RAY VELOCITY DERIVATIVES IN ANISOTROPIC ELASTIC MEDIA.
PART II – POLAR ANISOTROPY
Igor Ravve (corresponding author) and Zvi Koren, Emerson
[email protected] , [email protected]
ABSTRACT
In Part I of this study, we obtained the ray (group) velocity gradients and Hessians with respect to the ray locations, directions and the anisotropic model parameters, at nodal points along ray trajectories, considering general anisotropic (triclinic) media and both, quasi-compressional and quasi-shear waves. Ray velocity derivatives for anisotropic media with higher symmetries were considered particular cases of general anisotropy. In this part, Part II, we follow the computational workflow presented in Part I, formulating the ray velocity derivatives directly for polar anisotropic
(transverse isotropy with tilted axis of symmetry, TTI) media for the coupled qP and qSV waves and for SH waves. The acoustic approximation for qP waves is considered a special case. The medium properties, normally specified at regular three-dimensional fine grid points, are the five material parameters: the axial compressional and shear velocities and the three Thomsen parameters, and two geometric parameters: the polar angles defining the local direction of the medium symmetry axis. All the parameters are assumed spatially (smoothly) varying, where their gradients and Hessians can be reliably computed. Two case examples are considered; the first represents compacted shale/sand rocks (with positive anellipticity) and the second, unconsolidated sand rocks with strong negative anellipticity (manifesting a qSV triplication). The ray velocity derivatives obtained in this part are first tested by comparing them with the corresponding
Page 1 of 146 numerical (finite difference) derivatives. Additionally, we show that exactly the same results (ray velocity derivatives) can be obtained if we transform the given polar anisotropic model parameters
(five material and two geometric) into the twenty-one stiffness tensor components of a general anisotropic (triclinic) medium, and apply the theory derived in Part I. Since in many practical wave/ray-based applications in polar anisotropic media only the spatial derivatives of the axial compressional velocity are taken into account, we analyze the effect (sensitivity) of the spatial derivatives of the other parameters on the ray velocity and its derivatives (which, in turn, define the corresponding traveltime derivatives along the ray).
Keywords: Seismic anisotropy, Wave propagation
1. INTRODUCTION
Considering general anisotropic elastic media parametrized with the spatially varying stiffness properties and all wave types, in Part I of this study we presented a generic approach for explicitly
(analytically) formulating the first and second partial derivatives of the ray velocity magnitude, vray Cxr , , with respect to (wrt) the ray location vector x , direction vector r , and the components of the stiffness matrices Cx(). These gradients and Hessians are the corner stones of the sensitivity kernels of traveltimes and amplitudes of waves/rays in seismic modeling and inversion methods. In this part of the study, we apply the theory presented in Part I explicitly for polar anisotropic (transverse isotropy with tilted axis of symmetry, TTI) media, which considerably simplifies the implementations and improves the computational performance. Note that a similar approach can be further applied to lower anisotropic symmetry materials such as
Page 2 of 146 tilted orthorhombic and tilted monoclinic, but these topics are beyond the scope of this paper, to be further developed in our future research.
Starting with the kinematic and dynamic variational formulations for stationary rays, in Part I we introduced the arclength-related Lagrangian L C x, r , its corresponding (e.g., through the
Legendre transform) arclength-related Hamiltonian H C x p, , and an (auxiliary) time-scaled
(reference) Hamiltonian H C x, p , which are used in chain rule operations for computing the above mentioned gradients and Hessians of the ray velocity magnitudes. The slowness vector p implicitly depends on the ray coordinates x and the ray direction r . Hence, the workflow for computing the ray velocity derivatives includes four basic steps: a) performing the slowness inversion, implicitly stated as ppCxr (), , b) computing the derivatives of the reference
Hamiltonian H xp, , c) computing the derivatives of the arclength-related Hamiltonian
H xp, , and d) computing the ray velocity derivatives.
All four steps have been elaborated for general anisotropic media (up to triclinic) in Part I of this study, where the higher symmetries have been considered particular cases. In this part, Part II, we propose an approach, where we directly adjust the equations for the specific anisotropic symmetry.
The beauty of the proposed approach is that the last two steps in the above-mentioned workflow are generic and remain the same in both approaches, as the computational formulae related to these steps are independent of any particular anisotropic symmetry. Only the first two steps, the slowness inversion and the computation of the derivatives of the reference Hamiltonian, should be adjusted to the specific anisotropic medium type under consideration, and this adjustment essentially improves the implementation performance.
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There are eight types of anisotropic symmetries in continuous elastic materials (e.g., Cowin and
Mehrabadi, 1985; Bona et al., 2004, 2007, 2008), including the isotropic and triclinic media. In this study we consider polar anisotropic (TTI) model which is commonly used in seismic methods for approximating wave propagation in realistic structural sedimentary (e.g., shale and sand; compacted or unconsolidated) layers.
The slowness inversion (i.e., obtaining the set of feasible slowness vectors from a given unique ray direction vector) in anisotropic media has been studied by several researchers (e.g., Musgrave,
1954a, 1954b, 1970; Fedorov, 1968; Helbig, 1994; Vavryčuk, 2006, Grechka, 2017; Zhang &
Zhou, 2018). A brief overview of these studies is provided in the first part of Appendix A. Ravve and Koren (2021b) proposed a new efficient approach to the slowness inversion in polar anisotropy based on the inherent property of this symmetry class, where the three vectors: the phase velocity
(or the slowness), the ray velocity and the medium axis of symmetry are coplanar. This leads to a system of two bivariate polynomials of degrees 3 and 4, and under the commonly accepted assumption that the angle between the slowness and the ray direction vectors cannot exceed 90o , the system is then reduced to a univariate polynomial of degree six. The method is briefly described in the second part of Appendix A.
To the best of our knowledge, unlike the case of computing the ray velocity (first) derivatives wrt the model parameters (e.g., the Fréchet first derivatives used in seismic tomography), there are very few studies related to the (first and second) spatial and directional derivatives of the ray velocity magnitude of seismic body waves (e.g., for two-point ray bending methods), and in none of them the derivatives are presented directly in the ray (group) angle domain. These types of derivatives are required, for example, for the implementation of the Eigenray method (a special ray bending method) recently presented by Koren and Ravve (2021) and Ravve and Koren (2021a).
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For example, in the context of ray-based tomography in polar anisotropic media, Zhou and
Greenhalgh (2008) suggested a method to compute the ray velocity derivatives wrt the medium properties and the polar angles of the symmetry axis. To bypass the challenges of directly computing the ray velocity derivatives wrt the model parameters, they use the relation of the ray velocity components to the phase velocity magnitude vphs (that depends on the phase angle phs between the slowness vector and the medium symmetry axis), and its derivative wrt the phase
ax nr angle, vphs dv phs/ d phs . The axial and normal components of the ray velocity, vvrayand ray , respectively, are given by (e.g., Crampin, 1981; Tsvankin, 2012),
nr vvvrayphsphsphsphssincos, vvv 22 . (1.1) ax rayphsphs vvvrayphsphsphsphscossin,
The phase velocity magnitude, in turn, depends on the medium properties and the phase angle, and may be formulated for the coupled qP and qSV waves as (e.g., Tsvankin, 2012),
2 vphs phs f 1 2 1sin2 sin2sin222 2 ff , (1.2) 2 phsphsphs vP 22
22 where vP is the axial compressional wave velocity, f1/ vSP v is the shear wave velocity factor, with vS the axial shear wave velocity, and and the Thomsen polar anisotropic parameters. Note that the ( ) upper sign in equation 1.2 is related to qP waves and the ( ) lower sign to qSV waves.
In principle, one can use equations 1.1 and 1.2 to further compute the ray velocity derivatives wrt the model parameters, via the phase velocity and its phase-angle derivative, applying a chain rule.
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However, in this case, the ray velocity derivatives become functions of the phase (slowness) direction rather than the required ray direction. Our approach is different; it includes several important new items (aspects) with definite advantages. First, the derivatives of the ray velocity are computed directly in the ray angle or ray direction domain; second, we provide the first and second derivatives of the ray velocity magnitude not only wrt the model properties m , but also wrt the ray locations and directions xrand , respectively; third, we provide a complete set of the velocity gradients and Hessians, including the mixed location/direction Hessian, which are essential, for example, for kinematic and dynamic two-point ray bending methods (e.g., the
Eigenray method) and for ray-based tomography. Finally, all the derivatives are formulated in the global frame (Cartesian coordinate system) without the need to apply any global-to-local or local- to-global rotations.
In summary, given the anisotropic model parameters mx() and a ray direction unit vector r , the following objects are computed,
p ,vray slowness vector and ray velocity magnitude
xvray, r v ray , m v ray gradients (1.3) T . x xvray, r r v ray , x r v ray r x v ray , m m v ray Hessians mixed
We start with the derivation (definition) of the reference Hamiltonians for the coupled qP-qSV waves, and for SH waves. We then discuss the sign of the Hamiltonian for each wave type; this may be considered as a continuation of the sign-rule discussion for general anisotropy provided in
Part I. The next sections are devoted to the gradient and Hessian of the reference Hamiltonian wrt
the model properties, HHmand mm , and then to the spatial and slowness gradients and Hessians
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of the reference Hamiltonian, HHHHxpxxpp,,, , including the mixed Hessian, Hpx . We then formulate the relations between these derivatives and the corresponding derivatives of the arclength-related Hamiltonian, leading eventually to the required derivatives of the ray velocity magnitude.
Any high symmetry medium (i.e., any anisotropy excluding triclinic media), can be defined in its
“crystal” reference frame (a term suggested by Pšenčík and Farra, 2017; referred to also as a local frame). Generally, this frame is defined by the three Euler angles: zenith, azimuth and spin, or (in some cases preferably) by the quaternion rotations: Given the three Euler angles, one can compute the rotational matrix, or an equivalent single rotation angle about the tilted axis, which, in turn, yields the quaternion components. The inverse solution provides the Euler angles given the quaternion components. Overall, the Euler rotations result in trigonometric equations, which can lead to a “gimbal” lock, where the quaternion rotations result in algebraic equations, with no such issues. However, the polar anisotropy reference frame is insensitive to the spin rotation (about the crystal symmetry axis), and therefore, only the two angles, zenith and azimuth, suffice. Note that there are different kinds of material parameterizations for a polar anisotropic material in its crystal frame. In addition to the five independent stiffness tensor components, the most known and widely used are the Thomsen (1986) parameterization, Alkhalifah (1998), and the weak anisotropy (WA) parameterization (e.g., Farra and Pšenčík, 2016; Farra et al., 2016).
In this study, the polar anisotropic material parameters include the axial compressional velocity
22 vP , parameter f1/ vSP v (e.g., Tsvankin, 2012) , where vS is the axial shear velocity, and the three Thomsen parameters, ,, . Note that out of the five material properties, only four parameters, vfP ,,, are needed to describe the coupled qP and qSV waves, and three parameters,
Page 7 of 146 vP ,, (with f 1) are needed for the acoustic approximation of qP waves. For SH waves, it is suitable to use the axial shear velocity vS directly, as a single parameter, rather than as a combination of two parameters, vSP v f1 . Therefore, only two material properties, vS a nd
, are needed to describe SH waves.
Hence, in our study, the properties of the polar anisotropic (TTI) model (combined into array m ) are divided into two groups: Five material (elastic) properties of the model (combined into array c , which includes in addition to vvPSa n d , the three Thomsen parameters ,, ), and two geometric properties: the two angles, a x a xa n d (combined into array a ), that define the direction of the symmetry axis. It is convenient to present the symmetry axis direction as a unit- length vector k ; obviously, vector k of length 3 and array a of length 2 are dependent. Vector k is an auxiliary one, and its introduction simplifies the derivations. Unlike arrays cam,and
(where mca and symbol means concatenation of the two arrays; in this case with lengths nnca5and2 ), which are just collections of scalar values, the symmetry axis direction k is a real physical vector that allows vector operations. Each of the spatially varying model parameters of array m is normally given on a 3D regular fine grid and it is assumed that their spatial distributions are smooth enough, such that their spatial gradients and Hessians can be reliably
(numerically) computed. We also need to compute the gradient and the Hessian of the reference
Hamiltonian wrt the axis of symmetry direction vector, HHkand kk ; the two mixed gradients
related to the symmetry axis direction are also required, HHmkpkand . Note that a derivative wrt vector k means a set of partial derivatives wrt its three Cartesian components.
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For completeness, we also consider the particular (and widely used) case of acoustic approximation (AC) for qP waves and provide the corresponding ray velocity derivatives for this case. We show that although the accuracy of the AC ray velocity magnitude is very high (even in relatively strong polar anisotropic media), the accuracy of the above-mentioned derivatives (in particular, the second derivatives) based on the AC may become questionable in some extreme cases.
Remark: Since in this part of the study we work mostly with the reference Hamiltonian
H m x p, , the word “reference” will be omitted, with a few exceptions when both the reference and the arclength-related Hamiltonians are considered in the same paragraph.
The final part of this study is dedicated to the actual (analytical and numerical) computations of the above-mentioned ray velocity derivatives. We use two real case examples of a point along a ray traveling in an inhomogeneous polar anisotropic model (where in both examples, the material parameters and the symmetry axis direction angles vary smoothly in space), representing compacted shale/sand sediments (referred to as model 1) and unconsolidated shallow sand rocks
(model 2), where in the latter case, for the given ray direction, the slowness inversion yields a shear wave triplication. The ray velocity derivatives are obtained for the coupled qP and qSV waves (quasi compressional and quasi-shear waves, respectively, polarized in the axial symmetry plane), and independently for the pure shear waves polarized in the direction normal to the axial symmetry plane (SH; in the isotropic plane). Additionally, we formulate and compute the acoustic approximation for the qP ray velocity derivatives and discuss their accuracy (validity). The resulted analytic derivatives for all cases are first compared with the corresponding numerical
(finite-difference) derivatives. To demonstrate the compatibility of the two approaches (of Part I
Page 9 of 146 and II), we show that the exact analytic derivatives can be also obtained when considering the polar anisotropic model as a particular case of general anisotropy. This requires first converting the parameters of polar anisotropy with the tilted symmetry axis to the twenty-one stiffness components of general (triclinic) anisotropy, and then applying the generic formulae of Part I.
Note that in many practical wave/ray-based applications in polar anisotropic (TTI) media, the spatial derivatives of the Thomsen parameters, the ratio between the axial shear and compressional wave velocities (parameter f ), and/or those of the symmetry axis direction angles are ignored, and only the derivatives of the axial compressional velocity are accounted for. In our test, we analyze the relative contribution of the model parameters’ spatial derivatives to the ray velocity gradients and Hessians.
2. HAMILTONIANS OF POLAR ANISOTROPY FOR ALL WAVE TYPES
The crystal frame xxx123 is a local coordinate frame rotated wrt the global frame x y z by the polar angles of the symmetry axis: zenith ax and azimuth ax . In the crystal frame, where the local vertical axis is the medium axis of symmetry, the stiffness matrix of the polar anisotropic elastic material has five independent components,
CCC11 12 13 CCC12 11 13 CCC13 13 33 C ,2CCC12 11 66 . (2.1) C44 C 44 C66
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Without losing generality, assume that the propagation occurs (locally) in the xx13 plane, where p2 0 . With the density-normalized fourth-order stiffness tensor C , the Christoffel matrix reads,
CpCpCCp22 p 0 111443134413 22 ΓpCp 00CpCp661443 . (2.2) CCp pCpCp 0 22 134413441333
The Christoffel matrix splits into two blocks: a 22 block of coupled quasi-compressional (qP) and quasi-shear (qSV) waves, polarized in the “vertical” (axial) plane, and a scalar component (in the second row and column) for SH shear waves, polarized in the “horizontal” (normal; isotropic) plane. The vanishing reference Hamiltonian H can be defined from the Christoffel matrix Γ , whose components are listed in equation 2.2. According to this split, the Hamiltonian is factorized into two terms related to the coupled qP-qSV and to SH waves,
HHHdetΓI PSV SH 0 , (2.3) where,
4 4 2 2 2 HPSV CCpCCp11 44 1 33 44 3 CC 11 33 C 13 2 CCpp 13 44 1 3 (2.4) 22 C11 C 44 p 1 C 33 C 44 p 3 1 0 , and,
22 HCpCpSH 66 144 3 10 . (2.5)
In order to extend the validity of the reference Hamiltonians in equations 2.4 and 2.5 for any reference frame, we introduce the concept of the axial and the normal slowness components ppppax3nr1and , where ppaxnrand are physical scalar invariants, independent of any specific reference frame. For this, however, the crystal-frame stiffness matrix components Cij
Page 11 of 146 should be replaced by parameters which are also independent of the reference frame (physical invariants), like the Thomsen (1986) or Alkhalifah (1998) parameters. As indicated, the model properties’ array m in this study is represented as a concatenation of two arrays of lengths 5 and
2,
camcavfp ,and axax , (2.6) where the dependency of the Thomsen parameters on the stiffness tensor components and the inverse relationships for the stiffness components vs. the Thomsen parameters are,
a b C vC , 11 1 2 , P 33 2 vP vCC2 f 1, S 3344 C 2 13 f ff 21, vP C33 2 vP CC 1133 , C (2.7) 2C 33 1, 33 v2 22 P CCCC13443344 C , 44 1,f 2CCC333344 2 vP CC6644 C , 66 11 f 2. 2C44 2 vP
Equation set 2.7b has been obtained from 2.7a with the commonly used assumption CC13440
, which excludes anomalous polarizations in the axial plane (Helbig and Schoenberg 1987;
Schoenberg and Helbig 1997). This leads to the plus sign before the square root in the equation for
C13 .
Introduction of the Thomsen parameterization (equation 2.7b) into the Hamiltonians of PSV and
SH waves (equations 2.4 and 2.5, respectively), leads to,
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2 22 2 2 2 4 4 HPSVxp, 1 f pnr p ax 2 f p nr p nr p ax 2 f p nr v P (2.8) 2 f p2 p 2 2 p 2 v 2 1 0 , nr ax nr P and,
2222 HfpvfpvSHPPxp, 112110 nrax . (2.9)
For the SH-wave Hamiltonian, HSH , it is convenient to introduce the axial shear velocity, vS .
According to the definition of the shear velocity factor f (Tsvankin, 2012), vSP v f1 , and the SH Hamiltonian of equation 2.9 simplifies to,
2 2 2 2 HSHxp, 1 2 pnr v S p ax v S 1 0 . (2.10)
Thus, depends on two material properties only, vS and .
Remark: Although the Thomsen parameters at each grid point are physical scalars independent of the reference frame, still, they describe the medium properties in its crystal frame.
The Hamiltonians in equations 2.8 and 2.10 are arranged in coordinate-free notations. To compute the axial and normal slowness invariants, we introduce the unit-length axis direction vector k , whose components in the global frame are defined by the Euler direction angles: zenith ax and azimuth ax ,
k sinax cos ax sin ax sin ax cos ax . (2.11)
Hence, the slowness invariants, ppaxand nr , can be presented in terms of the scalar product of the physical vectors: the symmetry axis direction vector k and the slowness vector p ,
2 ppaxk p, nr p p k p . (2.12)
This leads to,
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Hfvfv x,12 pp p 2 44p pk pp p2 PSVPP (2.13) 2 2 fvp pk ppp 2 422210f , ppkvv p 2 2 PPP and,
22 Hvv xp, 1210 ppk kpp 22 . (2.14) SHSS
Equations 2.13 and 2.14 expand into,
2 442 HfvvPSVPPx,121p p p 21 ff k pp p (2.15) 4 4 2 2 2 212fvv k pp PPP 1210f , p k p v and,
222 HvvSHSSx,p 12102 pp kp , (2.16) where we recall that the material parameters of the medium c , and the symmetry axis direction k , are spatially varying. Equations 2.15 and 2.16 provide the polar anisotropic (TTI) Hamiltonians in the global frame, making it possible to obtain the ray velocity derivatives directly in the global frame as well.
3. SIGN OF THE HAMILTONIAN
The (reference) Hamiltonian is a vanishing function representing the Christoffel equation, and thus multiplying it by a positive or negative factor has no effect on the solution of this equation. For the inverse problem of finding the slowness vector, the sign of the Hamiltonian is inessential as well, because the governing equation of the inversion (along with the vanishing Hamiltonian) manifests the collinearity of the slowness gradient of the reference Hamiltonian and the ray
direction vector, Hp r 0 , which is equivalent to Hp r,0. Still, the two collinear
Page 14 of 146 vectors may prove to have the same or opposite directions, i.e., parameter , with the units of velocity, may be positive or negative.
However, the sign is important for the Hamiltonian derivatives. An improper sign (after scaling
the reference Hamiltonian H xp, to the arclength-related one, HHHHxp,/ pp equation 4.2 in Part I), yields Hp r instead of Hp r , which is an evidence of a wrong
Hamiltonian sign. In Appendix B, we analyze the sign of the Hamiltonians for all wave types
H HPSVSH,, H and conclude that,
HHHHHPPSVSVPSVP , , (3.1)
where H PSV is the Hamiltonian of the coupled qP-qSV waves listed in equation 2.15. The sign of
the Hamiltonian HSH in equation 2.16 is correct. Therefore, in this paper, the Hamiltonian derivatives are computed for qP and SH waves only, where for qSV waves, we consider
HH SVP , and the “nabla” symbol with no subscript means that this relation holds for all
types of first and second derivatives. Note, however, that the values obtained for HHPand SV are not just opposite numbers, they have different magnitudes because they refer to either qP or
the qSV slowness vectors introduced into HHPand SV , respectively.
4. HAMILTONIAN GRADIENT WRT THE MATERIAL PROPERTIES
The gradient of the (reference) Hamiltonians wrt the model parameters (that include both material and geometric counterparts) are written as,
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HHHHHHPPPPPP HnP,mc,26 , vfP axax (4.1) HHHHSHSHSHSH HnSH ,mc,24 , vS axax
where, H Hmca H . In this section, we only compute the derivatives with respect to the
material components, Hc , while in the next section, the derivatives are computed wrt the
geometric components Ha . The material gradient of the qP-qSV Hamiltonian can be obtained from equation 2.15, and the components of this gradient are,
HP 2 332 4 1 2 1 fvp p vPP 81 ff k p p p vP 423 8fv k p P 21 f 1 2 p p vPP 4 k p v , HP 2 442 12 pp vPP 22 k p p pv f (4.2) 4 4 2 2, k p vvPP p p H P 2 f k p22 p p k p v4 , P H 2 P 2 f p p k p2 v42p p k p2 p pvv 2 1 2 . P PP
The gradient of the SH Hamiltonian can be obtained from equation 2.16,
HHSH22 SH 2 2 1 2p p 2 k p vvPS , 2 p p k p . (4.3) vS
5. HAMILTONIAN GRADIENT WRT THE GEOMETRIC PROPERTIES
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To obtain the Hamiltonian gradient wrt the polar angles, we apply the chain rule for a multivariate function, multiplying the Hamiltonian gradient wrt the symmetry axis direction vector, and the gradient of the axis components wrt the two polar angles,
HHakak , (5.1)
where Hk is a vector of length 3, and ka is a matrix of dimension 32 containing the derivatives of the components of the symmetry axis direction vector k wrt its polar angles. The gradient of the qP-qSV Hamiltonian and that of the SH Hamiltonian wrt the axis direction are, respectively,
4 HPP,k x,p 41 f f k pp p pv (5.2) 3 42 8 f k ppk vv pPP p 4, and,
2 HvSHS,k xp, 4 kpp . (5.3)
Using the components of the axial direction vector k listed in equation 2.9, we obtain the gradient of this vector with respect to the symmetry axis direction angles,
coscossinsinaxaxaxax ka 3,2cossinsincos axaxaxax . (5.4) sin0ax
The first column of this matrix includes the derivatives of vector k wrt the zenith angle ax , and the second column – wrt the azimuth ax . The rows include the Cartesian components.
