Raypath and traveltime computations Main for 2D transversely isotropic media with Heading dipping symmetry axes Bing Zhou Authors Stewart Greenhalgh

Transcription

1 Exploration Geophysics (2006) 37, Raypath and traveltime computations Main for 2D transversely isotropic media with Heading dipping symmetry axes Bing Zhou Authors Stewart Greenhalgh Key Words: seismic waves, Key ray Words: tracing, key traveltime, words raypath, anisotropic media ABSTRACT This paper presents a simple method to calculate the traveltimes and corresponding raypaths for the first-arrival (refraction), reflected, diffracted, and converted waves of the three modes (qp, qsv, qsh) in a 2D transversely isotropic medium, whose symmetry axis may have an arbitrary orientation in the xz-plane. This method is a direct extension of the shortest path method to the anisotropic situation. In this extension, we employ analytic solutions for the group velocity of the three wave modes, and transform a 2D heterogeneous, transversely isotropic medium defined by five elastic moduli and the arbitrary orientation angle of the symmetry axis into the three group-velocity models, which correspond to the qp-, qsv- and qsh-waves. The three group velocities are functions of the spatial coordinates and the ray direction, as well as the orientation angle of the symmetry axis of the media. With these group-velocity models, the traveltimes of these waves and their corresponding raypaths are then simultaneously or individually calculated by a modified shortest path method. We present some numerical experiments to show the accuracy, efficiency, and capability of the method. The results demonstrate that a rotated (dipping) symmetry axis may significantly change the kinematic properties of the three wave modes in a transversely isotropic medium. INTRODUCTION Ultrasonic laboratory studies as well as seismic field investigations show that many geological materials and subsurface structures are elastically anisotropic. This is especially true in media having aligned cracks (Crampin, 1984), fine-layered rocks (Helbig, 1981; Schoenberg, 1983), and some sedimentary rocks like sandstone and shale (Thomsen, 1986). There is also abundant evidence to show that some portions of the crust and the upper mantle of the Earth are distinctly anisotropic (e.g., Anderson and Dziewonski, 1982). There are substantial differences in the kinematic and dynamic features of wave propagation in anisotropic as opposed to isotropic media. The former involves three modes of body wave, one quasicompressional wave (qp) and two quasi-shear waves (qs 1 and qs 2 ). Each propagates with its own wave speed and polarization direction. Unlike the isotropic case, the phase velocity and the group velocity of each wave mode are no longer equivalent, and both velocities, as well as the polarization directions, are functions of not only the elastic moduli but also the direction of wave propagation (Červeny, 1972; Crampin, 1981). An anisotropic medium is defined by up to 21 independent elastic moduli, which may be spatially Department of Physics Adelaide University North Terrace, Adelaide, SA 5005, Australia bing.zhou@adelaide.edu.au Manuscript received 21 March, Revised manuscript received 30 August, variable (heterogeneous). Specific classes of anisotropy include transversely isotropic (five moduli), orthorhombic (nine moduli), monoclinic (12 moduli), and so on (Musgrave, 1970; Helbig, 1994). In order to properly understand seismic observations, and to avoid erroneous interpretations of seismic data, one should pay careful attention to seismic anisotropy when dealing with wave propagation in many geological situations. Since the 1950s many scientists have investigated wave propagation in anisotropic media. The books by Hearmon (1961), Fedorov (1965), Musgrave (1970), and Helbig (1994) give comprehensive and valuable reviews. The seismic ray method is an effective way to simulate and interpret seismic data in practice (Červeny, 2001). It may be employed for non-linear seismic tomography, normal moveout calculations, seismic anisotropic migration, and synthetic seismogram generation. The main advantages of the ray method are the efficiency and flexibility in the computation of amplitudes, traveltimes, and raypaths for the different wave modes (qp, qs 1, qs 2 ), which include seismic events such as reflected, refracted, diffracted, and converted waves. Babich (1961) and Červeny (1972) made significant contributions to ray tracing for a general anisotropic medium. A number of quite specific approaches for computing traveltimes or tracing raypaths in anisotropic media have been devised in the last few decades. Červeny and Firbas (1984) describe a linearised approach for obtaining traveltimes in weakly anisotropic media. Gajewski and Psencik (1987) presented a 3D ray shooting tracing method for laterally varying layered anisotropic media. They traced rays for given trials of the phase-slowness vector in order to find the target receivers. Shearer and Chapman (1988) developed an analytic approach to raypath and traveltime computation for an anisotropic medium with a linear gradient. Most other researchers have focused on the simplified form of hexagonal anisotropy involving a vertical axis of symmetry, the so-called transversely isotropic or VTI medium. The reasons for this are twofold. The first is the structural similarity of minerals in sedimentary rock, like shale, to the hexagonal crystal. Finelayered rocks and those having aligned (parallel) cracks also have properties similar to the VTI medium. Second, such VTI media have only five elastic moduli, which leads to relatively simple and tractable analytic solutions that avoid the daunting algebraic complexity in the ray tracing. For example, Qin and Schuster (1993) demonstrated a first-arrival traveltime computation method using Huygens Principle for a 2D VTI medium. Faria and Stoffa (1994) presented an iterative approach to calculate the traveltime field for a 2D/3D VTI medium. Rüger and Alkhalifah (1996) demonstrated an efficient 2D ray tracing method for a VTI medium. In recent years, Ettrich and Gajewski (1998), starting with a model of elliptical anisotropy, developed a perturbation method to compute the traveltimes in VTI media. Cardarelli and Cerreto (2002) presented a ray tracing method in elliptical anisotropic media, using linear traveltime interpolation. Elliptical anisotropy is actually the degenerate case or rough model of a VTI medium, and results in a simple dip-angle dependent expression for the group velocity. Qian and Symes (2002) employed a finite difference solver to calculate qp traveltimes in 2D/3D VTI media. Alkhalifah (2002) gave an alternative traveltime computation 150 Exploration Geophysics (2006) Vol 37, No. 2

2 scheme for 2D/3D VTI media, by solving a linearised eikonal equation. In this paper we extend the raypath treatment to a more general class of heterogeneous, anisotropic media (dipping TI) and compute travel times and raypaths for all three modes (qp, qs 1, qs 2 ) using a modified shortest-path approach. GENERAL TI MEDIA AND THE SHORTEST PATH METHOD The VTI medium is an idealised model. Even though some rocks, like shales or sandstones, and ordered structures, like aligned cracks and thin layers, are known to be VTI media in the laboratory, in the field their symmetry axes may take any direction other than the vertical because of tilting and other geological movements subsequent to deposition. In addition to the spatial variation of the five elastic moduli of a transversely isotropic medium, the orientation of the symmetry axis may vary with location in the Earth depending upon the geological situation. Here, we simply call a transversely isotropic medium with a dipping symmetry axis a general TI medium. Such situations are often encountered in seismic applications. Jech (1988) demonstrated a linearised inversion routine for such media, and Jech and Psencik (1989, 1992) developed traveltime inversion formulae for the weakly anisotropic case of the general TI medium. In the following sections, we present a simple method to calculate the traveltimes and the raypaths in a 2D general TI medium (the dipping symmetry axis lies in the xz-plane). Actually, this method is a direct extension of the shortest path method (SPM) for an isotropic medium (Nakanishi and Yamaguchi, 1986; Moser, 1991, Gruber and Greenhalgh, 1998) to the anisotropic situation. There are various ray tracing methods available, such as shooting and bending methods, and the method of characteristics. The major advantages of the SPM over the other ray tracing procedures are its simplicity, ease of applicability to either 2D or 3D models, and ability to simultaneously yield both the traveltime and the raypath without missing the receiver targets in a complex velocity model. Some traveltime computation methods do not yield the raypaths at all, and those that do may fail to pick up all source-receiver pairs. The latter point is very important for practical kinematic inversion. SPM is a network scheme that explores all possible connections in the grid and guarantees that rays and traveltimes will be found (Klimes and Kvasnicka, 1994; Bai, 2004). Furthermore, in an anisotropic medium, the raypath direction may not coincide with the gradient direction of the traveltime (i.e., the phase-slowness vector, or wavefront normal). In general, the phase-slowness vector and the raypath vector at a given point lie at an angle of between 0 and 90. This disparity makes it more difficult to find the raypath along with the traveltime distribution from a given source than in the case of an isotropic medium. But using the SPM eases the difficulty, as illustrated below. To accomplish the extension, we firstly investigate the conversion of a general TI medium defined by five elastic moduli {a 11 (x), a 13 (x), a 33 (x), a 55 (x), a 66 (x) } and the orientation angle θ 0 (x) of the symmetry axis into three group velocity models {U m (x,θ 0,r 0 ), m = 1, 2, 3}, m = 1, 2, 3}, which correspond to the group velocities of the three wave modes (qp, qs 1, qs 2 ); they are generally functions of the ray direction r 0 and the orientation angle θ 0 (x). Here x represents the spatial coordinates, implying that the medium is heterogeneous as well as anisotropic. This model conversion is achieved by discretising the model and performing an elasticmoduli-to-group-velocity mapping with analytic formulae for the general TI medium. Based on the group velocity models, we adapted the shortest path method for calculating the first-arrival traveltimes and the raypaths for the three wave modes (qp, qs 1, qs 2 ), and then further modified the method for dealing with reflected, diffracted, and converted waves. Some numerical tests are given to show the accuracy, efficiency, and capability of the method. From these results, one can appreciate the difference in wave propagation behaviour between VTI and general TI media, and the robustness of the method. The only drawback