an svd in spherical surface wave tomographyqmi/presentations/2017rio.pdf · rio de janeiro, 2...

An SVD in Spherical Surface Wave TomographyChemnitz University of Technology, Faculty of Mathematics

An SVD in Spherical Surface Wave Tomography

Michael Quellmalz(joint work with Ralf Hielscher and Daniel Potts)

Chemnitz University of TechnologyFaculty of Mathematics

New Trends in Parameter Identification for Mathematical ModelsChemnitz Symposium on Inverse Problems on Tour

Rio de Janeiro, 2 November 2017

2 November 2017 · Michael Quellmalz 1 / 24 tu-chemnitz.de/∼qmi

An SVD in Spherical Surface Wave Tomography

Content

1. IntroductionMotivation

2. Arc transformDefinitionSingular value decomposition

3. Special casesArcs starting in a fixed pointRecovery of local functionsArcs with fixed length

Introduction

Content

Introduction

Funk–Radon transformI Sphere S2 = ξ ∈ R3 : ‖ξ‖ = 1I Function f : S2 → C

I Funk–Radon transform (a.k.a. Funktransform or spherical Radontransform)

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dλ(η)

Theorem [Funk 1911]

Any even function f can be reconstructed from Ff .

Introduction

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dλ(η)

Theorem [Funk 1911]

Introduction

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dλ(η)

Theorem [Funk 1911]

IntroductionMotivation

Spherical surface wave tomography

I Seismic waves propagate along the surface of the earthI Speed of propagation depends on the position on S2

Method

I Measure the traveltimes of surface waves between many pairs ofepicenter and detector

I Reconstruct the local speed of propagation

Assumption

A wave propagates along the arc of a great circle.

Method

Assumption

Method

Assumption

Method

Assumption

Method

Assumption

Method

Assumption

IntroductionSelected references

P. Funk.Uber Flachen mit lauter geschlossenen geodatischen Linien.Math. Ann., 74(2): 278 – 300, 1913.

J. H. Woodhouse and A. M. Dziewonski.Mapping the upper mantle: Three-dimensional modeling of earth structure byinversion of seismic waveforms.J. Geophys. Res. Solid Earth, 89(B7):5953–5986, 1984.

A. Amirbekyan, V. Michel, and F. J. Simons.Parametrizing surface wave tomographic models with harmonic spherical splines.Geophys. J. Int., 174(2):617–628, 2008.

R. Hielscher, D. Potts and M. Quellmalz.An SVD in spherical surface wave tomographyIn B. Hofmann, A. Leitao and J. Zubelli, Eds., New Trends in ParameterIdentification for Mathematical Models. Birkhauser, Basel, 2018.https://arxiv.org/abs/1706.05284

Arc transform

Content

Arc transformDefinition

The arc transformI Function f : S2 → R

I Surface waves: f = 1c

(c ... speed of sound)I ξ, ζ ∈ S2 not antipodalI γ(ξ, ζ) great circle arc

Definition

t(ξ, ζ) =

∫γ(ξ,ζ)

t is not continuous on S2 × S2

We choose a different parameterization

Definition

t(ξ, ζ) =

∫γ(ξ,ζ)

Definition

t(ξ, ζ) =

∫γ(ξ,ζ)

Definition

t(ξ, ζ) =

∫γ(ξ,ζ)

Definition

t(ξ, ζ) =

∫γ(ξ,ζ)

Definition

t(ξ, ζ) =

∫γ(ξ,ζ)

Definition

t(ξ, ζ) =

∫γ(ξ,ζ)

The arc transform: alternative parameterization

I ψ = arccos(ξ>ζ) ... length of γI Q ∈ SO(3) such that

I Qξ = e−ψ/2 andI Qζ = eψ/2,

where eψ = (sinψ, cosψ, 0)

DefinitionA : C(S2)→ C(SO(3)× [0, 2π]),

Af(Q,ψ) =

∫ ψ/2

−ψ/2f(Q−1eϕ) dϕ

Af(Q,ψ) =

∫ ψ/2

Af(Q,ψ) =

∫ ψ/2

Arc transformSingular value decomposition

Notation: On the sphere S2

I Spherical coordinates

ξ(ϕ, ϑ) = sin(ϑ)eϕ + cosϑ(001

I Orthonormal basis on L2(S2): spherical harmonics of degree n ∈ N

Y kn (ϕ, ϑ) =

√2n+ 1

(n− k)!

