ANISOtime : Traveltime computation software for laterally homogeneous, transversely isotropic, spherical media

概要

Supplemental Material for “ANISOtime: Traveltime computation
software for laterally homogeneous, transversely isotropic, spherical
media”
Kensuke Konishi, Anselme F. E. Borgeaud, Kenji Kawai, Robert J. Geller

1

Contents of this file

2

Text S1. Theory for laterally homogeneous, transversely isotropic, media

3

Text S2. Integration near the turning point

4

Text S3. “PolynomialStructure” file

5

Text S4. “Named Discontinuity” file

6

Text S5. Analytical solution for a spherically symmetric, TI, medium with constant velocity gradient

7

S1

8

We present a self-contained derivation of the traveltimes for laterally homogeneous, transversely isotropic (TI) media

9

with a vertical symmetry axis (sometimes referred to as VTI). As noted in the main body of the paper, results for a

10

flat-layered, laterally homogeneous, TI, medium were originally derived by Vlaar (1968), and results for a spherically

11

symmetric TI medium were derived by Vlaar (1969) and restated concisely by Woodhouse (1981). (References cited in

12

this supplemental material are listed in the bibliography in the main body of the paper.)

13

Theory for laterally homogeneous, transversely isotropic, media

We begin by stating the constitutive relation for a TI medium.

14


e
  xx 


 e 
  yy 




  ezz 


,



  2exz 




  2e 
  yz 


2exy
N


 
σ
 xx   A
 

 σ   H
 yy  
 

 

 σzz   F
 

=


 
 σxz  
 

 

 σ  
 yz  
 

σxy


H

F

A

F

F

C
L
L

(S1)

where
H = A − 2N.

(S2)

Thus there are five independent elastic constants, A, C, F, L, and N, in a TI medium. For an isotropic medium the
relations between the above five constants and λ and µ are as follows:

15

λ + 2µ = A = C

(S3)

µ=L=N

(S4)

λ = F = H.

(S5)

We first derive the theory for a flat-layered, transversely isotropic (TI), medium for SH- and pseudo-P-SV-waves, and

16

then extend the results to the spherical case.

17

S1.1

SH waves in a flat-layered, TI, medium

We consider an SH plane wave in a flat medium (the extension to spherical media is straightforward, and is given below
in subsection S1.3), for which the displacement in cartesian coordinates u = (ux , uy , uz )T is given by
ux = 0

(S6)

uy = D exp (i(ωt − kx x − kz z))

(S7)

uz = 0,

(S8)

1

18

where D is the amplitude of the displacement, ω is the angular frequency, and k = (kx , ky , kz )T is the wavenumber vector.
The non-zero strains ei j are
1
1
exy = uy,x = (−ikx uy )
2
2
1
1
eyz = uy,z = (−ikz uy ),
2
2

19

(S9)
(S10)

where the subscript , x denotes the spatial differentiation with respect to the x coordinate.
The explicit values of the non-zero components of the stress tensor for the 2-D SH case are as follows:
σxy = −Nikx uy

(S11)

σyz = −Likz uy .

(S12)

The strain energy is
1
PE = e∗i j σi j
2
1
= (2e∗xy σxy + 2e∗yz σyz )
2
1
= W∗ H W,
2

(S13)

where



H=

Nkx2 + Lkz2


W=
20

,

(S14)


,

D

(S15)

and W∗ is the conjugate transpose of W.
The kinetic energy is
1
KE = ω2 u∗i ρui
2
1
= ω2 (u∗x ρux + u∗z ρuz )
2
1
= ω2 W∗ T W,
2

21

where

T=

22

(S16)


ρ

.

(S17)

Because T and H are Hermitian, the Lagrangian, L (W), is given by

1
2

1
2

L = ω2 W∗ TW − W∗ HW.

2

(S18)

23

The equations of motion are obtained by setting the gradient of the Lagrangian to zero, which gives



ω2 T − H W = 0.
24

The dispersion relation is then obtained by solving the equation of motion (S19), giving:
s
ω=

25

26

(S19)

Nkx2 + Lkz2
.
ρ

(S20)

The group velocity U = (Ux , Uz ) is given as follows:

Ux =

∂ω
kx N/ρ
=p
∂kx
(Nkx2 + Lkz2 )/ρ

(S21)

Uz =

∂ω
kz L/ρ
=p
.
∂kz
(Nkx2 + Lkz2 )/ρ

(S22)

Traveltime and epicentral distance kernels
By analogy to eqs. (1) and (2) in the main body of the paper, the distance and time are given by
Z

X(p) =

qX (p, z)dz

(S23)

qT (p, z)dz,

(S24)

Z

T (p) =
27

28

29

where z is the depth, and X is the distance from the epicenter to the receiver along the Earth’s surface.
Given the group velocity U = (Ux ,Uz ), the kernels for the traveltime (qT ) and epicentral distance (qX ) for the flatlayered case (eqs. S23 and S24) are as follows:

qT (p) = Uz−1

(S25)

Ux
.
Uz

(S26)

qX (p) =

30

To compute qT (p), we first write Uz as a function of

L kz
Uz =
ρ kx
31

 2
kz
kx



:

N L kz2
+
ρ ρ kx2

−1/2
.

