Rotation parameters for model building and stable parameter inversion in orthorhombic media Cintia Lapilli* and Paul J. Fowler, WesternGeco.

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1 otation parameters for model building and stable parameter inversion in orthorhombic media Cintia Lapilli* and Paul J Fowler, WesternGeco Summary Symmetry groups commonly used to describe seismic anisotropy include transverse isotropy, orthorhombic, monoclinic, and triclinic For all but the last, the choice of particular orientations of the coordinate system can substantially simplify parameterization of the anisotropy Choosing a coordinate system requires defining a rotation in three-dimensional space relative to fixed world coordinates We discuss here two major families of rotation parameterizations: Euler angles and axis/angle quaternion representations Either method can work well for forward modeling However, for traveltime tomography or full-waveform inversion, the inverse problem formulated in Euler angles can become ill-posed because very different choices of angle parameters can yield nearly identical data In the worst case, there can be a complete loss of a degree of freedom, making it difficult to invert for important model parameters such as fracture orientation Quaternions provide an alternative representation for 3D rotations that avoid these inversion problems Quaternions also provide efficient and wellbehaved interpolation of rotation angles, as well as differentials of data with respect to rotation parameters Introduction Prospective sedimentary basins are usually seismically anisotropic Accounting for anisotropy properly is essential for good seismic imaging and inversion Symmetry groups commonly used to describe seismic anisotropy include transverse isotropy, orthorhombic, monoclinic, and triclinic Transversely isotropic media have a rotationally invariant symmetry axis, so one must specify the proper orientation of this axis in space Orthorhombic media have three orthogonal symmetry planes, so one wants to choose coordinates axes aligned with the intersections of these symmetry planes Monoclinic media have only a single symmetry plane, so one wants to choose a coordinate system with one axis normal to this plane, but the particular orientation of the other two axes within the symmetry plane still can simplify the parameterization (Helbig, 1994) Choosing a coordinate system requires defining a rotation in three-dimensional space relative to fixed world coordinates 3D rotations form the noncommutative group 3 3 SO(3), defined as all linear maps that preserve distance and orientation Elements of this group can be represented by 3x3 orthogonal matrices with determinant +1 Composition of rotations then corresponds to multiplication of these matrices otation matrices have nine elements, fully specified using only three independent angle parameters and corresponding eigenvectors However, there are many different ways to choose these three parameters and axes of rotation We discuss here two major families of rotation parameterizations: Euler angles and axis/angle representations For the latter, we derive quaternion representations purely from geometric considerations and a non-singular continuous map in terms of Euler angles Quaternions have a number of attractive properties, and they resolve the ambiguity present in the representation by Euler angles when there is a loss of a rotational degree of freedom, which makes inversion for rotation parameters become ill-posed 3D rotations using Euler angles Euler (1776) showed that any 3D rotation could be constructed by three successive rotations, each around preestablished axes of rotation This representation is not unique, because there are at least 1 such combinations of rotations that yield a given 3D rotation Whichever rotation sequence is chosen, one can then parameterize the resulting full rotation by three angles (φ, θ, ψ), known commonly as Euler angles We express the orientations of the principal axes with respect to the global coordinate system (x, y, z) The first rotation then yields new coordinate axes (x, y, z ), and so forth For tilted transverse isotropy (TTI), we only need to specify two rotations, because the rotational anisotropic symmetry implies that the third elemental rotation has no effect on the wave propagation Fletcher et al (8) used two cascaded rotations using Euler angles (φ, θ) to rotate around the vertical (z) and rotated y (y ) axes, respectively In this case, θ measures the angle between the rotated vertical axis (z ) with respect to the fixed vertical axis (z), increasing positive from z to x, and φ measures the angle between the rotated x axis (x ) and the fixed horizontal x axis, increasing positively from x to y Figures 1-a and 1-b illustrate these two rotations Tilted orthorhombic (TO) media require the specification of all three Euler angles and rotation axes Here, we want to specify not just a tilt axis as with TTI, but also the choice of a preferred direction within the plane perpendicular to the tilt axis This latter is usually taken to define the orientation of the fast velocity direction, or the dominant fracture direction Zhang and Zhang (11) suggested using a rotation (as for the TTI media) given by (φ, θ) SEG Houston 13 Annual Meeting Page 4656

