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Geophys. J. Int. (2022) 229, 605–637 https://doi.org/10.1093/gji/ggab468
Advance Access publication 2021 November 15
GJI Seismology
Ray velocity derivatives in anisotropic elastic media – Part I: general
anisotropy
Zvi Koren and Igor Ravve
Emerson, Houston, TX, USA. E-mail: igor.ravve@emerson.com
Accepted 2021 November 12. Received 2021 October 6; in original form 2021 July 1
SUMMARY
We present an original, generic and efficient approach for computing the first and second
partial derivatives of ray (group) velocities along seismic ray paths in general anisotropic
(triclinic) elastic media. As the ray velocities deliver the ray element traveltimes, this set of
partial derivatives constructs the so-called kinematic and dynamic sensitivity kernels which
are used in different key seismic modelling and inversion methods, such as two-point ray
bending methods and seismic tomography. The second derivatives are useful in the solution of
the above-mentioned kinematic problems, and they are essential for evaluating the dynamic
properties along the rays (amplitudes and phases). The traveltime is delivered through an
integral over a given Lagrangian defined at each point along the ray. In our approach, we
use an arclength-related Lagrangian representing a reciprocal of the ray velocity magnitude.
Although this magnitude cannot be explicitly expressed in terms of the medium properties
and the ray direction components, its derivatives can still be formulated analytically using the
corresponding arclength-related Hamiltonian that can be explicitly expressed in terms of the
medium properties and the slowness vector components; this requires first to obtain (invert
for) the slowness vector components, given the ray direction components. Computation of the
slowness vector and the ray velocity derivatives is considerably simplified by using an auxiliary
scaled-time-related Hamiltonian obtained directly from the Christoffel equation and connected
to the arclength-related Hamiltonian by a simple scale factor. This study consists of two parts. In
Part I, we consider general anisotropic (triclinic) media, and provide the derivatives (gradients
and Hessians) of the ray velocity, with respect to (1) the spatial location and direction vectors
and (2) the elastic model parameters. The derivatives are obtained for both quasi-compressional
and quasi-shear waves, where other types of media, characterized with higher symmetries, can
be considered particular cases. In Part II, we apply the theory of Part I explicitly to polar
anisotropic media (transverse isotropy with tilted axis of symmetry, TTI), and obtain the
explicit ray velocity derivatives for the coupled qP and qSV waves and for SH waves. The
derivatives for polar anisotropy are simplified (as compared to general anisotropy), obviously
yielding more effective computations. The ray velocity derivatives are tested by checking
consistency between the proposed analytical formulae and the corresponding numerical ones.
Key words: Body waves; Seismic anisotropy; Wave propagation.
1 INTRODUCTION
In seismology, the first and second partial derivatives of the traveltimes along ray trajectories in anisotropic elastic media construct the seismic-
based kinematic and dynamic sensitivity kernels, which are used in many seismic-driven modelling, imaging and tomographic methods. For
example, assuming fixed model parameters, the partial traveltime derivatives with respect to (wrt) the spatial coordinate vectors and the ray
direction unit vectors along approximated (non-stationary) ray paths, are used for solving two-point ray bending problems (converging to
stationary ray paths). Recently, Koren & Ravve (2021) and Ravve & Koren (2021a) proposed an original two-point ray bending optimization
method (referred to as the Eigenray method) for obtaining stationary ray paths and for computing the dynamic parameters along rays in
general anisotropic elastic media. A comprehensive list of references on ray bending methods, mainly existing for isotropic or transverse
isotropic media, is included in these papers (e.g. Julian & Gubbins 1977; Smith et al.1979;Pereyraet al.1980;Moseret al.1992;Snieder
& Spencer 1993;Pereyra1992,1996; Grechka & McMechan 1996; Bona & Slawinski 2003; Kumar et al.2004; Zhou & Greenhalgh 2005,
2006; Casasanta et al.2008; Wong 2010; Sripanich & Fomel 2014;Caoet al.2017, among others). Also, assuming fixed ray trajectories in
C
The Author(s) 2021. Published by Oxford University Press on behalf of The Royal Astronomical Society. 605
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606 Z. Koren and I. Ravve
a background anisotropic velocity model, the derivatives of their traveltime wrt the spatially varying elastic model parameters (the Fr´
echet
derivatives) are used in the solution of seismic tomography for updating the model parameters. (Among the many publications on this topic,
we mention the papers by Bishop et al.1985; Farra & Madariaga 1988;Farraet al.1989; Williamson 1990;Stork1992;Kosloffet al.
1996 and Koren et al.2008; Zhang et al.2013). The first derivatives, and optionally the second derivatives, are required for solving the
above-mentioned kinematic problems, and the second derivatives are essential for computing the dynamic parameters, such as amplitudes
and phases along the stationary rays, and their sensitivity to changes in the model parameters (e.g. Wang & Pratt 1997, for isotropic media).
The primary objective of this study is to provide explicit analytic formulae for the derivatives of the ray (group) velocity of quasi-
compressional (qP) and quasi-shear (qS) waves that construct the seismic-based kinematic and dynamic sensitivity kernels. It is assumed
that the spatial distributions of the medium elastic parameters, normally given on a fine 3-D grid, are preconditioned (smoothed) to reliably
provide their numerical first and second spatial derivatives. This study is motivated by the need to compute the spatial/directional gradients
and Hessians of the ray velocity magnitudes which are the core computational components in the Eigenray method mentioned above. The
kinematic Eigenray method is formulated as a solution of Fermat’s principle for obtaining a stationary ray path between two fixed endpoints,
t=
R
S
dτ=
R
S
L(x,r)ds→stationary →δt=
R
S
δL(x,r)ds=0,(1)
where tand δtare the global traveltime and the traveltime variation (change) computed along the ray, respectively, and dτis the integration
running time. Sand Rare the fixed endpoints of the path: the source and receiver positions, sis the arclength flow parameter, x(s)andr(s)≡
˙x ≡dx/ds are the ray locations and ray directions (also the ray velocity directions), respectively, and L(x,r) is the arclength-related
Lagrangian, proposed in the Eigenray method to be,
L(x,r)=√r·r
vray (x,r)=√r·r
vray ˜
C(x),r,(2)
where ˜
C(x) is the spatially varying density-normalized fourth-order stiffness tensor. Parameter vray is the magnitude of the ray velocity
which directly depends on the ray direction, and indirectly on the ray coordinates (via the spatially varying anisotropic elastic parameters).
Thus, traveltimes along the stationary ray paths are computed with the arclength-related Lagrangian (eq. 1.2) that contains the reciprocal ray
velocities along the ray elements, and the core computational part is reduced to evaluating the partial derivatives of the ray velocity magnitudes
at discretized nodes along the rays. However, using the Lagrangian formulation, these derivatives can only be obtained implicitly (there is no
explicit form for the ray velocity versus its direction in general anisotropy), while, using the Hamiltonian formulation (with the dependency
on the slowness vector prather than the ray direction vector r), the ray velocity derivatives can be obtained explicitly and analytically. The
arclength-related Hamiltonian H(x,p) can be defined through the scaled-time-related Hamiltonian H¯τ(x,p) which is obtained directly from
the Christoffel equation,
H¯τ(x,p)=det (−I).(3)
It will be considered in this study as the ‘reference Hamiltonian’. The slowness vector pis related to the ray direction unit vector r
through an inversion mechanism, implicitly formulated as p=p[˜
C(x),r], where a constraint holds between the two vectors, rand p,on
one side, and the ray velocity magnitude, on the other side, r·p=v−1
ray . Parameter is the Christoffel matrix (tensor), =p˜
Cp,Iis the
identity matrix, and the superscript above τ,¯τ, indicates that the reference Hamiltonian’s flow parameter is a scaled time rather than the actual
traveltime (e.g. Koren & Ravve 2021, to be explained later). This scaled-time-related (reference) Hamiltonian H¯τ(x,p) is connected to the
arclength-related Hamiltonian H(x,p)by,
H(x,p)=H¯τ(x,p)
H¯τ
p·H¯τ
p
;H¯τ
p≡∂H¯τ(x,p)
∂p,(4)
where the latter is also related to the arclength-related Lagrangian via the Legendre transform,
L(x,r)=r·p−H(x,p).(5)
The reference Hamiltonian H¯τ(x,p) is first used in the slowness inversion to obtain (invert for) the slowness vectors p,giventheray
direction r: A unique solution for quasi-compressional waves and multiple solutions for quasi-shear waves. Then, as indicated above, the ray
velocity magnitude vray can be expressed (and computed) as the reciprocal of the scalar product of the slowness vector and the ray direction
vector,
vray =(p·r)−1,p(x,r)=p[m(x),r],(6)
where, for general anisotropy, the spatially varying model parameters m(x)include the components of the density-normalized elastic (stiffness)
tensor ˜
C(x) or its equivalent matrix (Voight) representation C(x). For higher symmetries, the model parameters may include the material
‘crystal’ parameters and the local orientation angles of the symmetry axis or planes.
We explicitly obtain all the ray velocity derivative types for general anisotropy and for all wave types, as functions of the derivatives
of the arclength-related Hamiltonian. The latter, in turn, are computed via the derivatives of the reference Hamiltonian, which considerably
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Ray velocity derivatives in triclinic media 607
simplifies the computations. Note however, that although the actual derivatives of the ray velocity magnitude are computed via the Hamiltonian
derivatives, the ray velocity magnitude is still considered a ‘Lagrangian object’, where the independent degrees of freedom (DoF) are the
ray location and direction vectors, xand r, respectively, while the Hamiltonian DoF are the location and the slowness vectors, xand p.In
particular, we emphasize that the spatial derivatives of the ray velocity assume a constant ray direction and a varying slowness, while the
spatial derivatives of the Hamiltonian assume a constant slowness and a varying ray direction.
In this paper, we present a generic approach for efficiently (analytically) computing the different types of the first and second derivatives
of the ray velocities at specified ray nodes; they deliver the corresponding derivatives of traveltime ray elements in general anisotropic media:
The spatial and directional gradient vectors, the spatial, directional and mixed Hessian matrices, and the first and second derivatives wrt the
model parameters. These derivatives are computed at specific points x(nodes) along the ray trajectory (or its approximation, for example, in
the case of ray bending problems), given the local physical (elastic) properties of the medium (or the background medium, for example, in
the case of tomography), and the ray velocity direction r. The spatial/directional derivatives constitute two gradient vectors of length 3 and
three Hessian matrices of dimensions 3 ×3,
vray,∇xvray,∇rvray ,∇x∇xvray,∇r∇rvray ,∇r∇xvray =∇x∇rvrayT,(7)
where the symbols ∇xand ∇rare the gradients with respect to the Cartesian components of the spatial location and the ray direction vectors,
respectively. Double gradient symbol means a Hessian. Gradient of a scalar value (e.g. the ray velocity magnitude) is a vector; its Hessian is
a second-order tensor: symmetric if both vector subscripts are identical and non-symmetric otherwise.
The derivatives wrt the model parameters (the Fr´
echet derivatives) are,
∇mvray ,∇m∇mvray.(8)
These derivatives are included, for example, in the components of the sensitivity matrix used in traveltime tomography for updating the
model parameters. In this case, the ray trajectory points xand the ray directions rare assumed fixed, where each partial derivative describes
the ray velocity change wrt a change of a given medium’s property miat a given point, where all the other properties (at the given point and
at all other points along the ray) are fixed. The mixed (non-diagonal) second derivatives show the collateral effect of a simultaneous change
of two different medium properties mi,mj,i= j(at the same location) on the ray velocity magnitude in a fixed direction at that point. Note
that the Hessian matrix describes the local curvature of the multivariate ray velocity function.
The medium properties mi(at the nodal points) may be the stiffness tensor components or their normalized equivalents (e.g. parametriza-
tions suggested by Thomsen 1986 and Alkhalifah 1998, for polar anisotropy; Tsvankin 1997; Alkhalifah 2003,2012; Stovas et al.2021b,for
orthorhombic media; Tsvankin & Grechka 2011, for monoclinic media; Farra & Pˇ
senˇ
c´
ık 2016, for general anisotropy up to triclinic). These
‘normalized’ parametrizations often replace the stiffness tensor components in the governing relationships, providing a clearer insight on the
strength of the actual anisotropy.
We re-emphasize that a prerequisite for the ray velocity derivatives is the knowledge of the slowness vector pat the considered point;
otherwise, it should be computed (inverted).
The paper is structured as follows. We star t with important background theoretical sections which are essential for establishing the desired
derivatives. We begin with the variational formulations delivering the kinematic and dynamic differential equations of the Eigenray method
and provide the explicit relations between the derivatives of the Lagrangian and those of the ray velocity magnitude. We then introduce the
reference Hamiltonian, which will be used for the actual computations, and relate its derivatives to those of the arclength-related Hamiltonian.
Next, we provide the relationships between the arclength-related Lagrangian and its derivatives, on one side, and the corresponding arclength-
related Hamiltonian and its derivatives, on the other side. Additionally, since the actual computation of the set of the ray velocity derivatives
requires the knowledge of the slowness vector components, we review the method for their inversion.
Finally, we check and confirm our relationships for the ray velocity derivatives with a synthetic example, using a given point and its
small neighborhood in an inhomogeneous triclinic medium, where the input includes (1) the model parameters: The 21 density-normalized
elastic components of the stiffness matrix and their spatial gradients and Hessians and (2) a given ray direction vector. In the example,
we first invert for the slowness vectors, where we obtain a single solution for quasi-compression waves and (in this case) all 18 solutions
for quasi-shear waves (Grechka 2017), and compute the corresponding ray velocity magnitudes, vray =(p·r)−1. We then compute all the
partial derivatives indicated in eqs (7)and(8). We validate the correctness of the derivatives by comparing the analytic derivatives with
the corresponding numerical derivatives approximated by finite differences. We then discuss the importance of the analytic derivatives and
indicate the drawbacks in using numerical difference approximations (which, in this work, are only used for verification purposes).
Most of the mathematical proofs are moved to the appendices. In Appendix A we prove a key identity that relates the spatial gradient of the
arclength-related Hamiltonian, Hx, to its slowness gradient, Hp=r. In Appendices B, C and D we derive the directional gradient, directional
Hessian, and the mixed (spatial/directional) Hessian of the ray velocity magnitude vs. the corresponding derivatives of the arclength-related
Hamiltonian. In Appendix E we demonstrate that the sign of the reference Hamiltonian for qP waves is always plus, while for qS waves, the
sign is chosen such that the condition Hp=+ris fulfilled. In Appendix F we derive the relationships between the gradient and Hessian of
the ray velocity magnitude wrt the ray direction vector, and the corresponding first and second derivatives of this magnitude wrt the dual ray
direction polar angles: zenith and azimuth.
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608 Z. Koren and I. Ravve
2 KINEMATIC AND DYNAMIC VARIATIONAL FORMULATIONS
Following the theory behind the Eigenray method, in order to find the stationary ray path corresponding to the vanishing first variation of the
traveltime (kinematic problem), and to explore its second variation while solving the dynamic problem, we introduce the arclength-related
Lagrangian (eq. 2) to the Euler–Lagrange equation,
d
dsLr=Lx,(9)
and obtain the governing kinematic relationship,
d
dsr
vray −∇rvray
v2
ray =−∇xvray
v2
ray
,(10)
where the expression in brackets represents the slowness vector p, due to the ‘momentum’ equation, Lr=p. Similarly, the second traveltime
variation yields the dynamic equation in Lagrangian formulation, referred to as Jacobi equation (Bliss 1916;Ravve&Koren2021a),
d
ds (Lrx ·u+Lrr ·˙u)=Lxx ·u+Lxr ·˙u,(11)
where sis the arclength of the central ray, vector u(s) is a normal shift between the central ray and a paraxial ray, u·r=0. For a point source
(where the initial conditions are based on the ray-normal eigenvectors of matrix Lrr ), two independent solutions for u(u1and u2) are needed
to obtain the Jacobian, J(s)=u1×u2·r(the cross-section area of the ray tube), which, in turn, makes it possible to compute the (relative)
geometric spreading (e.g. ˇ
Cerven´
y2000;Ravve&Koren2021a; Stovas et al.2021a).