(S27)

The ray parameter, p, is given by

p=

ω
.
kx

(S28)

32

Note that we are dealing with a flat-layered model through the end of subsection S1.2, and that p thus has units of

33

s/km, but that for the spherical case (everywhere else in this paper) p has units of s/radian (see subsection S1.3 for details).
3

34

We then multiply both sides of the eq. (S28) by kz and divide both sides by kx to obtain

kz
kz
=p .
ω
kx
35

We now use the above two equations and the dispersion relation (20) to write

kx2 = p2 ω2 = p2
36

Nkx2 + Lkz2
,
ρ

(S30)

which can be re-written to obtain:

kz2
1
= 2
2
kx
p
37

where qτ is defined as in Woodhouse (1981).

38

Using eqs. (S25), (S27) and (S31), we find:

39

(S29)



ρ N p2
−
L
L


≡

qT (p) = Uz−1 =

ρ
L



N
qX (p) = p
ρ



ρ N 2
− p
L L

1 2
q ,
p2 τ

ρ N 2
− p
L L

(S31)

− 1
2

.

A similar derivation leads to
1

2

.

40

S1.2

41

We consider a pseudo-P-SV plane wave, for which the displacement in cartesian coordinates u = (ux , uy , uz )T is

42

43

(S32)

(S33)

Pseudo-P-SV waves in a flat-layered TI medium

ux = Dx exp(i(ωt − kx x − ky y))

(S34)

uy = 0

(S35)

uz = Dz exp(i(ωt − kx x − ky y)),

(S36)

where D = (Dx , Dy , Dz )T gives the amplitude of the displacement, and other quantities are the same as for the SH case.
The non-zero strains ei j are

exx = ux,x = −ikx ux

(S37)

ezz = uz,z = −ikz uz

(S38)

i
1
exz = (ux,z + uz,x ) = − (kx uz + kz ux ).
2
2

(S39)

4

44

45

The explicit values of the non-zero components of the stress tensor for the 2-D pseudo-P-SV case are as follows:

σxx = −i(Akx ux + Fkz uz )

(S40)

σyy = −i(Hkx ux + Fkz uz )

(S41)

σzz = −i(Fux kx +Cuz kz )

(S42)

σxz = −iL(kx uz + kz ux ).

(S43)

The strain energy is

1
PE = e∗i j σi j
2
1 ∗
= (exx σxx + e∗zz σzz + e∗xz σxz )
2
1
= W∗ HW,
2
46

(S44)

where W = (Ax , Az )T , W∗ is the conjugate transpose of W, and



2 + Lk2
Ak
k
k
(F
+
L)
x
z
x
z


H=
.
kx kz (F + L) Lkx2 +Ckz2
47

The kinetic energy is

1
KE = ω2 u∗i ρui
2
1
= ω2 (u∗x ρux + u∗z ρuz )
2
1
= ω2 W∗ TW,
2
48

(S45)

(S46)

where




 ρ
T=


.

(S47)

ρ
49

The two eigenvalues ω± for pseudo-P and pseudo-SV waves are, respectively, as follows:



q
2
1
2
2
2
2
2
2
2
=
k (A + L) + kz (C + L) +
kx (A − L) − kz (C − L) + 4kx kz (F + L)
2ρ x


q
2
1
2
2
2
2
2
2
2
2
ω− =
k (A + L) + kz (C + L) −
kx (A − L) − kz (C − L) + 4kx kz (F + L) .
2ρ x

ω2+

5

(S48)
(S49)

50

The components of the group velocity U± = (Ux± , Uz± )T are then given as follows:




 2

2
2
2
2kx (A − L) kx (A − L) − kz (C − L) + 4kx kz (F + L)
∂ω±
1 

q
2kx (A + L) ±
Ux± =
=
2
∂kx
4ρω
kx2 (A − L) − kz2 (C − L) + 4kx2 kz2 (F + L)2


 2

2 (C − L) + 4k2 k (F + L)2
2k
(L
−C)
k
(A
−
L)
−
k
∂ω
1
z
z
±
x
z
x
.
2kz (C + L) ±
q
Uz± =
=
2
∂kz
4ρω
2
2
2
2
2
kx (A − L) − kz (C − L) + 4kx kz (F + L)

(S50)

(S51)

51

Traveltime and epicentral distance kernels

52

As for the SH case, the kernels for the traveltime and epicentral distance integrals are computed from the group velocity.