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Now we can obtain the gradients of the qP-qSV and SH Hamiltonians wrt all model properties
(both material and geometric), whose components are listed in equation 4.1,
HHHHHmcaaka , k , (5.5)
12nc 1121232nc 13
where recall that symbol means concatenation of two arrays (in this case, with lengths nc a n d 2
); the resulting array includes all the components of the model at a given point and has length nc 2, where nc 4 for qP and qSV waves, nc 3 for the acoustic approximation of qP waves, and nc 2 for SH waves.
6. SPATIAL GRADIENT OF THE HAMILTONIAN
The Hamiltonian depends on the spatial coordinates indirectly, implicitly through the medium properties mx(); therefore, its gradient obeys the chain rule for a multivariate function,
HHxx, p m m x where m x c x a x . (6.1) n 3 2 3 13 1nc 2 n c 2 3 n c 2 3 c
Vector Hnmc1,2 is a gradient of the Hamiltonian wrt all (material and geometric) model properties, obtained in equation 5.5; caxn c,3 and x 2,3 are the two matrices containing the spatial gradients of the physical and geometric properties, respectively. We assume that the spatial gradients of the model parameters,
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vxvxvxPPP/// 123 fxfxfx/// 123 cx 5,3 ///xxx123 , (6.2) ///xxx123 ///xxx123 and,
ax1ax2ax3///xxx ax 2,3 , (6.3) ax1ax2ax3///xxx
can be reliably computed. Recall that all the five parameters involved in matrix cx are never used simultaneously. As mentioned, two, three or four parameters out of the five are applied in each
case. The resulting spatial gradient of the Hamiltonian, Hx , is a vector of length 3.
7. HAMILTONIAN HESSIAN WRT THE MATERIAL PROPERTIES
It is convenient to split the Hamiltonian Hessian into four blocks,
HH H ccca , (7.1) mm HHacaa
ττ where the upper and lower diagonal blocks, HHccaaand , respectively, are symmetric and related to the material and geometric properties, respectively, while the off-diagonal blocks,
HnHnacccac2,and,2 , are mixed (i.e., related to both, material and geometric properties of the model); the mixed blocks are transposed of each other. Numbers in brackets indicate the dimensions of the matrices. In this section, we derive only the Hamiltonian Hessian wrt the
Page 19 of 146 material properties (the upper left block), while the other blocks are derived in the next two sections. For the qP-qSV Hamiltonian, the Hessian can be obtained from equation 4.2 (the unspecified components are zeros),
2H P 12 1 21241 fvffvp pk22 pp22 p 2 PP vP 422 242fvf 11 24,k pp pkP p 2 HP 2233 4 1 28 2p pk vv ppPP p vfP 4 3 82,k pp vv pPP 2 HP 223 8,fv p pk pk p P vP 2H 2 P 8 f p pk pp pk 22 ppvvv32 p 421, PPP vP 2H P 2, p pk pk p22 v4 f P 2H 2 P 2. p pk p 2 v4 f P (7.2)
For the SH Hamiltonian, the Hessian can be obtained from equation 4.3 (again, the unspecified components are zeros),
22HH SHSH2 1 22,4 p pk pp p k22 p v . (7.3) 2 P vS vS
8. HAMILTONIAN HESSIAN WRT THE GEOMETRIC PROPERTIES
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Recall that the Hamiltonian depends on the symmetry axis direction vector, and the latter, in turn, depends on its two polar angles, a x a xa n d . Applying the chain rule for the second derivative of a multivariate function, we obtain,
T HHHaaakkakaakkk . (8.1) 222332322 3313
The result is a 22 symmetric matrix – the Hamiltonian Hessian wrt the polar angles axand ax
. The formula (equation 8.1) is generic for any wave type, but the gradient, Hk , and the Hessian,
Hkk , of the Hamiltonian wrt the symmetry axis direction vector differ for the qP-qSV and the SH wave Hamiltonians. The gradients are listed in equations 5.2 and 5.3. These equations also make it possible to establish the two Hessians,
4 HPP,kk xp, 41 f f p ppp v (8.2) 2 4 2 24 fvv k ppppp PP 4, and,
2 HvSHS,kk xp, 4 pp . (8.3)
Matrix ka of dimensions 32 is given in equation 5.4. Three-dimensional array kaa ,322 represents a “Hessian of vector k wrt array a ”, where a axax ; in other words, k 2k/ a a aa ijk i j k . This array is symmetric wrt the last two indices. It can be computed from equation 2.9 that lists the components of the axis direction vector k ,
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sincoscossinaxaxaxax cossinsincosaxaxaxax sinsincoscosaxaxaxax kaa . (8.4) coscossinsinaxaxaxax cos0ax 00
9. MIXED HESSIAN OF THE HAMILTONIAN WRT THE MATERIAL AND
GEOMETRIC PROPERTIES
In this section, we derive the mixed Hessian of the Hamiltonian, Hna c c2, related to both, the material and geometric properties of the Hamiltonian; this block is shown in equation 7.1. We start
from equation 5.1, HHakak . The left-hand side of this equation is a vector of length 2, and
the right-hand side is a product of vector Hk of length 3 and matrix ka of dimensions 32 . This matrix is given in equation 5.4; as we see, its components are independent of the model parameters.
We apply a general algebraic law for arbitrary vector a and matrix B ,
aBBaaB T , (9.1) second-order tensor
and in our case, this leads to,
T HHacakck . (9.2) 23nncc23
Thus, we need to compute the components of matrix Hkc ,
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2H H xp, iH 1, 4 for , k , P (9.3) k cc ii iH 1, 3 forSH .
For the qP-qSV Hamiltonian, we apply equation 5.2 and obtain,
HP,k x,p 3 161 ff k p p p p vP vP 3 3 32,fv k p pkPP p 8 pv H xp, P,k 8 p pp kp k2 pp vv444, p p k p f P P (9.4) H xp, P,k 4 f p pp 2, kp2 k p v4 P H x,p P,k 4 1 22fvfvvpp 222 p kk2 p 1 p . PPP
For the SH Hamiltonian, we apply equation 5.3 and obtain,
HHSHSH,,kk2 8,4 k p pkvvSS p p . (9.5) vS
Each item on the left-hand side of equation sets 9.4 and 9.5 is a column of a matrix.
10. SPATIAL HESSIAN OF THE HAMILTONIAN
To compute the spatial Hessian of the Hamiltonian, we apply the chain rule in a similar way it has been applied in equation 6.1, but now for the second derivatives (Faà di Bruno, 1855, 1857), and adjusted for a multivariate function,
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T HHHxxxmmxmxxx, pmmm , (10.1)
33 32222312233nnnnnncccccc where,
mcamcaxxxxxxxxx, . (10.2) nn32333233 nncc23233 cc
Note that cx, a x and m x are two-dimensional arrays (spatial gradients of one-dimensional arrays). They are not second-order tensors, but collections of vectors. Similarly, camxxxxxx,and are three-dimensional arrays (spatial Hessians of one-dimensional arrays). They are not third-order tensors, but collections of second-order tensors. However, their multiplication rules are identical to those of the second- and third-order tensors.
Throughout this study, we assume that the spatially varying model parameters are smooth and well-conditioned, and their spatial derivatives: cacaxxxxxx,,, can be reliably computed. The
computation of the gradient Hm and the Hessian Hmm is carried out according to equations 5.5 and 7.1, respectively.
11. SLOWNESS GRADIENT OF THE HAMILTONIAN
Applying the slowness gradient operator p to the Hamiltonians of the qP-qSV and SH-waves listed in equations 2.15 and 2.16, respectively, we obtain the corresponding slowness gradients of the two Hamiltonians,
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4 HfvPP,p x,4 pp 121 p p 41 f f kppkpp2 p k v4 (11.1) P 3 4 2 2 8212,fvkpkpkp PPP k 1 f vv4 and,
22 HvvSHSS,p xp, 214 2pkpk . (11.2)
12. SLOWNESS HESSIAN OF THE HAMILTONIAN
Applying the slowness gradient operator p to the gradients of the Hamiltonians of the qP-qSV and SH-waves listed in equations 11.1 and 11.2, respectively, we obtain the corresponding
Hessians of the Hamiltonians,
4 HP,pp xp, 4 1 2 1 fv 2p p p pI P 4 1ff 2k p p k k p p pk k k p2 I v4 (12.1) P 2 422 24f kpk k vPPP 21f 1 2 I v 4 k k v , and,
22 HvvSHSS,pp xp, 2 124 Ikk . (12.2)
13. MIXED SLOWNESS-SPATIAL HESSIAN OF THE HAMILTONIAN
Applying the spatial gradient operator x to the slowness gradient of the Hamiltonian Hp , we
obtain the mixed slowness-spatial Hessian of the Hamiltonian, Hpx . Recall that the Hamiltonian
Page 25 of 146 depends on the spatial coordinates indirectly, through the model properties, and thus, we compute the spatial derivatives with the use of the chain rule for a multivariate function,
HHpxpmxm , (13.1) 33 n 23 32nc c
where the model gradient array, mxcn 2,3, is a result of concatenation of the material properties’ gradient array, cxcn ,3 (where n is 2, 3 or 4, depending on the wave type), and the axis angles’ gradient array, ax 2 ,3 . In this study, arrays caxxa n d are considered the input data.
It is suitable to split array Hnpmc3,2 into two blocks that will be computed separately and then concatenated,
HHHpmpcpa . (13.2) 332n 32nc c
The mixed “slowness-material” Hessian Hpc can be computed for qP-qSV waves as,
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HP,p xp, 3 3 8411232p k kppvfvfvPP kk P vP 33 16 11 2116, fvffppppkpppkpk PP v HP,p xp, 412 p p pvv4 42 k p pp p kk p 4 f PP 3 42 8 k pkp vvP 2 P , (13.3) HP,p xp, 4 f p k pp p k p kpvfv44 kk 8, 3 PP HP,p xp, 48p k kpp kkvfv24 3 PP 44 8 14 12,fvfvp p ppPP kp p kp k and for the SH waves, this mixed Hessian can be computed as,
HHSHSH,,pp2 4 1 28,4pkvvvSSS p kpk p k . (13.4) vS
Each item on the left-hand side of equation sets 13.3 and 13.4 is a column of a matrix.
Eventually, we need the second block on the right-hand side of equation 13.2, Hpa 3,2 . Recall that the Hamiltonian depends on the material properties (array c ) directly, and on the polar angles
(array a ) indirectly, through the unit-length symmetry axis direction vector k , and this leads to an application of one more chain rule for a multivariate function,
HHpa pkk a . (13.5) 3 2 3 3 32
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The components of matrix ka 3,2 are given in equation 5.4. The mixed “slowness/axis of symmetry” Hessian can be obtained from equation 11.1 for the slowness gradient of qP-qSV
Hamiltonian,
HffP,pk xp,41 2pkppppkppkppI v4 P (13.6) 2344 248fvfv pkkppkI PP 22 44,kppkIvvPP and from equation 11.2 for the slowness gradient of the SH Hamiltonian,
2 HvSHS,pk xp, 4 kpkpI . (13.7)
Note that the Hamiltonian gradient wrt the model parameters, Hm , the corresponding Hessian,
Hmm , the slowness Hessian, Hpp , and the mixed Hessian, Hpm , are needed to compute the gradient and the Hessian of the ray velocity magnitude wrt the model, mmmvvrayrayand .
14. TTI MODEL 1: PARAMETER SETTING AND SLOWNESS INVERSION
Consider a polar anisotropic (TTI) model, whose properties mi and their relative (normalized) spatial gradients and Hessians are listed in Tables 1 and 2, respectively. The normalized gradient and Hessian are defined by,
mm mm xxii x , , (14.1) xxii x oo mmii
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o where mmii xo is the nodal value of each parameter mi . Thus, in the proximity of the reference node xo , the value of the model parameter is assumed a quadratic function of the
Cartesian coordinates,
m 1 i 1 mmxxxxxx . (14.2) o xxxii ooo mi 2
Remark 1: These relative derivatives are presented solely to provide a better “feeling” of how fast the values change wrt their reference point values. In this specific example the values of mi at the reference (nodal) point and its proximity are far from zero, which makes this normalization possible and reasonable; otherwise, the absolute values of the first and second derivatives,
xmmiiand x x should be specified directly.
Table 1. Properties of the polar anisotropy and their spatial gradients, model 1
Component Relative gradient, km1 (in global frame) Value # mi x1 x2 x3
1 vP 3.5 k m / s +0.0842 +0.0612 –0.0308 2 f 0.78 +0.1022 +0.1156 +0.0923 3 +0.10 +0.0914 +0.0718 +0.0310 4 +0.25 –0.0763 –0.1080 –0.0604 5 +0.08 +0.1123 +0.0330 –0.0431 o 6 ax 30 / 6 rad –0.0854 +0.1211 –0.1163 o 7 ax 45 / 4 rad +0.0221 –0.1099 +0.0529
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Table 2. Spatial Hessians of polar anisotropy properties, model 1
Component Relative Hessian, km2 (in global frame)
# mi xx11 xx12 xx13 xx22 xx23 xx33
1 vP +0.0290 +0.0368 –0.0972 +0.0623 –0.1077 –0.0319 3 f +0.0372 +0.0263 +0.1012 –0.0871 –0.0210 +0.0414
3 +0.0963 –0.1211 –0.0655 +0.0822 –0.0553 –0.0850 4 –0.0281 –0.0719 +0.0269 –0.0357 +0.0543 –0.0308 5 +0.0355 –0.0421 +0.0101 +0.0823 –0.0311 –0.0826
6 ax –0.1172 +0.1231 +0.0633 +0.0904 +0.0308 –0.0404
7 ax +0.0615 +0.0320 –0.0801 +0.0412 +0.0765 –0.0835
The symmetry axis direction angles lead to the following Cartesian components of the axis direction,
223 k 0.35355339 0.35355339 0.86602540 . (14.3) 442
The ray velocity direction is chosen to be,
r 0.36 0.48 0.80 . (14.4)
For the given model parameters and the ray direction, we first perform the slowness inversion and obtain a single solution for each wave type: quasi-compressional qP, quasi-shear SV, and shear
SH; the solutions are listed in Table 3. There is no qSV shear wave triplication for the given medium and ray velocity direction.
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In addition, we obtain a solution for the acoustic approximation of qP waves, q Pac , by setting f 1.
Table 3. Results of the slowness vector inversion, model 1
Mode p1 , s / k m p2, s / k m p3, s / k m vphs , k m/s vr a y, k m/s
qPac 0 . 1 0 2 5 4 2 9 1 0 . 1 3 0 9 2 7 5 1 0 . 2 3 1 8 2 5 1 2 3 . 5 0 4 9 9 9 3 3 . 5 0 6 0 5 6 3 qP 0.10254249 0.13091618 0.23183152 3.5050011 3.5060621
qSV 0 . 2 1 7 0 4 0 1 6 0 . 2 4 8 3 1 7 2 5 0 . 5 1 0 3 1 2 8 6 1 . 6 4 5 7 9 8 8 1 . 6 5 1 3 1 7 6
SH 0 . 2 1 8 7 5 3 9 3 0 . 2 8 1 8 5 7 2 7 0 . 4 9 2 8 1 1 5 0 1 . 6 4 3 6 3 4 6 1 . 6 4 3 9 4 70
Remark 2: Normally, the accuracy of the input model parameters and the computed results does not exceed three or four digits. We keep eight digits here, as it is a reproducible benchmark problem. For the same reason, along with the final results, we provide also the most important intermediate values. The units of distances are km, the units of velocities are km/s. The reference
Hamiltonian is unitless, and the arclength-related Hamiltonian has the units of slowness, s/km.
15. MODEL 1. RAY VELOCITY DERIVATIVES FOR qP WAVES
Gradients and Hessians of the Reference Hamiltonian
The gradient of the reference Hamiltonian wrt the material properties, c vfP ,,, , is computed with equation 4.2,
H 4.4373311 101 1.1765600 10 5 2.1314490 10 2 3.8040622 10 4 c , (15.1)
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where the first component of Hc has the units of slowness, and the others are unitless. The Hessian of the reference Hamiltonian wrt the medium properties is computed with equation 7.2,
1.9158416 102 5.6978709 10 1 2.4359417 10 2 1.5446185 10 2 5.6978709 101 0 2.7326269 10 2 3.8618753 10 4 H , (15.2) cc 22 2.4359417 10 2.7326269 10 0 0 24 1.5446185 10 3.8618753 1 0 0 0
where the component in the first row and column, Hcc 1,1 has the units of slowness squared, the other components in the first row and the first column have the units of slowness, and the
2222HHHH remaining components are unitless. The zeros are related to ,,,and . f 222
The gradient and Hessian of the reference Hamiltonian wrt the symmetry axis direction vector are computed with equations 5.2 and 8.2, respectively,
H 1.1636151 101.4856218111 10 2.63070190 1 unitless)( k , (15.3)
1.9497624 102.4893193111 104.4080529 10 111 Hkk 2.4893193 103.1781875 105 .6278914 10 (unitless) , (15.4) 4.4080529 105.6278914111 109.96579 3018 where the numerical components of the symmetry axis direction are given in equation 14.3. The
first and second derivatives of this direction wrt the symmetry axis direction angles (zenith and
azimuth), a axax , are computed with equations 5.4 and 8.4, respectively,
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6.1237244 103.53553391 10 1 11 1 ka 6.1237244 103.535533 9 10 radian , (15.5) 50101
3.5355339 106.123724411 10 11 6.1237244 103.5355339 10 1 1 3.5355339 106.1237244 1 0 2 k radian . (15.6) aa 11 6.122 4374103.5355339 10 8.66025400101 0 0
The reference Hamiltonian gradient and Hessian wrt the symmetry axis direction angles are obtained with equations 5.1 and 8.1, respectively,
H 3.0696020 102 1.138465 6 10 2 radi an1 a , (15.7)
3.3505997 101.468652112 10 H ra dian2 . (15.8) aa 22 1.4686521 109.5531063 1 0
The mixed Hessian of the reference Hamiltonian wrt medium properties and symmetry axis direction angles is computed with equation 9.2,
1.8565234 102 3.9010957 10 2 2.8393439 10 1 9.2103212 10 3 H . (15.9) ac 3 2 1 3 6.8855442 10 1.4468532 10 1.0530666 10 3.4159588 10
The first column has the units of slowness divided by radians, and the other components’ units are radian1. Next, we assemble the gradient and Hessian of the reference Hamiltonian wrt the model parameters (both material and geometric) according to equations 5.5 and 7.1, respectively.
The spatial gradient and Hessian of the reference Hamiltonian are computed with equations 6.1 and 10.1, respectively,
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H 1.3212974 101 9.4225640 10 2 4.6378730 10 2 km1 x , (15.10)
7.2751789 107.748856322 101.4229782 10 1 2112 Hxx 7.7488563 101.1866113 101.6670916 10 km . (15.11) 1.4229782 101.6670916112 105.5466724 1 0
The slowness gradient and Hessian of the reference Hamiltonian are computed with equations 11.1 and 12.1, respectively,
Hp +1.9602510+2.6137340+4.3560698km / s , (15.12)
19.6520924.74117507.3413575 2 Hpp 4.741175016.60443410.465764+km /s . (15.13) 7.341357510.4657647.5 176 820
The mixed Hessians “slowness-material properties”, “slowness-axis of symmetry”, and “slowness- geometric properties” of the reference Hamiltonian are computed with equations 13.3, 13.6 and
13.5, respectively,
4.7222015 102.51706151.425168112 106.6964429 1 0 12 Hpc 5.3574936 103.35470441.20890965.2182917 10 , (15.14) 1.11338425.59441093.7797376 101.7410672 1011 where the first column is unitless, and the three others have the units of velocity,
6.8424181 1011 5.7519181 10 1.0185419 1 1 1 Hpk 3.8915989 10 6.3790958 10 8.7981866 10 km / s , (15.15) 1.1453785 1.4623386 1.4547279
5.7605014 1011 4.4527703 10 11 Hpa 5.9223678 10 3.6312389 10 km / radian s . (15.16) 8.6953015 1011 1.1206231 10
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The mixed Hessian “slowness-model properties”, Hpm , is then assembled according to equation
13.2. The mixed “slowness-spatial” Hessian of the reference Hamiltonian, Hpx , is computed with equation 13.1,
3.7588038 102.5597898111 101.8533944 10 1 111 Hpx 4.5753299 104.2112410 102.0927746 10 s . (15.17) 7.3700854 107.9037307111 102.3590478 10
This completes the computation of the derivatives of the reference Hamiltonian. The resulting
gradients are HHHxpm,,and , and the resulting Hessians are
HHHHHxxppmmpxpm,,,, and ; all other vectors and matrices are intermediate results.
Gradients and Hessians of the arclength-related Hamiltonian
Next, we compute these three gradients and five Hessians for the arclength-related Hamiltonian,
H xp, , using the conversion formulae derived in Part I. Since these conversion formulae do not include the medium properties, they are universal and can be equally applied to all anisotropic symmetries.