of the shortest-path method is that it is unable to handle the triplications or cuspidal effects sometimes associated with the qs 1 (or qsv) wave which can arise for certain directions of propagation. GROUP VELOCITY MODEL To trace a ray, one needs knowledge of the group velocity variation, because in an elastic medium the raypath from a source to a receiver is the trajectory of (wave) energy flux whose propagation speed and direction defines the group velocity. In an isotropic medium, the group velocity U is identical to the phase velocity c (U = c) and can be directly obtained from the elastic moduli (λ, μ) and the density ρ. The velocities may vary with spatial position via or for P- or S-waves. In an anisotropic medium, the group velocity may differ from the phase velocity, and both are functions of the elastic moduli and the wave propagation direction. Expressions for the most general class of anisotropy are given by Zhou and Greenhalgh (2004, 2005). For a general TI medium, the modified VTI formulae given below can be used. They are relatively simple and lead to easy computation. We begin with the phase velocities c for the qp-, qsv- and qshwaves in a VTI medium (here qsv and qsh correspond to the more general qs 1 and qs 2 modes, respectively). Daley and Hron (1977) gave explicit solutions: where and, (1), (2) The angle ϑ is the incident angle (in the vertical plane) of the phase-slowness vector, measured from the symmetry axis of the medium. The quantities (a 11, a 13, a 33, a 55, a 66 ) are the five independent elastic moduli. The phase velocity subscripts 1, 2, 3 on the left side of equation (1) correspond to the three wave modes qp, qsv and qsh, respectively. The magnitude of the group velocity vector can be calculated using the expression that may be found in, for example, Berryman (1979): (3), (4) Here we have used the subscript m to denote the three wave modes. Equivalently, the individual horizontal and vertical (m) components U z of the group velocity vector can be written as (Crampin, 1981): Exploration Geophysics (2006) Vol 37, No

3 , (10) where the mode m is now written as a superscript to avoid confusion with the component subscript h or z. The derivatives in equation (4) or (5) can be calculated from equations (1) to (3) as follows: (5) (6) where a ijkl are the elements of the fourth-order elastic modulus tensor, τ is the travel time and are the Cartesian components of the eigenvectors of the three wave modes (m = 1,2,3; i, j = x, y, z or 1, 2, 3), it follows that if the elastic moduli are not varying in the (horizontal) strike or y-direction, then the phase-slowness vector p (m) has a zero x 2 - (or y-) component, i.e.,, which will remain so along the entire path for all the rays. Note that the traveltime field in the plane par allel to the central plane (x 2 = 0 or y = 0) is the same for a line source. In this case equation (9) does not involve the azimuthal angle because ϕ 0 = ϕ = 0 for the x 1 x 3 - or xz-plane. The ray tracing becomes two-dimensional (2D) and equation (8) reduces to:. In such situations, equation (9) becomes and Equations (1) to (7) show that the group velocity in a VTI medium depends on the five density-normalised elastic moduli (a 11, a 13, a 33, a 55, a 66 ) and the phase-slowness vector n = (cosϑ, 0, sinϑ). In order to make equations (1) (6) applicable to a general TI medium (having arbitrary dip), we can use standard trigonometric relations (co-ordinate rotation) to represent the cosine of the incident angle as follows: (7), (8) where the two pairs of spherical polar angles(θ 0, ϕ 0 ) and (θ, ϕ) are the orientation of the symmetry axis of the TI medium and the phase-slowness direction n = (cosϕ sinθ, sinϕ sinθ, cosθ), respectively. The angles θ and ϕ stand for the inclination angle, measured from the z-axis (downward), and the azimuthal angle, measured from the x-axis. They have the ranges of [0, 180 ] and [0, 360 ]. Substituting equation (8) for equations (1) to (7) yields the group velocities for a general TI medium. From the above formulation, one can see that the group velocities are functions of not only the phase-slowness direction (θ, ϕ) but also the orientation angles (θ 0, ϕ 0 ) of the symmetry axis, as well as the five elastic moduli. By rotating the symmetry axis from the vertical to (θ 0, ϕ 0 ), we obtain the following three components of the group velocity vector for a general TI medium: In this paper, we restrict our attention to the 2D case, where the density-normalised elastic moduli (a 11, a 13, a 33, a 55, a 66 ) do not change in the y- or x 2 -axis (strike) direction. According to the ray equation for the phase-slowness vector given by Červeny (1972): (9) (11) The discussion above suggests that a 2D general TI medium may be described by the model vector, (12) and all the components of the model vector vary with the spatial coordinates x = (x, z). Substituting (12) for (1) (7), one obtains the three group velocities {U m (x, θ 0, n), m = 1, 2, 3}, which correspond to the three wave modes (qp, qsv, qsh) and are functions of the spatial coordinates x, the orientation angle θ 0 of the symmetry axis and the phase-slowness direction n = (sinθ, cosθ). By such a procedure, we transform a general TI medium defined by equation (12) into three group velocity models {U m (x, θ 0, n), m = 1, 2, 3}. This elastic-moduli-to-group-velocity mapping is basic to the ray tracing method described in the next section. As an example, Figure 1 shows the transformation with the above equations