53

To compute the traveltime kernel, we express below the group velocity Uz as a function of the ray parameter p and the TI

54

elastic constants. We re-write the z component of the group velocity (eq. 49) as function of kz /kx :




h
i
kz2
2
(L −C) (A − L) − k2 (C − L) + 2(F + L) 
p kz 
x
(C + L) ± r
.
Uz± =

i2
h
2ρ kx 
2
kz2
kz
(A − L) − k2 (C − L) + 4 k2 (F + L)2
x

55



q
2
p2
2
2
2
2
2
2
2
kx (A − L) − kz (C − L) + 4kx kz (F + L) ,
=p ω =
k (A + L) + kz (C + L) ±
2ρ x
2 2

(S53)

which can be written as a second-order equation in (kz /kx )2 :



57

x

Using eqs. (S28) and (S29) and the dispersion relation (eq. S48), we obtain the relation between kx and kz :

kx2
56

(S52)

kz
kx

4


CL +

kz
kx


2 
ρ2
ρ
ρ
AC − F 2 − 2FL − 2 (C + L) + 4 − 2 (A + L) + AL = 0,
p
p
p

(S54)

whose solution is given by


kz2
ρ
1
F 2 + 2FL − AC + 2 (C + L) ±
=
2
kx
2CL
p
s

# (S55)
ρ2
ρ
ρ
2
2
2
2
(F + 2FL − AC) + 4 ((C + L) − 4CL) − 2(AC − F − 2FL) 2 (C + L) − 4CL − 2 (A + L) + AL .
p
p
p
58

Using the quantities s1 , s2 , s3 , s4 , s5 , and R defined in Woodhouse (1981),

6

C+L
2CL
C−L
s2 = ρ
2CL

s1 = ρ

−2LCs3 = F 2 + 2FL − AC
A
s4 = s23 −
C


1
A+L
s5 = ρ
− s1 s3
2
LC
q
R = s4 p4 + 2s5 p2 + s22 ,
59

(S56)
(S57)
(S58)
(S59)
(S60)
(S61)

eq. (S55) can be simplified to

s
kz2
s2
s1
1
s
ρ
2C2 s2 − 4L2CA + 4L2C2 2 − 4LCs 2LC 1 + 4CL
=
−s
+
±
4L
(A + L)
3
3
3
2
2
4
2
kx
p
2CL
p
p
p2
s
s2
s1
s5
= −s3 + 2 ± s4 + 24 + 2 2
p
p
p
= −s3 +

s1
R
± 2
2
p
p

≡ qτ (p),

(S62)

60

where the last equality comes from comparison of eq. (S62) with the integrand qτ for the intersect traveltime in Woodhouse

61

(1981).

62

63

The denominator in the equation for the vertical component of the group velocity Uz (eq. S52) can be expressed using
the dispersion relation (eq. S48),

q
2
± kx2 (A − L) − kz2 (C − L) + 4kx2 kz2 (F + L)2 = 2ρω2± − kx2 (A + L) − kz2 (C + L)


ω2±
kz2
2
= kx 2ρ 2 − A − L − 2 (C + L) .
kx
kx
64

65

66

Thus,

s

2
kz2
k2
k2
2ρ
(A − L) − 2 (C − L) + 4 z2 (F + L)2 = 2 − A − L − z2 (C + L),
±
kx
kx
p
kx

where we used eq. (S28).
Using eqs. (S62) and (S64) , we rewrite Uz as follows:

7

(S63)

(S64)

h
i
2
kz2
2ρ
2 + (L −C) (A − L) − kz (C − L) + 2(F + L)2
−
(A
+
L)(C
+
L)
−
(C
+
L)
(C
+
L)
2
2
2
qτ
p
kx
kx
Uz =
kz2
2ρ
2ρ
− A − L − (C + L)
p2

kx2

k2
− 4LCs3 − 4CL kz2
x
.
= qτ 2
kz2
4ρ
−
2ρ(A
+
L)
−
2ρ(C
+
L)
2
2
p
kx

(C + L) 2ρ
p2

67

Using eqs. (S56) and (S62), the numerator of eq. (S65) simplifies to

(C + L)
68

(S65)

k2
2ρ
4LC
− 4LCs3 − 4CL z2 = ∓ 2 R,
2
p
kx
p

and the denominator is simplified as



kz2
4ρ2
4
−
2ρ(A
+
L)
−
2ρ(C + L) = 2 ρ2 − LCs21 ± LCRs1 − 4LCs5
p2
kx2
p


4LC
= 2 −s22 ± Rs1 − 4LCs5 .
p
69

71

(S67)

Thus, the traveltime kernel is given by

qT (p) =
70

(S66)