The spatial gradient and Hessian of the arclength-related Hamiltonian are computed with equations
4.10 and 4.12 of Part I,
H 2.4265528 102 1.7304482 1023 8.5175169 10 s / km2 x , (15.18)
4.9423420 103 7.0998463 10 3 2.6241540 10 2 3 2 2 3 Hxx 7.0998463 10 1.5902803 10 3.0297691 10 s / km . (15.19) 2.6241540 102 3.0297691 10 2 9.07307850 1 3
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The slowness gradient and Hessian of the reference Hamiltonian are computed with equations 4.10 and 4.13 of Part I,
Hp +0.36+0.480.80 unitless r , (15.20)
3.75113316.3658517 101.06316311 1 Hpp 6.3658517 103.42127491.4424407 km / s . (15.21) 1.06316311.44244071.88365 60
We will need also the inverse matrix of the slowness Hessian,
4.4590566 102.7922631111 104.6549785 10 1 111 Hpp 2.7922631 106.0 649970 106.2203658 10 s /km . (15.22) 4.6549785 106.220365811 101 .2699511
The eigenvalues of the inverse Hessian are all positive, as the slowness surface of compressional waves is convex,
1.85264712.3644739 102.3326198 11 10 s / km . (15.23)
The mixed “slowness-space” Hessian of the arclength Hamiltonian is computed with equation 4.14 of Part I,
1.1369338 1023 1.0836831 10 2 8.8275821 10 2 3 3 1 Hpx 1.0158733 10 2.3586643 10 3.7607307 10 km . (15.24) 5.1607592 10322 1.5136887 10 1.197570 610
The gradient of the arclength-related Hamiltonian wrt the model properties is computed with equation 17.13 of Part I. It is presented here as consisting of two parts,
HHHm c a , (15.25) where the material part is,
Page 36 of 146
H 8.1491509 102 2.7493568 10 6 3.9136076 10 3 6.9849459 10 5 c , (15.26) and the geometric part is,
H 5.6367353 102.090571533 10s /kmradian a . (15.27)
The first component of the material part, Hc , has the units of slowness squared; the other components of this vector have the units of slowness. The Hessian of the arclength-related
Hamiltonian wrt the model properties is computed with equation 18.9 of Part I. In this part (Part
II), this matrix consists of the four blocks (material, geometric and mixed),
HHccca Hmm , (15.28) HHacaa where the two diagonal blocks are symmetric, and the two off-diagonal blocks are transposed of each other. The material block reads,
4.2964811 101.31609412635 101.4006158 103.34161 62 10 1.3160941 107.06066246665 107.6839587 101.88914 15 10 H , (15.29) cc 3644 1.4006158 107.6839587 104.7314960 101.31188 16 10 5 546 3.3416162 10 1.8891415 101.3118816 104.8335714 10 where the component Hcc 1,1 has the units of slowness to power three, the other elements of the first row and the first column have the units of slowness squared, and the rest of the components have the units of slowness. The mixed “geometric-material” block reads,
1.7212877 103523 7.4437213 10 5.1945246 10 1.8888263 10 H , (14.34) ac 4524 6.3839704 10 2.7607526 10 1.9265629 10 7.0053432 10 where the first column has the units of slowness squared divided by radians, and the other components have the units of slowness divided by radians. The geometric block reads,
Page 37 of 146
6.1955323 102 2.5403875 10 3 s H . (15.30) aa 322 2.5403875 10 1.7601993 10 km radian
Eventually, the 63 mixed “model-slowness” Hessian Hmp of the arclength-related Hamiltonian is computed with equation 18.10 of Part I,
1.5667574 101.3777136222 103.9666468 10 6.8679118 102.7799839644 101.6803458 10 311 5.1798463 101.9449259 101.1638594 10 H . (15.31) mp 433 1.7428814 107.0541632 104.2637990 10 1 11 1.2038848 10 1.2892618 101.2772736 10 223 7.6361026 107.4165347 108.7262162 10
The components of the first row have the units of slowness, the components of rows 2, 3, 4 are unitless, and the components of rows 5, 6 have the units r a d i a n1 .
Gradients and Hessians of the ray velocity magnitude
Until now we have computed all necessary derivatives of the arclength-related Hamiltonian, needed to establish the required gradients and Hessians of the ray velocity magnitude. The computation of the compressional ray velocity yields vray 3.5060621 km / s . The spatial gradient and Hessian of the ray velocity magnitude are computed with equations 7.6 and 8.6 of Part I, respectively,
v 2.9828330 101 2.1271485 10 11 1.0470133 10 s1 x ray , (15.32)
1.0734758 101 1.2109813 10 1 3.3941305 10 1 1 11 1 xxvray 1.2109813 10 2.1854053 10 3.8350808 10 ksm . (15.33) 3.3941305 1011 3.8350808 10 1.0637793 101
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The directional gradient and Hessian of the ray velocity magnitude are computed with equations
9.2 and 10.3 of Part I, respectively,
v 1.6817488 107.36264732 1 104.49326 69 10 2 km / s r ray , (15.34)
5.2064468 101.3019110111 101.5827763 10 111 rrvray 1.3019110 10 4.2233608 102.8684874 10 km / s , (15.35) 1.5827763 102.86848741 102.99500001121 and the mixed “spatial-directional” Hessian of the ray velocity magnitude is computed with equation 11.5 of Part I,
1.9621841 102.3573946222 102.2974196 10 2 312 xrvrya 4.1062108 102.9615883 101.6700996 10 s . (15.36) 3.0360352 101.262589322 102.12376942 10
The gradient of the ray velocity wrt the model parameters was computed with equation 17.13 of
Part I. We present the gradient as a concatenation of the material and geometric counterparts,
mcavvv rayrayray , (15.37) where the material part reads,
v 1.0017323.3796388 104.8107908524 108.5862246 1 0 c ray , (15.38) and the geometric part (symmetry axis direction angles) reads,
v 6.9289406 1022 2.569829 10 km / s radian a ray . (15.39)
Observations:
The first component of cvray is unitless, while the other components of this vector have the units of velocity. As expected, parameter vP (the axial compressional velocity) directly affects the ray
Page 39 of 146 velocity magnitude vr a y , and the derivative vvr a y / P in realistic (weakly and moderately anisotropic) cases is normally close to 1; in this example, despite the relatively strong anisotropy,
3 the value of the derivative differs from 1 by 1 . 7 3 1 0 . Parameter vfr a y / (the second component of array cvr a y ) is fairly small, which is also not a surprise, as it was shown by
Alkhalifah (1998) for polar anisotropy and Alkhalifah (2003) for orthorhombic media that the axial shear velocity vS (or alternatively parameter f ) has a negligible effect on the propagation of qP waves. It is suitable to apply the normalized characteristic, which proves to be a tiny fraction of percent,
1 v ray 9.6394154(unitless)106 . (15.40) vfray
The Hessian of the ray velocity wrt the model parameters mmvray is computed with equation
18.6 of Part I. This 66 matrix (for qP and qSV waves) consists of four blocks: two symmetric diagonal blocks (ccaa vvrayrayand of dimensions 44and22) related to either material or geometric properties of the model, respectively, and two mutually transposed off-diagonal blocks ( 24and42, accavv rayrayand , respectively) related to both, the material and geometric properties,
c cvvray c a ray mm vray . (15.41) a cvvray a a ray
The material block reads,
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5.5511151 109.656111016624 101.3745117 102.45320 70 10 9.6561110 108.64909006544 101.1589334 102.24556 77 10 v . ccray 2 431 1.3745117 101.1589334 101.5196709 103.72488 05 10 4434 2.4532070102.2455677 103.7248805 101.3509062 10
(15.42)
22 The component in the first row and the first column, vvr a y / P , has the units of slowness. To evaluate whether its magnitude is large or small, we normalize this number, using the ray velocity vray 3.5060621km / s as a scaling factor. This leads to a unitless characteristic number of
2215 vvvrayray /1.9510P , which can be considered comparable to a “machine epsilon” for
22 the eight-byte (double precision) arithmetic that we use; in other words, vvray /0P . This is an expected result (confirmed also by the finite-difference numerical evaluation of this derivative).
Parameter vP is a scaler for the ray and phase velocities. Should we keep fixed the other (material and geometric) parameters of the model, the wave type, and the ray velocity direction r , then both the ray and phase velocities become proportional to vP . The ray and phase velocities are first- degree homogeneous functions of the axial compressional velocity vP , and the Euler theorem holds (e.g., Buchanan and Yoon, 1999); for this simple case it reduces to,
vvrayray vAvAray P ,const , (15.43) vvPP where “const” means independent of vP (but still depending, of course, on all the other model parameters and the ray direction). The derivative vvray / P on the left-hand side is the first component of the array in equation 15.38, where vP 3.5 km / s . Introducing the numbers, we
Page 41 of 146 make sure that equality 15.43 is honored; for the phase velocity, a similar property follows directly from equation 1.2.
The other components in the first row and the first column of the matrix in equation 15.42 (except
22 the vanishing vvr a y / P ) are unitless; the rest of the components in this matrix have the units of velocity. The mixed block reads,
1.9796973 103 7.4958122 10 4 5.2399585 10 1 1.9020481 10 2 v . acray 3 4 1 3 7.3423686 10 2.7800723 10 1.9434136 10 7.0543807 10
(15.44)
The first row of the mixed block has the units r a d i a n1 , while the other components have the units km /sradian . Finally, the geometric block reads,
6.2753676 103.412697012 10 km v . (15.45) aaray 212 3.4126970 101.8373 2 9 10 s rad i an
Validation test
In order to check the correctness of all the ray velocity derivatives, obtained analytically, we compare their values with numerical values obtained by the finite differences method. The numerical directional derivatives of the ray velocity, approximating the gradient and the Hessians
rvvray, r r ray and the mixed Hessian xrvray , were computed wrt the zenith and azimuth angles, rayand ray , of the ray direction vector, and then were related analytically to the
Cartesian components of this direction, as explained in Appendix F of Part I. The numerical approximations for the ray velocity derivatives were computed with the spatial resolution 104 km
Page 42 of 146 and angular resolution 105 radians for the spatial and directional gradients and Hessians of the ray velocity magnitude, including the mixed Hessian. When computing the numerical ray velocity
4 derivatives wrt the model parameters, mmmvvrayrayand , the resolution was 8 1 0 times the absolute value of the parameter. The (unitless) relative errors of the numerical approximations of the ray velocity derivatives for qP waves are listed below.
For the spatial gradient and Hessian,
Ev 4.41 101.18111111 101.06 10 x ray , (15.46)
1.91 103.31679 101.87 10 779 Evxx ray 3.31 104.62 108.77 10 . (15.47) 1.87 108.77996 101.67 10
For the directional gradient and Hessian,
Ev 4.73 108 1.13 10 10 9.07 10 10 r ray , (15.48)
1.82 106 5.80 10 5 2.59 10 5 5 6 5 Evrr ray 5.80 10 2.14 10 1.37 10 . (15.49) 2.59 105 1.37 10 5 1.41 10 5
For the mixed Hessian,
1.20 101.97556 107.54 10 656 Evxr ray 3.25 101.59 105.29 10 . (15.50) 1.24 103.32556 103.89 10
For the material, cvray , and geometric, avray , parts of the model-related gradient
mvray ,
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Ev 1.51 1013 6.50 10 7 1.58 10 8 7.90 10 10 c ray , (15.51)
Ev 2.71 101.391077 a ray . (15.52)
For the material, ccvr a y , geometric, aavr a y , and mixed, acvr a y , parts of the
model-related Hessian mmvray ,
7.41 107.68686 101.11 10 7.41 107.316755 104.24 102.41 10 Ev , (15.53) ccray 8566 7.68 104.24 101.10 105.86 10 6566 1.11 102.41 105.86 109.72 10
2.72 109.177676 104.80 107.67 10 Ev , (15.54) acray 7576 1.47 101.09 103.42 105.43 10
1.07 101.0876 10 Ev . (15.55) aaray 68 1.08 104.77 10
A dash has been placed to the upper left corner of Evccray because the relative error cannot be estimated for a parameter whose exact theoretical value is zero; however, the numerical (finite difference) estimate was also exact zero (i.e., the result of subtracting close values in the finite-
22 difference approximation of the second derivative vvray / P proved to be below the machine epsilon for double-precision arithmetic).
We performed one more validation test. In the proposed workflow, we used the reference and arclength-related Hamiltonians to compute the spatial gradient and Hessian of the ray velocity.
However, this gradient and Hessian can also be obtained through a different chain of operations
Page 44 of 146 that involves the derivatives of the ray velocity wrt the model properties, mmmvvrayrayand , and the spatial derivatives of the model properties, mmx x xa n d ,
xmxvv rayray m , (15.56) T xxxmmxmxxvvrayray mmm
where (as mentioned in Part 1) mvr a y is a 1D array (not a physical vector), mmvr a y is a 2D array (not a tensor and not even a collection of vectors), mx is another 2D array representing a collection of vectors, and mxx is a 3D array representing a collection of second-order tensors.
However, the multiplications in the above equation set are performed as if all these items were vectors and tensors of the corresponding orders; this allows a shorthand notation where summations and multiplications are hidden. Applying the chain rule of equation 15.56 does not show any apparent advantages, because the gradient and Hessian of the ray velocity wrt the model parameters, mmmvvrayrayand , respectively, have also been computed with the use of the derivatives of the reference and the arclength-related Hamiltonians; this equation is therefore only used for double-checking the correctness of the resulting spatial derivatives. These tests revealed full coincidence of the spatial gradient and Hessian of the ray velocity, xxxvvrayrayand , computed with our proposed workflow, and with equation 15.56; the relative errors for all components were of order 1016 , i.e., within the machine accuracy.
Sensitivity of the ray velocity derivatives to the spatial derivatives of the model parameters
Page 45 of 146
Our next test checks the significance (the effect) of the spatial gradients and Hessians of the model parameters f ,,,and axax (all model parameters except the axial compressional velocity) on the derivatives of the ray velocity for compressional waves. For this, we repeat the computations, assuming mmxxx0and0 , except xxxvvPPand which are taken into account.
Obviously, this does not affect the directional derivatives, rrrvvrayrayand , and the derivatives wrt the medium properties, mvvrayand m m ray . We re-compute the spatial derivatives, xxxvvrayrayand and the mixed Hessian xrvr a y , and compare the vector and the two tensors obtained for three cases: 1) accounting for the spatial derivatives of all model parameters, 2) accounting for the spatial derivatives of the medium properties and ignoring those of the symmetry axis direction angles, and 3) accounting only for the spatial derivatives of the
ˆ axial compressional velocity. For cases 2 and 3, we use notations xxand , respectively. Case
3 is referred as FAI media – Factorized Anisotropic Inhomogeneous media (Červený, 2000). To compare the spatial gradients, we compute the ratio of their magnitudes and the angle between their directions; to compare the spatial Hessians, we compute the ratio of their corresponding eigenvalues. The non-symmetric mixed Hessian, , has a single zero eigenvalue, and the corresponding eigenvector is the ray direction r (Koren and Ravve, 2021); this property holds even if we ignore the spatial derivatives of the model, so we compare the two other eigenvalues, which prove to be real for the given example and for qP waves.
Accounting for the spatial derivatives of the medium properties, ccxand xx and ignoring those of the symmetry axis direction angles, aaxxxand , we obtained the following results,
v 2.9563106 10112 2.1489018 10 1.0785298 10 s1 x ray , (15.57)
Page 46 of 146
1.0219286 101.28470031 103.411053611 10 111 1 xx vray 1.2847003 102.1885227 103.7785968 10 kms , (15.58) 3.4110536 103.7785968111 101.1226839 10
2.4759031 101.083943243 106.61507472 10 4 331 xr vray 1.8013387 107.8862081 104.8127851 10 s . (15.59) 2.0652537 109.04162075 105.51790884 10 4
Accounting for the spatial derivatives of the axial compressional velocity only, xxxvvPPa n d , we obtained,
ˆ v 2.9521042 102.1457100111 101.07 98671 10 s1 x ray , (15.60)
1.0167580 101.29023081 103.407892311 10 ˆˆ 111 1 xx vray 1.2902308 102.1842767 1 03.7760288 10 km s , (15.61) 3.4078923 103.7760288111 101.11 84338 10
1.4160325 104 6.1993488 10 33 3.7833308 10 ˆ 4 3 3 1 xr vray 1.0292302 10 4.5059400 10 2.7498794 10 s . (15.62) 5.1797862 105 2.2676953 10 3 1.3839262 10 3
The magnitudes of the ray velocity gradients computed with complete and incomplete data are related as,
ˆˆ xxxxvvvvrayrayrayray 1.0000857 ,9.9885642 10 1 . (15.63) xxxxvvvvrayrayrayray
The angles between the corresponding gradient directions are,
Page 47 of 146
xxvvrayray 2 o xxvvrayray,arccos1.2224727rad0., 1070042524 xxxxvvvvrayrayrayray
vvˆ vv,arccos1.2727741rad0..ˆ xxrayray 1072924582 5o xxrayray ˆˆ xxxxvvvvrayr ayrayr ay
(15.64)
We compute the ratios of the eigenvalues (major, intermediate and minor) of the spatial Hessians,
vv ˆˆ majorraymajorrayx xx x 0.99791752 ,0.99763379 , majorraymajorrayx xxvv x vv ˆˆ intermrayintermrayx xx x 0.73369807 ,0.70452683 , (15.65) intermrayintermrayx xxvv x v ˆˆv minorrayxx 1.00004069 ,0.99852940minorrayxx . minorrxxv ayminorray xxv
Note that the obtained “errors”, associated with the incomplete ray velocity derivatives, highly depend on the magnitudes of the spatial derivatives of the model parameters. Of course, at ray points within transition zones between layers, the gradients and Hessians of the model parameters are higher and hence the “errors”.
ˆ The mixed Hessians, x rvvrayand x r ray , computed with the incomplete spatial model derivatives, have very special eigensystems. As mentioned, the mixed Hessian, xrvray , computed with all spatial derivatives of the model accounted for, has a zero eigenvalue, and the corresponding eigenvector is the ray direction, r . Even if all other eigenvalues are real, the eigenvectors are not necessarily mutually orthogonal because this matrix is not symmetric. In this specific case, the mixed Hessian computed with the “complete” data has a pair of complex- conjugate eigenvalues and a zero eigenvalue,
v 6.7286747 104 3.21352 5 6 10 2 i 0 xrray , (15.66)
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ˆ while the “incomplete” mixed Hessians, xrxrvvrayrayand , have two zero eigenvalues,
v 8.68558931003 0, xrray (15.67) ˆ v +6.031469.5103 0 0 xrrya
The eigenvectors corresponding to the two zero eigenvalues may be reduced to a form, where one of them is the ray direction, vr2 , and the other is (up to a constant factor) the cross product of the ray and the axis of symmetry directions, v k3 r. Thus, the two eigenvectors are mutually orthogonal, but they are (generally) not orthogonal to the first eigenvector corresponding to the single nonzero eigenvalue. In the case when only the spatial derivatives of parameter vP are taken
ˆ into account, the eigenvector corresponding to the nonzero eigenvalue of xrvray is (up to a constant factor) the spatial gradient of the axial compressional velocity, v1 xvP (listed in the first row of Table 1).
Summarizing the above, we conclude that when the spatial derivatives of the medium symmetry axis direction angles are ignored, the accuracy of the spatial ray velocity gradient, xvray , and the
Hessian, xxvray , is still adequate for some practical applications. The accuracy further decreases when the gradients and Hessians of all polar anisotropy parameters are ignored (except those of the axial compressional velocity vP ). However, even in this case, the accuracy may still meet the requirements, provided the ignored change of anisotropy is mild. However, we will show that for rapidly varying (in space) strong polar anisotropic media with a large negative anellipticity
(referred to below as model 2), ignoring the derivatives of the symmetry axis’ angles leads to a low accuracy of the ray velocity derivatives. When all spatial derivatives of the model parameters
Page 49 of 146 are discarded, except those of the axial compressional velocity, the results for such media may be plainly wrong, especially when the ray crosses transition regions, where the model parameters change rapidly. The eigensystem of the mixed Hessian, xrvr a y (whose accuracy is less crucial for the ray bending method), undergoes a principal change.
16. MODEL 1. ACOUSTIC APPROXIMATION FOR qP WAVES
When using the acoustic approximation (AC) for qP waves in polar anisotropic media, the slowness inversion equation reduces to the fourth-degree polynomial, presented in Appendix A
(equation A5). The reference Hamiltonian for qP waves in equation 2.15 reduces to,
Hvvv x,2p ppppkk pp 22 222 10 , (16.1) PP PP and all its derivatives simplify accordingly. We show that although the AC can still be applied with high accuracy for computing the ray velocity magnitude, the accuracy of the ray velocity derivatives degrades, in particular for the second derivatives.
In this section we skip the intermediate results, related to the derivatives of the reference and arclength-related Hamiltonians (as the procedure is similar to the case of actual (elastic) quasi- compressional waves described in detail in the previous section), and present only the required gradients and Hessians of the ray velocity magnitude.
Spatial gradient and Hessian of the ray velocity magnitude,
v 2.9828150 102.12722761 101.047039311 1 0 s1 x ray , (16.2)
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1.0734337 101.2111443111 103.3940997 10 1 111 xxvray 1.2111443 102.1853003 103.8349522 10 ksm . (16.3) 3.3940997 103.834952211 101. 0638498 101
Directional gradient and Hessian of the ray velocity magnitude,
v 1.6786248 107.34897032 3 104.48492 03 10 2 km / s r ray , (16.4)
5.1967675 101.2998389111 101.5796249 10 111 rrvray 1.2998389 10 4.2003550 102.8539068 10 km / s . (16.5) 1.5796249 102.85390681 102.98379001131
The mixed Hessian of the ray velocity magnitude,
1.9586285 102.3569268222 102.2955389 10 2 3 2 1 xrvrya 4.0982388 103.1068515 101.6577964 10 s . (16.6) 3.0304186 101.261257622 102.12044292 10
The gradient of the ray velocity wrt the model parameters, mvray , includes the medium
part: the derivatives wrt the material components, c vP ,, (where the shear factor
f has been eliminated), and the geometric part: the derivatives wrt the symmetry axis
direction angles (see equation 14.45). The material part is,
v 1.0017304 4.8127717 1024 8.2008573 10 c ray , (16.7)
and the geometric part is,
v 6.9160695 102.565055322 10 a ray . (16.8)
The Hessian of the ray velocity wrt the model parameters, mmvray , includes the material
, the geometric, and the mixed blocks (see equation 14.49). The material block reads,
Page 51 of 146
6.6613381 101.37507761624 102.3431021 10 213 ccvray 1.3750776 101.5130509 103.8439256 10 , (16.9) 2.3431021 103.8439256434 101.3308136 10 the mixed block reads,
1.9760199 105.2444022212 101.8167098 10 v , (16.10) acray 331 7.3287295 101.9450617 106.7378751 10 and the geometric block reads,
6.2502835 103.456156412 10 v . (16.11) aaray 21 3.4561564 101.8321319 10
Validation
The relative error for all gradient and Hessian components has been computed twice: comparing the numerical solution N for the AC with its analytical solution A (error E ), and comparing the exact analytical solution for compressional wave P with the analytical solution for the AC (error
E ),
NAAP EE, . (16.12) AP
Accuracy of the spatial gradient and Hessian of the ray velocity magnitude,
Ev4.69 1011 1.34 10 111 2.53 10 1 x ray , (16.13)
Ev6.05 103.72655 102.48 10 x ray , (16.14)
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1.56 103.65677 101.67 10 7 77 E xxvray 3.65 105.57 101.09 10 , (16.15) 1.67 101.0976 103.02 107
3.92 101.3556 109.10 104 455 E xx vray 1.35 104.81 103.36 10 . (16.16) 9.10 103.3665 106.63 105
Accuracy of the directional gradient and Hessian of the ray velocity magnitude,
E v 2.29 101.5089 101.86 10 9 r rya , (16.17)
E v 1.86 101.86333 101.86 10 r ray , (16.18)
3.47 105.80555 102.27 10 5 55 E rrvray 5.80 103.44 101.85 10 , (16.19) 2.27 101.8555 101.60 105
1.86 101.5933 101.99 103 333 E rr vray 1.59 105.45 105.08 10 . (16.20) 1.99 105.0833 103.74 103
Accuracy of the mixed Hessian of the ray velocity magnitude,
1.98 106 1.20 10 5 6.60 10 6 5 46 E xrvray 1.24 10 1.00 10 2.54 10 , (16.21) 9.58 1066 2.72 10 5 3.56 10
1.81 10344 1.98 10 8.19 10 33-2 E xr vray 1.94 10+4.90 10 7.37 10 . (16.22) 1.85 1033 1.05 10 1.57 10 3
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A large error component is marked with a bold number. Although the relative error of this component is large, the component itself is small (as compared to the other components; see equation 16.6), so that its absolute error is approximately the same as that of the other terms.
