the travel time tomography inverse problem for...

The Travel Time Tomography Inverse Problem forTransversely Isotropic Elastic Media

Joey Zou

Department of Mathematics, Stanford University

December 2, 2019MSRI Microlocal Analysis Graduate Student Seminar

Joey Zou (Stanford University) Elastic Travel Time Tomography Dec. 2, 2019 1 / 30

The Elastic Wave Equation

Consider the linear elastic wave equation

utt − Eu = 0, u : Rt × R3x → R3, (Eu)i = ρ−1

∑jkl

∂j(cijkl∂luk).

The tensor cijkl satisfies the symmetries cijkl = cjikl = cijlk = cklij .

The principal symbol of the elastic wave operator is a matrix given by−τ2Id + σ(−E )(x , ξ), with

σ(−E )(x , ξ) =

∑j ,l

cijkl(x)

ρ(x)ξjξl

ik

symmetric and positive definite for all (x , ξ).


Elastic Waves

The wavefront set of u must be contained in the set of(t, x , τ, ξ) ∈ T ∗(Rt × R3

x) where −τ2Id + σ(−E )(x , ξ) is notinvertible, i.e. the points where τ2 = Gj(x , ξ) where Gj(x , ξ)(j = 1, 2, 3) are the eigenvalues of σ(−E )(x , ξ).

A classical propagation of singularities result states that withinτ2 = Gj(x , ξ) the singularities of u will be invariant under theHamilton flow of τ2 − Gj(x , ξ) (assuming constant multiplicity of theeigenvalues).

Note that if G is the dual metric function of some metric g , then theHamilton flow of 1

2 (τ2 − G ) restricted to τ = 1 is exactly thegeodesic flow with respect to g , and the singularities would propagatein the same manner as the singularities for the scalar wave equationutt −∆gu = 0.

Thus, the Hamiltonian dynamics with respect to the Hamiltonianτ2 − Gj describe the dynamics of the elastic waves, with Gj called thewave speeds.


Figure 1: P wave propagation

Figure 2: S wave propagation

Animation courtesy of L. Braile, Purdue University: https://web.ics.purdue.edu/~braile/edumod/waves/WaveDemo.htm


https://web.ics.purdue.edu/~braile/edumod/waves/WaveDemo.htm

The Inverse Problem

We study the travel time tomography problem for transversely isotropicelasticity: given a bounded domain Ω ⊂ R3, suppose we know the traveltimes of the elastic waves between any two points on the boundary.

Problem

Does knowledge of the travel times of the wave speeds determine theelasticity tensor?

(i.e. the injectivity problem)In general, too many independent components to consider: focus first onsimpler (i.e. more symmetric) cases.


Isotropic Elasticity

The elasticity tensor a priori has up to 81 independent components(4-tensor in 3-d); however inherent symmetries reduce this to 21components.

In the case of (fully) isotropic elasticity this is reduced to just 2, oftendenoted by the Lame parameters λ and µ.

The wave speeds (eigenvalues) can be explicitly computed:

GP(x , ξ) =λ(x) + 2µ(x)

ρ(x)|ξ|2

GS(x , ξ) =µ(x)

ρ(x)|ξ|2 (multiplicity 2)


Note that the eigenvalues are (dual) Riemannian metric functions onT ∗R3: thus Hamilton trajectories are geodesics, and knowledge oftravel times is the same as knowledge of geodesic lengths w.r.t thismetric.

Hence injectivity of travel time data for isotropic elasticity wouldfollow from boundary rigidity results in Riemannian geometry.

Can use the work of Stefanov-Uhlmann-Vasy on boundary rigidity(’17) to prove injectivity for isotropic elasticity, assuming geometricconditions such as the “convex foliation” condition.

More generally, one can use the Dirichlet-to-Neumann (DN) map torecover the Lame parameters.


Transversely Isotropic Elasticity

Here there is a pointwise axisof isotropy around which thematerial behavesisotropically.

This models rock formationsin the Earth (and otherplanets), as well as the brain.