1
1
1 2
=
s1 ±
s2 + s5 p2 ,
Uz
qτ
R

(S68)

where the +, and - signs correspond to the pseudo-P, and pseudo-SV waves, respectively.
A similar derivation using the horizontal component of the group velocity Ux (eq. 48) leads to

qX (p) =





p
1
Ux
=
s3 ±
s5 + s4 p2 ,
Uz
qτ
R

(S69)

72

S1.3

Extension to a spherical TI medium

73

Integrands for a spherical TI medium are obtained from integrands in a flat TI medium using the following transformations

p
r
qX
q∆ =
.
r
p→

74

75

(S70)
(S71)

The traveltime and epicentral distance kernels for the spherical case for the SH, pseudo-SV, and pseudo-P waves are
thus given by, respectively,

8


− 12
ρ ρ N p2
−
qT,SH (p, r) =
L L L r2

−1 


r2
r2
1 2
p2
qT,pseudo−SV (p, r) = −s3 + s1 2 − R 2
s1 −
s + s5 2
p
p
R 2
r





−1
r2
r2
1 2
p2
qT,pseudo−P (p, r) = −s3 + s1 2 + R 2
s1 +
s2 + s5 2
,
p
p
R
r

(S72)
(S73)
(S74)

and

1
N ρ N p2 2
q∆,SH (p, r) = p
−
ρ L L r2

−1 


r2
r2
1
p2
p
s3 −
s5 + s4 2
q∆,pseudo−SV (p, r) = 2 −s3 + s1 2 − R 2
r
p
p
R
r

−1 


2
2
p
r
r
1
p2
q∆,pseudo−P (p, r) = 2 −s3 + s1 2 + R 2
s3 +
s5 + s4 2
,
r
p
p
R
r

(S75)
(S76)
(S77)

76

which are identical to the traveltime and epicentral distance kernels in Woodhouse (1981).

77

S2

78

In order to perform the integration accurately, we have to take special care in handling the integration near the turning

79

point of a raypath, where qτ = 0, and the integrands become singular but integrable. In ANISOtime, the integration for

80

the range between the turning point, at radius rturn , and radius rjeff > rturn , is computed using the relations in Jeffreys and

81

Jeffreys (1956, p. 288–290), which are also mentioned in Woodhouse (1981). The radius rjeff is defined as the radius ri

82

closest to rturn that satisfies the following inequality:

Integration near the turning point

qT (ri )/qT (ri+1 ) < threshold,
83

where ri+1 < ri is the node of the integration mesh adjacent to ri , and the threshold takes the default value of 0.9.
For SH waves, the computation for T and ∆ near a turning point can be written as:

84

Z

T=

85

Z

qT dr =

Z

87

1 ρ
dr =
qτ L

Z

1 ρ dr
dx,
L dx

(S79)

ρ − Lx dr
dx,
pLx1/2 dx

(S80)

x1/2

and
∆=

86

(S78)

Z

q∆ dr =

p N
dr =
2
r qτ L

Z

where x = q2τ near the turning point.
For the pseudo-P-SV case,
Z

T=

qT dr =




1
s1 − x ∓ R
dr
2
s
∓
s
,
+s
dx,
1
5
2
1/2
R
s3
dx
x
1

9

(S81)

88

and
Z

∆=

q∆ dr =




s1 − x ∓ R
1
s1 − x ∓ R
dr
s
±
s
+
s
dx,
3
4
5
R
s3
dx
ps3 x1/2

(S82)

89

where x = q2τ near the turning point. Note that the equations for the pseudo-P-SV case are for pseudo-P and pseudo-SV

90

when the upper and lower signs are taken, respectively.

91

√
We assume that all the integrals for T and ∆ can be approximated by α/ x and use the relations in Jeffreys and

92

Jeffreys (1956, p. 288–290).

93

S3

94

A “PolynomialStructure” completely specifies the Earth model parameters as piecewise functions using third order poly-

95

nomials within each layer. A given layer is specified as in Table S1,

“PolynomialStructure” file

Table S1: A single layer in a “PolynomialStructure” file
r1

r2

ρ0

ρ1

ρ2

ρ3

VPV,0

VPV,1

VPV,2

VPV,3

VPH,0

VPH,1

VPH,2

VPH,3

VSV,0

VSV,1

VSV,2

VSV,3

VSH,0

VSH,1

VSH,2

VSH,3

η0

η1

η2

η3

Qµ

Qκ

96

where the values at normalized radius x = r/rEART H of the density ρ (in g cm−3 ), velocities VPV , VPH , VSV , VSH (in

97

km s−1 ), and η (dimensionless) (see Panning and Romanowicz, 2006, for definitions) are given as

3

ρ(x) = ∑ ρi xi ,

(S83)

v(x) = ∑ Vi xi ,

(S84)

η(x) = ∑ ηi xi .