Accuracy of the material part of the ray velocity gradient wrt the model parameters,
Ev 3.75 1014 1.56 10 8 1.63 10 9 c ray , (16.23)
Ev1.65 104.1264 10 -4.4910 -2 c ray . (16.24)
Accuracy of the geometric part of the ray velocity gradient wrt the model parameters,
Ev2.56101.291077 c ray , (16.25)
Ev1.86 101.8633 10 c ray . (16.26)
We note that the relative discrepancies between the ray velocity derivatives for the acoustic approximation and elastic models wrt the zenith and azimuth angles of the medium symmetry axis are identical,
vvvvrayaxrayaxrayaxrayax //// ACPACP , (16.27) vv// rayaxrayax PP where subscripts AC and P denote the acoustic and elastic models for compressional waves, respectively. This equation can be also arranged as,
vvray// ax ray ax AC AC . (16.28) vv// ray ax PP ray ax
Accuracy of the material part of the ray velocity Hessian wrt the model parameters,
Page 54 of 146
7.58 101.6296 10 966 Evcc ray 7.58 101.28 101.70 10 , (16.29) 1.62 101.70664 101.75 10
4.12 104 -4.49 10-2 43-2 Evc c ray 4.12 104.36 10 +3.20 10 . (16.30) -4.49 10+3.20-2-2 10-1.49 10 -2
Accuracy of the mixed part of the ray velocity Hessian wrt the model parameters,
2.70 105.20776 107.84 10 Ev , (16.31) acray 767 1.41 103.74 105.92 10
1.86 108.4834 10 -4.49 10-2 Ev . (16.32) acray 34-2 1.86 108.48 10 -4.49 10
The first row of the mixed Hessian matrix in equation 16.30 is related to the zenith angle ax , while the second row – to the azimuth angle ax . The columns are related to the material properties c vP ,, . As we see, the two lines are identical.
Accuracy of the geometric part of the ray velocity Hessian wrt the model parameters,
8.60 1087 9.93 10 Ev , (16.33) aaray 7 8 9.93 10 3.75 10
4.00 103 +1.27×10-2 Ev . (16.34) a a ray -2 3 +1.27 10 2.86 10
As we see, the numerical errors E of the finite difference approximations are small for the gradients and acceptable for the Hessians, which indicates that the analytical formulae for the AC derivatives of the ray velocity are correct. However, the discrepancies E of the gradients and, in
Page 55 of 146 particular, the Hessians computed with the elastic and acoustic models are large, and for some components unacceptable (bold numbers). The reason is inherent to the nature of the differential operator. The AC for the ray velocity in itself is accurate; still, it is an approximation. The accuracy of the derivatives of an approximated function is much worse than the accuracy of the function itself, and the accuracy of the second derivatives is even worse. While the accuracy of the AC slowness inversion along with the computed phase and ray velocities is very high, using the AC for computing the qP ray velocity derivatives is questionable. The reduced computational complexity of the AC is accompanied by a declined accuracy of its results when the derivatives of the approximated values are involved.
17. MODEL 1. RAY VELOCITY DERIVATIVES FOR qSV WAVES
For the given model properties m and the ray direction r , there is no shear wave triplication: the qSV solution is unique, with the ray velocity, vray 1.6513176 km / s (Table 3). In this section, we list the derivatives of the ray velocity magnitude for this wave type and validate them numerically with the finite-difference approximations.
The spatial gradient and Hessian of the ray velocity magnitude are,
v 1.5524655 101 2.3999453 10 11 3.1525198 10 s1 x ray , (17.1)
1.5863467 10111 1.3552001 10 5.1585529 10 1 111 xxvray 1.3552001 10 2.4620300 10 1.8682279 10 ksm . (17.2) 5.1585529 1011 1.8682279 10 1.9252759 101
The directional gradient and Hessian of the ray velocity magnitude are,
Page 56 of 146
v 2.6384053 1031 1.1550861 10 7.0492451 10 2 km / s r ray , (17.3)
8.1678783 102.0528496111 102.4768156 10 111 rrvray 2.0528496 10 6.1726235 104.2236495 10 km / s . (17.4) 2.4768156 104.22364951 104.52991201131
The mixed Hessian of the ray velocity magnitude is,
3.0303315 101.5954905222 102.3209435 10 2 2 2 1 xrvrya 6.4841113 101.3784012 103.7448908 10 s . (17.5) 4.7343762 107.244973222 102.56516773 10
The ray velocity gradients wrt the material properties, c vfP, , , , and wrt the
geometric properties, a axax, , are,
v 4.7180503 103.73407152.778593412 102.774608 12 0 c ray , (17.6)
where the first component is unitless, and the others have the units of velocity, and
km v 1.0870443 1012 4.0316667 10 . (17.7) a ray sradian
The ray velocity Hessians wrt the material, mixed, and geometric properties of the model,
are,
4.4408921 1016 1.0668776 7.9388383 10 3 7.9274516 10 3 1.0668776 8.5424260 8.2246372 1022 8.1867757 10 v , ccray 3 2 1 1 7.9388383 10 8.2246372 10 2.1318676 10 2.1283892 10 3 211 7.9274516 10 8.1867757 10 2.1283892 10 2.1249209 10
(17.8)
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3.1058408 102 3.0683985 10 2 3.1807813 10 1 3.1716496 10 1 v , acray 2 2 1 1 1.1519048 10 1.1380181 10 1.1796990 10 1.1763122 10
(17.9)
9.4437246 1012 6.8425839 10 v . (17.10) aaray 21 6.8425839 10 2.8273764 10
Validation
The relative errors of the finite-difference approximations are listed below.
The relative errors of the spatial gradient and Hessian of the ray velocity magnitude are,
Ev1.59 106.439101 102.29 10 0 x ray , (17.11)
1.36 101.87688 102.39 10 8 77 Evxxray 1.87 105.86 102.71 10 . (17.12) 2.39 102.7187 102.27 107
The relative errors of the directional gradient and Hessian of the ray velocity magnitude
are,
Ev4.50 102.119101 101.32 10 0 r ray , (17.13)
1.55 106 3.00 10 5 1.26 10 5 5 66 Evrrray 3.00 10 5.50 10 1.73 10 . (17.14) 1.26 1056 1.73 10 6 4.07 10
The relative errors of the mixed Hessian of the ray velocity magnitude are,
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1.14 102.33556 102.90 10 6 58 Evxrray 5.16 101.79 107.64 10 . (17.15) 4.23 101.9967 101.47 105
The relative errors of the ray velocity gradients wrt the material and geometric properties
of the model are,
Ev2.19 101.591369 102.52 108.57 10 9 c ray , (17.16)
Ev5.49107.45101110 a ray . (17.17)
The relative errors of the ray velocity Hessians wrt the material, mixed, and geometric
properties of the model are,
1.93 102.22876 101.30 10 1.93 103.398867 101.94 103.11 10 Ev , (17.18) ccray 7657 2.22 101.94 102.90 103.10 10 677 7 1.30 103.11 103.10 101.16 10
1.10 102.727779 104.01 102.98 10 Ev , (17.19) acray 7677 1.24 106.35 101.07 102.20 10
1.68 102.5487 10 Ev . (17.20) aaray 7 8 2.54 106.04 10
18. MODEL 1. RAY VELOCITY DERIVATIVES FOR SH WAVES
The material parameters required for computing the ray velocity magnitude and its spatial derivatives for SH waves, are the axial shear velocity vS and the Thomsen parameter. However, the spatial gradients and Hessians in Tables 2 and 3, respectively, are given for the axial
Page 59 of 146 compressional velocity vP and the shear velocity factor f (rather than vS ). For consistency reason, we continue with these input spatial material derivatives, and apply the relation for the axial shear velocity, vSP v f1 , for computing its spatial gradient and Hessian,
xxxvvSSand ,
vfPx xxvfvSP1, 21 f (18.1) vfvffvffvPPPPxxxxxxx xxxx vfvSP 1. 2 12ff 1 41 f 3/2
We then use a normalization factor of vvfSP 11.6416455km / s (the values are taken from
Table 1), resulting in,
v 9.69727271.437272710101k2111 1.9442272 0 m x S , (18.2)
1.00278507.52926073.144408910101012 1 211 2 xxvS 7.52926071.49626261.0770544km1010 10 . (18.3) 3.14440891.07705441.210101111 9 1 8415 0
Unlike the case of qP-qSV waves, the slowness inversion for SH waves is performed analytically:
The slowness vector components are computed with equation A9; the magnitudes of the phase and ray velocities are computed with equation A11 (see results in Table 3), vray 1.6439470km / s .
In this section, we provide the ray velocity derivatives, computed analytically for SH wave, and validate them numerically with the finite-difference approximations.
The spatial gradient and Hessian of the ray velocity magnitude are,
Page 60 of 146
v 1.5820924 101 2.3702272 10 11 3.1853528 10 s1 x ray , (18.4)
1.6322424 101.2672607111 105.1671733 10 1 111 xxvray 1.2672607 102.4635332 101.7928789 10 ksm . (18.5) 5.1671733 101.792878911 101. 9901780 101
The directional gradient and Hessian of the ray velocity magnitude are,
v 6.2489860 102.73578742 8 101.66959 31 10 2 km / s r ray , (18.6)
1.9345225 104.8689821122 105.8620741 10 212 rrvray 4.8689821 10 1.4319010 109.8200986 10 km / s . (18.7) 5.8620741 109.82009862 101.06169802141
The mixed Hessian of the ray velocity magnitude is,
7.1878668 104.2435921333 105.7806953 10 2 3 3 1 xrvrya 1.5387345 104.5744269 109.6689613 10 s . (18.8) 1.1130627 101.900463823 103.86850393 10
The ray velocity gradient wrt the model parameters, m vS ,,,axax , is,
v 1.00140202.4853022 102.574632622 109.5488849 10 3 m ray . (18.9)
The ray velocity Hessian wrt the model parameters is,
2.220446 101.513909216223 10 1.5683244 10 5.816654 6 10 1.5139092 102211 8.4572901 10 2.7860655 10 1.0333054 10 v . mmray 2112 1.5683244 10 2.7860655 10 2.2100896 10 1.7194085 10 3122 5.8166546 10 1.0333054 10 1.7194085 10 6.6599291 10
(18.10)
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Validation
The relative errors of the finite-difference approximations are listed below.
The relative errors of the spatial gradient and Hessian of the ray velocity magnitude are,
Ev2.88 101.641111 101.50 10 21 x ray , (18.11)
3.00 104.797108 101.19 10 1078 Evxx ray 4.79 101.50 102.24 10 . (18.12) 1.19 102.24887 103.27 10
The relative errors of the directional gradient and Hessian of the ray velocity magnitude
are,
Ev5.37 103.769101 102.79 10 0 r ray , (18.13)
2.76 105.85555 101.18 10 5 55 Evrrray 5.85 104.17 102.34 10 . (18.14) 1.18 102.3455 101.59 105
The relative errors of the mixed Hessian of the ray velocity magnitude are,
1.17 103.68566 108.17 10 6 66 Evxrray 8.47 104.50 107.34 10 . (18.15) 7.04 105.3975 101.68 107
The relative errors of the ray velocity gradient wrt the model parameters are,
Ev7.01 101.721399 101.15 101.76 10 9 m ray . (18.16)
The relative errors of the ray velocity Hessian wrt the model parameters are,
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5.61 104.97787 102.67 10 5.61 106.077577 101.24 106.26 10 Ev . (18.17) mmray 8776 4.97 101.24 104.27 101.75 10 777 6 2.67 106.26 101.75 108.72 10
19. MODEL 1. TTI AS A PARTICULAR CASE OF GENERAL ANISOTROPY
In this section we show that the same results obtained in this part (Part II) for the gradients and
Hessians of the ray velocity (for the given inhomogeneous polar anisotropic model parameters and their spatial gradients and Hessians, presented in Tables 1 and 2, and the given ray direction), can be obtained by applying the theory presented in Part I for general anisotropy. For this, we first have to convert (transform) the polar anisotropic model parameters and their spatial gradients and
Hessians, into “equivalent model parameters” described with the full set of the twenty-one elastic properties and their spatial gradients and Hessians. Recall that the input model parameters are given by the array mxca() : The five material parameters cx(),,,, vfP (considered as a set of scalar invariants) and the two polar angles ax(), ax ax , as well as their spatial gradients and Hessians; all of them are given in the global frame x . The converted twenty-one elastic parameters are in the global frame as well. In this section, we consider quasi-compressional waves only (to be compared with the derivatives obtained in Section 14).
As mentioned, Tables 1 and 2 list the model parameters and their gradients and Hessians in the global frame. Recall that we store the relative gradients and Hessians, whose units are
12 kmand km , respectively, for all model parameters mi (five material parameters and two angles).
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The conversion of the set m c a into the twenty-one stiffness tensor components can be done in two stages:
First, converting set c into the five crystal stiffness components,
ˆˆˆˆˆˆ vfCCCCCS 1113334466 or c xC x , (19.1)
where the “hat” symbol emphasizes that these are the crystal stiffness components. We
reemphasize that in this section, we use the global frame x only.
Second, converting the five crystal stiffness components, Cxˆ , along with the two polar
angles, a axax , into the twenty-one stiffness components of the general
anisotropic matrix Cx ,
Cˆ xaxC x . (19.2) 5 items2items21 items
At each of the two stages, we also compute the corresponding spatial gradients and Hessians: at
ˆˆ stage one xxxCxCx and , and at stage two xxxCxCx and .
Remark: The crystal stiffness can be interpreted in two ways: a) A stiffness tensor in the crystal frame that relates the strains to the stresses; b) Another set of the medium invariants – re- parameterization of the normalized set of parameters cx into Cxˆ , where all the components have the units of velocity squared. We will use definition b).
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To implement the first stage of the conversion, we apply equation set 2.7b, and compute the crystal stiffness components,
CCˆˆxx 1113 12,21,xxxxx fff 22 vvPPxx (19.3) CCCˆˆˆ xxx 334466 1 ,1,112. ffxxx 222 vvvPPPxxx
The equations of this set make it possible to compute the spatial gradients and Hessians of the
ˆˆ crystal stiffness tensor components, xxxCxCx and . The spatial gradients are,
ˆ 2 xxxCvvv11 22PPP 1 2 ˆ 22ffffxxx xxxCvvff13 PPPP ff vv 222 1 ff 2 (19.4) ˆ xxCvv33 2 PP ˆ 2 xxxCf44 vvvf21 PPP ˆ 22 xxxxCfvvf66 2 vvf 11 22 11 2 PPPP
The spatial Hessians are,
ˆ x xC11 4 vPPP x v x x x v (19.5a) 2 2 1 2 xvPPPP x v 2 v x x 2 1 2 x x v ,
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ˆ xx C13 2 vP xxxxxxfffffvv 2 12 PP ff 2 3/2 22 2 f vP vvvffPPP xxxxxx f 2 ff3/2 2 3/2 f fv2 21 vvffv P PPP xxxxxx 3/2 ff 2 f 2 2 f 2 fvP 2 121fffvv PPxx vfPxxxx , ff 2 f 2
(19.5b)
ˆ xxxxxxCvvvv33 22PPPP , (19.5c)
Cˆ 2 v v f f v v2 f x x44 PPPP x x x x x x (19.5d) 2 1 f x vPPP x v 2 1 f x x v ,
ˆ x xxxCfvv 66 2 11 2 PP 22vfffvv2 11 2 PPP xxxxx x (19.5e) 2 4 11f 2 vvvvf PPPP xxxxx x 2 2 1 22 1.vvffvfPPPP xxxxx v x
Note that the spatial gradients and Hessians of the polar anisotropic model parameters and the crystal stiffness tensor components in equation sets 19.4 and 19.5 are absolute values (rather than relative). The computed relative gradients are presented in Table 4, and the relative Hessians in
Tables 5 and 6, where the normalization factors are the reciprocals of the (non-zero) crystal
ˆ 1 stiffness components, Cij .
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Table 4. “Crystal” stiffness components of the polar
anisotropy model 1 and their spatial gradients
1 ˆ 2 Relative gradient, km (in global frame) # Cij Value, k m / s x1 x2 x3 ˆ 1 2 2 1 C11 18.375 1.4296667 10 8.64 10 8.1733333 10 ˆ 1 1 1 2 C13 8 . 0 1 5 1 7 1 6 4.253249410 4.087052810 1.634085510 ˆ 1 1 2 3 C33 1 2 . 2 5 1 . 6 8 4 1 0 1 . 2 2 4 1 0 6 . 1 6 1 0 ˆ 1 1 1 4 C44 2.695 1.939454510 2.874545510 3.888454510 ˆ 1 1 1 5 C66 3 . 1 2 6 2 1 . 7 8 4 5 5 8 1 0 2.829028210 3.947902810
Table 5. Spatial Hessians of “crystal” stiffness tensor (diagonal components), model 1
Component Relative Hessian, km2 (in global frame)
# mi xx11 xx22 xx33 ˆ 2 1 2 1 C11 5.4246667 10 1.1137808 10 6.9688960 10 ˆ 1 3 3 2 C13 2.608229810 5.016480810 2.237035210 ˆ 2 1 2 3 C33 7.217928010 1.320908810 6.190272010 ˆ 1 5 1 4 C44 1.817495810 3.405675810 1.68367910 ˆ 1 1 1 5 C66 1.828613210 3.493024710 1.751377610
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Table 6. Spatial Hessians of “crystal” stiffness tensor (off-diagonal components), model 1
Component Relative Hessian, km2 (in global frame)
# mi xx12 xx23 xx13 ˆ 2 1 1 1 C11 5.076397310 2.013166410 1 . 9 2 4 4 3 8 1 1 0 ˆ 1 1 2 2 C13 2.0995847 10 2.6707485 10 5.5608440 10 ˆ 2 1 1 3 C33 8.390608010 2.191699210 1.995867210 ˆ 1 1 1 4 C44 1.227099610 1.595231810 1.683679010 ˆ 1 1 1 5 C66 1.338522210 1.638738810 5.946513810
Given the 33 rotation matrix Ar ot (equation C.4), we apply equations C8 – C12 to build the
66 matrix Rr ot for the global-to-local (crystal) rotation of the stiffness tensor in the Kelvin- matrix form (Appendix C). However, the rotation that we need now is local-to-global, therefore, we apply the first equation of set C13,
T ˆ CRCRCCCCglb rot loc rotwhere glb and loc . (19.6)
For polar anisotropy, the only difference between the Voigt and the less commonly used Kelvin forms is the factor 2 for the shear components CC4466and in the lower right block of the Kelvin form. Expanding the rotation in equation 19.6, we obtain the transform between the crystal and the global stiffness components in an analytic form. This form is needed to further compute analytically the gradients and Hessians of the global stiffness components vs. the gradients and
Hessians of the crystal components and the axis zenith and azimuth. The crystal-to-global stiffness conversion, CCˆ resulting from the Bond algorithm, reads,
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2422 ˆˆˆˆˆˆ 2 CCCCCCC1111coscosaxaxaxaxaxax 2sinsinsin1311131144 2 sin 44242 ˆˆˆˆ 2 coscos2cossaxaxaxaxaCCCC1114 3 sni 334 insin x 2 ax ,
(19.7a)
ˆˆˆˆ 222 44 CCCCC1211661344 2cossin 2sinaxaxaxaxaxsinsin cos2cossincossin42 CCCˆˆˆ 222 axaxaxxax 116613 a CCCCCCˆˆˆˆˆˆcossinc4 42222 os2sin4sin 2sin 2 /4 111133136644 axaxaxax axa x
(19.7b)
2 ˆˆˆˆ ˆˆˆˆ CCCCCCCCC131113334411133344cos6424cosaxax 4 /8 (19.7c) sin2CCCˆˆˆ co s(2)si22 n , aaxaxx 1311 66
2 ˆ ˆ ˆ ˆ CCCCC14 cosaxsin ax sin ax cos ax 11 33 4 44 4 66 (19.7d) CCCCCCCˆ 2 ˆ ˆ 4 ˆ cos 2 / 2 ˆ ˆ 2 ˆ sin2 , 11 13 334 4 ax 11 13 66 ax
2 C15 cosaxaxaxasincoscos x (19.7e) CCCCCCCCCˆˆˆˆˆˆˆˆˆ 24cos 2/ 22, sin2 113311334411134 31 axax 4
22 ˆˆ CCC16 sin ax cos ax sin ax cos ax 11 33 (19.7f) CCCCCCCˆ 2 ˆ ˆ 4 ˆ cos 2 2 ˆ ˆ 2 ˆ sin2 / 2 , 11 13 33 44 ax 11 13 44 ax
C CCˆˆcoscoscos 422 ˆˆC ˆ C2 C sin 22 11axax 11 13 11 13axax ˆˆ44224 CC11axaxcos 13axaxaxsin2 cos sin sin (19.7g) ˆˆˆ 224424 CCC44axaxsin 33axax sin 2sin 44axax sinsin 2 sin,
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ˆˆˆˆ 2 CCCCCCCCC231113334411133344axax6424cos 4sin /8 (19.7h) coscos2sin222CCCˆˆˆ , ax13ax1166ax
ˆˆˆ 2 CCCC24axaxax131144axcossinsin2cos (19.7i) CCCCCCˆˆˆˆˆˆ 24cos 2sin/ 2, 2 113311133344axax
ˆˆˆ 2 CCCC25axaxax131166axcossincos2cos (19.7j) CCCCCCCCˆˆˆˆˆˆˆˆ 4424cos 2sin/ 2, 2 1133446611133344axax
22ˆˆ CCCC26 sin ax cos ax sins ax 2 11 13 2 44 cos ax (19.7k) CCCCCCˆ ˆ ˆ 2 ˆ ˆ 4 ˆ cos 2 sin2 / 2 , 11 33 11 13 33 44 ax ax
ˆˆˆˆ 42242 CCCCC3333ax13axax11ax44axcos2cossinsinsin 2 , (19.7l)
(Note: the global stiffness component C33 is independent of the axial azimuth, ax ),
CCCCCCCˆˆˆˆˆˆ 24cos2sin2sin/ 4 , (19.7m) 34331111133344axaxax
CCCCCCCsin 2cos24cos 2/ 4ˆˆˆˆˆˆ , (19.7n) 35axax331111133344ax
CCCCCCCCCsin2 sin 2 ˆ ˆ 4 ˆ 4 ˆ ˆ 2 ˆ ˆ 4 ˆ cos 2 / 4 , 36 ax ax 33 11 44 66 11 13 33 44 ax
(19.7o)
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 CCCCCCCCC44 11 2 13 33 4 44 11 2 13 33 4 44 cos 4 ax sin ax / 8 (19.7p) ˆˆ2 2 2 CC44cos ax 66 sin ax cos ax ,
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2 ˆˆˆˆˆ CCCCCC45axax1113334466sinsin 222 (19.7q) CCCCˆˆˆˆ 24cos 2/ 4, 11133344ax
2 CCC46axaxax44ax66axcossincos2cos 222sin (19.7r) 2cosCCCC 22sin/ 2, 2 44ax111333ax
2 ˆˆˆˆˆˆˆˆ CCCCCCCCC55ax1113334411133344axcos2424cos 4/ 8 (19.7s) ˆˆ222 CC44ax66axaxcossinsin,
CCCCCCcos sin sin cos2 ˆ 2 ˆ ˆ 2 ˆ ˆ 56 ax ax ax ax 11 13 33 44 66 (19.7t) CCCCCCˆ 2 ˆ ˆ 4 cos 2 2 ˆ ˆ sin2 / 2 , 11 13 33 44 ax 44 66 ax
ˆˆˆˆˆ C66 3632032CCCCC1113334466 ˆˆˆˆˆˆˆˆˆ 4248cosCCCCCCCCC1113334466ax11133344 224 (19.7u) 4cos 23 cos 4224cos 4sin 2/CCCCˆˆˆˆ 64 . 2 axax11133344axa x
Equation set 19.7 defines the twenty-one components of the global stiffness tensor that depend in a linear way on the five polar anisotropy components of the “crystal” stiffness tensor, while the coefficients of these linear functions depend on the polar angles of the symmetry axis.