Thus, the spatial and directional gradients of the Lagrangian along the ray path, Lxand Lr, are needed for the kinematic ray bending,
and the spatial, directional and mixed Hessians of the Lagrangian along the stationary path, Lxx ,Lrr and Lxr =LT
rx, are needed for solving
the dynamic problem. Depending on the solution method (e.g. using the Newton optimization method), the Hessians may be useful in the
kinematic stage as well. Finally, the gradients and Hessians of the arclength-related Lagrangian defined in eq. (2) are arranged as (Koren &
Ravve 2021),
Lx=∂L
∂x=−∇xvray √r·r
v2
ray
,Lr=∂L
∂r=r
vray√r·r−∇rvray√r·r
v2
ray
,
Lxx =∂2L
∂x2=2∇xvray ⊗∇
xvray
v3
ray −∇x∇xvray
v2
ray
,
Lxr =LT
rx =∂2L
∂x∂r=−∇xvray ⊗r
v2
ray +2∇xvray ⊗∇
rvray
v3
ray −∇x∇rvray
v2
ray
,
Lrr =I−r⊗r
vray −r⊗∇
rvray +∇
rvray ⊗r
v2
ray +2∇rvray ⊗∇
rvray
v3
ray −∇r∇rvray
v2
ray
,(12)
where symbol ⊗indicates an outer product operation (tensor product). The derivatives of the Lagrangian in eq. (12) are expressed in terms of
the derivatives of the ray velocity magnitude (hence, the motivation for this study). Throughout the paper we provide this set of ray velocity
derivatives explicitly in terms of the corresponding derivatives of the arclength-related Hamiltonian, and eventually, we relate them to the
derivatives of the reference Hamiltonian. The reference Hamiltonian is more convenient for the slowness vector inversion, and its derivatives
are much simpler than those of the arclength-related one.
3 THE REFERENCE HAMILTONIAN
As briefly mentioned in the Introduction, by definition, we call the (vanishing and unitless) Hamiltonian arising from the Christoffel (1877)
equation,
H¯τ=det (−I),=p·˜
C·p,(13)
the reference Hamiltonian; we find this Hamiltonian the simplest and most convenient to perform the slowness inversion and it is also used
as an auxiliary Hamiltonian in the actual computation of the ray velocity derivatives. Recall that matrix is the Christoffel tensor, ˜
Cis the
fourth-order density-normalized stiffness tensor and Iis the identity tensor (matrix). The definition of the reference Hamiltonian is valid for
any type of anisotropy, and the scaled time ¯τis related to the actual running time τ,as(Koren&Ravve2021),
αsc (¯τ)=d¯τ
dτ=1
p·H¯τ
p
,ds
d¯τ=H¯τ
p·H¯τ
p,vray =vray r=H¯τ
p
p·H¯τ
p
,(14)
where αsc is the unitless scaling factor and vray is the ray velocity vector. The gradients and Hessians of the ray velocity magnitude are
related to those of the arclength-related Lagrangian, and the latter, in turn, are related to the derivatives of its corresponding arclength-related
Hamiltonian. However, the analytic expressions for the gradients and especially for the Hessians of the arclength-related Hamiltonian are
unwieldy, and it becomes easier to link the derivatives of the arclength-related Hamiltonian to the derivatives of the reference Hamiltonian.
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Ray velocity derivatives in triclinic media 609
The latter can be computed relatively easily. Note that the components of the Christoffel matrix (tensor) for triclinic media are quadratic
functions of the slowness vector components and linear functions of the stiffness tensor components (e.g. Auld 1990a,b; Slawinski 2015),
11 =C11 p2
1+C66 p2
2+C55 p2
3+2(C16 p1p2+C56 p2p3+C15 p1p3),
22 =C66 p2
1+C22 p2
2+C44 p2
3+2(C26 p1p2+C24 p2p3+C46 p1p3),
33 =C55 p2
1+C44 p2
2+C33 p2
3+2(C45 p1p2+C34 p2p3+C35 p1p3),
12 =C16 p2
1+C26 p2
2+C45 p2
3+(C12 +C66)p1p2+(C25 +C46 )p2p3+(C14 +C56)p1p3,
13 =C15 p2
1+C46 p2
2+C35 p2
3+(C14 +C56)p1p2+(C36 +C45 )p2p3+(C13 +C55)p1p3,
23 =C56 p2
1+C24 p2
2+C34 p2
3+(C25 +C46)p1p2+(C23 +C44 )p2p3+(C36 +C45)p1p3.(15)
Remark: The Christoffel eq. (13) defines the shear-wave Hamiltonian only up to its sign; we discuss this important issue later.
4 THE ARCLENGTH-RELATED HAMILTONIAN
In this section, we introduce the arclength-related Hamiltonian H(x,p) and relate its first and second derivatives to those of the reference
Hamiltonian, H¯τ(x,p). The term ‘arclength-related Hamiltonian’ has dual (complementary) meanings. First, with this Hamiltonian, the
kinematic ray tracing equation set is,
dp
ds=−Hx,r=dx
ds=Hp,dτ
ds=p·Hp=1
vray
,(16)
where the corresponding flow parameter is the arclength sand parameter τis the current traveltime. Secondly, as already mentioned in eq.
(5), it is also related to the arclength-related Lagrangian via the Legendre transform, L(x,r)=r·p−H(x,p).
The arclength-related Hamiltonian, H(x,p), can be obtained from the reference Hamiltonian, H¯τ(x,p), with the scaling operator,
H(x,p)=H¯τ(x,p)
H¯τ
p·H¯τ
p
,(17)
where
H¯τ
p·H¯τ
p=
3
i=1∂H¯τ
∂pi2
.(18)
The arclength-related Hamiltonian H(x,p), defined with the absolute value of the slowness gradient of the reference Hamiltonian H¯τ
pin
the denominator, is complicated, such that analytical tracking of its Hessians becomes unwieldy. The remedy is to work with H(x,p), but to
express its derivatives through those of H¯τ(x,p). The derivatives include two gradients and three Hessians, and we obtained the corresponding
relationships listed below.
The spatial gradient Hxis then written as,
Hx(x,p)=∇
x⎡
⎣H¯τ(x,p)
H¯τ
p·H¯τ
p⎤
⎦=H¯τ
x
H¯τ
p·H¯τ
p−H¯τ
px
T·H¯τ
pH¯τ
H¯τ
p·H¯τ
p3/2,(19)
which leads to,
Hx(x,p)=H¯τ
x
H¯τ
p·H¯τ
p−H¯τ
xp ·H¯τ
p
H¯τ
p·H¯τ
p3/2H¯τ(x,p).(20)
In a similar way we compute the slowness gradient of the arclength-related Hamiltonian Hp,
Hp(x,p)=∇
p⎡
⎣H¯τ(x,p)
H¯τ
p·H¯τ
p⎤
⎦=H¯τ
p
H¯τ
p·H¯τ
p−H¯τ
pp ·H¯τ
p
H¯τ
p·H¯τ
p3/2H¯τ(x,p).(21)
In eqs (20)and(21), the reference Hamiltonian H¯τ(x,p) in the second terms on the right-hand sides vanishes (since H¯τ=0). However,
we can set it to zero only after we derive the three Hessians of the Hamiltonian (spatial, directional and mixed), otherwise the results will be
wrong.
The spatial Hessian of the arclength-related Hamiltonian reads,
Hxx (x,p)=H¯τ
xx
H¯τ
p·H¯τ
p−H¯τ
x⊗H¯τ
xp ·H¯τ
p+H¯τ
xp ·H¯τ
p⊗H¯τ
x
H¯τ
p·H¯τ
p3/2.(22)
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610 Z. Koren and I. Ravve
The slowness Hessian reads,
Hpp (x,p)=H¯τ
pp
H¯τ
p·H¯τ
p−H¯τ
p⊗H¯τ
pp ·H¯τ
p+H¯τ
pp ·H¯τ
p⊗H¯τ
p
H¯τ
p·H¯τ
p3/2.(23)
The mixed Hessian reads,
Hxp (x,p)=H¯τ
xp
H¯τ
p·H¯τ
p−H¯τ
x⊗H¯τ
pp ·H¯τ
p+H¯τ
xp ·H¯τ
p⊗H¯τ
p
H¯τ
p·H¯τ
p3/2.(24)
Only now we set the Hamiltonian H¯τto zero in eqs (20) and (21) and obtain the final results for the spatial and the slowness gradients
of the Hamiltonian,
Hp=H¯τ
p
H¯τ
p·H¯τ
p
,Hx=H¯τ
x
H¯τ
p·H¯τ
p
.(25)
Note that for arbitrary vectors aand bof lengths naand nb, respectively, and matrix Aof size nb×na, the following dual algebraic
identities hold,
(A·a)⊗b=A(a⊗b),b⊗(A·a)=(b⊗a)AT.(26)
In both cases, the result is a square matrix of dimensions nb×nb. Applying this rule to eqs (22), (23)and(24), we obtain the spatial
Hessian,
Hxx (x,p)=H¯τ
xx
H¯τ
p·H¯τ
p−H¯τ
x⊗H¯τ
pH¯τ
px +H¯τ
xp H¯τ
p⊗H¯τ
x
H¯τ
p·H¯τ
p3/2,(27)
the slowness Hessian,
Hpp (x,p)=H¯τ
pp
H¯τ
p·H¯τ
p−H¯τ
p⊗H¯τ
pH¯τ
pp +H¯τ
pp H¯τ
p⊗H¯τ
p
H¯τ
p·H¯τ
p3/2,(28)
and the mixed Hessian,
Hxp (x,p)=H¯τ
xp
H¯τ
p·H¯τ
p−H¯τ
x⊗H¯τ
pH¯τ
pp +H¯τ
xp H¯τ
p⊗H¯τ
p
H¯τ
p·H¯τ
p3/2.(29)
Remark: Both, the reference and the arclength-related Hamiltonians vanish along the ray, thus, the definition of the reference Hamiltonian
in eq. (13) might equally be written as H¯τ(x,p)=±det(−I). However, although this sign is inessential for the Hamiltonian itself, it is
important for the Hamiltonian derivatives: an improper sign leads to wrong ray equations. The criterion of correctness is the gradient, Hp,
of the arclength-related Hamiltonian wrt the slowness vector: If this gradient coincides with the ray direction vector, Hp=r, the sign of the
Hamiltonian is correct, otherwise the opposite sign should be taken.
Overall, the relation Hp=rin eq. (13) is always consistent for quasi-compressional waves; however, for quasi-shear waves, we may
have (or may have not) to change the sign of the reference Hamiltonian, to avoid the case of Hp=−r. Since our derivations and proofs
are based on Hp=r, keeping this identity is a must for the proposed approach. The ray velocity derivatives include terms with both odd
and even powers of the Hamiltonian components, which means that an improper Hamiltonian sign leads not only to wrong signs, but also to
wrong magnitudes of the quasi-shear-wave ray velocity derivatives. The proper sign of the quasi-shear-wave reference Hamiltonian can be
established only after performing the slowness inversion (except the cases of qSV and SH waves of polar anisotropy, presented in Part II of
this study, where this sign is known ahead). This issue is further discussed in Section 6 (in particular, equation set 36) and in Appendix E.
5 RELATIONS BETWEEN LAGRANGIAN AND HAMILTONIAN DERIVATIVES
As mentioned, the ray velocity and its derivatives are the building stones of the arclength-related Lagrangian L(x,r); however, their explicit
forms can only be expressed in terms of the corresponding arclength-related Hamiltonian H(x,p). The gradients of the arclength-related
Lagrangian and those of the Hamiltonian are connected by the following relationships (e.g. Koren & Ravve 2021),
Lr=p(the generalized momentum equation),
Hp=r≡˙x =dx/ds,
Lx=−Hx=˙
p=dp/ds;also:Lx=Lxr r,
Lr·r=L=p·r=v−1
ray ,(30)
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Ray velocity derivatives in triclinic media 611
The Hessians of the arclength-related Lagrangian and those of the Hamiltonian are also connected. The following equation set makes it
possible to obtain the Lagrangian Hessians from the Hamiltonian Hessians (Ravve & Koren 2021a),
Lrr =H−1
pp −λrr⊗r,Lrx =−H−1
pp Hpx,
Lxr =−Hxp H−1
pp ,Lxx =Hxp H−1
pp Hpx −Hxx,(31)
where the double-letter subscript means ‘the gradient of the gradient’, that is the Hessian matrix. The reverse transform also exists,
H−1
pp =Lrr +λrr⊗r,Hpx =−Hpp Lrx ,
Hxp =−Lxr Hpp,Hxx =Hxp H−1
pp Hpx −Lxx,(32)
where λris the eigenvalue of the Hamiltonian’s inverse Hessian, H−1
pp , corresponding to the eigenvector r. Note that tensors Lrr and H−1
pp
have the same eigenvectors: One of them is the ray direction r, while the two others are in the plane normal to the ray (and they are also
normal to each other, of course). The eigenvalues, related to the eigenvectors in the normal plane, are the same for Lrr and H−1
pp ;however,the
eigenvalue related to rvanishes for Lrr and does not vanish for H−1
pp . Note also that since H−1
pp is the inverse matrix of Hpp, their eigenvectors
are identical and their eigenvalues are reciprocals of each other.
There are two more key relationships that will be used in our derivations,
Hx=Hxp H−1
pp Hp=Hxp H−1
pp r,(33)
proven in Appendix A, and,
Lrr =TH−1
pp ,where T=I−r⊗r,(34)
(Ravve & Koren 2021a).
6 SLOWNESS INVERSION: SLOWNESS VECTORS FROM RAY DIRECTION
The combined Lagrangian–Hamiltonian approach used in this study for obtaining the analytic derivatives of the ray velocity magnitude,
requires knowing both the ray direction vector rand the slowness vector pat the given ray location x. In cases where an initial-value ray
tracing is applied (e.g. in ray shooting methods), the solution is based on the derivatives of the Hamiltonian H(x,p), and it provides the values
of xand pat the ray nodes, where the ray directions r=Hpand the ray velocity magnitudes vray =(p·r)−1can be directly computed. In
cases where a two-point boundary-value ray bending method is used (like in the Eigenray method), the solution is based on optimizing (in
most, but not all, cases minimizing) an integral over the Lagrangian L(x,r) which provides the values of xand rat the ray nodes. Obtaining all
of the corresponding (multiple) slowness vector solutions pi(where the index iindicates one of the existing solutions) in general anisotropic
media becomes a challenging inverse problem. If the slowness vector components are not known, they should be computed first.
6.1 Obtaining the slowness vector from the ray direction vector
The slowness vectors can be obtained from the equation set which manifests that the slowness gradient of the (vanishing) Hamiltonian
H¯τ
p≡∂H¯τ/∂pis collinear with the ray direction r(Musgrave 1954a,b,1970; Fedorov 1968;Helbig1994; Grechka 2017;Koren&Ravve
2021;Ravve&Koren2021b), written in our notations as,
H¯τ
p[m(x),p]×r=0
use two equations from three
,H¯τ[m(x),p]=0
the third equation
.(35)
The three Cartesian components of the cross-product in the first equation of set 35 are linearly dependent, because the mixed product
r×H¯τ
p·rvanishes, so we use only two Cartesian components from the first equation, where the third one is the second equation of set
35. The solution for quasi-compressional waves is unique, while quasi-shear waves may have up to 18 solutions in triclinic media (Grechka
2017). The numerical example in this study uses the triclinic anisotropy model, suggested by Grechka (2017), and a specific ray direction,
where all the 19 solutions do exist. Any Hamiltonian can be used for the slowness vector inversion in equation set 35; however, for general
inhomogeneous anisotropic media, we find the reference Hamiltonian H¯τ(x,p) the simplest.
Note that the other option to formulate the slowness inversion is to arrange equation set 35 as,
H¯τ
p[m(x),p]=αr,H¯τ[m(x),p]=0,(36)
where αis an unknown scalar value with the units of velocity (positive for quasi-compressional waves, and aprioriindefinite sign for
quasi-shear waves). In this case we solve a set of four equations with four unknown variables, {p1,p2,p3,α}. The cost is an additional
variable, while the advantage is the revealed sign for the shear-wave Hamiltonian.