(S85)

i=0
3

i=0
3

i=0

98

Qµ , and Qκ are the shear, and bulk attenuation coefficients in each layer, respectively, and are not used in computations

99

by ANISOtime. They are included in the input parameter file so that the same input file can be used in both ANISOtime

100

and the DSM waveform calculation software.

101

The first line of the “PolynomialStructure” file has to contain the total number of layers. For the first layer, r1 should

102

be the radius of an Earth-like planet. For the last layer, r2 has to be 0. At present, the “PolynomialStructure” file must

10

103

contain, in this order, a “mantle” (i.e., a solid outer layer that can include a crust, upper mantle, transition zone, etc.), an

104

outer core with VSV = VSH = 0, and an inner core.

105

106

As an example, the polynomial structure file for the anisotropic PREM (Dziewonski and Anderson, 1981) is given in
Table S2.
Table S2: Complete ‘PolynomialStructure’ file for the anisotropic PREM

12
0.0

1221.5

3480.0

3630.0

5600.0

1221.5

3480.0

3630.0

5600.0

5701.0

13.0885

0.0000

-8.8381

0.0000

11.2622

0.0000

-6.3640

0.0000

11.2622

0.0000

-6.3640

0.0000

3.6678

0.0000

-4.4475

0.0000

3.6678

0.0000

-4.4475

0.0000

1.0000

0.0000

0.0000

0.0000

12.5815

-1.2638

-3.6426

-5.5281

11.0487

-4.0362

4.8023

-13.5732

11.0487

-4.0362

4.8023

-13.5732

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

1.0000

0.0000

0.0000

0.0000

7.9565

-6.4761

5.5283

-3.0807

15.3891

-5.3181

5.5242

-2.5514

15.3891

-5.3181

5.5242

-2.5514

6.9254

1.4672

-2.0834

0.9783

6.9254

1.4672

-2.0834

0.9783

1.0000

0.0000

0.0000

0.0000

7.9565

-6.4761

5.5283

-3.0807

24.9520

-40.4673

51.4832

-26.6419

24.9520

-40.4673

51.4832

-26.6419

11.1671

-13.7818

17.4575

-9.2777

11.1671

-13.7818

17.4575

-9.2777

1.0000

0.0000

0.0000

0.0000

7.9565

-6.4761

5.5283

-3.0807

29.2766

-23.6027

5.5242

-2.5514

29.2766

-23.6027

5.5242

-2.5514

22.3459

-17.2473

-2.0834

0.9783

22.3459

-17.2473

-2.0834

0.9783

1.0000

0.0000

0.0000

0.0000

11

84.6

1327.7

-1.0

57823.0

312.0

57823.0

312.0

57823.0

312.0

57823.0

Table S2: (continued)

5701.0

5771.0

5971.0

6151.0

6291.0

6346.6

6356.0

5771.0

5971.0

6151.0

6291.0

6346.6

6356.0

6371.0

5.3197

-1.4836

0.0000

0.0000

19.0957

-9.8672

0.0000

0.0000

19.0957

-9.8672

0.0000

0.0000

9.9839

-4.9324

0.0000

0.0000

9.9839

-4.9324

0.0000

0.0000

1.0000

0.0000

0.0000

0.0000

11.2494

-8.0298

0.0000

0.0000

39.7027

-32.6166

0.0000

0.0000

39.7027

-32.6166

0.0000

0.0000

22.3512

-18.5856

0.0000

0.0000

22.3512

-18.5856

0.0000

0.0000

1.0000

0.0000

0.0000

0.0000

7.1089

-3.8045

0.0000

0.0000

20.3926

-12.2569

0.0000

0.0000

20.3926

-12.2569

0.0000

0.0000

8.9496

-4.4597

0.0000

0.0000

8.9496

-4.4597

0.0000

0.0000

1.0000

0.0000

0.0000

0.0000

2.6910

0.6924

0.0000

0.0000

0.8317

7.2180

0.0000

0.0000

3.5908

4.6172

0.0000

0.0000

5.8582

-1.4678

0.0000

0.0000

-1.0839

5.7176

0.0000

0.0000

3.3687

-2.4778

0.0000

0.0000

2.6910

0.6924

0.0000

0.0000

0.8317

7.2180

0.0000

0.0000

3.5908

4.6172

0.0000

0.0000

5.8582

-1.4678

0.0000

0.0000

-1.0839

5.7176

0.0000

0.0000

3.3687

-2.4778

0.0000

0.0000

2.9000

0.0000

0.0000

0.0000

6.8000

0.0000

0.0000

0.0000

6.8000

0.0000

0.0000

0.0000

3.9000

0.0000

0.0000

0.0000

3.9000

0.0000

0.0000

0.0000

1.0000

0.0000

0.0000

0.0000

2.6000

0.0000

0.0000

0.0000

5.8000

0.0000

0.0000

0.0000

5.8000

0.0000

0.0000

0.0000

3.2000

0.0000

0.0000

0.0000

3.2000

0.0000

0.0000

0.0000

1.0000

0.0000

0.0000

0.0000

107

12

143.0

57823.0

143.0

57823.0

143.0

57823.0

80.0

57823.0

600.0

57823.0

600.0

57823.0

600.0

57823.0

108

S4

“Named Discontinuity” file

109

A “Named Discontinuity” structure file completely specifies the Earth model parameters as piecewise functions within