The gradient and the Hessian of the fourth-order stiffness tensor, xxxCC and (where the tilde emphasizes that this is the fourth-order stiffness tensor rather than its Voigt or Kelvin matrix representation), are tensors of the fifth and sixth orders, respectively. Working with these tensors is inconvenient and unnecessary. For the further analysis, we arrange the global stiffness components as an array of length 21, and its gradient and Hessian will represent 2D and 3D arrays, respectively (representing collections of vectors and second-order tensors) of dimensions
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213and2133 . Furthermore, due to the symmetry of the Hessians, the 3D array can be reduced to a 2D 2 1 6 array, with the Hessians of each component represented by vectors of length six.
With this in mind, the “crystal to global” transform can be arranged as,
5 ˆ CACix ijax x, ax x j x , (19.8) j1 21 1 21 5 5 1
where Aa x a, x is a conversion matrix of dimensions 2 1 5 whose components are defined in equation set 19.7. The order (sequence) of the crystal frame components is,
ˆˆˆˆˆˆˆˆˆˆˆ C C123451113334466,,,,,,,, CCCCCCCCC , (19.9) and the order of the global tensor components is given in Table 1. The local (crystal) and global stiffness components and the axis direction angles are spatially dependent.
The components of matrix follow from equation set 19.7,
2 2 2 2 A1,1cos ax cos ax sin ax , 2 2 2 2 2 A1,22 cos ax cos ax sin ax sin ax cos ax , 44 (19.10a) A1,3 sin ax cos ax , 2 2 2 2 2 A1,44 cos ax cos ax sin ax sin ax cos ax ,
A1,5 0 ,
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222222 A2,1axaxaxaxaxaxcoscossincossincos , 422242 A2,2axaxaxaxaxaxcos2coscossinsinsin , 422 (19.10b) A2,3axaxaxsincossin , 42 A2,4axaxsinsin2, 2 A2,5ax2cos ,
2222 A3,1axaxaxaxcoscossinsin , 222 A3,2axaxaxax3cos 4cos/ 4cossin , 222 (19.10c) A3,3axaxax cossincos , 222 A3,4axaxax4cossincos , 22 A3,5axax2sinsin,
222 A4,1axaxaxaxaxax coscossincossinsin , 22 A4,2axaxaxaxaxax cos 2cossi ncossinsin , 32 (19.10d) A4,3axaxaxax coss ncossini , 32 A4,4axaxaxax4cossicossinn , A4,5axax sin 2sin,
222 A5,1axaxaxaxaxax coscossincos sin cos , 22 A5,2axaxaxaxaxax cos 2 cossincos sin cos , 33 (19.10e) A5,3axaxax cos sincos , 22 A5,4axaxaxaxax cos 2 cossinsi n 2 cos ,
A5,5 0 ,
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2222 A6,1axaxaxaxaxax coscossinsincossin , 222 A6,2axaxaxaxaxax cos 2cossinsincossin , 43 (19.10f) A6,3axaxax sincossin , 222 A6,4axaxaxaxaxcos 2cossinsin sin2 ,
A6,5 0 ,
222 2 A7,1axaxax cossincos , 22222 A7,2axaxaxaxax2cossincossinsin , 44 (19.10g) A7,3axax sinsin, 22222 A7,4axaxaxaxax4cossincossinsin ,
A7,5 0 ,
2 2 2 2 A8,1cos ax sin ax cos ax sin ax , 2 2 2 A8,23 cos 4 ax sin ax / 4 cos ax cos ax , 222 (19.10h) A8,3 cos ax sin ax sin ax , 22 A8,4sin 2 ax sin ax , 22 A8,52sin ax cos ax ,
222 A9,1axaxaxaxaxax cossincoscos sin sin , 22 A9,2axaxaxaxaxax cos 2 sincoscos sin sin , 33 (19.10i) A9,3axaxax cos sinsin , 22 A9,4axaxaxaxaxax2 cos 2 sincoscoi ss n sn i ,
A9,5 0 ,
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2 2 2 A10,1 cos ax sin ax cos ax cos ax sin ax cos ax , 22 A10,2 cos 2 ax sin ax cos ax cos ax sin ax cos ax , 32 (19.10j) A10,3 cos ax sin ax cos ax sin ax , 32 A10,44cos ax sin ax cos ax sin ax , A10,5 sin 2 ax cos ax ,
2222 A11,1axaxaxaxaxax cossincossincoss in , 222 A11,2axaxaxaxaxax cos 2sincossincossin , 43 (19.10k) A11,3axaxax sincossin , 222 A11,4axaxaxaxaxax2cos 2sincossincossi n ,
A11,5 0 ,
4 A12,1ax sin , 22 A12,2axax 2cossin, 4 (19.10l) A12,3ax cos , 2 A12,4ax sin2 , A12,5 0 ,
3 A13,1axaxaxcossinsin , A13,2axaxsin 4sin/ 4 , 3 A13,3axaxaxcossinsin , (19.10m) A13,4axaxsin 4sin/ 2 , A13,5 0 ,
3 A14,1cos ax sin ax cos ax , A14,2sin 4 ax cos ax / 4 , 3 A13,3cos ax sin ax cos ax , (19.10n) A14,3sin 4 ax cos ax / 2 , A14,3 0 ,
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4 A15,1axaxaxsincossin , 2 A15,2axaxaxcos 2s insin 2/ 2 , 22 A15,3axaxax cossincossin , (19.10o) 22 A15,4axaxax4cossincos sin , 2 A15,5axax sisinn2,
222 A16,1axaxax cossinsin , 222 A16,2axaxax2cossinsin , 222 A16,3axax cossinsin , (19.10p) 2222 A16,4axaxaxaxcoscoscos2sin , 22 A16,5axax sincos,
22 A17,1 cos ax sin ax cos ax sin ax , 22 A17,22cos ax sin ax cos ax sin ax , 22 A17,3 cos ax sin ax cos ax sin ax , (19.10q) 2 A17,4 1 2cos 2 ax sin ax sin 2 ax / 2 , 2 A17,5sin ax scos ax in ax ,
32 A18,1axaxaxax cossincossin , 32 A18,2axaxaxax2cossincossin , 32 A18,3axaxaxax cossinc ossin , (19.10r) 2 A18,4axaxaxaxaxax 2cos 2sincos 2cosc ssin o ,
A18,5axaxaxcosinscos ,
2 2 2 A19,1 cos ax sin ax cos ax , 2 2 2 A19,22cos ax sin ax cos ax , 2 2 2 A19,3 cos ax sin ax cos ax , (19.10s) 2 2 2 2 A19,4cos 2 ax cos ax cos ax sin ax , 22 A19,5 sin ax sin ax ,
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32 A20,1axaxaxax cossincossin , 32 A20,2axaxaxax2cossincossin , 32 (19.10t) A20,3axaxaxax cossincossin , 33 A20,4axaxaxaxaxax cossinsincossinsin 3 , A20,5axaxaxcossinsin ,
422 A21,1axaxax sincossin , 422 A21,2axaxax2sincossin , 422 A21,3axaxax sincossin , (19.10u) 22 A21,4axaxaxaxax53cos 48cos 2cos22cos 4sin2/16 , 2 A21,5axcos .
Thus, the components of matrix A215 are the known trigonometric functions of the two polar angles, axand ax . We arrange equation 19.8 as,
C AC ˆ , (19.11) where CCˆ and are arrays of lengths 5 and 21, respectively. The gradient of the global stiffness is an array 21 3, representing a collection of vectors,
ˆAA ˆ ˆ xCACCC x xax x ax . (19.12) ax ax
The Hessian of the global stiffness is an array 21 3 3, representing a collection of symmetric second-order tensors,
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ˆ xxxx CAC AAT3,1,2 CCˆˆT xxxx axax axax AAT3,1,2 CCˆˆT xxxx axax axax 22AA (19.13) CCˆˆ 22 xxxxaxaxaxax axax 2A Cˆ x axaxaxaxxxx axax AAˆˆ CCxxxx axax . axax
Multi-dimensional arrays in this relationship obey tensor multiplication rules, even when they are not true physical tensors. Notation T 3,1,2 means that the 3D array is transposed: indices 1, 2, 3 of the original become 3, 1, 2 of the result.
We apply equations 19.11, 19.12, and 19.13 to compute the global stiffness components, their gradients, and their Hessians, respectively. The results are listed in Tables 7, 8, 9, where the units of the (density-normalized) stiffness components are km/s 2 , while the gradients and Hessians are presented in the “relative form”, with the units kmandkm12 , respectively.
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Table 7. Stiffness components and their spatial gradients for polar
anisotropy of model 1 presented as general (triclinic) anisotropy
Component Relative gradient, km1 (in global frame) Value, k m / s 2 # Cij x1 x2 x3 1 1 2 2 1 C11 1.7192147 10 1.6132821 10 6.8401473 10 5.8359797 10 1 1 1 1 2 C12 1.115534710 3.405867510 2.806899010 1.105400810 1 1 1 3 C13 8 . 8 8 6 2 2 4 2 3 . 7 5 4 3 5 8 1 1 0 3.814732210 1.189540210 2 2 2 4 C14 1 . 1 1 1 6 3 8 6 8.3840966 10 2.9826347 10 6.1125491*10 2 2 1 5 C15 1 . 3 7 5 6 9 3 6 1.078342510 9.7174204*10 2.209186810 1 2 1 1 6 C16 5.616245410 7.512492010 1 . 5 7 2 9 3 0 1 1 0 3.200010310 1 1 2 2 7 C22 1.719214710 1.567920610 9.095910610 6.921783810 1 1 1 8 C23 8.8862242 3.7421849 10 3.8752678 10 1.1604017 10 2 2 1 9 C24 1.3756936 3.812965110 3.881449610 1.554609810 2 1 1 10 C25 1.1116386 4.000773610 1.881497610 1.660475710 1 2 1 1 11 C26 5.616245410 8.249329210 1.939348310 3.376384510 1 1 1 2 12 C33 1.306600210 1.567139110 1.394202410 7.925536210 1 2 1 1 13 C34 6.456964510 3.386643210 1.505251710 2.202227010 1 4 1 1 14 C35 6.456964510 8.481672810 3.231556810 3.033178210 1 2 1 1 15 C36 1.558044710 1.688032510 6.725396210 2.360067510 1 1 1 16 C44 3.1065241 1.819819310 2.610426110 3.772022410 1 2 2 1 17 C45 3.037240810 9.117638310 6.564743410 2.626850310 2 1 1 18 C46 1.3971923 10 6.5595707 10 2.0529577 7.5963197 10 1 1 1 19 C55 3.1065241 1.8877000 10 2.2728652 10 3.9345062 10 2 1 20 C56 1.3971923 10 1.3467419 5.4881278 8.9387572 10 1 1 1 21 C66 3.0780040 1.7915491 10 2.7946435 10 3.9009910 10
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Table 8. Hessians of polar anisotropy stiffness components presented
as general (triclinic) anisotropy: diagonal elements, model 1
Component Relative Hessian, km1 (in global frame)
# Cij xx11 xx22 xx33 2 1 2 1 C11 8.0031180 10 1.0611903 10 8.1321886 10 1 2 2 2 C12 2.094960710 2.415516710 1.041627610 1 2 2 3 C13 2.147464910 1.147785610 1.141724310 2 3 1 4 C14 2.2296092 10 9.8673368 10 1.7823804 10 1 2 2 5 C15 1.221663410 1.962226510 4.159900910 1 1 1 6 C16 2.159895210 1.509684610 1 . 1 5 7 7 6 6 1 1 0 2 1 2 7 C22 6.812291810 1.055853510 5.704233410 1 2 3 8 C23 2.1131782 10 1.7350969 10 5.4424859 10 2 2 1 9 C24 4.118079610 7.496050810 1.59947810 1 2 2 10 C25 1.000369610 4.639124210 2.636095910 1 1 2 11 C26 2.317841510 1.623957810 6.672789910 2 1 2 12 C33 5.562824410 1.518585010 6.515337210 2 1 1 13 C34 6.135915510 2.208276510 1.951377210 1 1 2 14 C35 1.591093410 2.378826110 2.047256110 1 1 2 15 C36 1.683450310 3.646000510 6.293094210 1 1 1 16 C44 1.828459210 3.120102410 1.725469210 1 2 2 17 C45 2.179475710 7.007945310 6.971454710 1 1 18 C46 4.0074558 10 1.7685679 4.9305893 10 1 1 1 19 C55 2.0049799 10 3.0378755 10 1.3836326 10
20 C56 2.0786628 2.9371336 2.9371336 1 1 1 21 C66 1.8145087 10 3.4384906 10 1.7343105 10
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Table 9. Hessians of polar anisotropy stiffness components presented
as general (triclinic) anisotropy: off-diagonal elements, model 1
Component Relative Hessian, km1 (in global frame)
# Cij xx12 xx23 xx13 2 1 2 1 C11 1.4046489 10 1.3362002 10 8.9484845 10 1 2 2 2 C12 1.114306810 4.417363910 3.489906610 1 2 3 3 C13 1.531502410 2.629902510 9.713112210 1 1 3 4 C14 3.0689364 10 1.0652913 10 2.8941059 10 2 1 1 5 C15 6.723147210 1.742587610 1 . 5 7 0 7 5 3 1 0 2 2 1 6 C16 5.773595810 9.709164210 1.127093910 2 1 1 7 C22 4.285819410 1.109661410 1.250624610 1 3 2 8 C23 1.3141530 10 8.8555138 10 4.2471284 10 1 2 1 9 C24 1.249291810 7.321274110 1.315046210 1 10 C25 2.7641566 3.2199762 6.408131610 1 2 2 11 C26 1.149033010 6.630260110 3.218382610 2 2 1 12 C33 6.730551910 9.979997910 1.271490310 1 2 2 13 C34 7.858356210 5.116224310 8.514691710 1 1 14 C35 1.1825760 5.395928510 4.871824510 1 1 1 15 C36 1.903197210 3.726128810 4.017134510 1 1 1 16 C44 1.741961210 6.893352410 5.926264910 1 1 1 17 C45 1.922714210 3.583283010 1.295725710 2 1 2 18 C46 9.1433584 10 3.0765028 10 4.1552149 10 2 2 1 19 C55 5.3613438 10 3.4570020 10 3.8139665 10 1 1 1 20 C56 3.9373203 10 1.8517986 10 9.8498727 10 2 1 1 21 C66 1.8525098 10 3.6276162 10 3.1987921 10
Now we have all the necessary data to compute the ray velocity and its derivatives for a general triclinic medium applying the workflow and technique of Part I. Page 81 of 146
The gradients and Hessians of the reference Hamiltonian
The slowness gradient and Hessian are,
HPp +1.5292672+2.03902303.3983716km / s , (19.14)
1.3156487 106.58379781.05704871 10 1 1 2 HPpp 6.58379789.12684671.4589380 10 km /s . (19.15) 1.0570487 101.458938011 105.1864 173
The spatial gradient and Hessian are,
HP 1.0307910 107.3508827122 103.6182110 10 km1 , (19.16) x
6.7942998 107.2816551221 101.0171649 10 2 112 HPxx 7.2816551 101.0451836 101.2496685 10 km . (19.17) 1.0171649 101.0498619112 101 5.1176013 0
The mixed Hessian is,
3.0292332 101 3.7083701 10 1 5.9581842 10 1 1 21 2 HPxp 2.7168949 10 4.2522278 10 7.7610092 10 km . (19.18) 3.3735610 101 4.1993330 10 1 6.1264755 1 0 1
The gradients and Hessians of the arclength-related Hamiltonian
The slowness gradient and Hessian are,
HPp 0.36 0.48 0.80 r , (19.19)
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4.26328584.6285195 107.495419322 10 22 HPpp 4.6285195 104.33176887.5048987 10 km/s . (19.20) 7.4954193 107.522 048987 104.41280 5 5
The inverse of this matrix is,
2.3465669 102.4389839133 103.9443069 1 0 1 3 13 HPpp 2.4389839 102.3094597 103.8862917 10 s / km . (19.21) 3.9443069 103.8862917331 1020 .2674629 1
The spatial gradient and Hessian are,
HP 2.4265528 101.73044828.51751610223 9 s / km2 , (19.22) x
7.2692223 108.759214933 102.7058305 10 2 322 3 HPxx 8.7592149 101.7086149 103.0880151 10 s / km . (19.23) 2.7058305 103.0880151223 108.7863832 1 0
The mixed Hessian is,
4.5890609 105.618709422 108.1874695 10 2 222 3 HPxp 1.3781332 103.5182881 106.9843915 10 s / km . (19.24) 3.2898342 101.2395824322 103.8903298 1 0
Finally, we compute the two gradients, xrvray x,,, r v ray x r , and the three Hessians,
x xrvxrayrayray rx rvx r, rvx ,, r ,, of the ray velocity magnitude. As expected, the resulted ray velocity derivatives fully coincide with the derivatives listed in equations 14.37-14.41 for compressional waves, computed directly for polar anisotropic media (without its conversion to general anisotropy). We define the discrepancies E between the two methods as,
TP E , (19.25) P
Page 83 of 146 where TPa nd stand for the numbers related to triclinic and polar anisotropies, respectively.
These discrepancies are listed below.
For the inverted slowness vector,
E p 2.98 108.48151615 101.32 10 . (19.26)
For the ray velocity magnitude,
16 Ev ray 3.8910 . (19.27)
For the spatial gradient and Hessian,
Ev4.28 101.361514 102.19 10 41 , (19.28) x ray
4.78 104.01151515 109.65 10 151416 Evxx ray 4.01 101.61 102.89 10 . (19.29) 9.65 102.891516 105.61 10 15
For the directional gradient,
Ev2.43 102.451214 106.50 10 41 . (19.30) r ray
The first component of the relative error in this array essentially exceeds the other two, but
the first component of the directional gradient rvray is small (as compared to the other
components; see equation 14.39), thus, the corresponding absolute error is not particularly
large.
For the directional Hessian,
5.71 1014 1.47 10 13 7.07 10 14 13 14 14 Evrr ray 1.47 10 6.16 10 2.48 10 . (19.31) 7.07 1014 2.48 10 14 1.10 10 13
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For the mixed Hessian,
7.55 1014 4.14 10 14 5.60 10 14 14 13 13 Evxr ray 4.66 10 5.23 10 1.09 10 . (19.32) 5.23 1014 41.9 10 14 5.000 1 14
As expected, the discrepancies between the results of the two approaches are “almost” zero, where
12 the worst relative discrepancy does not exceed Emax 2.510 , and most of relative discrepancies are much smaller than that. Note that both methods are “almost” analytic and exact.
We say “almost”, because computing the ray velocity derivatives includes the slowness vector inversion, which is a numerical procedure. In addition to the round-off errors (which are the main source of the tiny discrepancies in equations 19.28-19.32), the governing equations for this inversion are formulated differently for polar and triclinic anisotropies. Still, the inverted slowness vectors with the two methods are pretty close: the relative discrepancy for the inverted slowness components does not exceed 3 1 0 15 (equation 19.26).
20. TTI MODEL 2: PARAMETER SETTING AND SLOWNESS INVERSION
Next, we consider another model representing an extreme (but still realistic) case of a large negative anellipticity. Unlike compacted sediment shale/sand rocks, this model, suggested by
Grechka (2013), is characterized with high positive and negative values,
vfP 3 km / s, 0.75, 0.3, 0.15 . (20.1)
These values represent polar anisotropic materials in unconsolidated shallow sand rocks (also used by Ravve and Koren, 2021b), and for the selected ray direction, the slowness inversion yields a
Page 85 of 146 shear wave triplication (i.e., the same ray velocity direction corresponds to three qSV slowness vectors with their corresponding ray velocity magnitudes; obviously, the triplication makes the problem more challenging).
Remark: The medium properties suggested by Grechka (2013) do not include the Thomsen parameter because this parameter is only related to SH shear waves. Since we also consider SH waves in this study, we assigned the value for imposing the stability condition: The stiffness tensor should be positive definite (positive semidefinite in the case of the acoustic approximation)
(Slawinski, 2015; Adamus, 2020). To find the eigenvalues, we arrange the polar anisotropic stiffness tensor in the Kelvin matrix form (rather than the commonly used Voight matrix form),
CCC111213 CCC121113 CCC131333 C ,2CCC121166 . (20.2) 2C44 2C 44 2C66
This polar anisotropy stiffness matrix has two double eigenvalues, 2and2CC4466 , and two simple eigenvalues (e.g., Bona et al, 2007), which are the roots of a quadratic equation. If the smaller root of this equation is positive, then of course, the larger root is also positive, and the material is stable.
Thus, we impose the condition min 0 , where,
2 min AAff A2 16 Bf 1 4 34 1, 2 vP (20.3) where1AffBf 4 2 f 4 1and2.
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Solving equation 20.2 for , we find its upper limit value, crit , where,
Bf 1/ 2, . (20.4) crit 1 f
For the polar anisotropic parameters c vfP , , , given in equation 20.1 and Table 10, the stability condition leads to a large negative critical parameter (equation 20.4),
crit 0.24376941, which means that a material with the given values of f ,, and a value of
c r it , does not exist; therefore, we assigned the value 0.25 . Equation 20.4 cannot be applied for the acoustic approximation (AC) due to the factor 1 f in the denominator (but equation 20.3 still can). Note that parameter is irrelevant for the AC. For f 1, equation 20.3 yields , i.e., only polar anisotropic (transversely isotropic) media with a non-negative anellipticity parameters, /12 , are stable under the AC conditions. However, the
AC can be considered an abstract model, introduced to decrease the number of medium parameters and to simplify the governing equations of compressional waves in elastic media; thus, to the best of our understanding, it can be (carefully) applied to polar anisotropy with negative anellipticities as well; we will show that even in this case it provides a reasonable accuracy for the quasi- compressional ray velocity; however the AC accuracy of the ray velocity derivatives (in particular the second derivatives) may be questionable for both positive and negative anellipticities .