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612 Z. Koren and I. Ravve
7 SPATIAL GRADIENT OF THE RAY VELOCITY
Starting from eq. (6), vray =(p·r)−1, we compute the spatial gradient of the ray velocity magnitude,
∇xvray =−(∇xp)Tr+(∇xr)Tp
(p·r)2.(37)
where we used a general formula for the gradient of a scalar product,
∇(a·b)=(∇a)Tb+(∇b)Ta.(38)
Recall that in the Eigenray method, the ray location and the ray direction at any given point are assumed independent parameters (degrees
of freedom; DoF). Therefore, tensor ∇xrvanishes, and eq. (37) simplifies to,
∇xvray =−(∇xp)Tr
(p·r)2=−v2
ray(∇xp)Tr.(39)
Thus, we need to compute the spatial gradient of the slowness vector, ∇xp,whichisa3×3 matrix. Recall the momentum equation,
Lr=p(equation set 30), where L(x,r)is the arclength-related Lagrangian. Therefore,
∇xp=∇
xLr=Lrx ,(∇xp)T=Lxr.(40)
The Hessians of the arclength-related Lagrangian and Hamiltonian are linked. Combining eqs (39) and (40) with the equation from set
31, Lxr =−Hxp H−1
pp , we obtain,
∇xvray =+v2
ray Hxp H−1
pp r.(41)
Next, we introduce identity 33 into eq. (41), and this leads to,
∇xvray =v2
ray Hx.(42)
Recall that the computation of Hxis simplified using the derivatives of H¯τ, given in equation set 25, Hx=H¯τ
x/H¯τ
p·H¯τ
p,Hp=
H¯τ
p/H¯τ
p·H¯τ
p.
8 SPATIAL HESSIAN OF THE RAY VELOCITY
The spatial Hessian of the ray velocity is the gradient of its gradient, so we start from eq. (42) and compute the gradient of its right-hand side,
∇x∇xvray =∇
xv2
ray (x,p)Hx(x,p),(43)
which is a gradient of a product of a scalar field and a vector field. The gradient of a product of an arbitrary scalar field c(x) and a vector field
d(x) represents a sum of two tensors,
∇[c(x)d(x)]=d(x)⊗∇c(x)+c(x)∇d(x).(44)
Subscript xis absent in this identity because it is valid for any gradient, not necessarily spatial.
Applying the rule of eq. (44) to the right-hand side of eq. (43), we obtain,
∇x∇xvray =2vray Hx⊗∇
xvray +v2
ray∇xHx.(45)
Note however, that ∇xHx= Hxx. The reason is that in the Lagrangian formulation, spatial gradient assumes constant ray direction,
while in the Hamiltonian formulation, it assumes constant slowness vector. The ray velocity vray(x,r) is an object of the Lagrangian; when
computing its spatial derivatives, it is assumed that the ray direction ris fixed, while the slowness vector pmay change as necessary. However,
when computing the spatial derivatives of the Hamiltonian, in their conventional notations, the slowness vector is preserved, while the ray
direction changes as necessary. This leads to,
∇xHx=dH
x
dx=∂Hx
∂x+∂Hx
∂p
∂p
∂x=Hxx +Hxp Lrx.(46)
Since p=Lr, then, for a fixed ray direction, ∂p/∂x=Lrx. We then apply a constraint from equation set 31, Lrx =−H−1
pp Hpx,which
leads to,
∇xHx=Hxx −HxpH−1
pp Hpx.(47)
Combining eq. (42) with 45 and 47, we obtain,
∇x∇xvray =2v3
ray Hx⊗Hx+v2
ray Hxx −HxpH−1
pp Hpx.(48)
The second term in the brackets accounts for the varying slowness components.
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Ray velocity derivatives in triclinic media 613
9 DIRECTIONAL GRADIENT OF THE RAY VELOCITY MAGNITUDE
To compute the directional gradient and Hessian of the ray velocity magnitude, we need to know both, the ray velocity vector vray and the
slowness vector p. Given the ray direction r, we compute the slowness vector pwith equation set 35 and the ray velocity magnitude vray
with eq. (6), so that the ray velocity vector becomes,
vray =vray r=r
p·r.(49)
The directional gradient of the ray velocity is then delivered by (Koren & Ravve 2021),
∇rvray =−vray ×p×vray =−v2
ray r×p×r.(50)
Note that vector r×p×ris the projection of the slowness vector ponto the plane normal to the ray direction r. The relationships in
this section are valid for any type of anisotropic symmetry (eq. 50 is proven in Appendix B).
10 DIRECTIONAL HESSIAN OF THE RAY VELOCITY MAGNITUDE
The directional Hessian of the ray velocity magnitude can be computed as,
∇r∇rvray =2v3
rayp⊗p−v2
ray (p⊗r+r⊗p)−vray vrayLrr −T,(51)
where matrix Tis defined in eq. (34). This relationship is proved in Appendix C. Due to the momentum equation, Lr=p,whereLris the
directional gradient of the Lagrangian L,tensorLrr in eq. (51),
Lrr =pr=∇
rp,(52)
represents the directional Hessian of the arclength-related Lagrangian. This value can be computed through the slowness Hessian of the
arclength-related Hamiltonian, listed in equation set 34, Lrr =TH−1
pp . Note that Lrr is a singular matrix (its eigenvalue, corresponding to
eigenvector r, is zero), while Hpp and its inverse, H−1
pp , are regular invertible matrices, with the exception of inflection points along the ray
path (Bona & Slawinski 2003).
Combining eqs (34), (51)and(52), we obtain the final relationship for the directional Hessian,
∇r∇rvray =2v3
rayp⊗p−v2
ray (p⊗r+r⊗p)−vrayT·vrayH−1
pp −I,(53)
where Iis the 3 ×3 identity matrix.
11 MIXED HESSIAN OF THE RAY VELOCITY MAGNITUDE
The two mixed Hessians, ∇x∇rvray and ∇r∇xvray are defined as,
∇x∇rvray =∂2vray
∂xi∂rj,∇r∇xvray =∂2vray
∂ri∂xj,i=1,2,3,
j=1,2,3,(54)
where iand jare the indices of the row and the column, respectively. The two tensors are transpose of each other,
∇x∇rvray =∇r∇xvrayT,(55)
and can be computed as,
∇r∇xvray =2
vray ∇rvray ⊗∇
xvray −v2
rayTp
x,
∇x∇rvray =2
vray ∇xvray ⊗∇
rvray −v2
ray pT
xT,(56)
where,
p=Lrand px=Lrx.(57)
Equation set 56 is proven in Appendix D. Tensor pxis the spatial gradient of the slowness vector, and it also represents the mixed
Hessian of the arclength-related Lagrangian. This matrix (tensor) can be computed through the Hessians of the arclength-related Hamiltonian
(equation from set 31), Lrx =−H−1
pp Hpx. The final formulae are,
∇r∇xvray =2
vray ∇rvray ⊗∇
xvray +v2
rayTH−1
pp Hpx,
∇x∇rvray =2
vray ∇xvray ⊗∇
rvray +v2
ray Hxp H−1
pp T.(58)
So far, we related the derivatives of the ray velocity magnitude with those of the arclength-related Hamiltonian, and we related the latter
with the derivatives of the reference Hamiltonian; these derivatives will be established in the following section.
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614 Z. Koren and I. Ravve
12 SLOWNESS GRADIENT OF THE REFERENCE HAMILTONIAN
The slowness gradient of the reference Hamiltonian reads,
H¯τ
p(x,p)=∇
pH¯τ(x,p).(59)
The reference Hamiltonian H¯τ(x,p) for triclinic media is given in eqs (13)and(15). It represents a polynomial in both the slowness vector
components and the stiffness tensor components. Computing the derivatives is easy, but the expanded determinant H¯τ(x,p)=det(−I)
includes 950 monomials. The direct approach becomes unwieldy, and we expand the derivative of the determinant using the approach
described, for example, in Bellman (1997) and Magnus & Neudecker (1999). With this approach, the derivative of the determinant (for a
square matrix of order n) is presented as a sum of ndeterminants, where in each of them one column (or row) is replaced by its derivative,
and the others are untouched,
H¯τ
pi=∂H¯τ(x,p)
∂pi=∂det (−I)
∂pi=det
∂a1
∂pi
a2a3+det a1
∂a2
∂pi
a3+det a1a2
∂a3
∂pi,(60)
where a1,a2,a3are the columns of matrix −I[see eq. (15) for components of the Christoffel matrix ]. The relationship yields a single
component of the gradient vector H¯τ
p. An advantage of this approach is its straightforward extension for obtaining the second derivatives of
the Hamiltonian.
13 SLOWNESS HESSIAN OF THE REFERENCE HAMILTONIAN
Next, we compute the slowness Hessian of the reference Hamiltonian. It represents the gradient of the gradient of the reference Hamiltonian,
H¯τ
pp (x,p)=∇
pH¯τ
p.(61)
We use eq. (60) for the gradient and apply the same rule for the derivative of the determinant,
H¯τ
ppij =∂2H¯τ(x,p)
∂pi∂pj=∂2det (−I)
∂pi∂pj=
+det
∂2a1
∂pi∂pj
a2a3+det
∂a1
∂pi
∂a2
∂pj
a3+det
∂a1
∂pi
a2
∂a3
∂pj
+det
∂a1
∂pj
∂a2
∂pi
a3+det a1
∂2a2
∂pi∂pj
a3+det a1
∂a2
∂pi
∂a3
∂pj
+det
∂a1
∂pj
a2
∂a3
∂pi+det a1
∂a2
∂pj
∂a3
∂pi+det a1a2
∂2a3
∂pi∂pj.(62)
This relationship yields a single component of the Hessian matrix. Eqs (60)and(62) make it possible to compute the slowness gradient
and Hessian, respectively, of the arclength-related Hamiltonian.
14 SPATIAL GRADIENT OF THE REFERENCE HAMILTONIAN
The Hamiltonian depends on the medium properties, which, in turn, are position dependent. We introduce a 1-D array m(x) whose components
represent spatially varying model parameters defined at any location, along with their spatial gradients and Hessians, ∇xmand ∇x∇xm. Recall
that we consider a single spatial location, where the model parameters are the medium properties at this point. In the case of a general (triclinic)
anisotropy, these are normally the stiffness tensor components, while in the case of higher symmetries, the medium properties may be presented
by the normalized unitless equivalents of the stiffness components. The Hamiltonian given by eqs (13)and(15) is spatially dependent,
H¯τ=H¯τ[m(x),p].(63)
The gradient of the Hamiltonian obeys the chain rule. Let mi(x),i=1, ...n, be the medium properties, then the spatial gradient can be
arranged as,
H¯τ
x(x,p)=
n
i=1
∂H¯τ
∂mi∇xmi,n=21.(64)
Eq. (64) can be arranged in a more suitable matrix/tensor form (rather than component-wise form), with all summations hidden inside
the ‘tensor’ operations. Let a 2-D array mxof dimensions n×3 be the gradient of the 1-D parameter array mwrt the spatial coordinates,
∇xm=mx=∂mi
∂xj,i=1,...n,j=1,2,3,(65)
where in the case of a triclinic medium, n=21. We assume that the components of array mand matrix mxare known or have been computed;
actually, these are the medium properties and their derivatives. Array mis not a physical vector; it represents a collection of scalar values.
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Ray velocity derivatives in triclinic media 615
Similarly, matrix mxis not a physical tensor, but a collection of nvectors of length 3. This means that when the components of matrix mxof
dimensions n×3 are transformed to a new Cartesian frame (with a different orientation), the matrix is not ‘rotated’ as a whole. Instead, each
of its column of length 3 is independently rotated as a physical vector.
We now introduce a 1-D array H¯τ
mof dimension n, which is the gradient of the Hamiltonian wrt the model parameters,
H¯τ
m=∂H¯τ[m(x),p]
∂mi,i=1,...n,
n=21 for triclinic media.(66)
The derivatives of the Hamiltonian wrt the medium properties are computed similarly to its derivatives wrt the slowness components, eq.
(60),
H¯τ
mi=∂H¯τ[m(x),p]
∂mi=∂det (−I)
∂mi=det
∂a1
∂mi
a2a3
+det a1
∂a2
∂mi
a3+det a1a2
∂a3
∂mi,i=1,...n.(67)
Combining eqs (64)–(66), we obtain,
H¯τ
x(x,p)
1×3
=H¯τ
m
1×n
mx
n×3
,(68)
where the components of array H¯τ
mare computed with eq. (67). We assume that the components of matrix mxare known or have been
computed.
15 SPATIAL HESSIAN OF THE REFERENCE HAMILTONIAN
The chain rule can be extended for the Hessian,
H¯τ
xx (x,p)=∇
xH¯τ
x=∇
xn
i=1
∂H¯τ
∂mi∇xmi=
n
i=1∇x∂H¯τ
∂mi∇xmi=
n
i=1∇x∂H¯τ
∂mi∇xmi+∂H¯τ
∂mi∇x∇xmi
=
n
i=1
n
j=1
∂2H¯τ
∂mi∂mj∇xmi⊗∇
xmj+
n
i=1
∂H¯τ
∂mi∇x∇xmi.(69)
Eq. (69) can be arranged in a more suitable array form, with all summations hidden inside the ‘tensor’ operations. Let a 3-D array
mxx(n,3,3), symmetric for the two last indices, be the Hessian of the 1-D array m,
mxx =∂2mi
∂xj∂xk,mxx (i,j,k)=mxx (i,k,j),
where i=1,...n,j=1,2,3,k=1,2,3,n=21.
(70)
We assume that the components of the 3-D array mxx are known or have been computed; actually, these are the Hessians of the medium
properties. We emphasize that array mxx is not a third-order physical tensor, but a collection of nsecond-order tensors of dimensions 3 ×3,
which still makes it possible to apply the tensor multiplication rules.
We now introduce also a symmetric square matrix H¯τ
mm(n,n), which is the Hessian of the Hamiltonian wrt the model parameters,
H¯τ
mm =∂2H¯τ[m(x),p]
∂mi∂mj,i=1,...n,
j=1,...n.(71)
The second derivatives of the Hamiltonian wrt the medium properties are computed similarly to its second derivatives wrt the slowness
components, eq. (62),
H¯τ
mmij =∂2H¯τ[m(x),p]
∂mi∂mj=∂2det (−I)
∂mi∂mj=
+det
∂2a1
∂mi∂mj
a2a3+det
∂a1
∂mi
∂a2
∂mj
a3+det
∂a1
∂mi
a2
∂a3
∂mj
+det
∂a1
∂mj
∂a2
∂mi
a3+det a1
∂2a2
∂mi∂mj
a3+det a1
∂a2
∂mi
∂a3
∂mj
+det
∂a1
∂mj
a2
∂a3
∂mi+det a1
∂a2
∂mj
∂a3
∂mi+det a1a2
∂2a3
∂mi∂mj,(72)
where i=1,...n,j=1,...n,and n=21 for triclinic media, and akare the columns of matrix −I. Note that according to eq. (15),
the Christoffel matrix components depend on the stiffness components in a linear way. Thus, if the model parameters are the components
of the stiffness tensor, the terms on the right-hand side of eq. (72), that include the second derivatives of the matrix columns wrt these
components, ∂2ak/(∂mi∂mj), vanish.
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616 Z. Koren and I. Ravve
With the notations of eqs (70)and(71), eq. (69) simplifies to,
H¯τ
xx [x(m),p]
3×3
=mT
x
3×n
H¯τ
mm
n×n
mx
n×3
+H¯τ
m
1×n
mxx
n×3×3
,(73)
where the components of n×nmatrix H¯τ
mm are computed with eq. (72). Considering the last item of the right-hand side of eq. (73), we
note that the result of the multiplication of arrays with dimensions (1,n)and(n,3,3) is a new array with dimensions (1,3,3), because the
adjacent indices collapse, but size 1 means actually that the resulting array is only 2-D, of the size (3,3). This array is a contribution to the
Hessian from the gradient of the reference Hamiltonian Hm. The multiplication of a 1-D array and a 3-D array is similar to vector–matrix
multiplication. Let abe a 1-D array, Aa 3-D array, and Ba matrix resulting from their multiplication (with the 1-D array on the left-hand
side). Then,
B=aA,Bjk =
n
i=1
aiAijk,(74)
where the first dimension of 3-D array Acoincides with the length of 1-D array a, and the other dimensions of Aare arbitrary. We assume that
the components of the 3-D array mxx are known or have been computed. In our numerical example, arrays m(21),mx(21,3) and mxx(21,3,3)
are the input data.