Figure 3: An example of transverselyisotropic elasticity. The normal vector tothe rock layers provides the axis ofisotropy. Picture from Wikipedia.


We denote the axis of isotropy by ξ (viewed as a covector field).There are additionally 5 independent components (“materialparameters”), denoted a11, a13, a33, a55, a66 (Voigt notation).

The wave speeds are labeled GqP , GqSH , GqSV , with

GqSH(x , ξ) = a66|ξ′|2 + a55ξ23

G±(x , ξ) = (a11 + a55)|ξ′|2 + (a33 + a55)ξ23

±√

((a11 − a55)|ξ′|2 + (a33 − a55)ξ23)2 − 4E 2|ξ′|2ξ2

3

where1E 2 = (a11 − a55)(a33 − a55)− (a13 + a55)2, + refers to the qPwave speed and − refers to the qSV wave speed, and |ξ′|2 = ξ2

1 + ξ22 .

The coordinates are taken so that ξ(x) aligns with the dx3 axis.

1Despite the notation, we don’t necessarily have E 2 ≥ 0.Joey Zou (Stanford University) Elastic Travel Time Tomography Dec. 2, 2019 9 / 30

Previous results in T.I. Elasticity

Mazzucato-Rachele ’07 showed that the DN map determined thetravel times assuming the “disjoint mode” assumption, and alsoshowed that the qSH travel times determined two material parametersand the axis of isotropy whenever boundary rigidity assumptions hold.

de Hoop-Uhlmann-Vasy ’19 also showed recovery of two materialparameters (a55 and a66) and the axis, assuming the convex foliationcondition as well as the assumption that ker ξ is an integrablehyperplane distribution (i.e. ξ is a smooth multiple of a closed form).They also showed recovery of the remaining parameters assuming thata33 and E 2 are known functions of a11.


From now we assume that the axis is known and satisfies the assumptionabove, and that a55 and a66 are known as well.We also assume Ω ⊂ R3 is a bounded domain with smooth boundary, andthat the boundary is strictly convex with respect to either qP or qSVHamiltonian dynamics. Assume as well that GqP and GqSV are strictlyconvex in the fiber variable, and that the Hamiltonian dynamics do nothave “conjugate points”.For ν one of the parameters a11, a33, or E 2, suppose aν and aν aretwo sets of parameters, and let rν = aν − aν , with rν supported in Ω.


Results

With these assumptions:

Theorem (Z. ’19)

Suppose for ν = a11, a33, or E 2 that the other parameters are known.Then we have

‖∇rν‖L2 ≤ C‖rν‖H1/2

if ν = a11 or a33 and the qP travel times agree, or if ν = E 2 and eitherthe qP or the qSV travel times agree. In particular, if rν is known to besupported in a set of sufficiently small width, then rν ≡ 0, i.e. we canrecover aν among parameters differing in sets of small width.

Remark: A priori assumptions on the support of rν are natural intime-lapse monitoring problems, where one wishes to keep track of elasticchanges in a relatively small “reservoir” region, outside of which theelasticity can be assumed to remain constant.


With the same assumptions as before:

Theorem (Z. ’19)

Suppose a11 is known. If the qP and the qSH travel times agree, then wehave

‖(∇r33,∇rE2)‖L2 ≤ C‖(r33, rE2)‖H1/2 .

In particular, if (r33, rE2) is known to be supported in a set of sufficientlysmall width, then r33, rE2 ≡ 0, i.e. we can recover a33 and E 2 amongparameters differing in sets of small width. Similarly if a33 is known andwe try to recover (a11,E

2).


With the same assumptions as before:

Theorem (Z. ’19)

Suppose there is a known functional relationship a33 = f (a11,E2) with

∂f∂a11≥ 0. If the qP and the qSH travel times agree, then we have

‖(∇r11,∇rE2)‖L2 ≤ C‖(r11, rE2)‖H1/2 .