110

each layer, where the functions are given by the Bullen law v(r) = ArB . The “Named Discontinuity” file format is similar

111

to that used in TauP (Crotwell et al., 1999). A given layer is specified as in Table S3,
Table S3: A single layer in a ‘Named Discontinuity’ file
depth1 ρ1 VPV,1 VPH,1 VSV,1 VSH,1 η1 Qκ,1 Qµ,1
depth2

112

ρ2

VPV,2

VPH,2

VSV,2

VSH,2

η2

Qκ,2

Qµ,2

where the density ρ is in g cm−3 , the velocities VPV , VPH , VSV , VSH are in km s−1 , and η, Qκ , and Qµ are dimensionless.

113

For the first layer, depth1 has to be 0. For the last layer, depth2 should be equal to the radius of an Earth-like planet. At

114

present, the “Named Discontinuity” structure file must contain, in this order a “mantle,” an outer core with VSV = VSH = 0,

115

and an inner core, as specified in Table S4. The “outer-core” and “inner-core” keywords in Table S4 are mandatory, but

116

the “mantle” keyword is optional. If the “mantle” keyword is used, the layer above it will be assumed as the crust. All

117

layers can contain additional named discontinuities, although more discontinuities result in higher memory requirements.
Table S4: Layers in a “Named Discontinuity” file.
0

structure parameters

...

...

rcrust (optional)

structure parameters+

mantle (optional)
rcrust (optional)

structure parameters−

...

...

rCMB

structure parameters+

outer-core
rCMB

structure parameters−

...

...

rICB

structure parameters+

inner-core
rICB

structure parameters−

...

...

r planet

structure parameters

118

Note that structure parameters+ , and structure parameters− specify the density, velocities, η, Qκ , and Qµ above, and

119

below the discontinuity, respectively, that lines specifying “outer-core,” and “inner-core” must be included, and that the

120

radii above and below these lines must be equal. Lines specifying “mantle” and the radius of the crust are optional. As

121

an example, a “Named Discontinuity” structure file for the isotropic PREM (Dziewonski and Anderson, 1981) is given in