for a homogenous TI medium having elastic moduli (a 11 = 5.2, a 13 = 0.93, a 33 = 4.0, a 55 = 1.0, a 66 = 1.0 ) and three different symmetry axis orientation angles (θ 0 = 0, 45, 90 ). The left panel in Figure 1 was directly obtained from equations (1) to (8). It shows the group velocities to be functions of the orientation angle θ 0 and the phase-slowness angle θ. The right panel in Figure 1 shows the group velocities as a function of the ray angle, which can be calculated with the aid of equation (11). These plots show the group velocity U(x, θ 0, r 0 ) varies with the ray direction, and that there are cusps (triplications) for the qsv-wave. Such cusps or multiple-values in the group velocity for a given ray direction, are hidden when plotted against phase-slowness direction θ (left panel); the plots show single-valued behaviour. This is one advantage of such a representation, although the singular directions have not been removed. They occur for those values of θ were c 2 = c 3 (for further discussion, see Zhou and Greenhalgh, 2004). From the figure one can see the different behaviour of the qp- and qsv-wave group velocities due to of the different orientation angles of the symmetry axis. The discrepancy between the phase-slowness angle and the ray angle in an anisotropic medium also leads to other interesting characteristics, like mixing of the polarization vectors (see Zhou and Greenhalgh, 2004, 2005). For the particular case illustrated, in which the numerical values for a 55 and a 66 are the same (a 55 = a 66 = 1), the qsh-wave group velocity turns out to be constant (see equation 1) and does not vary with the orientation of the symmetry axis, the phase-slowness angle or the ray direction. 152 Exploration Geophysics (2006) Vol 37, No. 2

4 RAY TRACING METHOD With a known raypath R(x), one can calculate the traveltime along the raypath by means of the line integral (13) where is the magnitude of the arc length, and x is the local ray vector. This equation shows that the fundamental steps for the traveltime computation are to determine the group velocity U and the raypath R(x) for a given elastic model m(x). The previous section has dealt with the conversion from an anisotropic model m(x) defined by equation (12) into a group velocity model U(x, θ 0, r 0 ). This section describes the method of how to find the raypath R(x) and calculate the traveltime τ with the directionallydependent group velocity U(x, θ 0, r 0 ). so-called shortest path method (SPM), developed by Nakanishi and Yamaguchi (1986) and Moser (1991) for isotropic media, to general TI media. Actually, the SPM implicitly follows Huygens Principle, which is described by the following picture of wave propagation: every point on a wavefront is a new source for the next wavefront, at which the wave solution is a summation of the contributions from all the point sources on the previous wavefront. Waves are allowed to evolve from neighbouring grid points or nodes, keeping track of times at each point. In kinematic terms, the first-arrival traveltime at a point in an elastic medium is the minimum of the traveltimes from all possible node connections from the source up to that point (Fermat s Principle). This is exactly described by equation (14). According to this principle, the modified SPM for the anisotropic medium involves the following steps: According to Fermat s Principle, the raypath R(x) is the trajectory that makes the integral (13) stationary (δτ = 0). This means that the first-arrival traveltime at a point x B can be written in the following form:, (14) where Ω B is the neighbourhood of the model around x B and r 0 is the unit vector of the ray direction from x A to x B. This equation presents the possibility for extending ray tracing in an isotropic medium to an anisotropic medium. It obviously shows that the extension requires consideration of the directional (r 0 ) dependence of the group velocity and an efficient way to find the minimum traveltime from x A to the point x B. This prompts us to extend the Fig. 2. Analytic (red dotted-lines) and numerical (white lines) results for the wavefronts in homogeneous TI media, which have vertical (the left panel) and dipping ( = 45, the right panel) symmetry axes. Fig. 1. Group velocities of the three wave modes (qp, qsv, qsh) for different orientations of the symmetry axis (θ 0 = 0, 45, 90 ). The density-normalised elastic moduli (a 11 = 5.2, a 13 = 0.93, a 33 = 4.0, a 55 = 1.0, a 66 = 1.0 ) were used in the computations. The left panel has group velocity plotted against the phase slowness direction (single-valued functions) whereas the right panel has U plotted against ray direction. The latter exhibits multiple values (triplications) or wavefront cusps (see Fig. 2) for certain directions with the qsv mode. Fig. 3. Maximum relative errors of the numerical traveltimes versus the secondary node density and the cell size. Exploration Geophysics (2006) Vol 37, No