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Table 10. Properties of the polar anisotropy and their spatial gradients, model 2
Component Relative gradient, km1 (in global frame) Value # mi x1 x2 x3
1 vP 3 km / s –0.0541 +0.0385 +0.0254 2 f 0.75 +0.0650 –0.1241 +0.0874 3 +0.30 –0.0633 –0.0352 +0.0759 4 –0.15 +0.0429 –0.1155 +0.0924 5 –0.25 –0.0212 +0.1170 +0.0763
6 ax 0.69509432 r a d –0.0405 +0.1252 –0.1635
7 ax 1.11832143 ra d +0.0815 –0.1544 +0.0916
Table 11. Spatial Hessians of polar anisotropy properties, model 2
Component Relative Hessian, km2 (in global frame)
# mi xx11 xx12 xx13 xx22 xx23 xx33
1 vP –0.0134 +0.0256 –0.0571 +0.0479 –0.0563 +0.0211 3 f +0.0186 +0.0098 +0.0692 –0.0315 +0.0559 –0.0687
3 –0.0196 +0.0473 –0.0347 +0.0292 –0.0112 –0.0812 4 –0.0405 –0.0651 +0.0137 –0.0321 +0.0543 –0.0301 5 +0.0244 –0.0105 –0.0289 +0.0913 +0.0487 –0.0507
6 ax –0.0272 –0.0921 +0.0177 +0.0599 –0.0141 –0.0614
7 ax +0.0719 +0.0208 –0.0255 +0.0414 –0.0362 +0.0413
The symmetry axis direction angles lead to the following Cartesian components of the symmetry axis direction,
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k 0.280 0.576 0.768 . (20.5)
We assume the following ray velocity direction,
r 0.56960.480.6672 , (20.6) and perform the slowness inversion. We obtain five solutions (also obtained in Ravve and Koren,
2021b) listed in Table 12: One for the quasi-compressional qP waves, three for the triplicated qSV shear waves ( qSV,qSV,qSV123 ) and one for the SH waves. In addition, we obtain a solution for the AC waves, q Pac , by setting f 1 .
Table 12. Results of the slowness vector inversion, model 2
Mode p1 ,s/km p2,s/km p3,s/km vphs ,km/s vray,km/s
qPac 0.23331742 0.20380847 0.25011372 2.5115500 2.5150808 qP 0.23355822 0.20428276 0.24952117 2.5114714 2.5152739
q SV1 0.39826410 0.35649389 0.39921519 1.4989371 1.5052881
qSV2 1.0331849 1.2499129 0.012121941 0.61663811 0.84719014
qSV3 0.45355759 0.057425633 1.5780707 0.60865734 0.74693784 SH 0.41700904 +0.35443873 0.47870791 1.3753134 1.3754158
21. MODEL 2. RAY VELOCITY DERIVATIVES FOR qP WAVES
The workflow and formulae used in this section are described in detail in section 15 for the parameters of model 1. Thus, in this section we only list the final results and discuss their accuracy.
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Gradients and Hessians of the ray velocity magnitude
After computing all necessary derivatives of the reference Hamiltonian and then the arclength- related Hamiltonian, we can establish the gradients and Hessians of the ray velocity magnitude.
The computation of the compressional ray velocity yields vray 2.5152739km / s . The spatial gradient and Hessian of the ray velocity magnitude are, respectively,
v 1.5304772 101.40644401 103.202297512 1 0 s1 , (21.1) x ray
1.9082789 109.6134592321 101.4220633 10 2 111 xxvray 9.6134592 101.6118634 101.9719449 10 ksm . (21.2) 1.4220633 101.971944911 101. 0376433 101
The directional gradient and Hessian of the ray velocity magnitude are, respectively,
v 4.4930092 108.50844022 9 109.95693 89 10 2 km / s , (21.3) r ray
2.3617230 104.3682610111 104.4854643 10 111 rrvray 4.3682610 10 7.6261840 107.9404761 10 km / s , (21.4) 4.4854643 107.94047611 108.0495430110 1 and the mixed “spatial-directional” Hessian of the ray velocity magnitude is,
1.8312703 104.8123132222 105.0254825 10 2 1 1 1 xrvrya 5.8889453 101.3407109 101.4672895 10 s . (21.5) 7.4386047 101.488732921 101.70607721 10
The gradient of the ray velocity wrt the model parameters is presented as a concatenation of the material and geometric counterparts, where the material part reads,
v 8.3842464 101 1.0988204 10 3 2.7597849 10 3 3.5790534 , (21.6) c ray and the geometric part (symmetry axis direction angles) reads,
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v 1.3194051 102.689763512 10km /sradian . (21.7) a ray
The 66 Hessian matrix of the ray velocity wrt the model parameters mmvr a y consists of four blocks: two symmetric diagonal blocks ( ccaavvrayrayand ) of dimensions 4 4 a nd 2 2 , and related to either material or geometric properties of the model, respectively, and two mutually transposed off-diagonal blocks 2 4 and 4 2 , a cvvrayand c a ray , respectively, related to both the material and geometric properties (equation 14.46). The material block reads,
1.1102230 103.66273461644 109.1992831 101.19301 78 3.6627346 103.71345834 104.7864340343 106.10411 48 10 v . ccray 4 432 9.1992831 104.7864340 107.3454004 102.15845 86 10 32 1.19301786.10 41148 102.1584586 105.2135064
(21.8)
The mixed block reads,
4.3980170 102.74065442221 106.9154265 101.67256 30 10 v acray 3322 8.9658784 105.5871486 101.4097916 103.40971 78 10
(21.9)
The first row of the mixed block has the units r a d i a n1 , while the other components have the units km / s radian . Finally, the geometric block reads,
1.6464628 3.0556052 101 km v . (21.10) aaray 112 3.0556052 10 1.2941038 10 s radian
Validation test
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As in section 15, in order to check the correctness of all the ray velocity derivatives, obtained analytically, we compare their values to numerical values obtained by the finite differences method. The numerical directional derivatives of the ray velocity, approximating the gradient and the Hessians rrrvvrayray, and the mixed Hessian xrvr a y , were computed wrt the zenith and azimuth angles, rayraya n d , of the ray direction vector, and then were related analytically to the Cartesian components of this direction. The numerical approximations for the ray velocity derivatives were computed with the spatial resolution 1 0 k4 m and angular resolution 105 radians for the spatial and directional gradients and Hessians of the ray velocity magnitude, including the mixed Hessian. When computing the ray velocity derivatives wrt the model
4 parameters, mmmvvrayrayand , the resolution was 10 times the absolute value of the parameter. The (unitless) relative errors of the numerical approximations of the ray velocity derivatives for qP waves are listed below.
For the spatial gradient and Hessian,
Ev 3.21 102.5011109 101.98 10 , (21.11) x ray
8.80 102.06677 102.85 10 777 Evxx ray 2.06 102.17 102.28 10 . (21.12) 2.85 102.28776 101.31 10
For the directional gradient and Hessian,
Ev 6.60 1011 3.87 10 10 2.13 10 10 , (21.13) r ray
2.02 105 1.80 10 6 1.03 10 5 666 Evrr ray 1.80 10 8.55 10 6.75 10 . (21.14) 1.03 105 6.75 10 6 9.71 10 6
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For the mixed Hessian,
1.29 102.03666 101.80 10 667 Evxrray 1.31 101.13 102.91 10 . (21.15) 5.94 101.37767 106.37 10
For the material, cvr a y , and geometric, avr a y , parts of the model-related gradient mvr a y
,
Ev5.14 101.6114678 101.59 102.30 10 , (21.16) c ray
Ev 6.20103.441086 . (21.17) a ray
For the material, ccvray , geometric, aavray , and mixed, acvray , parts of the
model-related Hessian mmvray ,
1.66 101.61678 102.29 10 1.66 101.676666 103.92 102.58 10 Ev , (21.18) ccray 7667 1.61 103.92 101.06 103.23 10 8678 2.29 102.58 103.23 105.98 10
6.89 101.778666 101.02 101.83 10 Ev , (21.19) acray 6666 3.44 105.15 103.75 102.81 10
2.95 108.3187 10 Ev . (21.20) aaray 77 8.31 106.43 10
As in section 15, a dash has been placed to the upper left corner of Evccray because the relative error cannot be estimated for a parameter whose exact theoretical value is zero; however,
2 2 11 the numerical (finite difference) estimate was vvray / P 4.93 10 s / km or
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1.24 1010 (unitless) after the normalization with the ray velocity magnitude, which is well below the numerical error for the other second derivatives and indicates the correctness of the vanishing value.
Sensitivity of the ray velocity derivatives to the spatial derivatives of the model parameters
As in section 15, our next test checks the significance (the effect) of the spatial gradients and
Hessians of the model parameters f ,,,and axax (all model parameters except the axial compressional velocity vP ) on the derivatives of the ray velocity magnitude for compressional waves. For this, we repeat the computations, assuming mmxxx0 and0 , except
xxxvvPPand which are taken into account. Obviously, this does not affect the directional derivatives, rrrvvrayrayand , and the derivatives wrt the medium properties,
mvvrayand m m ray . We re-compute the spatial derivatives, xxxvvrayrayand and the mixed Hessian xrvray , and compare the vector and the two tensors obtained for three cases: 1) accounting for the spatial derivatives of all model parameters, 2) accounting for the spatial derivatives of the medium properties and ignoring those of the symmetry axis direction angles, and 3) accounting only for the spatial derivatives of the axial compressional velocity. For cases 2
ˆ and 3, we use notations xxand , respectively. As mentioned in Section 15, case 3 is referred to as FAI media – Factorized Anisotropic Inhomogeneous media (Červený, 2000). To compare the spatial gradients, we compute the ratio of their magnitudes and the angle between their directions; to compare the spatial Hessians, we compute the ratio of their corresponding eigenvalues. The non-symmetric mixed Hessian, 33 , has a single zero eigenvalue, and the corresponding
Page 94 of 146 eigenvector is the ray direction r (Koren and Ravve, 2021); this property holds even if we ignore the spatial derivatives of the model, so we compare the two other eigenvalues, which prove to be real for the given example and for qP waves.
Accounting for the spatial derivatives of the medium properties, ccx x xa n d and ignoring those of the symmetry axis direction angles, aax x xa n d , we obtained the following results,
v 1.592135 101.5677084112 101.4273089 10 s1 , (21.21) x ray
9.6991852 109.57850353 101.494205621 10 211 1 xx vray 9.5785035 101.4092704 101.6999229 10 kms , (21.22) 1.4942056 101.6999229112 106.5688008 10
2.9663811 105.6174554333 106.5737849 10 333 1 xr vray 1.3971873 102.6458628 103.096301 10 s . (21.23) 1.855089 103.51299423 104.111053 10 5 1
Accounting for the spatial derivatives of the axial compressional velocity only, xxxvvPPand , we obtained,
ˆ v 1.3607632 109.6838046122 106.388 791058 s1 , (21.24) x ray
3.3704670 102 6.4391012 1021 1.4362214 10 ˆˆ 2111 xx vray 6.4391012 10 1.2048162 10 1.4160992 10 km s , (21.25) 1.4362214 10112 1.4160992 10 5.3072280 10
2.4307180 103 4.6030665 10 33 5.3867039 10 ˆ 3 3 3 1 xr vray 1.7298085 10 3.2757497 10 3.8334215 10 s . (21.26) 1.1412243 103 2.1611440 10 3 2.5290625 10 1
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The magnitudes of the ray velocity gradients computed with complete and incomplete data are related as,
ˆ ˆ xvray x v ray x v ray x v ray 1.0646079 , 0.85026491 . (21.27) xvray x v ray x v ray x v ray
The angles between the corresponding gradient directions are,
xxvvrayray 2 o xxvvrayray,arccosr 9.5432 004 10 ad5.4678511, xxxxvvvvrayrayrayray
vvˆ vv,arccosrad13.989148.ˆ xxrayray 2.4415670 101 o xxrayray ˆ ˆ xxxxvvvvrayrayrayray
(21.28)
We compute the ratios of the eigenvalues (major, intermediate and minor) of the spatial Hessians,
vv ˆˆ majorx x ray0.88930125 , major x x ray 0.73499 760 , majorx xvv ray major x x ray vv ˆ ˆ intermx x ray1.01261071 , interm x x ray 0.31614778 , (21.29) intermx xvv ray interm x x ray v ˆˆv minorxx ray 1.19452327 ,minorxx ray 1.32545392 . minorxxv ray minorxxv ray
The “complete” mixed Hessian, xrvray , has a zero eigenvalue, and the “incomplete” mixed
ˆ Hessians, xrxr vvrayray and , have two zero eigenvalues,
v 7.2334381 101.74850442 10 2 0, xrray v 3.7905367 1003 0, (21.30) xrray ˆ v 3.3740943 1003 0 . xrray
As also noted in section 15 for model 1, the obtained “errors” highly depend on the magnitudes of the spatial derivatives of the model parameters. Of course, at ray points within transition zones
Page 96 of 146 between layers, the gradients and Hessians of the model parameters are higher and the “errors” increase.
22. MODEL 2. ACOUSTIC APPROXIMATION FOR COMPRESSIONAL WAVES
Following the workflow and formulae described in Section 16, we compute and list the ray velocity derivatives using the acoustic approximation (AC) for qP waves, but now for the polar anisotropic parameters of model 2. Recall that the medium with the properties listed in Table 10 has a (huge) negative anellipticity; as mentioned, such a medium is formally unstable under the assumption of
AC. We show that although the AC can still be applied with high accuracy for computing the ray velocity magnitude, the accuracy of the ray velocity derivatives considerably degrades, in particular the second derivatives.
Skipping the intermediate results, related to the derivatives of the reference and arclength-related
Hamiltonians, we present only the required gradients and Hessians of the ray velocity magnitude.
Spatial gradient and Hessian of the ray velocity magnitude,
v 1.5321642 101.41139151 103.143003812 1 0 s1 , (22.1) x ray
2.0769204 10321 9.5915004 10 1.4219269 10 2 111 xxvray 9.5915004 10 1.6093135 10 1.9662911 10 ksm . (22.2) 1.4219269 1011 1.9662911 10 1.0273439 101
Directional gradient and Hessian of the ray velocity magnitude,
v 4.3289968 108.19784922 7 109.59347 19 10 2 km / s , (22.3) r ray
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2.2753472 104.2084925111 104.3213646 10 111 rrvray 4.2084925 10 7.3472115 107.6499309 10 km / s . (22.4) 4.3213646 107.64993091 107.7548990114 1
The mixed Hessian of the ray velocity magnitude,
1.8012127 104.7063162222 104.9235650 10 2 1 1 1 xrvrya 5.7478903 101.3057670 101.4301079 10 s . (22.5) 7.2197431 101.444363821 101.65547241 10
The gradient of the ray velocity wrt the model parameters, mvr a y , includes the medium
part: the derivatives wrt the material components, c vP ,, (where the shear factor
f has been eliminated), and the geometric part: the derivatives wrt the symmetry axis
direction angles. The material part is,
v 8.3836026 102.827927113 103.5800398 , (22.6) c ray
and the geometric part is,
v 1.2712417 102.591576712 10 . (22.7) a ray
The Hessian of the ray velocity wrt the model parameters, mmvray , includes the
material, the geometric, and the mixed blocks. The material block reads,
09.4264237 101.1933466 4 432 ccvray 9.4264237 107.0562750 102.0156696 10 , (22.8) 1.19334662.0156696 105.20656772 the mixed block reads,
4.2374722 102 7.0813587 10 2 1.4236922 10 1 v , (22.9) acray 32 2 8.638589 10 1.4436188 10 2.9023651 10
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and the geometric block reads,
1.58621932.9437760 10 1 v . (22.10) aaray 1 1 2.9437760 101.2468053 10
Validation
As in Section 16, the relative error for all gradient and Hessian components has been computed
twice: comparing the numerical solution N for the AC with its analytical solution A (error E ),
and comparing the exact analytical solution for compressional wave P with the analytical solution
for the AC (error E ),
NAAP EE, . (22.11) AP
Accuracy of the spatial gradient and Hessian of the ray velocity magnitude,
Ev 6.30 102.491110 102.04 10 9 , (22.12) x rya
Ev 1.10 103.52332 10.85 10 1 , (22.13) x ray
4.47 105 4.03 10 7 1.12 10 7 7 77 E xxvray 4.03 10 6.10 10 1.43 10 , (22.14) 1.12 1077 1.43 10 7 5.52 10
8.84 102.2825 109.59 103 333 E xx vray 2.28 101.58 102.87 10 . (22.15) 9.59 102.8753 109.93 103
Accuracy of the directional gradient and Hessian of the ray velocity magnitude,
E v 9.05 1010 1.64 10 101 4.49 10 0 , (22.16) r ray
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E v 1.07 102.21211 101.14 10 , (22.17) r ray
2.70 101.54556 101.36 10 5 66 E rrvray 1.54 108.26 101.52 10 , (22.18) 1.36 101.5267 104.29 106
3.6573 103.6575222 103.6585 10 2 22 E rr vray 3.6575 103.6581 103.6590 10 . (22.19) 3.6585 103.6590222 103.6604 10
Accuracy of the mixed Hessian of the ray velocity magnitude,
6.44 102.27666 103.57 10 6 66 E xrvray 2.20 101.34 101.64 10 , (22.20) 1.51 102.0176 101.21 106
1.64 102.2022 102.03 102 222 E xr vray 2.40 102.61 102.53 10 . (22.21) 2.94 102.9822 102.97 102
The matrices in equations 22.19 and 22.21 look like systematic discrepancies, rather than random ones.
Accuracy of the material part of the ray velocity gradient wrt the model parameters,
Ev7.57 1013 7.28 10 101 2.26 10 0 , (22.22) c ray
Ev7.68 102.47524 102.76 10 . (22.23) c ray
Accuracy of the geometric part of the ray velocity gradient wrt the model parameters,
Ev7.16 1010 3.45 10 8 , (22.24) c ray
Ev3.65 1022 3.65 10 . (22.25) c ray
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We re-emphasize that the relative discrepancies between the ray velocity derivatives for the acoustic approximation and elastic models wrt the zenith and azimuth angles of the medium symmetry axis are identical.
Accuracy of the material part of the ray velocity Hessian wrt the model parameters,
1.17 109.8559 10 555 Evcc ray 1.17 104.29 102.35 10 , (22.26) 9.85 102.35957 105.00 10
2.47 102.7624 10 2 22 Evcc ray 2.47 103.94 106.62 10 . (22.27) 2.76 106.624 101.33 1023
Accuracy of the mixed part of the ray velocity Hessian wrt the model parameters,
6.92 101.23866 101.49 10 Ev , (22.28) acray 766 5.01 106.31 103.61 10
3.65 102.40221 101.49 10 Ev . (22.29) acray 221 3.65 102.40 101.49 10
The first row of the mixed Hessian matrix in equation 22.9 and its acoustic approximation error in equation 22.29 is related to the zenith angle ax , while the second row – to the azimuth angle ax
. The columns are related to the material properties c vP ,, . As we see, the two lines are identical.
Accuracy of the geometric part of the ray velocity Hessian wrt the model parameters,
7.44 1098 6.35 10 Ev , (22.30) aaray 8 7 6.35 10 2.00 10
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3.6590 1022 3.6598 10 Ev . (22.31) aaray 2 2 3.6598 10 3.6549 10
The discrepancies in the last equation seem systematic, rather than random.
As we see, the numerical errors E of the finite difference approximations are small for the gradients and acceptable for the Hessians, which indicates that the analytical formulae for the AC derivatives of the ray velocity are correct. However, the discrepancies E of the gradients and, in particular, the Hessians computed with the elastic and acoustic models are large, and for some components unacceptable. The reason is inherent to the nature of the differential operator. The AC for the ray velocity in itself is accurate; still, it is an approximation. The accuracy of the derivatives of an approximated function is much worse than the accuracy of the function itself, and the accuracy of the second derivatives is even worse. While the accuracy of the AC slowness inversion along with the computed phase and ray velocities is very high, using the AC for computing the qP ray velocity derivatives is questionable. The reduced computational complexity of the AC is accompanied by a declined accuracy of its results when the derivatives of the approximated values are involved.
23. MODEL 2. COMPUTING RAY VELOCITY DERIVATIVES FOR qSV1 WAVES
For the given model properties m and the ray direction r , the shear waves have been triplicated; qSV1 is the fast shear wave with the highest ray velocity, vray 1.5052881km/s (Table 12). In this section, we list the derivatives of the ray velocity magnitude for this wave type and validate them numerically with the finite-difference approximations.
The spatial gradient and Hessian of the ray velocity magnitude are,
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v 2.2218753 103.22460691 101.416973611 1 0 s1 , (23.1) x ray
5.4139253 102.6664177221 102.5063221 10 2 111 xxvray 2.6664177 101.2459680 101.9422356 10 ksm . (23.2) 2.5063221 101.942235611 101. 8219036 101
The directional gradient and Hessian of the ray velocity magnitude are,
v 4.5011424 108.52384222 9 109.97496 30 10 2 km / s , (23.3) r ray
2.3882582 104.4183224111 104.5429142 10 111 rrvray 4.4183224 10 7.7198161 108.0482672 10 km / s . (23.4) 4.5429142 108.04826721 108.17343501111
The mixed Hessian of the ray velocity magnitude is,
1.3399489 103.8843283222 103.9384179 10 2 1 1 1 xrvrya 4.7186527 101.1195001 101.2082352 10 s . (23.5) 6.5600793 101.322511021 101.51149191 10
The ray velocity gradients wrt the material properties, c vfP,,, , and wrt the
geometric properties, a ax, ax, are,
v 5.0176269 1013 3.0139010 2.3708889 10 3 7.107072 10 , (23.6) c ray
where the first component is unitless, and the others have the units of velocity, and
km v 1.3217935 102.612946326 10 . (23.7) a ray sradian
The ray velocity Hessians wrt the material, mixed, and geometric properties of the model,
are,
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6.6613381 101.00463377.90296311643 102.3690240 10 1.00463376.01360971.5177692 105.5966855 10 22 v , ccray 422 2 7.9029631 101.5177692 101.2697265 102.75153 34 10 3 222 2.3690240 105.5966855 102.7515334 105.0866535 10
(23.8)
4.4059783 103.48333422121 105.9780979 101.79061 46 10 v , acray 322 2 8.9821085 107.1011892 101.2187060 103.65037 98 10
(23.9)
1.6686393.1002695 10 1 v . (23.10) aaray 1 1 3.1002695 101.3044241 10
Validation
The relative errors of the finite-difference approximations are listed below.
