v 0 ] + F [v 0 ]r ' S o (75) F [v 0 ]r ' S o S[v 0 ] (76) The perturbation of Green's function The Green's function, which represents the response of a system to an impulse source, for the acoustic wave equation satises:! 2 v 2 r2 G(x; t; x s ) = (t)(x x s ); (77) where G 0 for t < 0. Similar to the full wave equation, a perturbation in the velocity model given by r, where v = ( + r)v 0, results in a perturbation in the Green's function:! 2 v 0 2 r2 G(x; t; x s ) = 2 2 G 0 (x; t; x s ) r(x); (78) v where G = G 0 + G, in which G 0 satises equation 77 for a velocity model given by v 0. Thus, based on the our denition of the forward operator, F [v]r = Gj x bfx r with an integral solution to the perturbation in the Green's function satises the classical solution for the wave equation that includes the convolution of the source function (given by the right hand side of the previous equation) and the Green's function: Z G(x r ; t; x s ) = dx 2r(x) Z v 2 (x) dsg 0 (x r ; t s; x s ) 2 G 0 (x r ; s; x s ) 2 : (79)

15 The acoustic wave equation 5 Alkhalifah In representing the convolution part as a distribution kernel we can represent the perturbation in the Green's function in term of K(x r ; t; x s ; x), Z G(x r ; t; x s ) = dx r(x)k(x r ; t; x s ; x); (80) with K(x r ; t; x s ; x) = 2 Z v 2 (x) dsg 0 (x r ; t s; x s ) 2 G 0 (x r ; s; x s ) 2 : (8) An approximate high-frequency asymptotic representation of the Green's function is given by: G(x; t; x s ) = A(x; x s )(s (x; x s )): (82) Thus, the distribution kernel is approximated as follows: K(x r ; t; x s ; x) 2A(x r; x)a(x; x s ) v 2 (x) Z ds (t s (x r ; x)) 00 (s (x; x s ))(83) = 2A(x r; x)a(x; x s ) 00 (t (x; x v 2 s ) (x; x r )): (84) (x) This kernel provides the basis for Kirchho modeling and migration.