5 (1) Model parameterisation With a certain cell size (or element), one may divide a 2D anisotropic medium m(x) defined by equation (12) into a gridded model of primary nodes where N x and N z are the numbers of cells in the x- (= x 1 ) and z- (= x 3 ) directions. The cells are generally rectangular and bounded by four primary nodes (2D). In a regular scheme the cells are made quite small. Irregular schemes, involving larger cells with additional (secondary) nodes along each cell edge, can be used for computational efficiency (see Bai, 2004), as explained below. The gridded anisotropic model is converted into three group velocity models { U m (x k, θ 0, n), m = 1, 2, 3 for qp, qsv and qsh, k N 1 } Employing equation (11), one may obtain the group velocities as functions of the ray direction r 0 : { U m (x k, θ 0, r 0 ) } (see the right panel of Figure 1). Where triplications exist for the qsv mode, we can only use the minimum values of group velocity so as to, ensure continuity with adjacent regions. For the other two modes, the group velocity is single valued and no problem arises. After the model conversion, the group velocity within each cell may be approximated by means of bi-linear interpolation, or using the Lagrangian interpolation function: (15) Obviously, the corner points coincide with the nodes of the gridded model {x k, k N 1 }, which are called the initial nodes or primary nodes. In order to trace a raypath in the cell, some additional nodes (secondary nodes) are required to provide alternative ray branching points throughout each cell (rather than just at its four corners). This is essential for accurate computations when using larger cell sizes. These secondary nodes are normally distributed in a regular way along the common boundaries of the cells (Moser, 1991; Fischer and Lees, 1993; Gruber and Greenhalgh, 1998). We simply call the number of secondary nodes on each boundary of the cell the secondary node density. Fig. 4. CPU times on a Pentium 4, 3.2GHZ PC for the numerical experiments with different secondary node densities and cell sizes: (a) cell size of 50 m was used, the bracketed numbers are total number of nodes of the network, (b) the secondary node density of 7 points was applied, the bracketed numbers are the total number of cells of the network. Fig. 5. Raypaths and first-arrival traveltimes for the three wave modes (qp, qsv, qsh) in a three-layered anisotropic model to which three orientation directions of the symmetry axis are applied: (a) the three layers all have vertical symmetry axes (0 ); (b) the top two layers have dipping symmetry axes of 45 ; (c) the top two layers have horizontal symmetry axes (90 ). 154 Exploration Geophysics (2006) Vol 37, No. 2

6 Fig. 6. Group velocity vectors for the three-layered model: a) the first layer (a 11 = 9.08, a 13 = 2.98, a 33 = 7.54, a 55 = 2.27, a 66 = 3.84 ), b) the second layer (a 11 = 20.31, a 13 = 9.58, a 33 = 22.29, a 55 = 8.35, a 66 = ), and c) the third layer (a 11 = 13.86, a 13 = 4.31, a 33 = 10.93, a 55 = 3.31, a 66 = 4.34 ). Vertical symmetry axes for the three layers were applied to the computations. If the positions of sources and receivers do not coincide with the regularly created nodes, the nodes for these positions are also added. In SPM, the primary nodes and the secondary nodes are all considered as candidate points of a raypath through the medium. If N 2 stands for the total number of the secondary nodes, after such a discretising procedure we obtain a network for ray tracing. This network has in total N 1 + N 2 nodes. All of them are identified or discriminated by the node number l and the spatial coordinates {x l = (x l, z l ), l = 1, 2,, N 1 + N 2 }. The secondary node density is crucial for the method (Moser, 1991; Gruber and Greenhalgh, 1998; Bai, 2004). A dense node grid generally gives a more accurate solution for the traveltime and raypath than a sparse node grid, but it increases computer time, particularly for 3D applications. A suitable secondary node density may be chosen as a compromise between the computational accuracy and cost in computer resources (Fischer and Lees, 1993). Gruber and Greenhalgh (1998) give formulae to estimate the maximum errors for a given secondary node density. (2) Traveltime initialisation Zero traveltime is assigned to the source point and a relatively large value (10 7 ) assigned to all of the other grid nodes. Starting with the source point, one calculates the traveltimes (see step (3)) from the source to all the neighbouring points, also from one neighbouring point to another, and then records the minimum traveltimes for all these neighbouring points. They become the computed frontal points of the source. (3) Minimum traveltime computation In order to calculate the minimum traveltimes for all the grid nodes, one may gradually expand the area of the computed frontal points by continually adding the undetermined neighbouring points to the computed frontal points. The traveltime between any two points in the cell may be simply approximated by the following formula (distance divided by the average group velocity between the two nodes i and j) due to the relatively small dimension of the cell: (16) Here N j is a subset of the computed frontal points from prior computation of the evolving wavefield near the undetermined point j and and are both calculated using equation (15). From equation (16), one can see that if the medium is isotropic, the group velocity is not dependent upon the ray direction rij 0 and equation (16) reduces to the standard SPM. The modified formula (16) may be considered as a general form of SPM and it is applicable for composite media, which may involve acoustic as well as elastic media, such as water, isotropic solids, general TI media and even more complicated anisotropic media. In tracing the first arrival for the undetermined points, one should always start with the point that has the minimum traveltime in the subset N j, that is,. Meanwhile, the node number of the incident point i* giving the minimum traveltime to the point j is