122

Table S5.
13

123

Table S5: Complete “Named Discontinuity” file for the isotropic PREM

0.00

2.60000

5.80000

5.80000

3.20000

3.20000

1.0

57823.0

600.0

15.00

2.60000

5.80000

5.80000

3.20000

3.20000

1.0

57823.0

600.0

15.00

2.90000

6.80000

6.80000

3.90000

3.90000

1.0

57823.0

600.0

24.40

2.90000

6.80000

6.80000

3.90000

3.90000

1.0

57823.0

600.0

24.40

3.38076

8.11061

8.11061

4.49094

4.49094

1.0

57823.0

600.0

40.00

3.37906

8.10119

8.10119

4.48486

4.48486

1.0

57823.0

600.0

60.00

3.37688

8.08907

8.08907

4.47715

4.47715

1.0

57823.0

600.0

80.00

3.37471

8.07688

8.07688

4.46953

4.46953

1.0

57823.0

80.0

115.00

3.37091

8.05540

8.05540

4.45643

4.45643

1.0

57823.0

80.0

150.00

3.36710

8.03370

8.03370

4.44361

4.44361

1.0

57823.0

80.0

185.00

3.36330

8.01180

8.01180

4.43108

4.43108

1.0

57823.0

80.0

220.00

3.35950

7.98970

7.98970

4.41885

4.41885

1.0

57823.0

80.0

220.00

3.43578

8.55896

8.55896

4.64391

4.64391

1.0

57823.0

143.0

265.00

3.46264

8.64552

8.64552

4.67540

4.67540

1.0

57823.0

143.0

310.00

3.48951

8.73209

8.73209

4.70690

4.70690

1.0

57823.0

143.0

355.00

3.51639

8.81867

8.81867

4.73840

4.73840

1.0

57823.0

143.0

400.00

3.54325

8.90522

8.90522

4.76989

4.76989

1.0

57823.0

143.0

400.00

3.72378

9.13397

9.13397

4.93259

4.93259

1.0

57823.0

143.0

450.00

3.78678

9.38990

9.38990

5.07842

5.07842

1.0

57823.0

143.0

500.00

3.84980

9.64588

9.64588

5.22428

5.22428

1.0

57823.0

143.0

550.00

3.91282

9.90185

9.90185

5.37014

5.37014

1.0

57823.0

143.0

600.00

3.97584

10.15782

10.15782

5.51602

5.51602

1.0

57823.0

143.0

635.00

3.98399

10.21203

10.21203

5.54311

5.54311

1.0

57823.0

143.0

670.00

3.99214

10.26622

10.26622

5.57020

5.57020

1.0

57823.0

143.0

670.00

4.38071

10.75131

10.75131

5.94508

5.94508

1.0

57823.0

312.0

721.00

4.41241

10.91005

10.91005

6.09418

6.09418

1.0

57823.0

312.0

771.00

4.44317

11.06557

11.06557

6.24046

6.24046

1.0

57823.0

312.0

871.00

4.50372

11.24490

11.24490

6.31091

6.31091

1.0

57823.0

312.0

971.00

4.56307

11.41560

11.41560

6.37813

6.37813

1.0

57823.0

312.0

1071.00

4.62129

11.57828

11.57828

6.44232

6.44232

1.0

57823.0

312.0

1171.00

4.67844

11.73357

11.73357

6.50370

6.50370

1.0

57823.0

312.0

1271.00

4.73460

11.88209

11.88209

6.56250

6.56250

1.0

57823.0

312.0

1371.00

4.78983

12.02445

12.02445

6.61891

6.61891

1.0

57823.0

312.0

1471.00

4.84422

12.16126

12.16126

6.67317

6.67317

1.0

57823.0

312.0

1571.00

4.89783

12.29316

12.29316

6.72548

6.72548

1.0

57823.0

312.0

1671.00

4.95073

12.42075

12.42075

6.77606

6.77606

1.0

57823.0

312.0

1771.00

5.00299

12.54466

12.54466

6.82512

6.82512

1.0

57823.0

312.0

1871.00

5.05469

12.66550

12.66550

6.87289

6.87289

1.0

57823.0

312.0

1971.00

5.10590

12.78389

12.78389

6.91957

6.91957

1.0

57823.0

312.0

2071.00

5.15669

12.90045

12.90045

6.96538

6.96538

1.0

57823.0

312.0

2171.00

5.20713

13.01579

13.01579

7.01053

7.01053

1.0

57823.0

312.0

2271.00

5.25729

13.13055

13.13055

7.05525

7.05525

1.0

57823.0

312.0

2371.00

5.30724

13.24532

13.24532

7.09974

7.09974

1.0

57823.0

312.0

2471.00

5.35706

13.36074

13.36074

7.14423

7.14423

1.0

57823.0

312.0

mantle

14

Table S5: (continued)