Similar results hold if we have a functional relationship E 2 = f (a11, a33)

with∣∣∣ ∂f∂a33

∣∣∣ > 0, or a11 = f (a33,E2) with the derivatives ∂f

∂a33and ∂f

∂E2

constant and ∂f∂a336= 0.

In particular, if the differences are known to be supported in a set ofsufficiently small width, then rν ≡ 0, i.e. we can recover aν amongparameters differing in sets of small width.


Travel time vs. lens data

We will work with the lens data: in addition to the travel times betweenany pair of points on the boundary, we suppose we also know the incomingand exiting (co)vectors of the corresponding Hamilton trajectory.

Lemma

If Ω has strictly convex smooth boundary, and every pair of points on theboundary are connected by a unique Hamilton trajectory, and the materialparameters’ values (and hence the Hamiltonian) is known at the boundary,then the travel time data determines the lens data.

Proof sketch: the exiting covector on the trajectory from x0 to x1 equalsdτx0 |x1 where τx0 = dist (x0, ·). (Generalization of argument first presentedin Michel ’81 on boundary rigidity.)


Stefanov-Uhlmann Pseudolinearization

Used to solve problems in boundary rigidity.Given two vector fields V and V , and given their corresponding flowsZ (t, z) and Z (t, z), we have

Z (t, z)− Z (t, z) =

∫ t

0

∂Z

∂z(t − s,Z (s, z)) · (V − V )|Z(s,z) ds.

The proof follows from an application of the fundamental theorem ofcalculus to the function s 7→ Z (t − s,Z (s, z)).


We apply the result to V and V corresponding to the Hamilton vectorfields of G , where G is either the qP or qSV wave speed. Note

V − V = −∂x(G − G ) · ∂ξ + ∂ξ(G − G ) · ∂x

and also that we can write

(G − G ) =∑ν

Eνrν , Eν(x , ξ) =

∫ 1

0

∂G

∂ν(a11 + sr11, a33 + sr33,E

2 + srE 2 ; ξ) ds

(so e.g. for rν small we have E ν(x , ξ) ≈ ∂G∂ν (aν(x); ξ)).

V − V = −

(∑ν

E ν∇rν + ∂xEνrν

)· ∂ξ +

(∑ν

∂ξEνrν

)· ∂x .


Substituting this into the Stefanov-Uhlmann pseudolinearization formula

and keeping the bottom three rows (i.e. the rows corresponding to ∂Ξ∂(x ,ξ) )

thus gives~03 =

∑ν

I ν [∇rν ](x , ξ) + I ν [rν ](x , ξ)

for all (x , ξ), where

I ν [f1, f2, f3](x , ξ) = −∫REν(X (t),Ξ(t))

∂Ξ

∂ξ(τ(X (t),Ξ(t)), (X (t),Ξ(t))) ·

f1f2f3

(X (t)) dt

and

I ν [f ](x , ξ) =

∫R

(−∂xEν

∂Ξ

∂ξ(τ(·), ·) + ∂ξE

ν ∂Ξ

∂x(τ(·), ·)

)(X (t),Ξ(t))f (X (t)) dt

with (X (t),Ξ(t)) = (X (t, x , ξ),Ξ(t, x , ξ)) (i.e. the trajectorycorresponding passing through (x , ξ)). In other words, we haveconstructed operators I ν and I ν , which depend on the unknownparameters aν and aν , for which the differences rν satisfy a linear equation.


Formal adjoint

We compose with a formal adjoint L to have the operators map fromfunctions on R3 back to functions on R3. This L has the form

Lv(x) =

∫S2

χ(x , ω)B(x , ω)v(x , ξ(ω)) dω,

where χ is smooth, B is smooth and matrix-valued, and ξ(ω) denotes theinverse of the Hamilton map ξ 7→ ∂ξG (so that the Hamiltonbicharacteristic with momentum ξ(ω) has x-tangent vector ω).We will take our χ to vanish outside a small neighborhood of the equatorialsphere ξ ∩ S2 and to be identically one in a smaller neighborhood, andchoose B to make the composed operators principally scalar.