The relative errors of the spatial gradient and Hessian of the ray velocity magnitude are,
Ev1.37 102.991110 108.73 10 01 , (23.11) x ray
2.38 101.08667 101.26 10 6 77 Evxxray 1.08 101.49 101.01 10 . (23.12) 1.26 101.0177 101.66 107
The relative errors of the directional gradient and Hessian of the ray velocity magnitude
are,
Ev1.28 109 1.56 10 10 3.98 10 10 , (23.13) r ray
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3.86 104.93565 101.39 10 6 56 Evrrray 4.93 101.37 107.14 10 . (23.14) 1.39 107.1455 101.16 106
The relative errors of the mixed Hessian of the ray velocity magnitude are,
5.51 102.88667 104.43 10 7 67 Evxrray 1.89 101.40 109.96 10 . (23.15) 1.15 101.3367 104.11 106
The relative errors of the ray velocity gradients wrt the material and geometric properties
of the model are,
Ev3.15 101.12131 108.79 107.1289 10 0 , (23.16) c ray
Ev5.09 1098 3.53 10 . (23.17) a ray
The relative errors of the ray velocity Hessians wrt the material, mixed, and geometric
properties of the model are,
7.61 109 4.57 10 6 2.30 10 6 7.61 109 4.94 10 8 2.88 10 6 2.09 10 6 Ev , (23.18) ccray 6 6 5 6 4.57 10 2.88 10 7.89 10 6.68 10 6 6 6 5 2.30 10 2.09 10 6.68 10 3.03 10
4.11 108 9.69 10 9 7.69 10 7 8.14 10 7 Ev , (23.19) acray 77 76 1.04 10 3.88 10 7.22 10 2.59 10
8.92 105.3388 10 Ev . (23.20) aaray 8 7 5.33 107.19 10
24. MODEL 2. COMPUTING RAY VELOCITY DERIVATIVES FOR qSV2 WAVES
Page 105 of 146
As in the previous section, here we list the derivatives of the ray velocity magnitude for the wave type qSV2 , with the intermediate ray velocity vray 0.84719014km/s (Table 12) and validate them numerically with the finite-difference approximations.
The spatial gradient and Hessian of the ray velocity magnitude are,
v 4.7331235 101 1.0222045 7.074902 7 10 1 s1 , (24.1) x ray
3.4929669 107.228094811 101.1479050 11 1 xxvray 7.2280948 101.52569729.3881815 10 km s . (24.2) 1.14790509.3881815 104.70667731110
The directional gradient and Hessian of the ray velocity magnitude are,
v 2.5898950 104.90450111 2 105.73945 56 10 1 km / s , (24.3) r ray
4.8775432 108.6362904111 106.4954631 10 11 rrvray 8.6362904 101.26307599.1089538 10 km / s . (24.4) 6.4954631 109.10895381 103.496 111901 10
The mixed Hessian of the ray velocity magnitude is,
1.1907525 10112 1.4799723 10 2.0812939 10 1 1 1 1 xrvrya 2.2605512 10 2.9808753 10 4.0743857 10 s . (24.5) 1.1198717 1011 1.6591081 10 1 2.149656510
The ray velocity gradients wrt the material and geometric properties are,
v 2.8239671 101.009977811 103.6513276 6.2152554 , (24.6) c ray
where the first component is unitless, and the others have the units of velocity, and
km v 7.6054167 1011 1.5504543 10 . (24.7) a ray sradian
Page 106 of 146
The ray velocity Hessians wrt the material, mixed, and geometric properties of the model,
are,
1.1102230 103.36659261.21710922.071751815 3.36659261.5011095 105.7055458211 108.5804428 10 v , ccray 1 11 1.21710925.7055458 101.8203190 103.2156455 10 1 11 2.07175188.5804428 103.2156455 105.8866481 10
(24.8)
2.5351389 106.80070562.33496444.58074291 v , (24.9) acray 211 5.1681809 101.38640444.7601015 109.3383868 10
1.95713932.2553655 10 1 v . (24.10) aaray 1 1 2.2553655 104.3286648 10
Validation
The relative errors of the finite-difference approximations are listed below.
The relative errors of the spatial gradient and Hessian of the ray velocity magnitude are,
Ev 2.43 101.0398 102.87 10 9 , (24.11) x rya
3.54 107 4.74 10 8 3.47 10 8 8 88 Evxxray 4.74 10 3.38 10 3.01 10 . (24.12) 3.47 1087 3.01 10 8 2.73 10
The relative errors of the directional gradient and Hessian of the ray velocity magnitude
are,
Ev4.08 1010 9.54 10 111 2.16 10 1 , (24.13) r ray
Page 107 of 146
6.94 105.86676 103.89 10 7 66 Evrrray 5.86 104.10 103.61 10 . (24.14) 3.89 103.6165 101.29 106
The relative errors of the mixed Hessian of the ray velocity magnitude are,
4.73 101.62867 108.51 10 7 77 Evxrray 5.47 101.43 101.84 10 . (24.15) 2.67 101.5377 109.68 106
The relative errors of the ray velocity gradients wrt the material and geometric properties
of the model are,
Ev4.13 105.201379 109.31 107.56 10 9 , (24.16) c ray
Ev5.97 106.3199 10 . (24.17) a ray
The relative errors of the ray velocity Hessians wrt the material, mixed, and geometric
properties of the model are,
5.21 107 7.43 10 9 1.04 10 8 5.21 107 1.09 10 6 2.16 10 6 2.24 10 6 Ev , (24.18) ccray 9 6 8 8 7.43 10 2.16 10 4.11 10 8.97 10 8 6 8 7 1.04 10 2.24 10 8.97 10 1.02 10
6.00 109 8.99 10 7 1.86 10 8 2.66 10 8 Ev , (24.19) acray 89 77 1.10 10 8.73 10 7.64 10 1.55 10
9.84 102.6188 10 Ev . (24.20) aaray 8 7 2.61 101.96 10
25. MODEL 2. COMPUTING RAY VELOCITY DERIVATIVES FOR qSV3 WAVES
Page 108 of 146
As in the two previous sections, here we list the derivatives of the ray velocity magnitude for the wave type qSV3 (slow shear), with the lowest ray velocity vray 0.74693784km/s (Table 12) and validate them numerically with the finite-difference approximations.
The spatial gradient and Hessian of the ray velocity magnitude are,
v 4.8313847 101 1.0710803 8.021996 3 10 1 s1 , (25.1) x ray
2.9134706 104.0865666111 108.6968745 10 1 111 xxvray 4.0865666 109.5623840 105.0625030 10 ksm . (25.2) 8.6968745 105.062503011 101. 9745030 101
The directional gradient and Hessian of the ray velocity magnitude are,
v 1.7240870 103.26491411 8 103.82074 19 10 1 km / s , (25.3) r ray
2.5584494 108.4846890231 102.3046031 10 311 rrvray 8.4846890 102.3182771 106.4888474 10 km / s . (25.4) 2.3046031 106.488847411 101.26 36 224
The mixed Hessian of the ray velocity magnitude is,
1.2775429 101 1.9033999 10 1 2.4600125 10 1 1 1 1 1 xrvrya 2.7575306 10 4.3565832 10 5.4883833 10 s . (25.5) 2.0661035 1011 3.6053075 10 1 4.357614110
The ray velocity gradients wrt the material and geometric properties are,
v 2.4897928 101 9.1541263 3.3229890 5.5803112 , (25.6) c ray
where the first component is unitless, and the others have the units of velocity, and
km v 5.0629078 1011 1.0321337 10 . (25.7) a ray sradian
Page 109 of 146
The ray velocity Hessians wrt the material, mixed, and geometric properties of the model,
are,
2.2204460 103.05137541.10766301.860103715 3.05137541.2749144 104.8166500211 107.2655075 10 v , ccray 1 11 1.10766304.8166500 101.4995226 102.6931307 10 1 11 1.86010377.2655075 102.6931307 104.9992637 10
(25.8)
1.6876359 105.01327881.75742433.36093181 v , acray 211 3.4404457 101.02201633.5827176 106.8516574 10
(25.9)
1.27652173.7569897 10 1 v . (25.10) aaray 1 1 3.7569897 101.8095986 10
Validation
The relative errors of the finite-difference approximations are listed below.
The relative errors of the spatial gradient and Hessian of the ray velocity magnitude are,
Ev 1.57 107.1199 101.03 10 9 , (25.11) x rya
5.88 107 1.77 10 7 3.61 10 8 7 88 Evxxray 1.77 10 6.72 10 6.42 10 . (25.12) 3.61 1087 6.42 10 8 2.19 10
The relative errors of the directional gradient and Hessian of the ray velocity magnitude
are,
Page 110 of 146
Ev3.71 102.191010 101.29 10 11 , (25.13) r ray
2.25 105.78446 106.02 10 4 56 Evrrray 5.78 103.75 103.20 10 . (25.14) 6.02 103.2066 102.17 106
The relative errors of the mixed Hessian of the ray velocity magnitude are,
5.91 103.90777 104.79 10 6 87 Evxrray 1.39 104.90 106.24 10 . (25.15) 6.03 108.1877 101.96 108
The relative errors of the ray velocity gradients wrt the material and geometric properties
of the model are,
Ev3.68 104.891379 108.85 107.36 10 9 , (25.16) c ray
Ev4.96 108.5399 10 . (25.17) a ray
The relative errors of the ray velocity Hessians wrt the material, mixed, and geometric
properties of the model are,
4.89 101.05789 105.20 10 4.89 101.107666 102.20 102.28 10 Ev , (25.18) ccray 8688 1.05 102.20 102.58 107.67 10 968 8 5.20 102.28 107.67 108.14 10
4.86 109 7.44 10 7 4.08 10 8 5.34 10 9 Ev , (25.19) acray 98 77 6.37 10 7.35 10 7.26 10 2.23 10
1.98 1078 1.45 10 Ev . (25.20) aaray 8 7 1.45 10 4.70 10
Page 111 of 146
26. MODEL 2. COMPUTING RAY VELOCITY DERIVATIVES FOR SH WAVES
Recall that the spatial gradients and Hessians in Tables 10 and 11, respectively, are given for the axial compressional velocity vP and the shear velocity factor f (rather than axial shear velocity vSP v f1 ). Applying equation set 19.1, we compute the spatial gradient and Hessian of the axial shear velocity, xxxvvSSand . We then use a normalization factor of vS 1 . 5k m / s (the values are taken from Table 10), resulting in,
v 1.516 102.24651111 101.057 10km , (26.1) x S
4.0256750 101.522516021 101.69066242 10 221 2 xxvS 1.5225160 107.4831727 101.1606488 10 km . (26.2) 1.6906624 101.1606488111 101.0030291 10
As already mentioned, unlike the case of qP-qSV waves, the slowness inversion for SH waves is performed analytically: The slowness vector components are computed with equation A9; the magnitudes of the phase and ray velocities are computed with equation A11 (see results in Table
12), vray 1.0622130km/ s .
In this section, we provide the ray velocity derivatives, computed analytically for SH wave, and validate them numerically with the finite-difference approximations.
The spatial gradient and Hessian of the ray velocity magnitude are,
v 1.4799718 101 1.7194039 10 11 1.4747621 10 s1 , (26.3) x ray
Page 112 of 146
5.7271685 103.5328847221 101.5749660 10 2 311 xxvray 3.5328847 102.6555509 101.6012227 10 ksm . (26.4) 1.5749660 101.601222711 106. 2688069 103
The directional gradient and Hessian of the ray velocity magnitude are,
v 1.3173817 102 2.4947343 10 22 2.919444 10 km / s , (26.3) r ray
6.9153621 101.2790294211 101.3130920 10 111 rrvray 1.2790294 10 2.2326886 102.3242689 10 km / s . (26.6) 1.3130920 102.32426891 102.35557801131
The mixed Hessian of the ray velocity magnitude is,
4.0882908 101.1683970322 101.1895977 10 2 2 2 1 xrvrya 1.3905876 103.2945978 103.5573825 10 s . (26.7) 2.0344076 104.087369622 104.67736202 10
The ray velocity gradient wrt the model parameters, m vS ,,, ax ax, is,
v 7.0814201 102.1182013.868587912 107.8865746 1 0 3 . (26.8) m ray
The ray velocity Hessian wrt the model parameters is,
5.5511151 1016 1.4121344 2.5790586 10 2 5.2577164 10 3 1.4121344 4.2736428 7.8051907 1022 1.5911806 10 v . mmray 2 2 1 2 2.5790586 10 7.8051907 10 4.8194597 10 8.9427675 10 3 2 2 2 5.2577164 10 1.591180610 8.9427675 10 3.7910430 10
(26.9)
Validation
Page 113 of 146
The relative errors of the finite-difference approximations are listed below.
The relative errors of the spatial gradient and Hessian of the ray velocity magnitude are,
Ev6.81 104.981110 101.46 10 01 , (26.10) x ray
2.28 102.82778 103.78 10 7 67 Evxxray 2.82 103.18 101.32 10 . (26.11) 3.78 101.3287 104.77 107
The relative errors of the directional gradient and Hessian of the ray velocity magnitude
are,
Ev9.63 105.461112 103.55 10 11 , (26.12) r ray
1.68 105 2.26 10 5 8.30 10 6 5 58 Evrrray 2.26 10 1.55 10 7.07 10 . (26.13) 8.30 1066 7.07 10 8 3.90 10
The relative errors of the mixed Hessian of the ray velocity magnitude are,
5.69 105.47666 102.19 10 6 66 Evxrray 2.51 101.32 101.72 10 . (26.14) 1.67 101.5766 101.61 106
The relative errors of the ray velocity gradient wrt the model parameters are,
Ev4.62 1013 1.26 10 98 5.66 10 11 3.42 10 . (26.15) m ray
The relative errors of the ray velocity Hessian wrt the model parameters are,
Page 114 of 146
1.65 101.82877 103.01 10 1.65 108.248886 108.22 102.03 10 Ev . (26.16) mmray 7888 1.82 108.22 105.20 106.10 10 767 8 3.01 102.03 106.10 104.65 10
27. MODEL 2. TTI AS A PARTICULAR CASE OF GENERAL ANISOTROPY
This section is similar to section 19, but model 2 is applied instead of model 1. We show that the same results obtained in this part (Part II) for the gradients and Hessians of the ray velocity (for the given inhomogeneous polar anisotropic model parameters, their spatial gradients and Hessians
(Tables 10 and 11), and the given ray direction), can be obtained by applying the theory presented in Part I for general anisotropy. For this, as mentioned in Section 18, we first have to convert
(transform) the polar anisotropic model parameters and their spatial gradients and Hessians, into
“equivalent model parameters” described with the full set of the twenty-one elastic properties and their spatial gradients and Hessians. We perform the analysis for quasi-compressional waves only; the technique for the other wave types does not differ.
The computed crystal stiffness components (equation set 2.7b), and their relative gradients
(equation set 19.4) are presented in Table 13, and the relative Hessians (equation set 19.5) in Tables
14 and 15, where the normalization factors are the reciprocals of the crystal stiffness components,
ˆ 1 Cij .
Page 115 of 146
Table 13. “Crystal” stiffness components of the polar
anisotropy model 2 and their spatial gradients
1 ˆ 2 Relative gradient, km (in global frame) # Cij Value, k m / s x1 x2 x3 ˆ 1 1 2 1 C11 6.3 1 . 2 6 5 8 5 7 1 1 0 1 . 2 4 7 8 5 7 1 1 0 1.12 10 ˆ 3 1 1 2 C13 6 . 8 0 6 0 7 5 3 4.816105610 1.849169010 2.503728310 ˆ 1 2 2 3 C33 9 1 . 0 8 2 1 0 7 . 7 1 0 5 . 0 8 1 0 ˆ 1 1 1 4 C44 2.25 3 . 0 3 2 1 0 4 . 4 9 3 1 0 2 . 1 1 4 1 0 ˆ 1 1 1 5 C66 1 . 1 2 5 2 . 8 2 1 0 3 . 3 2 3 1 0 2 . 8 7 7 1 0
Table 14. Spatial Hessians of “crystal” stiffness tensor (diagonal components), model 2
Component Relative Hessian, km2 (in global frame)
# mi xx11 xx22 xx33 ˆ 4 1 2 1 C11 3.8943143 10 1.1988064 10 5.2366960 10 ˆ 2 3 2 2 C13 1.458421010 2.704171810 9.948315310 ˆ 2 2 2 3 C33 2.094638010 9.876450010 4.349032010 ˆ 2 1 1 4 C44 3.454838010 2.50598710 2.229508010 ˆ 2 2 1 5 C66 7.180406010 5.316250010 3.059104410
Page 116 of 146
Table 15. Spatial Hessians of “crystal” stiffness tensor (off-diagonal components), model 2
Component Relative Hessian, km2 (in global frame)
# mi xx12 xx23 xx13 ˆ 2 1 1 1 C11 6.834818610 1 . 3 4 5 3 7 3 1 1 0 1.194689810 ˆ 1 3 3 2 C13 1.1867264 10 1.4634276 10 2.9130849 10 ˆ 2 1 1 3 C33 4.703430010 1.106442010 1.169482810 ˆ 2 1 1 4 C44 3.766356010 2.796207610 3.060842410 ˆ 2 1 1 5 C66 1.783600010 3.378685510 2.585317610
We apply equations 19.11, 19.12, and 19.13 to compute the global stiffness components, their gradients, and their Hessians, respectively. The results are listed in Tables 16, 17, 18, where the units of the (density-normalized) stiffness components are km/s 2 , while the gradients and
Hessians are presented in the “relative form”, with the units kmandkm12 , respectively.
Page 117 of 146
Table 16. Stiffness components and their spatial gradients for polar
anisotropy of model 2 presented as general (triclinic) anisotropy
Component Relative gradient, km1 (in global frame) Value, k m / s 2 # Cij x1 x2 x3 1 1 2 1 C11 7 . 0 4 0 0 0 8 0 1.6808611 10 2.0186365 10 4.4157486 10 2 2 1 2 C12 4 . 9 9 0 2 7 7 9 1.115418310 8 . 8 4 5 3 8 7 7 1 0 +1.911183310 2 1 1 3 C13 5 . 5 5 3 5 4 5 7 2.224355510 2.033792910 2.989790710 1 1 1 1 4 C14 9.6560206 10 2.7196262 10 8.894508 10 6.0723304 10 1 1 1 2 5 C15 9.532298910 2.452692910 2.621281510 2.587945310 1 1 1 1 6 C16 7.149224210 2.581964910 3.551193510 2.071344910 1 2 2 7 C22 8 . 8 1 6 9 0 4 0 1.154162410 9.478147710 1.355176310 2 1 1 8 C23 5.1590883 3.6870921 10 2.3560459 10 3.0269800 10 2 1 1 9 C24 1.1413445 7.813673410 1.055113110 3.096038910 2 10 C25 7.098023710 1.1368019 3.8903093 3.0239231 1 1 1 1 11 C26 4.161151810 3.228642310 4.266205710 1.321776110 1 2 2 12 C33 9.6615655 1.116672410 6.458547110 7.143284910 1 1 1 13 C34 3.066465710 2.696878110 7.77556310 1.0188193 1 1 14 C35 1.490643110 5.014881110 1.2166970 1.2793458 1 1 1 1 15 C36 2.510817710 5.443153310 9.289936810 1.714121410 1 1 1 16 C44 7.308892910 7.352465810 1.3953900 8.449673410 1 1 2 2 17 C45 5.141416010 1.977326310 9.254630710 5.152117410 1 1 2 1 18 C46 2.7976620 10 1.0538391 10 1.4452145 10 2.0860524 10 1 1 1 19 C55 1.5386221 2.5607238 10 3.3823298 10 1.5292887 10 1 2 1 1 20 C56 2.4406654 10 4.0914502 10 1.4080937 10 3.1499622 10 1 1 1 21 C66 1.3962499 2.8271919 10 3.9222835 10 2.6664964 10
Page 118 of 146
Table 17. Hessians of polar anisotropy stiffness components presented
as general (triclinic) anisotropy: diagonal elements, model 2
Component Relative Hessian, km1 (in global frame)
# Cij xx11 xx22 xx33 2 1 2 1 C11 2.5217007 10 1.4355551 10 2.3598753 10 3 2 1 2 C12 7.414082210 3.273550110 1.589681210 3 2 2 3 C13 9.865780610 3.067962610 4.337799210 2 1 1 4 C14 3.0155432 10 3.5600743 10 1.4861618 10 1 1 1 5 C15 1.923446910 1.914444810 2.472061510 1 2 1 6 C16 1.960458510 6.937487810 2 . 9 1 8 5 1 0 1 1 0 2 1 4 7 C22 1.415466410 1.134102310 6.083804110 3 2 2 8 C23 3.9988965 10 2.1873359 10 7.1977109 10 2 2 1 9 C24 4.943524710 1.174337610 1.059885210
10 C25 1.2077583 6.2171055 1.5083418 1 2 1 11 C26 1.991962810 1.041309510 2.186668810 2 2 2 12 C33 1.812752510 8.699713510 2.749767310 1 1 1 13 C34 2.186334410 3.706067310 7.324540810 1 1 14 C35 2.775620710 1.0094907 2.931131810 2 1 1 15 C36 5.659523310 8.9039331 10 1.493504310 2 1 1 16 C44 7.155515710 4.699238810 8.870194010 1 1 1 17 C45 2.246104010 4.85133910 4.462751810 1 1 1 18 C46 2.9434881 10 5.8251629 10 6.1932426 10 4 1 1 19 C55 3.9593281 10 2.0201302 10 3.8098269 10 1 1 1 20 C56 2.3572450 10 3.4014649 10 4.1660650 10 2 1 1 21 C66 2.6949765 10 2.2013427 10 3.0077695 10
Page 119 of 146
Table 18. Hessians of polar anisotropy stiffness components presented
as general (triclinic) anisotropy: off-diagonal elements, model 2
Component Relative Hessian, km1 (in global frame)
# Cij xx12 xx23 xx13 2 1 2 1 C11 1.4046489 10 1.3362002 10 8.9484845 10 1 2 2 2 C12 1.114306810 4.417363910 3.489906610 1 2 3 3 C13 1.531502410 2.629902510 9.713112210 1 1 3 4 C14 3.0689364 10 1.0652913 10 2.8941059 10 2 1 1 5 C15 6.723147210 1.742587610 1 . 5 7 0 7 5 3 1 0 2 2 1 6 C16 5.773595810 9.709164210 1.127093910 2 1 1 7 C22 4.285819410 1.109661410 1.250624610 1 3 2 8 C23 1.3141530 10 8.8555138 10 4.2471284 10 1 2 1 9 C24 1.249291810 7.321274110 1.315046210 1 10 C25 2.7641566 3.2199762 6.408131610 1 2 2 11 C26 1.149033010 6.630260110 3.218382610 2 2 1 12 C33 6.730551910 9.979997910 1.271490310 1 2 2 13 C34 7.858356210 5.116224310 8.514691710 1 1 14 C35 1.1825760 5.395928510 4.871824510 1 1 1 15 C36 1.903197210 3.726128810 4.017134510 1 1 1 16 C44 1.741961210 6.893352410 5.926264910 1 1 1 17 C45 1.922714210 3.583283010 1.295725710 2 1 2 18 C46 9.1433584 10 3.0765028 10 4.1552149 10 2 2 1 19 C55 5.3613438 10 3.4570020 10 3.8139665 10 1 1 1 20 C56 3.9373203 10 1.8517986 10 9.8498727 10 2 1 1 21 C66 1.8525098 10 3.6276162 10 3.1987921 10
The data in Tables 16, 17 and 18 make it possible to compute the ray velocity and its derivatives for a general triclinic medium applying the workflow and technique of Part I. Page 120 of 146
The gradients and Hessians of the reference Hamiltonian
The slowness gradient and Hessian are,
HPp +1.5166714+1.27809391.7765505km / s , (27.1)
8.2513733 101 3.5936251 1.1136010 10 1 1 2 HPpp 3.5936251 6.7229251 1.4142784 10 km /s . (27.2) 1.1136010 1011 1.4142784 10 8.9619569
The spatial gradient and Hessian are,
HP 6.4413638 105.9193417222 101.3477602 10 km1 , (27.3) x
2.3359340 106.2310053222 105.9045123 10 2 212 HPxx 6.2310053 105.8444300 101.0498619 10 km . (27.4) 1.1994543 101.0498619212 101 8.2031297 0
The mixed Hessian is,
2.4180773 101 1.0552098 1011 5.6442094 10 12 2 HPxp 2.5525685 10 6.0778214 10 1.1286547 km . (27.5) 5.0489656 1021 2.6507915 10 1.02994 83
The gradients and Hessians of the arclength-related Hamiltonian
The slowness gradient and Hessian are,
HPp 0.5696 0.48 0.6672 r , (27.6)
Page 121 of 146
4.0256052 1.9821015 4.7604599 101 HPpp 1.9821015 5.5014458 1.9534099 km / s . (27.7) 1 4.7604599 10 1.9534099 6.9500392
The inverse of this matrix is,
3.0273776 101.1298551112 101.1020071 1 0 1 1 12 HPpp 1.1298551 102.4408944 106.0865906 10 s / km . (27.8) 1.1020071 106.0865906221 1010 .6023651 1
The spatial gradient and Hessian are,
HP 2.4191138 102.22306365.06163810223 2 s / km2 , (27.9) x
1.4928374 101.078686732 101.7903156 10 2 222 3 HPxx 1.0786867 103.6464051 104.2688646 10 s / km . (27.10) 1.7903156 104.2688646221 102.1351968 1 0
The mixed Hessian is,
1.0894744 101.372234211 105.4292379 10 3 111 3 HPxp 1.6259604 102.4845444 101.4635067 10 s / km . (27.11) 9.0398797 101.8992910211 102.5279577 1 0
Finally, we compute the two gradients, xrvxrayray rvx ,,, r , and the three Hessians,
x xrvxrayrayray rx rvx r, rvx ,, r ,, of the ray velocity magnitude. As expected, the resulted ray velocity derivatives fully coincide with the derivatives listed in equations 14.40-14.44 for compressional waves, computed directly for polar anisotropic media (without its conversion to general anisotropy). We define the discrepancies E between the two methods as,
TP E , (27.12) P
Page 122 of 146 where TPa nd stand for the triclinic and polar anisotropies, respectively. These discrepancies are listed below.