(n+ k)!P kn (cosϑ) eikϕ, k = −n, . . . , n

I P kn ... associated Legendre function

Notation: On the sphere S2

I Spherical coordinates

ξ(ϕ, ϑ) = sin(ϑ)eϕ + cosϑ(001

I Orthonormal basis on L2(S2): spherical harmonics of degree n ∈ N

Y kn (ϕ, ϑ) =

√2n+ 1

(n− k)!

(n+ k)!P kn (cosϑ) eikϕ, k = −n, . . . , n

I P kn ... associated Legendre function

Notation: On the rotation group SO(3)

I Rotation group

SO(3) = Q ∈ R3×3 : Q−1 = Q>, det(Q) = 1

I Orthogonal basis on L2(SO(3)): rotational harmonics (WignerD-functions)

Dj,kn (Q) =

∫S2Y kn (Q−1ξ)Y j

n (ξ) dξ

Notation: On the rotation group SO(3)

I Rotation group

SO(3) = Q ∈ R3×3 : Q−1 = Q>, det(Q) = 1

I Orthogonal basis on L2(SO(3)): rotational harmonics (WignerD-functions)

Dj,kn (Q) =

∫S2Y kn (Q−1ξ)Y j

n (ξ) dξ

Theorem [Dahlen & Tromp 1998]

Let n ∈ N and k ∈ −n, . . . , n. Then

AY kn (Q,ψ) =

n∑j=−n

P jn(0)Dj,kn (Q) sj(ψ),

sj(ψ) =

ψ, j = 02 sin(jψ/2)

j , j 6= 0

P jn(0) =

(−1)

√2n+14π

(n−j−1)!!(n+j−1)!!(n−j)!!(n+j)!! , n+ j even

0, n+ j odd.

Singular value decomposition [Hielscher, Potts, Q. 2017]

The operatorA : L2(S2)→ L2(SO(3)× [0, 2π]) is compact with the singularvalue decomposition

AY kn = σnE

kn, n ∈ N, k ∈ −n, . . . , n,

with singular values

√32π3

√√√√π2

∣∣∣P 0n(0)

∣∣∣2 +n∑j=1

∣∣∣P jn(0)∣∣∣2 ∈ O( 1√

and the orthonormal functions in L2(SO(3)× [0, 2π])

Enk = σ−1n

n∑j=−n

P jn(0)Dj,kn (Q) sj(ψ).

Special cases

Content

Special casesArcs starting in a fixed point

Arcs from the north pole

I Fix one endpoint of the arcs as the northpole e3:

Bf(ξ(ϕ, ϑ)) =

∫γ(e3, ξ(ϕ,ϑ))

I If f is differentiable, it can be recoveredfrom Bf by

f(ξ(ϕ, ϑ)) =d

dϑBf(ξ(ϕ, ϑ)).

Arcs from the north pole

I Fix one endpoint of the arcs as the northpole e3:

Bf(ξ(ϕ, ϑ)) =

∫γ(e3, ξ(ϕ,ϑ))

I If f is differentiable, it can be recoveredfrom Bf by

f(ξ(ϕ, ϑ)) =d

dϑBf(ξ(ϕ, ϑ)).

Arcs between two sets

More general Theorem [Amirbekyan 2007]

Let S be an open subset of S2 and A,B ⊂ Snonempty sets with A ∪B = S. If f ∈ C(S2)and ∫

γ(ξ,ζ)f dγ = 0 ∀ξ ∈ A, ζ ∈ B,

then f ≡ 0 on S.

Special casesRecovery of local functions

Theorem [Hielscher, Potts, Q. 2017]

Let f ∈ C(S2) and Ω be a convex subset of S2 whose closure Ω is strictlycontained in a hemisphere, i.e., there exists a ζ ∈ S2 such that 〈ξ, ζ〉 > 0 forall ξ ∈ Ω. If ∫

γ(ξ,η)f dγ = 0 for all ξ,η ∈ ∂Ω, (1)

then f = 0 on Ω.