Write Nν± = L I ν± (ν for the parameter in question, ± for the

qP/qSV wave speed).


Stability Estimates

We thus have~03 =

∑ν

Nν±[∇rν ] + Nν

±[rν ].

For the one parameter recovery problem, it suffices to show a stabilityestimate of the following form:

Theorem

For u ∈ H1(R3) with compact support, we have

‖∇u‖L2 ≤ C (‖f ‖H2 + ‖u‖H1/2)

where f = Nν±∇u + Nν

±u.


Stability Estimates

We thus have~03 =

∑ν

Nν±[∇rν ] + Nν

±[rν ].

Similarly for the two parameters recovery problem (say recovering a11 andE 2 knowing a33), it suffices to show a stability estimate of the following:

Theorem

For u1, u2 ∈ H1(R3) with compact support, we have

‖∇(u1, u2)‖L2 ≤ C (‖f ‖H2 + ‖(u1, u2)‖H1/2)

where f =

(N11

+ NE2

+

N11− NE2

−

)(∇u1

∇u2

)+

(N11

+ NE2

+

N11− NE2

−

)(u1

u2

).

Similarly for the functional relationship recovery problem.


Symbol calculation

To show the estimates, we show Nν± and Nν

± are (matrix-valued) ΨDOs oforder −1, and we compute (qualitatively) some of their symbols:

Proposition

For ν = 11, 33,E 2, we have that Nν± and Nν

± are matrix-valued ΨDOs oforder −1, with Nν

± having scalar-valued (i.e. multiples of the identity)

principal symbols. Furthermore N11+ is elliptic, while for N = N33

+ ,NE2

± wecan write their left-reduced symbols in the form

σL(N)(x , ξ) = |ξ|−3(p2(x , ξ) + ip1(x , ξ) + P1(x , ξ) + P0(x , ξ))

where pi ,Pi ∈ S i (T ∗R3), p2 is nonnegative and vanishes only onΣ = span ξ, where it vanishes nondegenerately quadratically, p1 is real andnonzero on Σ (assuming certain geological assumptions), and P1 vanisheson Σ. Finally, all operators Nν

± except N11± have (matrix-valued) principal

symbols which vanish on Σ.


To show the statement, we can use oscillatory testing to write the symbolas an oscillatory integral. Then by stationary phase we have

σ(Nν±)(x , ξ) = a−1|ξ|−1 + a−2|ξ|−2 + . . .

where

a−1 = 2π

∫ξ⊥∩S2

χ(x , ω)E ν±(x , ξ(ω)) dS1(ω).

For E 11− , E 33

± , and EE2

± , we have E ν±(x , ξ) = f ν±(x , ξ) · (ξ · ξ(x))2 with f ν±having a definite sign, and ξ(ω) · ξ = 0 ⇐⇒ ξ annihilates ω.Thus the integral, taken over some circle S1, is nonzero unless that circle

is ξ⊥ ∩ S2; hence the principal symbol is nonvanishing except along Σ,

where it vanishes quadratically.The term a−2 can be computed via stationary phase as well. Underreasonable geological assumptions this term is never vanishing near Σ.


Parabolic parametrices: Boutet de Monvel’s calculus

A prototypical example of an operator with symbol of the form p2 + ip1

where p2 ∈ S2 vanishes quadratically on a fiber-dimension 1 subset and p1

is real-valued and is elliptic on that subset is the heat operator: ∂t −∆has symbol |ξ|2 + iτ which is of the above form. In this case its inversewill satisfy (1/2, 0) symbol estimates:∣∣∣∣Dβ

(ξ,τ)Dαx

(1

|ξ|2 + iτ

)∣∣∣∣ ≤ C |(ξ, τ)|−1−|β|/2.

This is enough to prove parabolic regularity results.We use a symbol calculus first developed by Boutet de Monvel ’74 whichgeneralizes the above observation.