2 otations in orthorhombic media around axes (z, y ) (Figures 1-a and 1-b), followed by a third rotation by an angle ψ around the direction z (Figure 1-c) These rotations in matrix representation are given by: x y 1 = cos sin ; ( α ) ( α ) ( α ) sin ( α ) cos( α ) cos( α ) sin( α ) ( α ) sin( α ) cos( α ) cos( α ) sin ( α ) ( α ) = ( α ) ( α ) = 1 ; z sin cos, 1 giving the cumulative rotation matrix zy z cos( ψ ) cos( θ ) cos( φ ) sin ( φ ) sin( ψ ) ' '' ( φ, θ, ψ ) = sin( ψ ) cos( θ ) cos( φ ) sin ( φ ) cos( ψ ) sin ( θ ) cos( φ ) ( ψ ) ( θ ) ( φ ) + ( φ ) ( ψ ) ( θ ) ( ψ ) ( ψ ) ( θ ) ( φ ) ( φ ) ( ψ ) ( ψ ) ( θ ) sin ( θ ) sin( φ ) cos( θ ) cos cos sin cos sin sin cos sin cos sin + cos cos sin sin (1) () It is, in general, difficult to predict the final orientation after the successive rotations It is also possible to generate a set of rotations in which one degree of freedom is lost; that is, where two rotations are specified around the same axis For specifying anisotropic rotations, the representation in terms of rotation matrices is ambiguous, which may not hamper forward modeling, but make inversion for the rotation angles as model parameters difficult or impossible 3D rotations using quaternions Euler also proved that any 3D rotation could be represented by an invariant direction vector, and a single rotation around that direction This axis-angle representation again requires a scalar rotation and a unit vector If one maps the rotation angle and three vector components into a normalized unit 4-vector, then, by applying the quaternion multiplication rules defined by Hamilton (1844), one obtains an alternative representation of computing successive 3D rotations Figure 1: Euler angles for a choice of rotations around the z, y, and z axes Arrows in dashed lines represent the rotated coordinate system and the solid arrows are the fixed global reference frame Panel a shows the definition of the angle φ, measured positive increasing from x to y; panel b shows the second rotation around the y axis by an angle θ defined positive increasing from the z to x Finally, panel c shows the third rotation in the plane perpendicular to z by an angle ψ measured positive from x to y SEG Houston 13 Annual Meeting Page 4657

3 otations in orthorhombic media The 4-element quaternion and rules for finding norms, inverses, and defining multiplication of two quaternions, can be written as (Shoemake, 1985) q q1 q = [ q, q1:3 ]; q q 3 (3) 1 [ q, q1:3 ] q = ; q = q + q1:3 q1:3 q [ q, 1:3 ] [ p, 1:3 ] [ q p q p q p p q q p ] q p = q p =, + + 1:3 1:3 1:3 1:3 1:3 1:3 Quaternions with unit norm rotate a vector in the threedimensional space r 1 = q q (4) 3 r r A matrix representation for the corresponding 3D rotation can be written as (Diebel, 6) 1 ; r q ( q) r q + q1 q q3 q1 q + qq 3 q1q 3 qq (5) q ( q) = q1 q qq3 q q1 + q q3 qq3 + qq1 q1 q3 + qq qq3 qq 1 q q1 q + q 3 Finally, the multiplication of quaternions in matrix form can be written as p p1 p p3 q p1 p p3 p q1 q p = p p3 p p1 q (6) p3 p p1 p q 3 Quaternions corresponding to a rotation by an angle α around the direction given by the unit vector q1:3 can be conveniently represented by α α q = cos,sin q 1:3 (7) otations of a given three-dimensional vector to any given desired direction thus can be represented by a simple quaternion multiplication An attractive property of quaternions is the simplicity of interpolating between rotations using the great circle arc on a unit sphere in four dimensions Euler angles can be interpolated as direction cosines treating each component independently, but may run into the gimbal lock problem caused by the loss of a rotational degree of freedom whenever two rotations are specified around the same axis (Shoemake, 1985) Figure : otation by quaternion multiplication The rotation is around the invariant axis v by an angle α orienting the initial unit vector u along the final unit direction w The corresponding quaternion leading to this transformation is derived in equation 3 Quaternions can be built without the use of trigonometric functions, derived from pure geometric considerations, as illustrated in Figure This work presents a method for this construction as follows: With the quaternion multiplication defined in equation 3, rotation of unit vector u to an new vector u in the direction of a unit vector w, is done by a quaternion multiplication with the definition of the components as 1 u w u = quq ; q = q, q1:3 q = ( 1 + u w), (8) ( u w) if u w 1 q1:3 = v such that v u =, v =1 if u w = 1 Note the quaternion in expression (8) is normalized It is also possible to have a representation of quaternions in terms of Euler angles A continuous map for Euler angles to quaternions without singularities is constructed for the choice of rotations around axis z, y, z as cascading q = q q q with ψ θ φ rotations ( ) z, y, z, ( φ ) ( θ ) ( ψ ) cos cos cos qz, φ = ; y, θ ; z, ψ, q = sin ( θ ) q = (9) sin ( φ ) sin ( ψ ) giving θ ψ + φ cos cos θ ψ φ sin sin = θ ψ φ sin cos θ ψ + φ cos sin q ( θ, φ, ψ ) (1) SEG Houston 13 Annual Meeting Page 4658