recorded so as to give the coordinates of the raypath at this point. The gradually expanding (evolving) computation continues until the computed frontal points (corresponding to minimum times) reach all the receivers. (4) Raypath Recovery After the minimum traveltime computations have been computed for all the receivers, one may pick the traveltimes at the points coinciding with the receivers and the incident node numbers in a backwards fashion from the receiver to the source. The sequence of the incident node numbers gives the coordinates of the raypaths linking the source to the receivers. This procedure gives the firstarrival traveltimes and corresponding raypaths for each common shot gather. For the next shot, one repeats steps (2), (3), and (4) until all the sources have been considered. To accomplish the above procedure, we wrote computer code that has two levels of looping. The outer loop is over all the cells of the network. Each cell has the group velocity distribution given by equation (15). For this loop we always start with the node that has the minimum traveltime from the given source. The inner loop calculates the minimum traveltimes using equation (16) for all independent pairs of nodes (including the primary nodes and the secondary nodes) in the cell. After finishing the outer loop, we obtain the first arrival times and their corresponding raypaths for all the nodes of the network, and then we pick up the traveltimes and the raypaths for all the receivers. MODIFICATION FOR REFLECTED AND DIFFRACTED WAVES If there are interfaces in the complex (heterogeneous) anisotropic model which can give rise to reflections or diffractions, then we can represent each interface with a dense string of points, which are called the interface nodes. Adding the interface nodes to the previous nodes, we obtain a new network, which contains all the primary nodes, secondary nodes, and interface nodes. The node framework for most of the cells in the network is the same, except Exploration Geophysics (2006) Vol 37, No

7 for those cells containing the interface nodes, which are only a small part of the entire number of model cells. The traveltime τ of the reflected or diffracted wave from the interface may be calculated from, (17) where τ s ) and τ g ) are the first-arrival times from a source and a receiver to the interface point x int, respectively, and Ω int denotes the set of the interface nodes in the model range. Equation (17) shows that there are two possible ways to calculate τ. One approach, called the upward method, involves taking the interface node xint as a pseudo-source and applying the modified SPM to simultaneously obtain the traveltimes τ s ) and τ g ). This method is an efficient algorithm only when there are a few interface nodes or when we know where the true reflection point is on the interface. Otherwise, in order to find the correct τ, the upward processing has to be repeated for all the points along the interface and is consequently expensive in computer time. The other approach, called the downward method, calculates τ s ) and τ g ) individually by taking in turn as the computational source, first the true source and then the receiver, and successively applying the modified SPM to reach all the interface nodes. Actually, τ s ) is already calculated as part of the normal process of SPM (first-arrivals). The additional computational work is to calculate τ g ) for all the receivers. In order to compute τ g ), we repeat the modified SPM with the receivers as pseudo-sources and then find the traveltimes and the raypaths of the reflected or diffracted waves in terms of equation (17). It should be noted, however, that the traveltime of the reflected or diffracted wave from an interface is not dependent on the group velocity distribution below the interface. This means that we may truncate the model to include only those nodes above the interface when computing τ s ) and τ g ), thus saving much computer time. On the other hand, if a cell contains interface nodes, we need to change the group velocities below the interface with the values above the interface in equation (15) in order to correctly compute τ s ) or τ g ). Due to the relatively small number of cells involving interface nodes, computing τ s ) or τ g ) for a dense interface node string does not significantly increase the computer time over a sparse interface node string. This fact shows that the downward method allows us to use dense nodes for the interface and obtain satisfactory levels of accuracy for the reflected or diffracted waves. A similar secondary node density may be used on the interface as for the normal cells. Furthermore, the downward method can be used to easily compute the traveltimes and the raypaths for converted waves, such as the qp-qsv wave or the qsv-qp wave. For example, the qp-qsv wave is calculated simply by using the qp-wave group velocity in equation (16) for τ s ) and the qsv-wave group velocity for τ g ). NUMERICAL EXPERIMENTS The initial experiments were carried out for a homogenous, transversely isotropic medium having two (separate cases) orientation angles (θ 0 = 0, 45 ) of the symmetry axis and whose group velocities are shown in Figure 1. The wavefronts in such homogeneous models can be calculated exactly using equation (11), by exploiting the equivalence between the distance-adjusted polar diagram of the group velocity vector and the wavefront at a certain time. The analytic solutions are very helpful to check the accuracy of the