For the inverted slowness vector,
E p 05.43103.34101616 , (27.13)
For the ray velocity magnitude,
15 Ev ray 1.6710 . (27.14)
For the spatial gradient and Hessian,
Ev5.44 10161 2.17 10 5 6.07 10 15 , (27.15) x ray
3.60 1014 1.01 10 15 5.86 10 16 15 16 Evxx ray 1.01 100 5.63 10 . (27.16) 5.86 1016 5.63 10 16 1.07 10 15
For the directional gradient and Hessian,
Ev1.79 101.17141 101 415 .12 10 , (27.17) r ray
3.76 1015 1.14 10 15 2.97 10 15 16 16 15 Evrr ray 6.35 10 1.46 1 0 1.0 4 10 . (27.18) 1.24 1015 2.10 101 5 2.48 10 15
For the mixed Hessian,
5.68 1016 5.77 10 16 4.14 10 16 15 16 Evxr ray 2.59 10 0 7.57 10 . (27.19) 4.2 9 1015 1.49 10 15 6.51 1 0 16
Page 123 of 146
As expected, the discrepancies between the results of the two approaches are “almost” zero, where
14 the worst relative discrepancy does not exceed Emax 3.610 . Note that both methods are
“almost” analytic and exact. We say “almost”, because computing the ray velocity derivatives includes the slowness vector inversion, which is a numerical procedure. In addition to the round- off errors (which are the main source of the tiny discrepancies in equations 27.15-27.19), the governing equations for this inversion are formulated differently for polar and triclinic anisotropies.
28. CONCLUSIONS
In this paper (Part II) we apply the theory and workflow, presented in Part I of this study for computing analytically the gradients and Hessians of the ray velocity magnitudes in general anisotropic media, specifically for polar anisotropic (TTI) media. The derivatives are formulated and computed wrt the ray locations and directions and wrt the model (material and geometric) parameters, considering both the coupled qP-qSV and the decoupled SH waves. The particular case of acoustic approximation for qP waves is considered as well.
The polar anisotropic model presented in this part is particularly important since it is widely used in the seismic method for describing the propagation of waves/rays in structural compacted (and also unconsolidated) rock layers. A similar approach can be applied to lower symmetry materials, such as tilted orthorhombic or tilted monoclinic; however, these topics are beyond the scope of this study and will be presented in our future research.
The obtained set of ray velocity partial derivatives constructs the traveltime and amplitude sensitivity kernels (matrices) which are the corner stones in many seismic anisotropic modeling and inversion applications, in particular, two-point ray bending methods and tomographic
Page 124 of 146 applications. Our formulae are obtained directly in the ray (group) angle domain (unlike previous works in which the ray velocity (first) derivatives are formulated in the phase angle domain).
The correctness of the analytic ray velocity derivatives was tested against numerical finite differences derivatives, for two different sets of model parameters. The first model characterizes compacted sedimentary shale/sand layering and the second one characterizes a special (extreme) case where the polar anisotropic material has a large negative anellipticity, and for the selected ray direction, the slowness inversion yields a shear wave triplication. We also analyzed the sensitivity and the correctness of the ray velocity derivatives for two approximation (simplification) cases
(which are carried out in many practical applications): a) When only the spatial derivatives of the material axial compressional velocity are accounted for (ignoring all other model parameters derivatives), and b) When the spatial derivatives of all the material parameters are accounted for
(ignoring only the spatial derivatives of the symmetry axis direction angles). The accuracy of the derivatives in both cases considerably decreases; the latter (case b) can be still acceptable, however, the former (case a) can yield wrong results. We further provide more insight on the feasibility and the correctness of the acoustic approximation for qP waves in polar anisotropic media, which has an excellent accuracy for the inverted slowness vector and the ray velocity magnitude, but insufficient accuracy for the ray velocity derivatives, especially for the second derivatives. Finally, we show that the same results for the ray velocity derivatives obtained directly for the polar anisotropic model, can be obtained by transforming the model parameters into the twenty-one stiffness tensor components of a general anisotropic (triclinic) medium and applying the theory described in Part I.
ACKNOWLEDGMENT
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The authors are grateful to Emerson for the financial and technical support of this study and for the permission to publish its results. Gratitude is extended to Anne-Laure Tertois, Alexey Stovas,
Yuriy Ivanov, Yury Kligerman, Michael Slawinski, Mikhail Kochetov, and Beth Orshalimy for valuable remarks and comments that helped to improve the content and the style of this paper.
APPENDIX A. SLOWNESS VECTOR INVERSION FOR POLAR ANISOTROPY
In this appendix we briefly review the existing studies on the slowness inversion for polar anisotropy, in particular, the approach proposed by Ravve and Koren (2021b) which is used in the present work. Vavryčuk (2006) suggested an inversion approach, based on the cofactors of the difference between the Christoffel matrix and the identity matrix, ΓI , to compute the slowness vector in anisotropic elastic media characterized by the following symmetries: general anisotropy, orthorhombic, and transverse isotropy with a vertical axis of symmetry (VTI). The Hamiltonian in this approach is the unit eigenvalue of the Christoffel matrix (i.e., the solution of the Christoffel equation, rather than the equation itself). This method leads to a set of three polynomial equations of degree six, which means a higher algebraic complexity, 666216 , than the classical approach suggested and used by Musgrave, 1954a, 1954b, 1970; Fedorov, 1968; Helbig, 1994;
Grechka, 2017, with the algebraic complexity 556150 . In a particular case of coupled qP- qSV waves in polar anisotropy, and under a commonly accepted assumption that the angle between
o the slowness and the ray direction vectors, nrand , respectively, cannot exceed 90 ,
0nr vvphs / ray 1 (where vphs is the phase velocity magnitude), the method suggested by
Vavryčuk (cited above) reduces to a set of two quartic equations (algebraic complexity 16) and a constraint that cuts off one half of the roots. For polar anisotropic media (TTI), the method has
Page 126 of 146 been later improved and extended by Zhang & Zhou (2018), where the axial and the normal (in the plane normal to the axial direction) slowness components were replaced by implicit functions of the phase angle between direction of the symmetry axis k and the phase direction n . Song &
1 Every (2000) derived approximate formulae for the reciprocal ray velocity magnitude ( vr a y pr
), for qP and qS waves in weakly anisotropic orthorhombic media vs. the ray direction r . The approximations are arranged as a truncated series in r , where the coefficients depend on the elastic properties of the media.
The inverse solutions for qP and SH waves in polar anisotropy are unique. However, for qSV waves, the inversion is challenging due to qSV triplications; the effect of qSV triplications was studied by Dellinger (1991), Thomsen and Dellinger (2003), Schoenberg and Daley (2003),
Vavryčuk (2003), and Roganov and Stovas (2010). Stovas (2016), Xu and Stovas (2018), and Xu et al. (2021) studied triplications in orthorhombic media. Xu et al. (2020) explored triplications of converted waves in TTI media. Stovas et al. (2021) studied phase and ray velocity surfaces and triplications in elastic and acoustic models of different symmetry classes.
Ravve and Koren (2021b) proposed an alternative method to solve the slowness vector inversion in polar anisotropic media. Taking into account that the three vectors: the ray velocity, the slowness and the polar axis direction are coplanar, and applying the common constraint that the angle
o between the phase and ray velocity vectors does not exceed 90 , Ravve and Koren (2021b) reduced the slowness inversion equation set 6.1 in Part I (for general anisotropy) to a sixth-degree univariate polynomial (for polar anisotropy) with a single real root for qP waves and either one or three real roots for qSV waves, depending on the triplications (e.g., Thomsen and Dellinger, 2003;
Grechka, 2013; Zhang and Zhou, 2018),
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6 5 4 3 2 dc6 dc 5 dc 4 dc 3 dc 2 dcd 1 o 0 , (A1)
where the unknown parameter c is related to the phase angle phs between the slowness vector and the polar axis of symmetry direction k ,
c cotphs . (A2)
The coefficients of the polynomial depend on the ray angle r a y between the ray velocity direction r and the axis of symmetry direction,
2 2 do 1 2 12 f fm 2 2 dff1 2211 m 1 m d 411 22 1ff 2 m 2 2 2ffffff 2 1248 f 103 2 m ffffff1 21 284 21 2 22m 2 2 dfff3 4 m 2 121 3 2 21 1 m d 221 f ff2 fff2 ff 2412 fm 4 2222 2 4 2 2 d5 211ff m 1 m 22 df6 fm11
(A3) where vfP x, x , x and x are the medium properties, and,
2 mmkr cosray , sin ray 1 , 0 ray . (A4)
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The real and complex roots of the univariate polynomial are computed as the eigenvalues of a non- symmetric companion matrix (e.g., Johnson & Riess, 1982; Edelman & Murakami, 1995; Press et at., 2002).
Applying the acoustic approximation, we reduce the resolving equation to a fourth-degree polynomial equation,
mm1 2 c41 2 c 3 4 c 2 1 2 c 12 2 0 , (A5) 1 m2 m where only the “compressional” root is used.
After the phase angle has been established, we compute the unit vector n of the phase (slowness) direction. The ray and phase direction vectors and angles, and the polar axis direction vector k satisfy the constraint,
sinrayn sin phs r sin ray phs k , (A6) which yields the phase direction,
sinray phskr sin phs n . (A7) sinray
Thus, to establish the slowness vector, we compute separately its magnitude p and its direction vector n . The magnitude is delivered by the Christoffel equation adjusted for polar anisotropy; in terms of the Thomsen (1986) parameters, it accepts the following form,
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1fffvp2sin2 2444 sin phsphs P (A8) 222 f 20 sni phs vpP 1 ,2 which is equivalent to equation 2.6.
For SH waves, an analytic solution exists where the slowness vector is an explicit function of the medium properties and the ray direction,
22mmkrkr p , (A9) 22 vfmvmPS112121212 while the magnitudes of the ray and phase velocities read,
vv1fm 1 2 1 2 2 1f 1 2 phs, ray , (A10) vvPP1 4 1 mm22 1 2 or equivalently,
vv1212m2 12 phsray, . (A11) vvSS14112 mm22
APPENDIX B. SIGNS OF THE HAMILTONIANS
To check for the right sign, consider an isotropic case where the Thomsen parameters vanish; due to continuity wrt these parameters, the conclusions will be valid for anisotropic media as well. For isotropic media, equation 2.15 separates into compressional and shear factors,
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2222 HpvpvPSV xp,110 PS , (B1)
where vvPSa n d are compressional and shear velocities of isotropic media, and p is the slowness magnitude. Of course, each factor in equation B1 can be now considered separately, but let’s pretend that the medium is very close to isotropic, with the infinitesimal Thomsen parameters, so we keep both multipliers as a limit case of equation 2.15 that converges to equation B1. The slowness gradient of the PSV reference Hamiltonian becomes,
22222 HvvpvvPSV,p xpp,22 PSPS . (B2)
The arclength-related Hamiltonian, HPSV xp, , is obtained from the reference Hamiltonian,
HPSV xp, ,
p2 vp 22 v 211 HHPSVPSVx,, px p 1 PS HPSV xp, , (B3) ddsp /2 vvp222 v 2 v 2 2 HHPSV,PSV,pp PSP S where d / ds is the metric, and the slowness gradient of the “isotropic” reference Hamiltonian is given by,
2 446 222 24 2 2222 2 2 22vvpP SP S vvv P SP vp SP SP S v vvvp v v H x, pp . (B4) PSV,p 32 2 2222 2 222 222pv PSPSPSPS v vvpv v vvp
For compressional waves in isotropic media,
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1 r pHv ,, ppr P,p P . (B5) vvPP
For shear waves in isotropic media,
1 r p,,p HSV,p vS p r . (B6) vvSS
Compute the scaler of the reference Hamiltonian,
d 1 , (B7) sc d p HPSV,p
v2 1 P qP 2 vv22 2 f 1 PS , (B8) scsc 222 2 22 2 22vvpPSP v S vp vS 1 f qSV 2 vv22 2 f PS
Thus, we see that the sign of the Hamiltonian in equation 2.15 is correct for compressional waves, but should be changed for the shear waves,
HHHHHPPSVSVPSVP , , (B9)
where H PSV is the Hamiltonian of the coupled qP-qSV waves listed in equation 2.15.
With this correction, for both wave types, HHP,,pp rrand SV , while the scaler sc is positive as well. Introduction of additional correction multipliers 1/ 2andfff 1/ 2 for qP and qSV waves, respectively, results in sc 1 when the anisotropic parameters become
Page 132 of 146 infinitesimal. However, this does not lead to apparent advantages, so we account only for the sign.
In the further derivations, the sign for the Hamiltonian HPSV and all types of its derivatives will be specified for qP waves only. This sign should be changed to the opposite for qSV waves.
Next, consider the sign of the SH Hamiltonian. In the case of isotropic media, equation 2.16 converges to,
2 2 2 HSHx,p p vSS 1, H SH,p x, p 2p v , (B10) which results in,
pv221 HHv x, pxS , ,p p , (B11) SHSH, 2 p S 2pvS and finally leads to,
1 r pH, prSH,p . (B12) vvSS
Thus, the sign of the SH Hamiltonian HSH in equation 2.16 is correct.
APPENDIX C. KELVIN-FORM ROTATION OF STIFFNESS TENSOR
In this appendix, we recall the technique that makes it possible to rotate the coordinate axes and obtain the components of the fourth-order stiffness tensor in a new frame, using only matrix algebra. We emphasize that the reference frame is rotated, not the tensor itself; “rotating a tensor” is just a (convenient) slang. Generally, given the three Euler angles, one can compute the rotation
Page 133 of 146 matrix that relates vector or tensor components between the two frames. Note that different sets of
Euler angles may exist. In this study, we apply the 푍푌푍 scheme shown in Figure 1, with the Euler angles rotrotrot ,, (zenith, azimuth spin), where the first two angles define the orientation of the local vertical axis in the global frame, and the third angle defines an additional rotation of the local frame about this axis.
Rotation 1 Rotation 2 Rotation 3
Figure 1: Euler angles and their corresponding rotations
In Figure 1, xyzxyzand ˆˆˆ are the global and local (crystal) reference frames, respectively, while xyzxyzvvvwwwand are the two intermediate frames, where zzyyzzvvww,and ˆ . The global-to-local rotation is performed in three stages,
About global axis z for azimuth rot
About axis yv of the first intermediate frame for zenith (tilt) rot
About axis zw of the second intermediate frame for spin rot
The three rotations that convert the global axes to the local are described by the three matrices.
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Combining these matrices, one can compute the resulting local-to-global rotation matrix Ar ot , whose columns are,
coscoscossinsinrotrotrotrotrot Arot,1rotrotrotrotrotcoscossinsincos , (C1) sincosrotrot
cosrot sin rot cosrot cos rot sinrot Arot,2 cosrot sin rot sinrot cos rot cosrot , (C2) sinrot sin rot
sinrot cosrot Arot,3 sinrot sinrot . (C3) cosrot
In the case where the local axes are the crystal frame of polar anisotropy, the zenith and azimuth angles are the polar angles of the axis of symmetry, and the spin angle may be assumed zero,
rot ax, rot ax , rot 0, thus, the rotation matrix simplifies to,
coscoscossinsinaxaxaxaxax Arotaxaxsincos0 . (C4) sincossinsincosaxaxaxaxax
The classical formula that allows the frame rotation for the fourth-order tensor can be written the component-wise form,
loc glb CAAAACijkl rot, im rot, jn rot, ko rot, lp mnop , (C5)
Page 135 of 146 where Ar ot is the 33 rotation matrix (equation C4) that converts the tensor components to the local reference frame. However, it is easier to rotate the frame for the matrix-form presentations of the stiffness tensor, applying only matrix algebra, rather than for the actual fourth-order tensor.
Rotation of a reference frame for Voigt-form stiffness tensor was originally proposed by Bond
(1943). It was later “upgraded” for more suitable Kelvin-form stiffness tensor (e.g., Bona et al.,
2008; Slawinski, 2015),
CCCCCC 222 111213141516 CCCCCC122223242526 222 CCCCCC132333343536 222 C , (C6) 222222CCCCCC142434444546 222222CCCCCC 152535455556 222222CCCCCC162636465666 and in this appendix we use the Kelvin-form rotation only. Assume the “global to local” rotation matrix Ar ot has been defined or computed,
aaa111213 Arot212223 aaa , (C7) aaa313233 where in a general case the spin angle does not vanish. The original 6 × 6 Bond rotation matrix consists of four 3 × 3 blocks,
R A RB Rrot , (C8) RC RD
Page 136 of 146 where each block is related to the components of the standard 3 × 3 rotation matrix Ar ot . The blocks read as follows,
2 2 2 a11 a12 a13 2 2 2 R A a21 a22 a23 , (C9) a2 a2 a2 31 32 33
222aaaaaa 121311131112 RB 222aaaaaa222321232122 , (C10) 222aaaaaa323331333132
222aaaaaa 21 3122 3223 33 RC 222aaaaaa11 3112 3213 33 , (C11) 222aaaaaa11 2112 2213 23
a22a33 a23a32 a21a33 a23a31 a21a32 a22a31 RD a12a33 a13a32 a11a33 a13a31 a11a32 a12a31 . (C12) a12a23 a13a22 a11a23 a13a21 a11a22 a12a21
This stiffness tensor rotation reads,
TT CRCRCRCRglb rot loc rot, loc rot glb rot . (C13)
We emphasize that rotation matrices ARrotand rot represent “global to local” rotations. If the strain and stress tensors are presented in the vector form of factor 2 for off-diagonal components, the same 6 × 6 rotation matrix can be used to transform these vectors,
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T εRεεRεglbrotloclocrotglb,, (C14) T σRσσRσglbrotloclocrotglb,, where,
T ε 112233231312 222, (C15) T σ 112233231312 222.
Of course, the Hook’s law holds in either frame, and can be arranged in the matrix (rather than tensor) form,
σ Cε , (C16) where σεand are vectors of stress and strain (equation C14), and C is the Kelvin-form stiffness matrix (equation C6).
The Kelvin mapping has a number of advantages over the Voigt mapping. The Kelvin mapping preserves the tensor character of vectors and matrices, while the Voigt mapping does not (Nagel
1 T et al., 2016). In particular, the Kelvin-form rotation matrix Rr ot is orthogonal, RRrotrot . In addition, the 6 × 6 compliance matrix SC 1 can be rotated in the same way as the stiffness matrix. Both properties do not hold for the Voigt-form rotation.
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LIST OF TABLES
Table 1. Properties of the polar anisotropy and their spatial gradients, model 1
Table 2. Spatial Hessians of polar anisotropy properties, model 1
Table 3. Results of the slowness vector inversion, model 1
Table 4. “Crystal” stiffness components of the polar anisotropy model 1 and their spatial gradients
Table 5. Spatial Hessians of “crystal” stiffness tensor (diagonal components), model 1
Table 6. Spatial Hessians of “crystal” stiffness tensor (off-diagonal components), model 1
Table 7. Stiffness components and their spatial gradients for polar anisotropy of model 1 presented as general (triclinic) anisotropy
Table 8. Hessians of polar anisotropy stiffness components presented as general (triclinic) anisotropy: diagonal elements, model 1
Table 9. Hessians of polar anisotropy stiffness components presented as general (triclinic) anisotropy: off-diagonal elements, model 1
Table 10. Properties of the polar anisotropy and their spatial gradients, model 2
Table 11. Spatial Hessians of polar anisotropy properties, model 2
Table 12. Results of the slowness vector inversion, model 2
Table 13. “Crystal” stiffness components of the polar anisotropy model 2 and their spatial gradients
Table 14. Spatial Hessians of “crystal” stiffness tensor (diagonal components), model 2
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Table 15. Spatial Hessians of “crystal” stiffness tensor (off-diagonal components), model 2
Table 16. Stiffness components and their spatial gradients for polar anisotropy of model 2 presented as general (triclinic) anisotropy
Table 17. Hessians of polar anisotropy stiffness components presented as general (triclinic) anisotropy: diagonal elements, model 2
Table 18. Hessians of polar anisotropy stiffness components presented as general (triclinic) anisotropy: off-diagonal elements, model 2
LIST OF FIGURES
Figure 1: Euler angles and their corresponding rotations
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