16 MIXED HESSIAN OF THE REFERENCE HAMILTONIAN
The Hamiltonian has two mixed Hessians, H¯τ
xp and H¯τ
px, but they are transpose of each other, so we can compute only one of them. In this
section, we compute the mixed HessianH¯τ
px, which can be viewed as a spatial gradient of the slowness gradient,
H¯τ
px =∇
p∇xH¯τ=∇
xH¯τ
p.(75)
From eq. (62), we conclude that,
H¯τ
pmij =∂2H¯τ(x,p)
∂pi∂mj=∂2det (−I)
∂pi∂mj=
+det
∂2a1
∂pi∂mj
a2a3+det
∂a1
∂pi
∂a2
∂mj
a3+det
∂a1
∂pi
a2
∂a3
∂mj
+det
∂a1
∂mj
∂a2
∂pi
a3+det a1
∂2a2
∂pi∂mj
a3+det a1
∂a2
∂pi
∂a3
∂mj
+det
∂a1
∂mj
a2
∂a3
∂pi+det a1
∂a2
∂mj
∂a3
∂pi+det a1a2
∂2a3
∂pi∂mj,(76)
where i=1,2,3,j=1,...n,and n=21 for triclinic media. Thus, we found the derivatives of the mixed Hessian of the reference
Hamiltonian wrt the slowness vector of length 3 and the medium properties array of length up to n=21, H¯τ
pm. This set can be considered a
matrix of dimensions 3 ×n. This matrix, H¯τ
pm,wheremis a set of the spatially varying model parameters, is not a physical tensor, but each
column of this matrix, with a fixed mi, is a physical vector.
Recall that matrix mxof dimensions n×3 is the gradient of the 1-D parameter array mwrt the spatial coordinates, with all components
available (e.g. the derivatives are computed numerically). We multiply the two matrices of dimensions 3 ×nand n×3, and obtain the mixed
Hessian of the reference Hamiltonian, whose dimensions are 3 ×3,
H¯τ
px
3×3
=H¯τ
pm
3×n
mx
n×3
.(77)
17 GRADIENT OF THE RAY VELOCITY WRT THE MEDIUM PROPERTIES
In this section we derive the first partial derivatives of the ray velocity magnitude along fixed ray elements wrt the medium properties
(the Fr´
echet derivatives). These derivatives construct the traveltime derivatives, which are, for example, the components of the traveltime
sensitivity matrix used in traveltime tomography for updating the model parameters. The ray trajectories and hence the ray directions rare
assumed fixed, where each partial derivative describes the ray velocity change wrt a given medium’s property mi, (index irefers to the model
parameter type at a given point), while all the other properties are fixed. It is also assumed that the slowness vectors pkfor the wave mode(s)
under consideration are known (have been computed).
Again, starting from eq. (6), vray =(p·r)−1, we obtain,
∂vray
∂mi=−∂p/∂ mi
(p·r)2·r=−v2
ray
∂p
∂mi·r,i=1,...n.(78)
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Ray velocity derivatives in triclinic media 617
Thus, to compute the ray velocity magnitude derivative wrt a specific model parameter, we need first to establish the corresponding
derivative of the slowness vector.
Consider the general equation set for the slowness inversion (valid for any anisotropic symmetry),
Hp(m,p)×r=0,H(m,p)=0,(79)
where His the arclength-related Hamiltonian, and m=mo(x)+δm(x) is a set of model parameters (e.g. the stiffness components); mois a
background model. Consider a fixed point and a fixed ray direction; thus, both xand rare kept constant, but the ray velocity may vary due
to a change of the parameters at the given point. For the arclength-related Hamiltonian, we replace the first equation of set 79 by Hp=r,
which means that the slowness gradient of this Hamiltonian is not only collinear to the normalized ray direction vector r, but also equal to
this vector.
As the medium property mislightly changes, the slowness vector varies accordingly, such that both the updated Hamiltonian, H(m,p)=
0, and its slowness gradient, Hp(m,p)=r, remain constant. This means that the total derivatives of each of them vanish,
dH
p
dmi=∂Hp
∂mi+∂Hp
∂p·∂p
∂mi=0,dH
dmi=∂H
∂mi+∂H
∂p·∂p
∂mi=0,i=1,...n,(80)
and the equation set can be arranged as,
Hpp ·∂p
∂mi=−∂Hp
∂mi
,∂H
∂mi+Hp·∂p
∂mi=0,i=1,...n.(81)
Note that array ∂p/∂mi(of length 3) can be computed from the first equation of set 81,
∂p
∂mi=−H−1
pp ·∂Hp
∂mi
,i=1,...n.(82)
Next, we introduce this result into the second equation of set 81,
∂H
∂mi=HpH−1
pp ·∂Hp
∂mi
,i=1,...n.(83)
Recall that the ray direction, Hp=r, is the eigenvector of the inverse Hessian H−1
pp , and the corresponding eigenvalue is λr,
rH−1
pp =H−1
pp r=λrr.(84)
This leads to,
∂H
∂mi=λr
∂Hp
∂mi·r,i=1,...n.(85)
Next, we introduce eq. (82)into(78),
∂vray
∂mi=v2
rayrH−1
pp ·∂Hp
∂mi=v2
rayλr
∂Hp
∂mi·r,i=1,...n.(86)
Finally, we combine eqs (85)and(86) to obtain the derivative of the ray velocity wrt the model parameter,
∂vray
∂mi=v2
ray
∂H(m,p)
∂mi
,i=1,...n.(87)
We emphasize that eq. (87) is only valid for the arclength-related Hamiltonian, and we use the reference Hamiltonian due to its simplicity.
The relationship between the two Hamiltonians is given in eq. (17), and it leads to,
∂H(m,p)
∂mi=1
H¯τ
p·H¯τ
p
∂H¯τ(m,p)
∂mi−H¯τ
p(m,p)·∂H¯τ
p(m,p)
∂mi
H¯τ(m,p)
H¯τ
p·H¯τ
p3/2,i=1,...n.(88)
Since the Hamiltonian vanishes, the second item on the right-hand side of eq. (88) vanishes, and this relationship simplifies to,
∂H(m,p)
∂mi=1
H¯τ
p·H¯τ
p
∂H¯τ(m,p)
∂mi
,i=1,...n.(89)
However, we still need expression 88 to further derive the second derivatives of the ray velocity. We arrange eqs (87)and(89)ina
concise vector form,
∇mvray =v2
ray Hm(m,p)where Hm(m,p)=H¯τ
m(m,p)
H¯τ
p·H¯τ
p
.(90)
Note that both Hamiltonians depend on the medium properties and the slowness vector, thus operators Hm(m,p)andH¯τ
m(m,p)resultin
model-related gradients, where the slowness vector pis assumed fixed, ∂H¯τ/∂ m|p=const ,∂H/∂m|p=const . However, the resulting array,∇mvray,
is the model-related ray velocity gradient, computed for the fixed ray direction r(and actually, for the fixed location as well, but due to its
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618 Z. Koren and I. Ravve
direct dependency on m, the dependency on xbecomes pointless). This gradient shows how the ray velocity varies with the change of the
medium properties, while the ray direction is kept constant, and the slowness components are modified as necessary in the updated medium.
The same is true for the model-related ray velocity Hessian, ∇m∇mvray, established in the next section.
18 HESSIAN OF THE RAY VELOCITY WRT THE MEDIUM PROPERTIES
We start from the resulting relationship (90) for a gradient component, which leads to a Hessian component,
∂2vray
∂mi∂mj=2vray
∂vray
∂mj
∂H
∂mi+v2
ray
d
dmj
∂H
∂mi
,i=1,...n,
j=1,...n.(91)
where operator d/dmjdenotes the full derivative, resulting from both, the explicit dependency of ∂H/mion parameter mj, which leads to
∂2H/(∂mi∂mj), and the implicit dependency of ∂H/mion mjthrough the slowness components,
d
∂mj
∂H
∂mi=∂2H
∂mi∂mj+∂2H
∂mi∂p
∂p
∂mj=∂2H
∂mi∂mj+∂Hp
∂mi·∂p
∂mj
,i=1,...n,
j=1,...n.(92)
The (arclength-related) Hamiltonian H(m,p) and its slowness gradient Hpare explicit functions of the medium properties, and their
derivatives wrt these parameters can be computed directly. The slowness vector derivative, ∂p/∂mjis given in eq. (82), which leads to,
d
∂mj
∂H
∂mj=∂2H
∂mi∂mj+∂2H
∂mi∂p
∂p
∂mj=∂2H
∂mi∂mj−∂Hp
∂mi·H−1
pp ·∂Hp
∂mj
,i=1,...n,
j=1,...n.(93)
Finally, we introduce eq. (93) into eq. (91), and the Hessian of the ray velocity, wrt the model parameters, simplifies to,
∂2vray
∂mi∂mj=2v3
ray
∂H
∂mi
∂H
∂mj+v2
ray
∂2H
∂mi∂mj−v2
ray
∂Hp
∂mi·H−1
pp ·∂Hp
∂mj
,i=1,...n,
j=1,...n.(94)
Recall that the arclength-related Hamiltonian has the units of slowness. We still need to convert the Hessian components ∂2H/(∂mi∂mj)
into the Hessian components of the reference Hamiltonian, ∂H¯τ/∂miand ∂2H¯τ/(∂mi∂mj). Applying eq. (88), we obtain,
∂2H(x,p)
∂mi∂mj=
∂2H¯τ(x,p)
∂mi∂mj
H¯τ
p·H¯τ
p−
∂H¯τ
∂miH¯τ
p·∂H¯τ
p
∂mj+∂H¯τ
∂mjH¯τ
p·∂H¯τ
p
∂mi
H¯τ
p·H¯τ
p3/2,(95)
where i=1,...n,j=1,...n.Eqs(94)and(95) can be arranged in a concise tensor form,
∇m∇mvray =2v3
ray Hm⊗Hm+v2
ray Hmm −v2
ray Hmp ·H−1
pp ·Hpm,(96)
and,
Hmm (x,p)=H¯τ
mm
H¯τ
p·H¯τ
p−H¯τ
m⊗H¯τ
mp ·H¯τ
p+H¯τ
mp ·H¯τ
p⊗H¯τ
m
H¯τ
p·H¯τ
p3/2.(97)
One more auxiliary relationship is needed to complete the computations,
Hmp (x,p)=H¯τ
mp
H¯τ
p·H¯τ
p−H¯τ
m⊗H¯τ
pp ·H¯τ
p+H¯τ
mp ·H¯τ
p⊗H¯τ
p
H¯τ
p·H¯τ
p3/2.(98)
Applying the auxiliary algebraic identities of eq. (26), we arrange eq. (97) as,
Hmm (x,p)=H¯τ
mm
H¯τ
p·H¯τ
p−H¯τ
m⊗H¯τ
p·H¯τ
pm +H¯τ
mp ·H¯τ
p⊗H¯τ
m
H¯τ
p·H¯τ
p3/2,(99)
and eq. (98)as,
Hmp (x,p)=H¯τ
mp
H¯τ
p·H¯τ
p−H¯τ
m⊗H¯τ
p·H¯τ
pp +H¯τ
mp ·H¯τ
p⊗H¯τ
p
H¯τ
p·H¯τ
p3/2.(100)
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Ray velocity derivatives in triclinic media 619
Tab l e 1 . Elastic properties of the triclinic model and their spatial
gradients.
Stiffness tensor component
(km2s−2) Relative gradient (km−1)
CiCij value x1x2x3
1C11 +45.597 +0.4647 +0.1201 +0.2126
2C12 +0.935 +0.2993 +0.0965 +0.1592
3C13 +0.895 +0.1517 −0.0708 +0.2103
4C14 +0.103 +0.0062 +0.3696 −0.0399
5C15 +0.074 −0.0308 −0.0730 +0.0514
6C16 +0.070 +0.3953 +0.2842 +0.3468
7C22 +14.442 −0.0786 +0.0767 +0.3028
8C23 +0.887 +0.1143 +0.1624 +0.3564
9C24 −0.083 +0.1668 −0.0605 +0.4459
10 C25 +0.018 +0.3221 +0.3742 +0.3664
11 C26 −0.059 +0.4380 +0.1161 +0.4072
12 C33 +44.303 +0.1007 +0.2677 +0.0293
13 C34 −0.049 +0.1362 +0.1640 +0.1998
14 C35 −0.040 −0.0412 +0.2176 +0.0890
15 C36 −0.026 +0.2618 +0.2117 +0.4530
16 C44 +0.459 −0.0788 +0.4569 +0.0150
17 C45 +0.090 +0.0019 +0.3758 +0.0294
18 C46 +0.052 +0.1136 +0.2295 +0.0055
19 C55 +0.374 +0.1294 +0.1885 +0.4181
20 C56 +0.101 +0.1619 +0.4534 +0.2185
21 C66 +0.450 +0.1679 +0.3936 +0.4576
19 COMPUTATIONAL TEST 1
19.1 Input data at a reference point
An example of a medium with triclinic symmetry is given by Grechka (2017). It is presented below in the Voigt form,
C=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
+45.597 +0.935 +0.895 +0.103 +0.074 +0.070
+0.935 +14.442 +0.887 −0.083 +0.018 −0.059
+0.895 +0.887 +44.303 −0.049 −0.040 −0.026
+0.103 −0.083 −0.049 +0.459 +0.090 +0.52
+0.074 +0.018 −0.040 +0.90 +0.374 +0.101
+0.070 −0.059 −0.026 +0.52 +0.101 +0.900
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,(101)
where the stiffness tensor components are density-normalized, and their units are (km s)−2. However, it becomes more suitable to present the
21 stiffness components as a 1-D array, rather than a tensor or a matrix. Table 1gives the correspondence between two indices of the Voigt
matrix and a single index of the array. For each stiffness component, we introduce the normalized spatial gradient and the normalized spatial
Hessian, whose units are km−1and km−2, respectively,
∇xCi=∇xCi
Co
i
,∇x∇xCi=∇x∇xCi
Co
i
,i=1,...n,
n=21,(102)
where Co
iis the value of the stiffness component Ciat a reference node. Thus, in the proximity of the reference node, each stiffness component
can be expanded into a (truncated) Taylor series,
Ci
Co
i=1+ ∇xCi·x+1
2x·∇x∇xCi·x,i=1,...n,
n=21.(103)
For each medium property (component of the triclinic stiffness tensor), we provide the normalized (relative) spatial gradient in Table 1
and the normalized spatial Hessian in Table 2. Normally, these derivatives are computed numerically, given the parameter field, but in this
example, we consider a single reference point; therefore, the spatial derivatives of the model parameters are assumed input data. Table 1
represents a normalized (divided by mi)matrixmx(21,3), and Table 2is a normalized 3-D array mxx(21,3,3), symmetric wrt the last two
indices (and therefore stored as a matrix of dimensions 21 ×6).
Normally, the accuracy of the resolving parameters does not exceed 3–4 significant decimal digits; in these examples we keep the
accuracy up to eight digits to allow reproducibility of the benchmark problems. We accept the ray velocity direction used by Grechka (2017),
r=0.54812444 0.55112512 0.62914283 .(104)
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620 Z. Koren and I. Ravve
Tab l e 2 . Hessians of elastic properties for triclinic model.
Stiffness tensor
component Normalized spatial Hessian components (km−2)
CiCij x1x1x1x2x1x3x2x2x2x3x3x3
1C11 +0.1312 +0.0956 +0.2023 −0.1279 +0.0801 +0.1125
2C12 +0.1241 +0.1150 −0.1544 +0.0905 −0.1351 −0.0607
3C13 +0.1420 −0.1518 +0.1108 −0.1373 +0.1648 +0.0142
4C14 +0.0562 +0.1298 −0.0895 +0.1630 −0.1255 +0.0348
5C15 −0.1311 −0.0842 +0.0514 +0.1220 +0.0792 −0.1346
6C16 +0.1942 +0.1832 +0.1981 +0.1257 −0.1114 +0.1634
7C22 −0.1536 +0.0233 +0.1011 −0.1488 −0.0711 +0.0823
8C23 +0.1053 +0.1124 +0.0564 −0.1196 −0.1291 +0.1066
9C24 +0.0877 −0.1008 +0.1240 +0.1321 −0.1190 −0.1572
10 C25 +0.1209 +0.1554 +0.1365 +0.0240 +0.1355 −0.0960
11 C26 +0.1370 +0.1255 +0.0062 −0.1371 +0.1220 +0.1342
12 C33 +0.0050 +0.0289 +0.0233 −0.1458 +0.0630 +0.0432
13 C34 +0.1362 +0.1640 +0.1998 +0.1265 −0.1481 −0.0455
14 C35 −0.0412 +0.1176 +0.0890 −0.0902 +0.0239 −0.1030
15 C36 +0.1418 +0.1917 +0.1530 −0.1499 +0.2370 +0.0723
16 C44 −0.0788 +0.1571 +0.0150 +0.0914 +0.1200 −0.1650
17 C45 +0.1020 +0.1762 +0.0184 +0.0912 +0.0631 +0.1271
18 C46 +0.1136 +0.2295 +0.0055 +0.1199 −0.1235 +0.1080
19 C55 +0.1062 +0.0380 +0.2011 +0.0731 +0.1655 +0.0887
20 C56 +0.0810 +0.1234 +0.1185 −0.1452 −0.1300 −0.0988
21 C66 +0.1679 +0.1936 +0.1576 +0.2149 +0.1425 −0.1241
Tab l e 3 . Inverted slowness vector for example 1.