I Extend f to zero outside Ω

I (1) implies that the Funk–Radon transform of f must vanishI f must be oddI Because supp f ⊂ Ω is contained in a hemisphere, f must vanish

then f = 0 on Ω.

Special casesArcs with fixed length

Arcs with fixed length

We fix the arclength ψ ∈ [0, 2π] and define

Aψ = A(·, ψ) : L2(S2)→ L2(SO(3)).

Arcs with fixed length

We fix the arclength ψ ∈ [0, 2π] and define

Aψ = A(·, ψ) : L2(S2)→ L2(SO(3)).

Singular Value Decomposition [Hielscher, Potts, Q. 2017]

Let ψ ∈ (0, 2π) be fixed. The operatorAψ : L2(S2)→ L2(SO(3)) has theSVD

AψY kn = µn(ψ)Zkn,ψ, n ∈ N, k ∈ −n, . . . , n,

with singular values

µn(ψ) =

√√√√ n∑j=−n

∣∣∣P jn(0)∣∣∣2 sj(ψ)2

and singular functions

Zkn,ψ =1

µn(ψ)

n∑j=−n

P jn(0) sj(ψ)Dj,kn ∈ L2(SO(3)).

HenceAψ is injective.2 November 2017 · Michael Quellmalz 19 / 24 tu-chemnitz.de/∼qmi

Singular values µn(ψ): dependency on nψ = 0.02π

0 5 10 15 20 25 30 35 40 45 50

polynomial degree n

1 2)µn(ψ

0 5 10 15 20 25 30 35 40 45 500

polynomial degree n

1 2)µn(ψ

0 5 10 15 20 25 30 35 40 45 50

polynomial degree n

1 2)µn(ψ

0 5 10 15 20 25 30 35 40 45 50

polynomial degree n

1 2)µn(ψ

Singular values µn(ψ): dependency on nψ = 1.00π (half circle)

0 5 10 15 20 25 30 35 40 45 50

polynomial degree n

1 2)µn(ψ

0 5 10 15 20 25 30 35 40 45 50

polynomial degree n

1 2)µn(ψ

0 5 10 15 20 25 30 35 40 45 50

polynomial degree n

1 2)µn(ψ

Singular values µn(ψ): dependency on nψ = 2.00π (Funk–Radon transform)

0 5 10 15 20 25 30 35 40 45 50

polynomial degree n

1 2)µn(ψ

Singular values µn(ψ): dependency on arc-length ψ

0 π 2π0

1 2)µn(ψ

n even

limitn = 0

0 π 2π0

limitn = 1

0 π 2π0

1 2)µn(ψ

n even

limitn = 2

0 π 2π0

limitn = 3

0 π 2π0

1 2)µn(ψ

n even

limitn = 4

0 π 2π0

limitn = 5

0 π 2π0

1 2)µn(ψ

n even

limitn = 6

0 π 2π0

limitn = 7

0 π 2π0

1 2)µn(ψ

n even

limitn = 8

0 π 2π0

limitn = 9

0 π 2π0

1 2)µn(ψ

n even

limitn = 100

0 π 2π0

limitn = 101

Singular values: asymptotic behavior

The singular values µn(ψ) ofAψ satisfy for odd n = 2m− 1

limm→∞

4m− 1

4µ2m−1(ψ)2 =

2πψ, ψ ∈ [0, π]

4π2 − 2πψ, ψ ∈ [π, 2π],

and for even n = 2m

limm→∞

4µ2m(ψ)2 =

2πψ, ψ ∈ [0, π]

12πψ − 2π2, ψ ∈ [π, 2π].

ψ = 2π: Great circles [Funk 1911]

I Funk–Radon transform

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dγ(η)

I Injective only for even functions

ψ = π: Half-circle transform

I Injective for all functions [Groemer 1998], [Goodey & Weil 2006]I [Rubin 2017] half circles in one hemisphere

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dγ(η)

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dγ(η)

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dγ(η)

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dγ(η)

F : C(S2)→ C(S2),

Ff(ξ) =

∫〈ξ,η〉=0

f(η) dγ(η)

The end

\endinput

an svd in spherical surface wave tomographyqmi/presentations/2017rio.pdf · rio de janeiro, 2...

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