To do so, choose local coordinates such thatΣ = span dxn = ξ′ = 0 ⊂ T ∗Rn where ξ′ = (ξ1, . . . , ξn−1), and let

dΣ =(|ξ′|2|ξ|2 + 1

|ξ|

)1/2.

Definition

Let m, k ∈ R. The space Sm,k(T ∗M,Σ) is the set of all a ∈ C∞(T ∗M;C)satisfying the property that whenever W α = W α1 . . .W α|α| is a product ofvector fields on T ∗M\o homogeneous of degree 0, and V β = V β1 . . .V β|β|

is a product of vector fields hom. of degree 0 and tangent to Σ, that wehave the local estimate

|W αV βa| ≤ C |ξ|mdk−|α|Σ .

Roughly speaking Sm,k are symbols of order m whose principal partvanishes to order k on Σ, with the subprincipal symbols of order less thank/2 lower also vanishing as well.


Some properties of this symbol calculus:

For k ≤ 0, we have Sm,k(T ∗M,Σ) ⊂ Sm−k/21/2 (T ∗M).

If a ∈ Sm1,0(T ∗M) and σm(a) vanishes to order k on Σ (k = 0, 1, 2),

then a ∈ Sm,k(T ∗M,Σ).

If a ∈ Sm,k(T ∗M,Σ) and |a| ≥ c |ξ|mdkΣ, then

a−1 ∈ S−m,−k(T ∗M,Σ).

The symbol class is invariant under diffeomorphisms.

We can quantize the symbol calculus to an operator calculus Ψm,k(M,Σ).Some properties:

The operator class is also invariant under diffeomorphisms, and onecan associate a principal symbol to operators in Ψm,k(M,Σ).

The calculus is closed under composition, with the principal symbol ofthe composition the product of the principal symbols.


Proof of Stability Estimate (one parameter)

For example, suppose a11 and E 2 is known, and we aim to recover a33

from the qP travel times. Recall we want to prove

‖∇u‖L2 ≤ C (‖f ‖H2 + ‖u‖H1/2)

where f = N33+ ∇u + N33

+ u.


Proof of Stability Estimate (one parameter)

From symbol calculations, we have N33+ ∈ Ψ−1,2(R3,Σ) and

N33+ ∈ Ψ−1,1(R3,Σ). Moreover, N33

+ is “elliptic” in this symbol class.Thus q = σ(N33

+ )−1 ∈ S1,−2(T ∗R3,Σ) quantizes to Q ∈ Ψ1,−2(R3,Σ)with Q N33

+ = I + R, R ∈ Ψ−1,−1(R3,Σ)⊗Mat3×3(C). Applying Q to

both sides of f = N33+ ∇u + N33

+ u gives

Qf = ∇u + R[∇u] + (Q N33+ )u.

Note Q ∈ Ψ21/2(R3), R ∈ Ψ

−1/21/2 (R3)⊗Mat3×3(C), and

Q N33+ ∈ Ψ0,−1(R3,Σ)⊗Mat3×3(C) ⊂ Ψ

1/21/2(R3)⊗Mat3×3(C), so

‖∇u‖L2(Rn) ≤ ‖Qf ‖L2(Rn) + ‖R[∇u]‖L2(Rn) + ‖(Q N)u‖L2(Rn)

≤ C (‖f ‖H2(Rn) + ‖∇u‖H−1/2(Rn) + ‖u‖H1/2(Rn))

≤ C (‖f ‖H2(Rn) + ‖u‖H1/2(Rn)).


Future directions

The argument is made globally, and to have injectivity we need globalassumption that the differences a priori have small width of support.

Would like to make “local” argument in the style ofde Hoop-Uhlmann-Vasy ’19, using the scattering calculus

I One challenge is that the corresponding scattering operator will havesubprincipal symbol vanishing at the boundary. Thus we can’t directlyapply this calculus.

I Note that the Boutet de Monvel calculus essentially “blows up Σ atfiber infinity parabolically”; possible adaptation is to perform furtherblowup at the spatial boundary.


Thanks for your attention!


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