#p1(s km–1)p2(s km–1 )p3(s km–1)vphs (km s–1 )vray (km s–1) ϑ (◦)
10.13555828 0.25145731 0.14025204 3.1422707 3.3208711 18.876378
20.14145161 0.26175272 0.15494351 2.9810194 3.1321140 17.869186
30.15324689 0.26250236 0.14535503 2.9679970 3.1238375 18.174221
40.14621400 0.27368703 0.14881941 2.9058187 3.0806395 19.394976
5.020462473 1.3739451 .028826692 0.72759035 1.2713463 55.089296
61.3261367 .069294564 .042152101 0.75266367 1.2632681 53.429836
7.082601563 1.4915586 .021901670 0.66934197 1.1349570 53.860664
8.017962805 1.5077454 .071948739 0.6624412 1.1285813 54.057851
9.026411709 1.5436997 .036288735 0.64752077 1.1260255 54.896878
10 .025072060 .058899768 1.3518256 0.73891242 1.1152063 48.503128
11 1.4998649 .056791944 0.14900379 0.66299032 1.0557915 51.100538
12 1.6495928 0.20673786 0.20673786 0.60145923 0.96994027 51.676618
13 1.7319248 0.12539465 .023070089 0.57583402 0.96811716 53.501776
14 0.15535027 .053104697 1.4831328 0.67015508 0.95463482 45.412193
15 1.1315446 0.86296430 .015418487 0.70266966 0.90454434 39.029513
16 .012640499 0.22331913 1.6154554 0.61317076 0.87232820 45.338822
17 .016424568 0.1147058 1.7137075 0.58220075 0.86927289 47.951763
18 .0076622051 0.74119655 1.2320229 0.69549985 0.84188530 34.297604
19 0.90607175 .043187752 1.0649368 0.71484731 0.84002605 31.681327
19.2 Inversion for the slowness vector
We compute analytically the reference Hamiltonian with eqs (13)and(15), and solve equation set 35 to invert for the slowness vector. In
Tab l e 3, the solutions are sorted in the order of decreasing ray velocity magnitudes. The first solution is for a compressional wave, and the
others are for shear waves. For each solution, the table includes the Cartesian slowness components, pk, the magnitudes of the phase and ray
velocities, vphs and vray and the angle ϑ between the phase and ray directions. We compute the ray velocity derivatives for the first three
solutions (out of the 19): compressional and shear with the highest and second-highest ray velocities; we call these solutions P,SAand SB.
For both shear waves, the correct sign before the reference Hamiltonian proved to be minus (see Appendix D).
Tab l e 4includes the minor, major and mean curvatures of the slowness surface. The last column includes the coefficient α(eq. 36)that
defines the sign of the shear-wave reference Hamiltonian H¯τ. The slowness surface is the constant (in our case, vanishing) Hamiltonian.
Both, the reference and the arclength-related Hamiltonians can be used, but the arclength-related Hamiltonian His more convenient because
Hp·Hp=r·r=1; it becomes clear from the formulae for computing the curvature, listed below. For a fixed location x, the Hamiltonian
can be considered an implicit function of the three slowness components,
H(p1,p2,p3)=0,(105)
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Ray velocity derivatives in triclinic media 621
Tab l e 4 . Signed ratio αbetween the lengths of the collinear vectors H¯τ
pand r,and
curvatures of the slowness surface for example 1.
# Curvature (km s–1 ) Ratio α(km s–1)
Minor Major Mean
1+61.795621 +100.49936 +81.147488 +0.12995592
2−139.14815 +93.142899 −23.002627 −0.07025568
3−184.87913 +76.061900 −54.408616 −0.06537122
4−1060.1390 +62.168078 −498.98546 −0.01406616
5−24.293369 −5.6765690 −14.984969 −14.3592906
6−15.626902 −3.1727608 −9.3998313 −79.0563923
7−4.7748834 +16.982291 +6.1037036 +28.2306563
8−3.8466348 +18.203224 +7.1782947 +23.8592781
9+5.2220325 +11.396992 +8.3095121 +20.7148975
10 −14.381405 −4.5066681 −9.4440365 −68.782846
11 −5.9287231 +7.4664983 +0.7688876 +193.30857
12 −0.71235548 +23.417681 +11.352663 +218.41032
13 +1.03549108 +20.138339 +10.586915 +194.78277
14 −5.17687070 +9.1493832 +1.9862563 +192.63753
15 −0.33563937 +29.765275 +15.050457 +1066.0033
16 −0.79302121 +32.195990 +15.701484 +188.14619
17 +1.49472127 +26.710559 +14.102640 +150.09615
18 +0.33667408 +37.726961 +19.031818 +834.74199
19 +0.46726311 +12.200988 +6.3341253 +3074.6920
The minor and major curvatures of an implicit surface are the two non-zero eigenvalues of the matrix,
M=I−Hp⊗Hp
Hp·Hp·Hpp
Hp·Hp
,(106)
and for Hp=±r, the matrix simplifies to,
M=(I−r⊗r)Hpp =THpp.(107)
Multiplication by matrix Tnullifies the eigenvalue related to the eigenvector rand does not alter the two other eigenvalues and the three
eigenvectors. The eigenvectors of matrix M, corresponding to the non-zero eigenvalues, are the principal directions in the tangent (to the
slowness surface) plane, normal each other and to the gradient Hp. The latter, in turn, is the third eigenvector, collinear with the ray direction
rand corresponding to the zero eigenvalue. To obtain the mean curvature only, there is no need to solve the eigenvalue problem,
kmean =kmin +kmax
2=tr M
2.(108)
Note that for all 19 solutions of the inverse problem, the signs of the coefficient αin eq. (36) and the mean curvature, listed in the last
two columns of Table 4, are identical. This is, however, not a must, as shown in the other example, Section 22.
19.3 Gradients and Hessians of the reference Hamiltonian
We compute the slowness gradients and Hessians of the reference Hamiltonian using eqs (60) and (62), respectively; its spatial gradients
with eqs (67)and(68), the spatial Hessians with eqs (72) and (73), and the mixed Hessians with eqs (76)and(77). The results are,
(i) the slowness gradient,
H¯τ
p(P)=7.1232017 ×10−27.1621973 ×10−28.1760837 ×10−2km s−1,
H¯τ
p(SA)=3.8508853 ×10−23.8719668 ×10−24.4200855 ×10−2km s−1,
H¯τ
p(SB)=3.5831561 ×10−23.6027719 ×10−24.1127832 ×10−2km s−1,(109)
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622 Z. Koren and I. Ravve
(ii) the slowness Hessian,
H¯τ
pp (P)=⎡
⎢
⎣
−1.1819972 ×10−1−8.8313085 −1.3030042 ×10+1
−8.8313085 −5.6423112 ×10−1−1.0655657 ×10+1
−1.3030042 ×10+1−1.0655657 ×10+1−2.4011328 ×10−1⎤
⎥
⎦km s−12,
H¯τ
pp (SA)=⎡
⎢
⎣
+1.3973083 ×10−1−9.6178410 +8.8403500 ×10−1
−9.6178410 −7.9895182 ×10−2+3.4468498
+8.8403500 ×10−1+3.4468498 +6.276404 ×10−1⎤
⎥
⎦km s−12,
H¯τ
pp (SB)=⎡
⎢
⎣
+4.5439111 ×10−1+2.5317289 −2.3447280 ×10−1
+2.5317289 −1.4788900 ×10−1−1.1774930 ×10+1
−2.3447280 ×10−1−1.1774930 ×10+1+1.6699778 ×10−1⎤
⎥
⎦km s−12,(110)
(iii) the spatial gradient,
H¯τ
x(P)=2.0292344 ×10−33.1343522 ×10−33.9306528 ×10−3km−1,
H¯τ
x(SA)=1.2277450 ×10−31.7432780 ×10−32.1761336 ×10−3km−1,
H¯τ
x(SB)=1.2017822 ×10−31.6136573 ×10−32.1131523 ×10−3km−1,(111)
(iv) the spatial Hessian,
H¯τ
xx (P)=⎡
⎢
⎣
+7.2522389 ×10−4−9.0927236 ×10−3−1.1994543 ×10−2
−9.0927236 ×10−3−1.3759214 ×10−2−1.7000955 ×10−2
−1.1994543 ×10−2−1.7000955 ×10−2−1.1904140 ×10−2⎤
⎥
⎦km−2,
H¯τ
xx (SA)=⎡
⎢
⎣
+6.6642191 ×10−3−1.3858581 ×10−3−8.8621517 ×10−3
−1.3858581 ×10−3−7.5007740 ×10−4−1.7354923 ×10−3
−8.8621517 ×10−3−1.7354923 ×10−3−1.0995600 ×10−2⎤
⎥
⎦km−2,
H¯τ
xx (SB)=⎡
⎢
⎣
−4.4251729 ×10−4+3.3951758 ×10−3+1.4315441 ×10−3
+3.3951758 ×10−3−5.7939013 ×10−3−8.2410466 ×10−3
+1.4315441 ×10−3−8.2410466 ×10−3+2.6664265 ×10−3⎤
⎥
⎦km−2,(112)
(v) the mixed Hessian,
H¯τ
px (P)=⎡
⎢
⎣
+1.8112914 ×10−2−3.5206988 ×10−1−3.4982070 ×10−1
−3.5096106 ×10−1−2.9061771 ×10−1−1.5876352 ×10−1
−3.0335156 ×10=1−2.2712095 ×10=1−5.9803835 ×10−1⎤
⎥
⎦s−1,
H¯τ
px (SA)=⎡
⎢
⎣
+1.2808909 ×10−1−7.1417659 ×10−2−3.6938443 ×10−1
−3.0040773 ×10−1−1.5095586 ×10−2−1.4296670 ×10−1
−1.4853915 ×10−3+7.3969663 ×10−2+1.5029337 ×10−1⎤
⎥
⎦s−1,
H¯τ
px (SB)=⎡
⎢
⎣
−6.6815737 ×10−3+3.8371842 ×10−2+1.1460374 ×10−1
+6.7151471 ×10−3−2.1783209 ×10−1+2.6220593 ×10−2
+1.2915470 ×10−1−1.1076245 ×10−1−4.7146619 ×10−1⎤
⎥
⎦s−1,(113)
19.4 Gradients and Hessians of the arclength-related Hamiltonian
Applying relationships of Section 4, we convert the gradients and Hessians of the reference Hamiltonian into those of the arclength-related
Hamiltonian. The results are,
(i) the slowness gradient of the arclength Hamiltonian is the same for all wave modes and equal to the ray direction r,
Hp(P,SA,SB)=0.54812444 0.55112512 0.62914283 =r,(114)
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Ray velocity derivatives in triclinic media 623
(ii) the slowness Hessian,
Hpp (P)=⎡
⎢
⎣
+109.84663 +37.729064 +18.828577
+37.729064 +96.213771 +31.234835
+18.828577 +31.234835 +125.62845⎤
⎥
⎦km s−1,
Hpp (SA)=⎡
⎢
⎣
+74.824798 −75.726044 +32.702021
−75.726044 +48.240324 +55.444553
+32.702021 +55.444553 −40.839958⎤
⎥
⎦km s−1,
Hpp (SB)=⎡
⎢
⎣
−18.150556 +77.272371 +36.616808
+77.272371 +100.62455 −66.489855
+36.616808 −66.489855 +127.91723⎤
⎥
⎦km s−1,(115)
(iii) the inverse of this matrix,
H−1
pp (P)=⎡
⎢
⎣
+1.0562259 ×10−2−3.9464905 ×10−3−6.0180896 ×10−4
−3.9464905 ×10−3+1.2780661 ×10−2−2.5861583 ×10−3
−6.0180896 ×10−4−2.5861583 ×10−3+8.6931698 ×10−3⎤
⎥
⎦skm
−1,
H−1
pp (SA)=⎡
⎢
⎣
+1.0745347 ×10−2+2.7256217 ×10−3+1.2304504 ×10−2
+2.7256217 ×10−3+8.7877394 ×10−3+1.4112787 ×10−2
+1.2304504 ×10−2+1.4112787 ×10−2+4.5264324 ×10−3⎤
⎥
⎦skm
−1,
H−1
pp (SB)=⎡
⎢
⎣
−5.9163758 ×10−3+8.6246585 ×10−3+6.1765808 ×10−3
+8.6246585 ×10−3+2.5641719 ×10−3−1.1360162 ×10−3
+6.1765808 ×10−3−1.1360162 ×10−3+5.4589971 ×10−3⎤
⎥
⎦skm
−1,(116)
(iv) the spatial gradient,
Hx(P)=1.5614790 ×10−22.4118579 ×10−23.0246046 ×10−2skm
−2,
Hx(SA)=1.7475385 ×10−22.4813340 ×10−23.0974488 ×10−2skm
−2,
Hx(SB)=1.8383965 ×10−22.4684524 ×10−23.2325424 ×10−2skm
−2,(117)
(v) the spatial Hessian,
Hxx (P)=⎡
⎢
⎣
+9.5539445 ×10−2+5.9108444 ×10−2+7.3589490 ×10−2
+5.9108444 ×10−2+7.8243216 ×10−2+1.0628065 ×10−1
+7.3589490 ×10−2+1.0628065 ×10−1+2.1351968 ×10−1⎤
⎥
⎦skm
−3,
Hxx (SA)=⎡
⎢
⎣
+1.4275799 ×10−1+1.4512470 ×10−2−3.7248733 ×10−2
+1.4512470 ×10−2−1.0020997 ×10−2+4.1648252 ×10−2
−3.7248733 ×10−2+4.1648252 ×10−2+8.1216863 ×10−3⎤
⎥
⎦skm
−3,
Hxx (SB)=⎡
⎢
⎣
−5.2493707 ×10−2+6.8683405 ×10−2+4.3385941 ×10−2
+6.8683405 ×10−2+3.8777515 ×10−2+4.0186250 ×10−2
+4.3385941 ×10−2+4.0186250 ×10−2+2.5772346 ×10−1⎤
⎥
⎦skm
−3,(118)
(vi) the mixed Hessian,
Hpx (P)=⎡
⎢
⎣
+3.2958795 +1.8197670 +3.1287060
+3.1143439 ×10−1+2.0676191 +4.3174630
+1.0599475 +3.0971727 +1.6357372 ⎤
⎥
⎦km−1,
Hpx (SA)=⎡
⎢
⎣
+3.7354925 +6.3932346 ×10−1−1.7430933
−2.7377385 +9.0397583 ×10−1+8.1723673 ×10−1
+1.498525 ×10−1+7.9639035 ×10−2+2.5859398 ⎤
⎥
⎦km−1,
Hpx (SB)=⎡
⎢
⎣
−1.2048035 +1.4363292 +2.8521662
+1.1333531 +3.9419000 ×10−1+5.2677282
+3.0249022 +2.3885996 −1.8804907 ⎤
⎥
⎦km−1.(119)
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624 Z. Koren and I. Ravve
19.5 Gradients and Hessians of the ray velocity
Computing the spatial gradient and Hessian of the ray velocity magnitude is explained in Sections 7 and 8, the directional gradient and
Hessian—in Sections 9 and 10, and the mixed Hessian in Section 11. Recall that we consider a quasi-compressional wave and two shear