4 otations in orthorhombic media It is easy to verify that the quaternion rotation 1 represents the rotation by means of the transformation 5 This particular design of quaternion components in terms of Euler angles behaves smoothly for continuously varying angular fields Advantages and drawbacks Modeling and migration with a given anisotropic model can be successfully implemented with either quaternions or Euler angles However, inversion for rotation parameters works is better posed if formulated in terms of quaternions The numerical implementation of rotation matrices in terms of Euler angles is well known and straightforward The major weaknesses involve the non-intuitive final orientation as a composition of rotations and the ambiguity in the determination of the rotation axes and angles that generated a given rotation Let us see an example of such ambiguity Suppose that we have a rotation specified in terms of Euler angles and represented by the matrix in equation For simplicity, let us assume we have a tilted orthorhombic (TO) system with smoothly varying dip (θ ), including points with vertically oriented orthorhombic (VO) symmetry planes -corresponding to θ = (Figure 3) same vertical axis (z) Thus, the rotation is parameterized by a single variable (φ +ψ), and the resolution of φ and ψ in this context is underdetermined In this case, the first angle φ has no effect on the data, but the second angle may represent an important physical property such as fracture orientation, so the inability to unscramble the two angles is a serious problem for parameter inversion Quaternions, on the other hand, are not very commonly used in geophysics They do, however, have a very compact representation and the geometric interpretation is very intuitive They are unambiguous in the description of rotations, and provide a well-defined and very efficient method for numerically computing concatenation or interpolation of rotations It is clear from our previous example, that rotations with quaternions for zero dip angle (θ ) are given by a quaternion of the form α cos q =, (1) α sin which specifies a rotation around the z axis by an angle α (see equation 7) The ambiguity does not exist in this case because we invert for a rotation around a single axis, and both parameters are well defined in all cases Conclusions Figure 3: Smoothly varying orthorhombic orientations along a surface The illustrated Cartesian axes align along the orthorhombic symmetry planes The vertically oriented orthorhombic (VO) symmetry planes correspond to the point with θ = We described two of the major choices for representing orientations of principal axes of symmetry groups under rotations in three dimensions We first reviewed a particular choice of Euler rotation angles and directions, and then showed an alternative representation using quaternions Quaternion representations of 3D rotations offer a number of computational advantages They can be interpolated without suffering from ambiguities and do not require cumbersome constructions of rotations around arbitrary axes, which is particularly appealing in anisotropic media with lower symmetries They also make the problem of inverting for rotation parameters wellposed, without the instability and ambiguity present with Euler angle representations It is clear that rotations for points with θ = are represented by cos( ψ + φ ) sin ( ψ + φ ) zy ' z '' ( φ, θ, ψ ) = sin ( ψ + φ ) cos ( ψ + φ ) (11) 1 There is a loss of a degree of freedom due to the fact that two rotations (around y and z ) are specified around the SEG Houston 13 Annual Meeting Page 4659

5 EDITED EFEENCES Note: This reference list is a copy-edited version of the reference list submitted by the author eference lists for the 13 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web EFEENCES Diebel, J, 6, epresenting attitude: Euler angles, unit quaternions, and rotation vectors: eport, Stanford University Euler, L, 1776, Novi Commentarii academiae scientiarum: Petropolitanae,, Fletcher, P, X Du, and P J Fowler, 8, everse time migration in tilted transversely isotropic (TTI) media: Geophysics, 74, no 6, WCA179 WCA187, Hamilton, S W, 1844, On quaternions; or on a new system of imaginaries in algebra: Philosophical Magazine, XXV, 1 13 Helbig, K, 1994, Foundations of elastic anisotropy for exploration seismic: Pergamon Press Shoemake, K, 1985, Animating rotation with quaternion curves: Computer Graphics, 19, no 3, 45 54, Zhang, H, and Y Zhang, 11 everse time migration in vertical and tilted orthorhombic media: 81 st Annual International Meeting, SEG, Expanded Abstracts, SEG Houston 13 Annual Meeting Page 466