2571.00

5.40681

13.47742

13.47742

7.18892

7.18892

1.0

57823.0

312.0

2671.00

5.45657

13.59597

13.59597

7.23403

7.23403

1.0

57823.0

312.0

2741.00

5.49145

13.68041

13.68041

7.26597

7.26597

1.0

57823.0

312.0

2771.00

5.50642

13.68753

13.68753

7.26575

7.26575

1.0

57823.0

312.0

2871.00

5.55641

13.71168

13.71168

7.26486

7.26486

1.0

57823.0

312.0

2891.00

5.56645

13.71660

13.71660

7.26466

7.26466

1.0

57823.0

312.0

2891.00

9.90349

8.06482

8.06482

0.00000

0.00000

1.0

57823.0

0.0

2971.00

10.02940

8.19939

8.19939

0.00000

0.00000

1.0

57823.0

0.0

3071.00

10.18134

8.36019

8.36019

0.00000

0.00000

1.0

57823.0

0.0

3171.00

10.32726

8.51298

8.51298

0.00000

0.00000

1.0

57823.0

0.0

3271.00

10.46727

8.65805

8.65805

0.00000

0.00000

1.0

57823.0

0.0

3371.00

10.60152

8.79573

8.79573

0.00000

0.00000

1.0

57823.0

0.0

3471.00

10.73012

8.92632

8.92632

0.00000

0.00000

1.0

57823.0

0.0

3571.00

10.85321

9.05015

9.05015

0.00000

0.00000

1.0

57823.0

0.0

3671.00

10.97091

9.16752

9.16752

0.00000

0.00000

1.0

57823.0

0.0

3771.00

11.08335

9.27867

9.27867

0.00000

0.00000

1.0

57823.0

0.0

3871.00

11.19067

9.38418

9.38418

0.00000

0.00000

1.0

57823.0

0.0

3971.00

11.29298

9.48409

9.48409

0.00000

0.00000

1.0

57823.0

0.0

4071.00

11.39042

9.57881

9.57881

0.00000

0.00000

1.0

57823.0

0.0

4171.00

11.48311

9.66865

9.66865

0.00000

0.00000

1.0

57823.0

0.0

4271.00

11.57119

9.75393

9.75393

0.00000

0.00000

1.0

57823.0

0.0

4371.00

11.65478

9.83496

9.83496

0.00000

0.00000

1.0

57823.0

0.0

4471.00

11.73401

9.91206

9.91206

0.00000

0.00000

1.0

57823.0

0.0

4571.00

11.80900

9.98554

9.98554

0.00000

0.00000

1.0

57823.0

0.0

4671.00

11.87990

10.05572

10.05572

0.00000

0.00000

1.0

57823.0

0.0

4771.00

11.94682

10.12291

10.12291

0.00000

0.00000

1.0

57823.0

0.0

4871.00

12.00989

10.18743

10.18743

0.00000

0.00000

1.0

57823.0

0.0

4971.00

12.06924

10.24959

10.24959

0.00000

0.00000

1.0

57823.0

0.0

5071.00

12.12500

10.30971

10.30971

0.00000

0.00000

1.0

57823.0

0.0

5149.50

12.16634

10.35568

10.35568

0.00000

0.00000

1.0

57823.0

0.0

5149.50

12.76360

11.02827

11.02827

3.50432

3.50432

1.0

57823.0

85.0

5171.00

12.77493

11.03643

11.03643

3.51002

3.51002

1.0

57823.0

85.0

5271.00

12.82501

11.07249

11.07249

3.53522

3.53522

1.0

57823.0

85.0

5371.00

12.87073

11.10542

11.10542

3.55823

3.55823

1.0

57823.0

85.0

5471.00

12.91211

11.13521

11.13521

3.57905

3.57905

1.0

57823.0

85.0

5571.00

12.94912

11.16186

11.16186

3.59767

3.59767

1.0

57823.0

85.0

5671.00

12.98178

11.18538

11.18538

3.61411

3.61411

1.0

57823.0

85.0

5771.00

13.01009

11.20576

11.20576

3.62835

3.62835

1.0

57823.0

85.0

5871.00

13.03404

11.22301

11.22301

3.64041

3.64041

1.0

57823.0

85.0

5971.00

13.05364

11.23712

11.23712

3.65027

3.65027

1.0

57823.0

85.0

6071.00

13.06888

11.24809

11.24809

3.65794

3.65794

1.0

57823.0

85.0

6171.00

13.07977

11.25593

11.25593

3.66342

3.66342

1.0

57823.0

85.0

6271.00

13.08630

11.26064

11.26064

3.66670

3.66670

1.0

57823.0

85.0

6371.00

13.08848

11.26220

11.26220

3.66780

3.66780

1.0

57823.0

85.0

outer-core

inner-core

15

124

125

126

S5

Analytical solution for a spherically symmetric, TI, medium with constant
velocity gradient

A simple analytical solution can be found for the medium defined below:

A(r) = A0 r2
C(r) = C0 r2
L(r) = L0 r2
N(r) = N0 r2
F(r) = F0 r2
ρ = constant.
127

For the SH case, using the notation of Woodhouse (1981), we have
s
ρ

qτ =

qT =
128

r2 L

0

N0 p2
qτ0
≡
,
2
L0 r
r

r ρ
1 ρ
qT 0
=
≡
,
2
qτ0 r L0
r qτ0 L0
r

Z r2
r1

qT (r, p)dr = qT 0 ln

r2
.
r1

Similarly, for the pseudo-P-SV case, the quantities s1 , s2 , s3 , s4 , s5 , and R are given as follows:



ρ
1
1
s10
+
≡ 2
2
2r
L0 C0
r


ρ
1
1
s20
s2 = 2
−
≡ 2
2r
L0 C0
r


1
s3 =
A0C0 − F02 − 2L0 F0 ≡ s30
2L0C0
A0
s4 = s230 −
≡ s40
C0


1 ρ
A0
s10
s50
s5 =
1
+
− 2 s30 ≡ 2
2
2 C0 r
L0
r
r
s
p4
p2 s2
R0
R = s40 4 + 2s50 4 + 20
≡ 2
r
r
r4
r
s1 =

130

(S86)

(S87)

and the traveltime between radius r1 and r2 for the SH case is given as follows:

T (p) =
129

−

Using the relations defined above, the integrands for the one-way intercept can be written as
16

(S88)

r
qτ =
131

p2 R0
qτ0
s10
− s30 2 ∓ 2 ≡
,
2
r
r
r
r

and for the traveltime eq. (S90)




r s10 r2
p2 s220
qT =
∓
s50 4 + 4
qτ0 r2
R0
r
r




1 1
1
2
2
=
s10 ∓
s50 p + s20
r qτ0
R0
qT 0
.
≡
r
132

(S89)

(S90)

The traveltime between radius r1 and r2 for the quasi-P-SV case is thus given as follows:

T (p) =

Z r2
r1

qT (r, p)dr = qT 0 ln

17

r2
.
r1

(S91)