waves (from 18 shear solutions total), with the highest and second-highest ray velocity magnitudes,
vray (P)=3.3208711 km s−1,vray (SA)=3.1321140 km s−1,
vray (SB)=3.1238375 km s−1.(120)
The results are,
(i) the spatial gradients of the ray velocity,
∇xvray (P)=+1.7220279 ×10−1+2.6598414 ×10−1+3.3355897 ×10−1s−1,
∇xvray (SA)=+1.7143594 ×10−1+2.4342230 ×10−1+3.0386401 ×10−1s−1,
∇xvray (SB)=+1.7939737 ×10−1+2.4088049 ×10−1+3.1544315 ×10−1s−1,(121)
(ii) the spatial Hessians of the ray velocity,
∇x∇xvray (P)=⎡
⎢
⎣
−1.6067446 ×10−1+6.7148368 ×10−2+1.5383046 ×10−1
+6.7148368 ×10−2−2.3482982 ×10−2+3.8049694 ×10−3
+1.5383046 ×10−1+3.8049694 ×10−3+4.4258402 ×10−2⎤
⎥
⎦1(km s)−1,
∇x∇xvray (SA)=⎡
⎢
⎣
−1.7347332 ×10−1+5.0560836 ×10−2+1.4939360 ×10−1
+5.0560836 ×10−2−2.3741840 ×10−1+1.3151670 ×10−2
+1.4939360 ×10−1+1.3151670 ×10−2+4.3038140 ×10−2⎤
⎥
⎦1(km s)−1,
∇x∇xvray (SB)=⎡
⎢
⎣
−1.8224066 ×10−1+5.9695079 ×10−2+1.7338675 ×10−1
+5.9695079 ×10−2−2.6115926 ×10−1+1.6666942 ×10−2
+1.7338675 ×10−1+1.6666942 ×10−2+6.3539163 ×10−2⎤
⎥
⎦1(km s)−1,(122)
(iii) the directional gradients of the ray velocity
∇rvray (P)=+3.2528882 ×10−1−9.4290215 ×10−1+5.4257681 ×10−1km s−1,
∇rvray (SA)=+3.2912842 ×10−1−8.4164359 ×10−1+4.5052980 ×10−1km s−1,
∇rvray (SB)=+2.1681322 ×10−1−8.3996740 ×10−1+5.4691318 ×10−1km s−1,(123)
(iv) the directional Hessians of the ray velocity,
∇r∇rvray (P)=⎡
⎢
⎣
+1.9333529 −7.8716336 ×10−1−1.5118686
−7.8716336 ×10−1+3.7657751 −1.1142901
−1.5118686 −1.1142901 +1.4308811⎤
⎥
⎦km s−1,
∇r∇rvray (SA)=⎡
⎢
⎣
+1.8754254 −7.8803278 ×10−1−1.4667420
−7.8803278 ×10−1+3.5568527 −1.0914648
−1.4667420 −1.0914648 +1.5178764⎤
⎥
⎦km s−1,
∇r∇rvray (SB)=⎡
⎢
⎣
+2.0643234 −7.7448787 ×10−1−1.4646589
−7.7448787 ×10−1+3.5567363 −1.1058277
−1.4646589 −1.1058277 +1.3754454⎤
⎥
⎦km s−1,(124)
(v) and the mixed Hessian of the ray velocity,
∇r∇xvray (P)=⎡
⎢
⎣
+3.0267021 ×10−1+7.7427806 ×10−3+4.8190430 ×10−2
−3.2247291 ×10−1−1.7374209 ×10−1+5.2464846 ×10−2
+1.8790615 ×10−2+1.4545127 ×10−1−8.7943538 ×10−2⎤
⎥
⎦s−1,
∇r∇xvray (SA)=⎡
⎢
⎣
+2.8071700 ×10−1+1.8910353 ×10−2+4.7558410 ×10−2
−3.0200560 ×10−1−1.5892682 ×10−1+5.1091963 ×10−2
+1.9987545 ×10−2+1.2274370 ×10−1−8.6190271 ×10−2⎤
⎥
⎦s−1,
∇r∇xvray (SB)=⎡
⎢
⎣
+2.7383543 ×10−1−4.3757573 ×10−3+3.6219836 ×10−2
−3.0192001 ×10−1−1.5802643 ×10−1+4.9214288 ×10−2
+2.5907962 ×10−2+1.4224241 ×10−1−7.4667000 ×10−2⎤
⎥
⎦s−1.(125)
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Ray velocity derivatives in triclinic media 625
Next, we compute the gradients and Hessians of the ray velocity wrt the medium properties, ∇mand ∇m∇mvray, and the auxiliary arrays
that need to be precomputed to obtain these gradients and Hessians. To avoid long arrays and large matrices, let’s assume, for example that
we are looking for the derivatives wrt only four density-normalized stiffness tensor components: m={C11 C22C13 C44 }. The other stiffness
components are assumed fixed. The ‘upgrade’ of the test from four parameters to 21 is straightforward. To establish ∇mand ∇m∇mvray,we
compute the following values,
(i) the model-related gradients of the reference Hamiltonian,
H¯τ
m(P)=+9.0042207 ×10−5+2.0345354 ×10−4+1.1492761 ×10−4+1.0494571 ×10−3km s−1−2,
H¯τ
m(SA)=+5.6365232 ×10−5−1.3562183 ×10−4+8.1580786 ×10−5+1.8508376 ×10−5km s−1−2,
H¯τ
m(SB)=+6.5020628 ×10−5−1.2952212 ×10−4+6.4502530 ×10−5+5.9571028 ×10−4km s−1−2,(126)
(ii) the model-related Hessians of the reference Hamiltonian,
H¯τ
mm (P)=⎡
⎢
⎢
⎢
⎣
00−2.798578 ×10−5−1.8357794 ×10−4
0+5.5971560 ×10−50+4.3095661 ×10−5
−2.7985780 ×10−500−4.3176653 ×10−5
−1.8357794 ×10−4+4.3095661 ×10−5−4.3176653 ×10−50
⎤
⎥
⎥
⎥
⎦km s−1−4,
H¯τ
mm (SA)=⎡
⎢
⎢
⎢
⎣
00−3.3020085 ×10−7+3.6575709 ×10−5
0+6.6040169 ×10−70−5.9059236 ×10−5
−3.3020085 ×10−700+1.8005321 ×10−5
+3.6575709 ×10−5−5.9059236 ×10−5+1.8005321 ×10−50
⎤
⎥
⎥
⎥
⎦km s−1−4,
H¯τ
mm (SB)=⎡
⎢
⎢
⎢
⎣
00−3.3064158 ×10−6+9.1837521 ×10−5
0+6.6128316 ×10−60−6.5246235 ×10−5
−3.3064158 ×10−600−5.6412911 ×10−5
+9.1837521 ×10−5−6.5246235 ×10−5−5.6412911 ×10−50
⎤
⎥
⎥
⎥
⎦km s−1−4,(127)
(iii) the model-related gradients of the arclength-related Hamiltonian,
Hm(P)=+6.9286728 ×10−4+1.5655580 ×10−3+8.8435839 ×10−4+8.0754853 ×10−3km s−1−3,
Hm(SA)=+8.0228723 ×10−4−1.9304038 ×10−3+1.1611985 ×10−3+2.6344315 ×10−4km s−1−3,
Hm(SB)=+9.9463697 ×10−4−1.9813326 ×10−3+9.8671149 ×10−4+9.1127306 ×10−4km s−1−3,(128)
(iv) the model-related Hessians of the arclength-related Hamiltonian,
Hmm (P)=⎡
⎢
⎢
⎢
⎣
+1.9316689 ×10−4+2.2864973 ×10−4+1.1767691 ×10−5+2.9318832 ×10−4
+2.2864973 ×10−4+4.7776726 ×10−4+2.4792321 ×10−4+1.7637944 ×10−3
+1.1767691 ×10−5+2.4792321 ×10−4+2.6507562 ×10−4+1.6184596 ×10−3
+2.9318832 ×10−4+1.7637944 ×10−3+1.6184596 ×10−3+1.3522533 ×10−2
⎤
⎥
⎥
⎥
⎦km s−1−5,
Hmm (SA)=⎡
⎢
⎢
⎢
⎣
+1.7804213 ×10−4−2.7198404 ×10−4+6.9522805 ×10−5+2.1679589 ×10−4
−2.7198404 ×10−4+2.8749120 ×10−4+4.7788876 ×10−5−5.8262593 ×10−5
+6.9522805 ×10−5+4.7788876 ×10−5−1.5811755 ×10−4−2.4368816 ×10−4
+2.1679589 ×10−4−5.8262593 ×10−5−2.4368816 ×10−4−2.1872015 ×10−4
⎤
⎥
⎥
⎥
⎦km s−1−5,
Hmm (SB)=⎡
⎢
⎢
⎢
⎣
−6.6522865 ×10−5−1.8043291 ×10−5+8.3964057 ×10−5+1.4101125 ×10−3
−1.8043291 ×10−5+4.3701438 ×10−4−4.1737027 ×10−4−2.3879390 ×10−3
+8.3964057 ×10−5−4.1737027 ×10−4+3.3240908 ×10−4+9.7952971 ×10−4
+1.4101125 ×10−3−2.3879390 ×10−3+9.7952971 ×10−4+5.6801289 ×10−3
⎤
⎥
⎥
⎥
⎦km s−1−5,(129)
(v) the model-related gradients of the ray velocity,
∇mvray (P)=+7.6410683 ×10−3+1.7265263 ×10−2+9.7528675 ×10−3+8.9057943 ×10−2km s−1−1,
∇mvray (SA)=+7.8705485 ×10−3−1.8937528 ×10−2+1.1391518 ×10−2+2.5844136 ×10−3km s−1−1,
∇mvray (SB)=+9.7060263 ×10−3−1.9334558 ×10−2+9.6286866 ×10−3+8.8925312 ×10−2km s−1−1,(130)
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626 Z. Koren and I. Ravve
(vi) the model-related Hessians of the ray velocity,
∇m∇mvray (P)=⎡
⎢
⎢
⎢
⎣
−2.0556231 ×10−4−1.8372495 ×10−4+5.4808073 ×10−5+5.1314962 ×10−4
−1.8372495 ×10−4−3.6063794 ×10−4−1.7293516 ×10−4−1.1951304 ×10−4
+5.4808073 ×10−5−1.7293516 ×10−4−2.6533219 ×10−4−7.8999324 ×10−4
+5.1314962 ×10−4−1.1951304 ×10−4−7.8999324 ×10−4−4.7887107 ×10−3
⎤
⎥
⎥
⎥
⎦km s−1−3,
∇m∇mvray (SA)=⎡
⎢
⎢
⎢
⎣
−2.1472900 ×10−4+2.1390164 ×10−4+5.3453083 ×10−5+2.2838676 ×10−5
+2.1390164 ×10−4−4.1649702 ×10−4+1.9147937 ×10−4−8.2522857 ×10−5
+5.3453083 ×10−5+1.9147937 ×10−4−3.0772757 ×10−4+6.6224391 ×10−5
+2.2838676 ×10−5−8.2522857 ×10−5+6.6224391 ×10−5+5.8754274 ×10−4
⎤
⎥
⎥
⎥
⎦km s−1−3,
∇m∇mvray (SB)=⎡
⎢
⎢
⎢
⎣
−2.6517448 ×10−4+2.0915783 ×10−4+5.4738655 ×10−5+4.9249128 ×10−4
+2.0915783 ×10−4−4.1840506 ×10−4+2.0924354 ×10−4+3.0226709 ×10−4
+5.4738655 ×10−5+2.0924354 ×10−4−2.6271071 ×10−4−7.8962743 ×10−4
+4.9249128 ×10−4+3.0226709 ×10−4−7.8962743 ×10−4−4.6702001 ×10−3
⎤
⎥
⎥
⎥
⎦km s−1−3,(131)
20 VALIDATION WITH NUMERICAL TESTS
We assume that each component of the stiffness tensor varies quadratically in a proximity of the reference point, where all of them are defined,
along with their spatial gradients and Hessians (normally computed numerically). Thus, we can compute the ray velocity values (inverting for
the slowness vector and then applying the first equation of set 6) in a stencil of neighbor points, where the medium properties differ slightly
from those at the central (reference) point, and establish the spatial gradient and Hessian of the ray velocity by the second-order differences.
Similarly, we can compute the ray velocity magnitude for slightly different ray directions and establish numerically the directional gradient
and Hessian. Finally, we can change the location and direction simultaneously for numerical computation of the mixed Hessian components.
In all cases, we present the (signed) relative error,
E=N−A
A.(132)
For the first derivatives, diagonal and mixed (off-diagonal) second derivatives, the second-order central difference approximation read,
respectively,
∂f
∂x=f(x+)−f(x−)
2+O2
∂2f
∂x2=f(x+)−2f(x)+f(x−)
2+O2
∂2f
∂x1∂x2=f(x1−1,x2−2)−f(x1+1,x2−2)−f(x1−1,x2+2)+f(x1+1,x2+2)
412
.(133)
We used the spatial resolution =5×10−5km and the directional (angular) resolution 10−5radian for all cases except the mixed
Hessian of the ray velocity, where the angular resolution was 10−4radian. Note that the goal is not to achieve the best numerical accuracy (for
this, the fourth-order differences will work). The aim of this section is to make sure that our analytic relationships are correct. The computed
relative numerical errors are as follows:
(i) for the spatial gradients of the ray velocity,
E∇xvray (P)=+4.84 ×10−8+1.27 ×10−6+4.24 ×10−7,
E∇xvray (SA)=+9.19 ×10−8+7.07 ×10−7+5.45 ×10−9,
E∇xvray (SB)=+1.25 ×10−7−7.39 ×10−7−1.78 ×10−7,(134)
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Ray velocity derivatives in triclinic media 627
(ii) for the spatial Hessian of the ray velocity,
E∇x∇xvray (P)=⎡
⎢
⎣
−4.18 ×10−5−8.12 ×10−5+3.95 ×10−5
−8.12 ×10−5+4.25 ×10−6+1.70 ×10−3
+3.95 ×10−5+1.70 ×10−3−6.97 ×10−4⎤
⎥
⎦,
E∇x∇xvray (SA)=⎡
⎢
⎣
+2.61 ×10−4+2.89 ×10−4−8.94 ×10−5
+2.89 ×10−4+1.54 ×10−4−1.41 ×10−3
−8.94 ×10−5−1.41 ×10−3+5.15 ×10−4⎤
⎥
⎦,
E∇x∇xvray (SB)=⎡
⎢
⎣
−4.65 ×10−5+1.06 ×10−4−2.86 ×10−5
+1.06 ×10−4+1.23 ×10−5−8.04 ×10−5
−2.86 ×10−5−8.04 ×10−5−1.07 ×10−3⎤
⎥
⎦,(135)
(iii) for the directional gradients of the ray velocity (with the angular resolution 10−5rad),
E∇rvray (P)=+1.26 ×10−9+1.27 ×10−9+1.26 ×10−9,
E∇rvray (SA)=−2.94 ×10−9+1.32 ×10−9+4.03 ×10−9,
E∇rvray (SB)=−5.70 ×10−10 +2.63 ×10−9+3.74 ×10−9,(136)
(iv) for the directional Hessians of the ray velocity,
E∇r∇rvray (P)=⎡
⎢
⎣
+4.96 ×10−4+1.06 ×10−3+7.01 ×10−5
+1.06 ×10−3+2.82 ×10−4+1.83 ×10−4
+7.01 ×10−5+1.83 ×10−4+1.89 ×10−4⎤
⎥
⎦,
E∇r∇rvray (SA)=⎡
⎢
⎣
−9.93 ×10−4−1.99 ×10−3−1.72 ×10−4
−1.99 ×10−3−6.30 ×10−4−5.49 ×10−4
−1.72 ×10−4−5.49 ×10−4−4.91 ×10−4⎤
⎥
⎦,
E∇r∇rvray (SB)=⎡
⎢
⎣
−4.64 ×10−4−1.61 ×10−3+1.75 ×10−4
−1.61 ×10−3−2.66 ×10−4+2.31 ×10−4
+1.75 ×10−4+2.31 ×10−4+3.25 ×10−4⎤
⎥
⎦,(137)
(v) for the mixed Hessians of the ray velocity,
E∇r∇xvray (P)=⎡
⎢
⎣
+1.77 ×10−5+2.50 ×10−6−2.11 ×10−4
−3.93 ×10−5+5.65 ×10−6+7.74 ×10−6
+6.07 ×10−5−3.42 ×10−5+1.11 ×10−5⎤
⎥
⎦,
E∇r∇xvray (SA)=⎡
⎢
⎣
−4.24 ×10−5−1.50 ×10−5+3.20 ×10−4
−1.14 ×10−4−2.54 ×10−5−1.35 ×10−5
−1.72 ×10−4+9.63 ×10−5−3.25 ×10−5⎤
⎥
⎦,
E∇r∇xvray (SB)=⎡
⎢
⎣
−5.34 ×10−5−7.51 ×10−6+4.15 ×10−4
+8.80 ×10−4−2.53 ×10−5−1.01 ×10−6
−2.57 ×10−4+5.29 ×10−5−7.81 ×10−5⎤
⎥
⎦.(138)
(vi) for the model-related gradients of the ray velocity,
E∇mvray (P)=+4.40 ×10−7−4.40 ×10−10 +4.28 ×10−7+8.58 ×10−10 ,
E∇mvray (SA)=+4.51 ×10−7−1.31 ×10−9+4.19 ×10−7−3.04 ×10−8,
E∇mvray (SB)=+4.49 ×10−7−2.90 ×10−9+4.29 ×10−7−8.13 ×10−10 ,(139)
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628 Z. Koren and I. Ravve
Tab l e 5 . Inverted slowness vector for example 2.