presented method. Figure 2 presents the traveltime results for these experiments. The left and right panels are for a vertical (θ 0 = 0 ) and a dipping symmetry axis (θ 0 = 45 ) respectively. They show the first-arrival traveltime contours for the three wave modes (qp, qsv, qsh). The shot is located at the centre of the surface line (0,0), the cell size of the network is 50 m (in total 5000 cells), and a secondary node density of 7 was applied to the computations. The red dotted lines in the diagrams are the analytic solutions and the white lines are the numerical results obtained by the extended SPM. From these results, one can see that the numerical results are consistent with the analytic solutions (maximum relative error < 0.71%), except for the cusps of the qsv-wave. Unfortunately, in the numerical results we actually lose the cusps for the qsv wavefronts, because the SPM (equation (16)) cannot handle multiple values of the group velocity in a single ray direction and it only picks the minimum from the triplications of the group velocity of the qsv-wave, as shown in Figure 1. However, the remaining parts of the wavefront, on either side of the cusps, are recovered very well. These results clearly show that each wave mode has its own wavefront due to differences in the group velocity. The qp- and qsv-wavefronts change with the orientation direction of the symmetry axis of the medium, and the qsh-wavefronts always appear as circles because of the constant group velocity for this particular mode in this particular rock (set of elastic constants) see Figure 1. As mentioned above, the cell size and the secondary node density are the two controlling parameters of the SPM. In order to investigate the influence of these two parameters on the accuracy and the efficiency of the method, we repeated the above experiments with different secondary node densities (3, 5, 7, 9, 11 points) and cell sizes (10 m, 25 m, 50 m, 100 m), and calculated the relative errors of the numerical traveltimes by comparison with the exact solutions, as well as recording the CPU times on a Pentium 4, 3.2 GHz PC for all these experiments. Results are plotted in Figures 3 and 4. The left panel of Figure 3 gives the maximum relative errors for a constant cell size (50 m) but for different secondary node densities. One can see that by increasing the secondary node density, greater accuracy is achieved. When the secondary node density exceeds 5, satisfactory results (relative errors < 1.3%) are obtained for all three wave modes. But, as the node density increases, so does the computation time (see Figure 4a). Figure 4a indicates that a secondary node density of between 5 to 9 points is a good compromise between high accuracy and efficient computational effort. One finds that even with a relatively large secondary node density (>5 points), the method may give different computational accuracies for the three wave modes. This seems to be related to the complexity of the plots of group velocity versus ray angle. The qsh-wave has the simplest group velocity versus ray angle relationship of all three modes, being constant, and as a result yields the highest accuracy for the traveltimes. The qsv-wave, which has the triplication of the group velocity, does not produce results as accurate as the qp- or qsh-wave. The complexity of the group velocity possibly affects the interpolation in equation (15). The right panel of Figure 3 shows the maximum relative errors for a fixed secondary node density ( = 7 points) but for four different cell sizes. These diagrams show that when an appropriate secondary node density is chosen the maximum relative error does not change much with cell size because we are dealing with a simple homogeneous model. In general, an appropriate cell size should be the one that adequately describes the main features of the heterogeneous model structure but it should be relatively large for reasons of computational efficiency. Figure 4b gives the CPU time for four different cell sizes with a constant secondary node density (7 points). One can see that employing a small cell size with a secondary node density of 7 points significantly increases the computer time but leads to almost no improvement in the accuracy (see the right panel of Figure 3). Consequently, an appropriate cell size, with either 5 to 7 points for the secondary node density, is a suitable choice for modelling. 156 Exploration Geophysics (2006) Vol 37, No. 2

8 Left Running Heading Zhou and Greenhalgh Raypath and traveltime Right Running computations Heading Fig. 7. Raypaths and traveltimes for the reflected qp-wave in a three-layered anisotropic model: (a) the three layers have vertical symmetry axes (0 ); (b) the top two layers have dipping symmetry axes of 45 ; (c) the top two layers have horizontal symmetry axes (90 ). Fig. 8. Raypaths and traveltimes for the reflected qsv-wave in a three-layered anisotropic model: (a) the three layers have vertical symmetry axes (0 ); (b) the top two layers have dipping symmetry axes of 45 ; (c) the top two layers have horizontal symmetry axes (90 ). Fig. 9. Raypaths and traveltimes for the reflected qsh-wave in a three-layered anisotropic model: (a) the three layers have vertical symmetry axes (0 ); (b) the top two layers have dipping symmetry axes of 45 ; (c) the top two layers have horizontal symmetry axes (90 ). Exploration Geophysics (2006) Vol 37, No