#p1(s km–1)p2(s km–1 )p3(s km–1)vphs (km s–1 )vray (km s–1)ϑ )deg
1 0.34085086 0.23796514 0.27892780 1.99757467 2.08453574 16.607983
2 0.45478451 0.17682680 0.32080596 1.71243639 1.89454546 25.327521
3 0.53132424 0.11778196 0.31926646 1.58488933 1.88938649 32.982374
4 0.27891578 0.59341395 0.45936882 1.24906955 1.29551828 15.388950
5 0.25878957 0.65852196 0.42718589 1.20991361 1.29064979 20.373135
6 0.42767426 0.71341951 0.28412679 1.13768610 1.28629372 27.813832
7 0.95822553 −0.24471372 1.14712665 0.66024503 0.89443731 42.424408
Tab l e 6 . Signed ratio αbetween the lengths of the collinear vectors H¯τ
pand r,and
curvatures of the slowness surface for example 2.
# Curvature (km s–1 ) Ratio α(km s–1)
Minor Major Mean
1+5.8756786 +1.38104105 +3.62835984 +0.60480963
2−1.4790732 +1.39905354 −0.04000981 −0.75725244
3−1.39432774 −0.65548554 −1.02490664 −1.26735659
4−5.23881096 +1.03870922 −2.10005087 +0.75965662
5+2.07729872 −1.00756251 +0.53486811 +1.68395842
6+3.00571428 +0.13393940 +1.56982684 +7.47070975
7+6.14183727 +0.31905916 +3.23044821 +90.3238680
(vii) for the model-related Hessians of the ray velocity,
E∇m∇mvray (P)=⎡
⎢
⎢
⎢
⎣
+3.44 ×10−6+2.78 ×10−6+4.54 ×10−6+2.58 ×10−6
+2.78 ×10−6−6.56 ×10−5+2.74 ×10−6−2.87 ×10−5
+4.54 ×10−6+2.74 ×10−6+3.38 ×10−6+2.50 ×10−6
+2.58 ×10−6−2.87 ×10−5+2.50 ×10−6−1.29 ×10−5
⎤
⎥
⎥
⎥
⎦km s−1−3,
E∇m∇mvray (SA)=⎡
⎢
⎢
⎢
⎣
+3.62 ×10−6+3.10 ×10−6+4.41 ×10−6−3.19 ×10−6
+3.10 ×10−6+1.42 ×10−4+2.75 ×10−6+9.51 ×10−5
+4.41 ×10−6+2.75 ×10−6+3.38 ×10−6−7.00 ×10−7
−3.19 ×10−6+9.51 ×10−5−7.00 ×10−7−2.27 ×10−4
⎤
⎥
⎥
⎥
⎦km s−1−3,
E∇m∇mvray (SB)=⎡
⎢
⎢
⎢
⎣
+2.96 ×10−3+6.63 ×10−4−3.88 ×10−4+8.94 ×10−4
+6.63 ×10−4−1.68 ×10−4−1.38 ×10−3+1.90 ×10−3
−3.88 ×10−4−1.38 ×10−3−3.68 ×10−3−2.10 ×10−3
+8.94 ×10−4+1.90 ×10−3−2.10 ×10−3−1.54 ×10−3
⎤
⎥
⎥
⎥
⎦km s−1−3,(140)
21 COMPUTATIONAL TEST 2
In this test we do not establish the derivatives of the ray velocity magnitude; we only perform the slowness inversion and compare the signs
of coefficient αin eq. (36) and the mean curvature. Consider another triclinic medium, introduced by Igel et al.(1995) and later used by
Saenger & Bohlen (2004)andK
¨
ohn et al.(2015). Its density-normalized stiffness matrix in the Voigt notation reads,
C=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
+5.0000 +1.7500 +1.2500 −2.5000 +0.0500 +0.1500
+1.7500 +4.0000 +0.7500 +0.1000 −0.0500 −0.0750
+1.2500 +0.7500 +3.0000 +0.5000 +0.2000 +0.1200
−2.5000 +0.1000 +0.5000 +2.5000 +0.1750 +0.2625
+0.0500 −0.0500 +0.2000 +0.1750 +2.0000 −0.5000
+0.1500 −0.0750 +0.1200 +0.2625 −0.5000 +1.5000
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(141)
Assume the ray direction vector,
r=568
5√5=0.4472136 0.53665631 0.71554175 .(142)
The slowness inversion for these data leads to a qP solution and six qS solutions, total seven, listed in Table 5.InTable6, we list the
minor, major, and mean eigenvalues, along with the coefficient αfor all solutions. The inversion solutions are sorted by descending ray
velocity magnitude. The first solution corresponds to a compressional wave, the others—to shear waves.
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Ray velocity derivatives in triclinic media 629
Figure 1. Stencil for directional finite differences of the ray velocity.
The case, where the mean curvature of the slowness surface and coefficient αrelating the slowness gradient of the reference Hamiltonian,
H¯τ
p, and the ray direction vector, r, have opposite signs (solution # 4), is highlighted in yellow. Recall that the sign of the Hamiltonian has to
be changed when αproves to be negative.
22 DISCUSSION: DO WE NEED TO COMPUTE DERIVATIVES ANALYTICALLY?
As we show in the numerical test section, the numerical difference schemes are simple and deliver the ray velocity derivatives directly, without
the need to compute first analytically the derivatives of the reference and arclength-related Hamiltonians. However, any finite difference
approach requires computing the function at the neighbor points (at the nodes of the stencil). The simplest, second-order central difference
scheme for the first and second derivatives requires a spatial stencil with 15 nodes (eight at the vertices of the box, six at the face centres and
one at the central node), and an angular stencil with eight additional nodes (see Fig. 1), and a number of stencil nodes to approximate the
derivatives of the ray velocity wrt the medium properties. At each node, the slowness vector inversion should be performed to establish the
ray velocity.
However, the low efficiency of the numerical approach is not the most essential issue. The slowness inversion is an iterative procedure
resulting in an approximate solution. While the slowness components and the ray velocity magnitude can be obtained with a very good
accuracy, and the first derivatives of the ray velocity can be still estimated with a reasonable accuracy as well, this is not so for the second
derivatives of the ray velocity, especially for the shear waves with non-convex ray (group) velocity surfaces. Approximating derivatives
with finite differences is not a ‘healthy’ operation in itself, and in particular, in this problem, where we apply an approximation to another
approximation. The choice of the finite difference resolution (spatial, xi, directional θray and ψray, and medium properties, mi) becomes
challenging as well. The absolute error of the second-order finite difference scheme is O(2) but this is only the case when there are no
round-off errors due to a limited number of machine digits. In reality, keeping small requires subtracting very close values (and even closer
for the second derivatives), resulting in a considerable accuracy decrease. On the other hand, increasing obviously reduces the ‘theoretic’
accuracy of the scheme. Moreover, the most essential problem is that by increasing (spatial, directional or related to medium properties),
we can switch to ‘an alien event’, which may be a different leaf of the shear-wave ray velocity surface leading to completely wrongful values.
As a result, each parameter or pair of parameters related to the values of the off-diagonal second derivatives has a narrow ‘window’ of its
feasible resolution rangemin ≤≤max. For different components of the Hessian matrix, these windows may not necessarily overlap, and
different resolutions should be applied when computing these components. In the most ‘accidental’ case, the resolution window may not
exist at all, or the result may become dependent on the sequence of equal priority operations of the difference scheme. An example of such
dependency is the numerator of the divided difference operator in the third equation of set 133, which can be arranged as,
f(x1−1,x2−2)−f(x1+1,x2−2)−f(x1−1,x2+2)+f(x1+1,x2+2),
or
f(x1−1,x2−2)−f(x1−1,x2+2)−f(x1+1,x2−2)+f(x1+1,x2+2).
(143)
Thus, the analytical derivatives of the ray velocity are essentially more accurate and stable than their finite difference approximations. The
accuracy of the analytical derivatives depends only on the accuracy of the slowness inversion and the spatial derivatives of the medium (model)
properties, which, although in real case applications are computed numerically, in this study they are considered the input data (mxand mxx).
In summary, the proposed analytical derivatives are computed in four steps: (i) slowness inversion, (ii) derivatives of the reference
Hamiltonian, (iii) derivatives of the arclength-related Hamiltonian and (iv) derivatives of the ray velocity. In Part II of this study, we formulate
the derivatives for models with a higher anisotropic symmetry (polar anisotropy—TTI) where we adjust the reference Hamiltonian in the first
two steps and considerably improve the computational performance of the proposed approach. The two other steps remain the same, as they
are generic, governed by identical relationships for any anisotropic media.
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630 Z. Koren and I. Ravve
23 CONCLUSIONS
We have presented an original and generic approach for computing all types of first and second partial derivatives of the ray velocity at
any point along a ray path in general anisotropic elastic media. The derivatives are obtained for both quasi-compressional and quasi-shear
waves. This set of partial derivatives construct the traveltime sensitivity kernels (matrices) which provide a deep insight to the sensitivity
and reliability of seismic traveltimes and amplitudes to changes in the ray locations and directions, and to perturbations of the anisotropic
model parameters. These derivatives are intensively used in different key modelling and inversion methods, such as ray bending and seismic
tomography.
We start with an arclength-related Lagrangian representing the reciprocal ray velocity magnitude, which, in turn, is a function of the
spatially varying anisotropic model parameters and the ray direction at each ray node. Since there is no explicit form for the ray velocity (and
hence the arclength-related Lagrangian) as a function of the ray direction components, we reduce the problem to computing the derivatives
of the corresponding arclength-related Hamiltonian that explicitly depends on the medium properties and the slowness vector components
(rather than the ray direction components). We then connect the arclength-related Hamiltonian (vanishing, with the units of slowness) to
a simpler, reference Hamiltonian (vanishing, unitless, related to a scaled traveltime flow parameter) obtained directly from the Christoffel
equation. The reference Hamiltonian is used to invert for the slowness vector components, while the arclength-related Hamiltonian—for the
actual computation of the required ray velocity gradients and Hessians.
This part (Part I) provides the derivatives (gradients and Hessians) of the ray velocity magnitude for general anisotropic media, and
it represents the theoretical basis for providing the same partial derivatives for the higher symmetry anisotropic media. The advantage of
the proposed scheme is that the final derivative expressions are solely based on the derivatives of the reference Hamiltonian which can be
explicitly expressed wrt the slowness components for any type of anisotropic media. Hence, for obtaining the ray velocity derivatives for
a specific anisotropic medium (e.g. monoclinic, orthorhombic, or polar anisotropic, where all of them may be tilted), only the reference
Hamiltonian and its derivatives should be adjusted. In the next part (Part II), we provide the explicit derivatives for polar anisotropic media
(TTI), for the coupled qP and qSV waves and for SH waves. The correctness of the derivations has been tested and confirmed by checking
the consistency between the proposed analytical derivatives and the corresponding numerical ones.
ACKNOWLEDGEMENTS
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.
DATA AVAILABILITY
Data sharing is not applicable to the paper as no new data were created or analysed in this study.
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APPENDIX A: PROPERTY OF THE ARCLENGTH-RELATED HAMILTONIAN
In this appendix, we prove eq. (33), a property of the arclength-related Hamiltonian, used in our derivations, that connects its spatial gradient,
Hx, to the slowness gradient, Hp=r,
Hx=Hxp H−1
pp Hp.(A1)
Consider the inversion equation set 36, where for the arclength related Hamiltonian, the collinearity of the slowness gradient and the ray
direction has been replaced by their equality,
Hp(x,p)=r,H(x,p)=0.(A2)
Since both the Hamiltonian and its slowness gradient are constant, the corresponding full derivatives wrt the location vector vanish,
dHp(x,p)
dx=∂Hp
∂x+∂Hp
∂p
∂p
∂x=0,
dH(x,p)
dx=∂H
∂x+∂H
∂p
∂p
∂x,(A3)
or, equivalently,
Hpx +Hpp ·∂p
∂x=0,Hx+Hp·∂p
∂x=0.(A3)
From the first equation of set (A3), we compute the matrix ∂p/∂x
∂p
∂x=−H−1
pp Hpx.(A4)
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632 Z. Koren and I. Ravve
Next, we introduce eq. (A4) into the second equation of set (A3),
Hx=HpH−1
pp Hpx.(A5)
In this study, we use the tensor algebra convention that does not distinguish between the row and column vectors: a 1-D array is assumed
equal to its transpose. Applying this rule to eq. (A5), we obtain,
Hx=HT
x=Hp·H−1
pp HpxT=Hxp ·H−1
pp Hp=Hxp ·H−1
pp r.(A6)
APPENDIX B: DIRECTIONAL GRADIENT OF THE RAY VELOCITY
Partial derivatives ∂vray/∂ rcharacterize a change in the ray velocity magnitude due to infinitesimal change of a ray direction component ri
while the two other components remain fixed. However, changing a single component of the ray direction ruins the normalization r·r=1.
The remedy is to keep the direction fixed after a change in a single component and adjust all three components, keeping the vector at a unit
length. As a result, the gradient vector obtains a correction,
∇rvray =T∂vray
∂r,(B1)
where the transformation matrix T=I−r⊗ris defined in eq. (34). The proof of eq. (B1)isgivenbyRavve&Koren(2019)andKoren&
Ravve (2021). The key idea is that we distinguish between the normalized, ∇rvray, and the non-normalized, ∂vray/∂ r, directional gradients of
the ray velocity. The non-normalized gradient reads,
∂vray
∂r=∂
∂r
1
p·r=−(∇rp)Tr+(∇rr)Tp
(p·r)2=−v2
ray pT
r·r+Ip
,(B2)
where
p=Lr,pr=Lrr,Lrr =LT
rr,Lrr ·r=0.(B3)
Thus, Lrr is a symmetric matrix (tensor), and the ray direction ris one of its eigenvectors, with the corresponding eigenvalue zero. Eq.
(B2) simplifies to,
∂vray
∂r=−v2
ray p.(B4)
Combining eqs (B1)and(B4), we obtain the normalized directional gradient,
∇rvray =−v2
ray (I−r⊗r)p=−v2
ray p+(r⊗r)p.(B5)
Next, we apply a general algebraic formula,
(a⊗b)c=(b·c)a,(B6)
and obtain,
∇rvray =−v2
ray p+v2
ray (p·r)r=−v2
ray p+vrayr=vray −v2
ray p.(B7)
It follows from eq. (B7) that the directional gradient of the ray velocity is normal to the ray, ∇rvray ·r=0. Eq. (B7) can be arranged as,
∇rvray =−v2
ray p−r
vray =−v2
ray [p−r(p·r)].(B8)
The second item in brackets is the component of the slowness vector along the ray direction. The whole expression in brackets is the
difference between the full slowness vector and the ray-tangent component; it represents the slowness component in a plane normal to the
ray,
p−r(p·r)=r×p×r.(B9)
Combining eqs (B8)and(B9), we obtain the directional gradient of the ray velocity,
∇rvray =−v2
ray r×p×r=−vray ×p×vray.(B10)
Obviously, for isotropic media, the slowness and the ray direction vectors are collinear and the cross products in eq. (B10)vanish.
APPENDIX C: DIRECTIONAL HESSIAN OF THE RAY VELOCITY
Like the directional gradient, the directional Hessian of the ray velocity magnitude is also computed in two stages, obtaining first the
non-normalized and then the normalized tensors. The non-normalized directional Hessian reads,
∂2vray
∂r2=−∂
∂rv2
ray p=−2vrayp⊗∂vray
∂r−v2
ray
∂p
∂r.(C1)
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Ray velocity derivatives in triclinic media 633
Combining eqs (B4)and(C1), and taking into account that,
Lr=p,Lrr =∂p
∂r,(C2)
we obtain,
∂2vray
∂r2=2v3
rayp⊗p−v2
ray Lrr.(C3)
Next, we apply equation from set 31, Lrr =H−1
pp −λrr⊗r=TH−1
pp ,whereλris the eigenvalue of the inverse Hessian of the arclength-
related Hamiltonian, H−1
pp corresponding to the eigenvector r.
The normalization of the Hessian differs from the normalization of the gradient. To obtain the normalized Hessian, we need not only
the non-normalized Hessian, but also the non-normalized gradient, ∂vray/∂r. Still, it is a linear operator, and the components of the matrices
ˆ
Eand Tin this operator depend on the ray direction alone,
∇r∇rvray =ˆ
E∂vray
∂r+T∂2vray
∂r2T,(C4)
where Tis the second-order symmetric transformation tensor defined in eq. (1.9), and ˆ
Eis a third-order supersymmetric tensor (i.e. its three
indices can be swapped in any order), defined as,
ˆ
E=−T⊗r−r⊗T−(T⊗r)T{1,3,2}.(C5)
Notation T{1,3,2}means that the third-order tensor is transposed, the first index is unchanged, while the second and the third are
swapped.
Remark: The linear operator normalizing the directional Hessian in eq. (C4), along with the definition of its gradient-related third-order
tensor ˆ
Ein eq. (C5), are equivalent to those of eqs (42)–(47) in Ravve & Koren (2019), but this operator is presented here in a compact form,
involving only physical vectors and tensors.
Taking into account that T=I−r⊗r,eq.(C5) simplifies to,
ˆ
E=3r⊗r⊗r−I⊗r−r⊗I−(I⊗r)T{1,3,2},(C6)
where Iis the second-order identity tensor (matrix), resulting in,
ˆ
Eijk =3rirjrk−δij rk−δjk ri−δki rj,(C7)
where δlm is the Kronecker delta.
Due to its complete symmetry (obvious from eq. C7), tensor ˆ
Ehas only ten distinct components (out of the 27), defined, for example,
by three orientation parameters (e.g. Euler’s angles) and seven rotational invariants (e.g. the non-negative eigenvalues).
The first term on the right-hand side of eq. (C4) leads to a symmetric matrix,
ˆ
E∂vray
∂r=−v2
ray ˆ
Ep=−v2
ray [3r⊗r(p·r)−I(p·r)−p⊗r−r⊗p]
=v2
ray (p⊗r+r⊗p)−2vrayr⊗r+vrayT.(C8)
The second term on the right-hand side of eq. (C4), with the use of eqs (B4)and(C3), simplifies to,
T∂2vray
∂r2T=2v3
rayT(p⊗p)T−v2
rayTLrr T
=2v3
ray (I−r⊗r)p⊗p(I−r⊗r)−v2
ray (I−r⊗r)Lrr (I−r⊗r)=2v3
rayp⊗p
−2v2
ray (p⊗r+r⊗p)+2vrayr⊗r−v2
ray [Lrr −r⊗(rLrr)−(Lrr r)⊗r+(rLrr r)r⊗r],(C9)
which is a symmetric matrix as well. Recall that the ray direction is the eigenvector of the symmetric matrix Lrr, with the corresponding zero
eigenvalue, rLrr =Lrr r=0. Thus, only the first item in the square brackets on the right-hand side of eq. (C9) does not vanish. Combining eqs
(C4), (C8)and(C9), we obtain the final expression for the normalized directional Hessian of the ray velocity,
∇r∇rvray =2v3
rayp⊗p−v2
ray (p⊗r+r⊗p)−vray vray Lrr −T,(C10)
where Lrr can be computed with the equation set 34.
For isotropic media, where p=r/vray and Lrr =T/vray, this directional Hessian vanishes.
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634 Z. Koren and I. Ravve
APPENDIX D: MIXED HESSIAN OF THE RAY VELOCITY MAGNITUDE
We start from the non-normalized mixed Hessians of the ray velocity magnitude,
∂2vray
∂x∂r=2
vray
∂vray
∂x⊗∂vray
∂r−v2
raypT
x,
∂2vray
∂r∂x=2
vray
∂vray
∂r⊗∂vray
∂x−v2
raypx.(D1)
The normalized mixed Hessians are,
∇x∇rvray =∂2vray
∂x∂rT,∇r∇xvray =T∂2vray
∂r∂x.(D2)
This leads to,
∇x∇rvray =2
vray ∇xvray ⊗∇
rvray −v2
ray LxrT,
∇r∇xvray =2
vray ∇rvray ⊗∇
xvray −v2
rayTLrx ,(D3)
where
∇xvray =v2
ray Hx,Lxr =−Hxp H−1
pp ,Lrx =−H−1
pp Hpx.(D4)
Note that,
(T·p)⊗Hx=T·(p⊗Hx)and Hx⊗(T·p)=(Hx⊗p)·T,(D5)
so that equation set D3 can be arranged as,
∇x∇rvray =−v2
ray 2vray Hx⊗p+LxrT,
∇r∇xvray =−v2
rayT2vrayp⊗Hx+Lrx .(D6)
As mentioned, the two mixed Hessians of the ray velocity are transpose of each other, ∇r∇xvray =(∇x∇rvray)T.
APPENDIX E: THE SIGN OF THE REFERENCE HAMILTONIAN
In a general case, the Hamiltonian has to be defined as,
H¯τ=+det (−I)quasi −compressional wave,
H¯τ=±det (−I)quasi −shear wave.(E1)
Otherwise (if we do not pick the right sign in the second equation of this set), the shear-wave direction of the slowness gradient H¯τ
p
may become opposite to the ray direction r. Although this is generally a legitimate direction,, in this study it is unacceptable because our
derivations are based on the assumption of Hp=+r. To explain where this rule comes from and when the upper or lower sign should be used
in the second equation of set E1, consider a particular case of isotropic media; due to continuity of the Christoffel matrix, the conclusions
will be valid for anisotropic media as well. For isotropic media, the Hamiltonian reduces to,
H¯τ=C11 p2−1C44 p2−12,(E2)
or alternatively,
H¯τ=p2v2
P−1p2v2
S−12,where vS=vP√1−f,
f≡1−v2
P/v2
S.(E3)
Of course, for isotropic media, we might apply the Hamiltonian H¯τ=p2v2
P−1 for compressional waves and H¯τ=p2v2
S−1forshear
waves, but our goal now is to explore what happens to the Hamiltonian that we use in the case of ‘infinitesimal anisotropy’.
For compressional waves, p2=v−2
P, and the slowness gradient becomes,
H¯τ
p=2v2
P−v2
S2r
v3
p=2vPf2r,and Hp=r.(E4)
Fac tor 2 vPf2is positive; thus, for compressional waves, the sign is plus in the first equation of set E1. However, there is no definite
conclusion for shear waves, where in the isotropic case, the gradient of the reference Hamiltonian becomes infinitesimal, while for the
arclength-related Hamiltonian, the limits for the gradient are different from the left and from the right, Hp=±r, depending on whether the
slowness magnitude papproaches v−1
Sfrom above or from below,
Hp=sgn p2v2
S−1r,lim
p→(1/vS)−Hp=−r,lim
p→(1/vS)+Hp=+r.(E5)
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Ray velocity derivatives in triclinic media 635
Thus, for the shear wave, the sign of the Hamiltonian is undefined, and to pick the right sign, we need to check vector H¯τ
pthat may
point in the same direction or opposite direction with r. Another option is to modify the slowness inversion set as in equation set 36,
H¯τ
p(x,p)=αr,H¯τ(x,p)=0.
Set 36 consists of four equations and four unknown variables: three slowness components and a scalar coefficient α. If in the shear-wave
solution we obtain a negative alpha, we pick the minus sign in the second equation of set E1. Then the slowness gradient of the arclength
Hamiltonian is Hp=+rfor all cases.
APPENDIX F: COMPUTING DIRECTIONAL DERIVATIVES WITH FINITE
DIFFERENCES
To compute the directional derivatives, we follow the approach suggested by Ravve & Koren (2019). The ray velocity magnitude, vray(x,r)
can be computed numerically for any direction. The direction is normalized, r·r=1. In this appendix, we compute the first and second
directional derivatives, and the mixed derivatives of the ray velocity,
∂vray
∂ri
,∂2vray
∂ri∂rj
and ∂2vray
∂xi∂rj
,i=1,2,3,
j=1,2,3,(F1)
approximating them by finite differences. Regular partial derivative means that one of the arguments varies while the other stays unchanged.
This is not the case for the normalized ray (group) velocity direction, because a change of only one direction component ruins the
normalization, r·r= 1. The remedy is to introduce the two components of the polar angle—zenith θray and azimuth ψray, related to the ray
direction components ri, to compute the finite differences of the ray velocity,
vray (r)=vray θray (r),ψ
ray (r),(F2)
with respect to zenith and azimuth, and to convert then the derivatives of the ray velocity wrt the polar angle components to those wrt the
Cartesian components of the ray direction, applying the chain rule. The components of the polar angle are,
θray =arccos r3
r2
1+r2
2+r2
3
,ψ
ray =arctan (r2,r1),(F3)
where the inverse tangent of two arguments means,
cos ψray =r1
r2
1+r2
2
,sin ψray =r2
r2
1+r2
2
.(F4)
The ray direction is normalized, r2
1+r2
2+r2
3=1, but this normalization should be taken into account only after all derivatives are
computed. Applying normalization in eq. (F3) before computing derivatives leads to wrongful results.
The Cartesian components of the ray velocity direction are related to the polar angle components θray and ψray,
r1=|r|sin θray cos ψray,r2=|r|sin θray sin ψray,r3=|r|cos θray,|r|=1.(F5)
Eq. (F4) may be also arranged as,
ψray =arctan r2
r1
for r1>0,
ψray =arctan r2
r1+πfor r1<0.
(F6)
For computing derivatives, the constant additive factor πdoes not matter, and therefore we can assume that for any signs of the horizontal
direction components r1and r2, the first equation of set F6 is applied.
F1 The first derivatives
The vector form of the chain rule for the first derivatives reads,
∂vray
∂r=∂vray
∂θray
∂θray
∂r+∂vray
∂ψray
∂ψray
∂r.(F7)
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636 Z. Koren and I. Ravve
The derivatives of zenith θray are,
∂θray
∂r1=+ 1
r2
1+r2
2
r1r3
r2
1+r2
2+r2
3=+cos ψray cos θray,
∂θray
∂r2=+ 1
r2
1+r2
2
r2r3
r2
1+r2
2+r2
3=+sin ψray cos θray,
∂θray
∂r3=− r2
1+r2
2
r2
1+r2
2+r2
3=−sin θray,(F8)
The derivatives of azimuth ψray are,
∂ψray
∂r1=− r2
r2
1+r2
2=−sin ψray
sin θray
,
∂ψray
∂r2=+ r1
r2
1+r2
2=+cos ψray
sin θray
.(F9)
F2 The second derivatives
The tensor form of the chain rule for the second derivatives reads,
∂2vray
∂r2=∂2vray
∂θ2
ray
∂θray
∂r⊗∂θray
∂r+∂2vray
∂ψ2
ray
∂ψray
∂r⊗∂ψray
∂r+∂2vray
∂θray ∂ψray
×∂θray
∂r⊗∂ψray
∂r+∂ψray
∂r⊗∂θray
∂r+∂vray
∂θray
∂2θray
∂r2+∂vray
∂ψray
∂2ψray
∂r2.(F10)
The ‘pure’ (non-mixed) second derivatives of zenith θray are,
∂2θray
∂r2
1=− r3
r2
1+r2
23/2
2r2
1+r2
1r2
2−r4
2−r2
2r2
3
r2
1+r2
2+r2
32=cos 2θraycos2ψray −cos 2ψraycot θray,
∂2θray
∂r2
2=− r3
r2
1+r2
23/2
2r2
2+r2
1r2
2−r4
1−r2
1r2
3
r2
1+r2
2+r2
32=cos 2θraysin2ψray +cos 2ψraycot θray,
∂2θray
∂r2
3=+
2r2
1+r2
2r3
r2
1+r2
2+r2
32=sin 2θray.(F11)
The mixed second derivatives of zenith θray are,
∂2θray
∂r1∂r2=− r1r2r3
r2
1+r2
23/2
3r2
1+r2
2+r2
3
r2
1+r2
2+r2
32=−2−cos 2θray
2cot θray sin 2ψray,
∂2θray
∂r1∂r3=+ r1
r2
1+r2
2
r2
1+r2
2−r2
3
r2
1+r2
2+r2
32=−cos 2θray cos ψray,
∂2θray
∂r2∂r3=+ r2
r2
1+r2
2
r2
1+r2
2−r2
3
r2
1+r2
2+r2
32=−cos 2θray sin ψray.(F12)
The second derivatives of azimuth ψray are,
∂2ψray
∂r2
1=+ 2r1r2
r2
1+r2
22=+sin 2ψray
sin2θray
,
∂2ψray
∂r2
2=− 2r1r2
r2
1+r2
22=−sin 2ψray
sin2θray
,
∂2ψray
∂r1∂r2=− r2
1−r2
2
r2
1+r2
22=−cos 2ψray
sin2θray
.(F13)
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Ray velocity derivatives in triclinic media 637
The finite differences will be applied to approximate the derivatives of the ray velocity with respect to the polar angles only. The first
finite differences read,
∂vray
∂θray =vray θray +θray,ψ
ray−vray θray −θray,ψ
ray
2θray
,
∂vray
∂ψray =vray θray,ψ
ray +ψray−vray θray,ψ
ray −ψray
2ψray
.(F14)
The second ‘pure’ finite differences read,
∂2vray
∂θ2
ray =vray θray +θray,ψ
ray−2vray θray,ψ
ray+vray θray −θray,ψ
ray
θ2
ray
,
∂2vray
∂ψ2
ray =vray θray,ψ
ray +ψray−2vray θray,ψ
ray+vray θray,ψ
ray −ψray
ψ2
ray
.(F15)
The second mixed finite difference reads,
∂2vray
∂θray ∂ψray =1
4θray ψray
×vray θray +θray,ψ
ray +ψray−vray θray +θray,ψ
ray −ψray
−vray θray −θray,ψ
ray +ψray+vray θray −θray,ψ
ray −ψray.(F16)
F3 The mixed ‘location-direction’derivatives of the ray velocity
The mixed derivative is approximated by finite differences as,
∇x∇rvray =∂2vray
∂xi∂rj=∇
r∇xvray (x,r)=∇
r⎡
⎢
⎢
⎢
⎢
⎣
vray (x1+)−vray (x1−)
2
vray (x2+)−vray (x2−)
2
vray (x3+)−vray (x3−)
2
⎤
⎥
⎥
⎥
⎥
⎦
.(F17)
The items inside the brackets form a column. After computing the directional finite differences, each item evolves in a row, and all of
them become a matrix. Applying the chain rule of eq. (F7), we obtain each component of this matrix,
∇rvray (xi+)−vray (xi−)
2=∂
∂θray
vray (xi+)−vray (xi−)
2·∂θray
∂r+∂
∂ψray
vray (xi+)−vray (xi−)
2
∂ψray
∂r.(F18)
This yields the components of the mixed Hessian,
∇x∇rvrayij =∂2vray
∂xi∂rj=1
4dstang Ai
∂θray
∂rj+Bi
∂ψray
∂rj,i=1,2,3,
j=1,2,3.(F19)
where,
Ai=vrayxi+dst ,θ
ray +ang,ψ
ray−vrayxi−dst ,θ
ray +ang,ψ
ray
−vrayxi+dst ,θ
ray −ang,ψ
ray+vrayxi+dst ,θ
ray −ang,ψ
ray,
and
Bi=vrayxi+dst ,θ
ray,ψ
ray +ang−vrayxi−dst ,θ
ray,ψ
ray +ang
−vrayxi+dst ,θ
ray,ψ
ray −ang+vrayxi+dst ,θ
ray,ψ
ray −ang.
(F20)
where i=1,2,3.Thus, to approximate the directional gradient and Hessian and the mixed Hessian of the ray velocity, we need two first and
three second derivatives, approximated by finite differences,
∂vray
∂θray
,∂vray
∂ψray
,∂2vray
∂θ2
ray
,∂2vray
∂ψ2
ray
,∂2vray
∂θray ∂ψray
.(F21)
The stencil for directional derivatives is shown in Fig. 1. Each of the first derivatives requires the function values at two nodes. The
pure second derivatives require in addition the central node value. These nodes are shown in blue. The mixed second derivative requires four
additional node values, where the corresponding nodes are shown in red.
The advantage of the analytical approach proposed in this paper is the necessity to solve the nonlinear equation set (that yields the
slowness vector components) only at the central node, where with the numerical approach, we must solve this set at all nodes of the stencil.
For the second-order central differences, approximating the first and second derivatives of the ray velocity with respect to the polar angle
components, this stencil includes nine nodes, as shown in the figure. For more accurate fourth-order finite differences, the stencil includes
25 nodes. Note that both analytical and numerical azimuthal derivatives become unstable at or near spherical grid singularities θray =0and
θray =π.
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