9 In order to test the capability of the method for ray tracing in heterogeneous, anisotropic media, we used a three-layered model, depicted in the upper row in Figure 5. Rüger and Alkhalifah (1996) used this model for presenting their method to calculate the traveltimes of elastic waves in VTI media (θ 0 = 0 ). The model involves three anisotropic shale rock units with a dipping interface between the second and third layers. The elastic moduli for each rock unit, expressed in terms of the equivalent Thomsen anisotropic parameters (α, β, δ, γ, ε), are given in Thomsen s (1986) paper. For comparison of the computational results, we conducted the experiments with different orientation directions for the symmetry axes of the shale units in this model, i.e., we sequentially used the three dips (θ 0 = 0, 45, 90 ) of the symmetry axes for the first and second shale layers and calculated the traveltimes for the three wave modes (qp, qsv, qsh). Figure 6 shows the group velocities for the three layers having vertical symmetry axes. From this figure one sees that the qsv-wave in the second layer has a small triplication (singular directions or cusps - Figure 6b). The three panels indexed a), b), and c) in Figures 5 give the first arrival traveltimes (head wave or refracted wave) and the raypaths in the model for the three separate orientation directions of the symmetry axis. From these results, one can observe the variation of the raypaths and the first arrival times with the changing axes of symmetry, i.e., the refraction raypaths are symmetric about the midpoint of the shot-geophone pair in the vertical and horizontal orientations (see Figure 5a and 5c), because of the symmetric property of the group velocity about the ray angle of 90 (see Figure 5). When the symmetry axes of the top two layers dip at 45, the raypaths of the refracted waves become asymmetric because of the non-symmetrical property of the group velocity in this case (rotating the diagrams (a) and (b) in Figure 6). One also finds that there is no critical refraction evident from the second interface because the group velocity contrasts are small (see Figure 6) and the first arrival from this interface only occurs at distances considerably larger than those depicted. The refracted qsv-wave is the slowest of all three wave modes in the models. All these plots are consistent with the group velocities shown in Figure 6. Figures 7, 8, and 9 depict the raypaths (upper row) and the traveltimes (bottom row) of the reflected or diffracted qp-, qsvand qsh-waves from the three interfaces. The different figures correspond to the different orientation angles of the symmetry axes of the top two layers. Comparing results we see how the raypaths and the traveltimes change for all these modes. Figure 10 gives the raypaths (Figure 10a) of the converted waves (qp-qsv or qsv-qp) and the traveltimes (coloured lines in Figure 10b) obtained by the extended SPM. It can be compared with the synthetic seismograms (background to Figure 10b) calculated by Rüger and Alkhalifah (1996). The raypath diagram clearly shows that the reflection angle on any interface is not equal to the incident angle due to the wave mode conversion. Figure 10b shows that our results for the reflected and converted waves match very well with the seismograms given by Rüger and Alkhalifah (1996). This consistency shows the capability of the presented method for application to a general TI medium. CONCLUSIONS We have presented a method that can be employed to calculate the traveltimes and raypaths of the three wave modes (qp, qsv, qsh) in 2D heterogeneous, transversely isotropic media having dipping symmetry axes. This method is simple, robust, and capable of simulating the kinematics of refracted, reflected, diffracted and converted waves in a complex anisotropic model. This method is an extension of the shortest-path approach, which can also be used for 3D situations, and is applicable to composite Fig. 10. Raypaths of the converted waves (a), synthetic seismograms (background in b) calculated by Rüger and Alkhalifah (1996) and the numerical traveltimes (coloured lines in b) obtained by the presented method. PP1, PP2 and PP3 stand for the reflected qp waves from the first, second and third interfaces; PS1 and PS2 denote the reflected qpqsv waves from the first and second interfaces; SP1 and SP2 represent the reflected qsv-qp waves from the first and second interfaces; SS1 and SS2 represent the reflected qsv waves from the first and second interfaces. geological models. These may involve acoustic, isotropic, or anisotropic media. The cell size and the secondary node density are the key controlling parameters for computational accuracy and efficiency. An appropriate cell size should give a good description of the main features of the model structure and be relatively large for reasons of computational efficiency. A secondary node density somewhere between 5 and 9 points will yield satisfactory results with reasonable cost of computer time for general applications. The only real drawback of the extended SPM is that it cannot handle the triplications and cuspidal effects sometimes associated with the qsv wave. The numerical examples show that the orientation of the symmetry axis of a transversely isotropic medium is an important factor that may significantly affect the kinematic properties of all the wave modes (qp, qsv, qsh). This fact should be taken into account in practical applications. ACKNOWLEDGMENTS This research was supported by a grant from the Australian Research Council. 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