velocity inversion using the quadratic wasserstein metric...oct 03, 2020 · -8 -6 -4 -2 0 2 4 6...

Velocity Inversion Using the Quadratic WassersteinMetric

Srinath MahankaliMentor: Prof. Yunan Yang (NYU)

Stuyvesant High School

October 17, 2020MIT PRIMES Conference

Srinath Mahankali Velocity Inversion October 17, 2020 1 / 17

Motivation

Goal: find materials lying underground and their positions

Motivation

Goal: find materials lying underground and their positions

Approach

1C(x)2

∂2ψ∂t2 −

(∂2ψ∂x1

2 + ∂2ψ∂x2

2 + · · ·+ ∂2ψ∂xn2

)= f (x, t),

ψ(x, 0) = 0, ψt(x, 0) = 0 on Ω,

∇ψ · n = 0 on ∂Ω

C∗(x) = argmin J(g(C), h) where J is the objective function

Convexity with respect to the velocity is beneficial

Approach

1C(x)2

∂2ψ∂t2 −

(∂2ψ∂x1

2 + ∂2ψ∂x2

2 + · · ·+ ∂2ψ∂xn2

)= f (x, t),

ψ(x, 0) = 0, ψt(x, 0) = 0 on Ω,

∇ψ · n = 0 on ∂Ω

Approach

1C(x)2

∂2ψ∂t2 −

(∂2ψ∂x1

2 + ∂2ψ∂x2

2 + · · ·+ ∂2ψ∂xn2

)= f (x, t),

ψ(x, 0) = 0, ψt(x, 0) = 0 on Ω,

∇ψ · n = 0 on ∂Ω

Problems With the Squared L2 Norm

Beydoun-Tarantola, 1988

Nonconvexity of squared L2 norm and local minima

Brossier et al, 2010

Sensitivity of L2 norm to noise

-8 -6 -4 -2 0 2 4 6 8

-2: < x < 2:

0.25Graph of f(x)

-10 -5 0 5 10

-4: < s < 4:

Squared L2 Norm of f(x) - f(x-s)

An Alternative Objective Function

Engquist-Froese, 2013 introduced the Wasserstein metric for FWI

Yang, 2019

Convexity in translations and dilations of the dataInsensitivity to noise

Engquist et al, 2020

Low-frequency bias

-8 -6 -4 -2 0 2 4 6 8

-2: < x < 2:

0.25Graph of f(x)

-10 -5 0 5 10

-4: < s < 4:

Squared W2 Distance of f(x) and f(x-s)

What is the Wasserstein Metric?

Optimal transport introduced by Monge

Definition

The pth Wasserstein metric is defined as

Wp(f , g) =

T∈M(µ,ν)

∫Ω|x − T (x)|pdµ

What is the Wasserstein Metric?

Explicit formula when data is in one dimension

Theorem

Let g and h be two probability distributions defined on R, and letG (t) =

∫ t−∞ g(s) ds and H(t) =

∫ t−∞ h(s) ds. Then

W 22 (g , h) =

0(G−1(s)− H−1(s))2 ds.

The Constrained Optimization Problem

1C(x)2

∂2ψ∂t2 −

(∂2ψ∂x1

2 + ∂2ψ∂x2

2 + · · ·+ ∂2ψ∂xn2

)= f (x, t),

ψ(x, 0) = 0, ψt(x, 0) = 0 on Ω,

∇ψ · n = 0 on ∂Ω

C∗(x) = argmin W 22 (g(C), h)

Previous results involve convexity in changes of the data

1C(x)2

∂2ψ∂t2 −

(∂2ψ∂x1

2 + ∂2ψ∂x2

2 + · · ·+ ∂2ψ∂xn2

)= f (x, t),

ψ(x, 0) = 0, ψt(x, 0) = 0 on Ω,

∇ψ · n = 0 on ∂Ω

1C(x)2

∂2ψ∂t2 −

(∂2ψ∂x1

2 + ∂2ψ∂x2

2 + · · ·+ ∂2ψ∂xn2

)= f (x, t),

ψ(x, 0) = 0, ψt(x, 0) = 0 on Ω,

∇ψ · n = 0 on ∂Ω

Challenges and Solutions

Wave data is generally not a probability distribution

Normalize wave data k(t) by k(t) = k(t)+γ∫ T0

k(t)+γ dt

Computing the W2 distance is difficult for higher dimensions

Data is only a function of time → apply one dimensional formula

No explicit solution for wave equation in general

k(t)+γ dt

Velocity Models in One Dimension

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Position

Velocity Model 1

0 0.5 1 1.5 2 2.5 3 3.5 4

Position

Velocity Model 2

0 0.5 1 1.5 2 2.5 3 3.5 4

Position

Velocity Model 3

Theorem

Suppose f (t) is nonnegative and compactly supported, and let k be thevelocity parameter in these three cases. Then W 2

2 (g(k), h) is convex in kover the interval (0, k∗].

Ray Tracing

Velocity function of the form C(X , z) = a + bz

Distance and traveltime formulas:

X = 2p

∫ zp

dz√u2(z)− p2

∫ zp

u2(z)√u2(z)− p2

For source wave data f (t), the predicted wave data is approximatedby Apredf (t − Tpred)

Ray Tracing

X = 2p

∫ zp

dz√u2(z)− p2

∫ zp

u2(z)√u2(z)− p2

Ray Tracing

X = 2p

∫ zp

dz√u2(z)− p2

∫ zp

u2(z)√u2(z)− p2

Ray Tracing

X = 2p

∫ zp

dz√u2(z)− p2

∫ zp

u2(z)√u2(z)− p2

Velocity Model in Two Dimensions

Theorem

Assume f (t) is nonnegative and compactly supported. Then, W 22 (g , h) is

jointly convex in a, b over the following region U:

U := (a, b) ∈ R2 : a, b > 0,bXr

2a≥ S0, T (Xr , a, b) ≥ T (Xr , a

∗, b∗)

for some positive constant S0.

For large Xr , the convex region contains (a∗, b∗)

Nonuniqueness of solution fixed by adding multiple receiver locations

Theorem

U := (a, b) ∈ R2 : a, b > 0,bXr

2a≥ S0, T (Xr , a, b) ≥ T (Xr , a

∗, b∗)

Theorem

U := (a, b) ∈ R2 : a, b > 0,bXr

2a≥ S0, T (Xr , a, b) ≥ T (Xr , a

∗, b∗)

Future Work

Study models with a larger number of parameters

Possible to study convexity using frequency instead of time domain

Future Work

Study models with a larger number of parameters

Possible to study convexity using frequency instead of time domain

Acknowledgements

I am extremely thankful to:

My mentor, Prof. Yunan Yang

The PRIMES program

My parents and my brother

Acknowledgements

I am extremely thankful to:

My mentor, Prof. Yunan Yang

The PRIMES program

My parents and my brother

Link to my paper (posted on PRIMES website):https://arxiv.org/abs/2009.00708

References

Wafik B. Beydoun and Albert Tarantola. First Born and Rytovapproximations: Modeling and inversion conditions in a canonicalexample. The Journal of the Acoustical Society of America,83(3):1045–1055, 1988.

Romain Brossier, Stephane Operto, and Jean Virieux. Which residualnorm for robust elastic frequency-domain full waveform inversion.Geophysics, 75, 05 2010.

Bjorn Engquist and Brittany D. Froese. Application of theWasserstein metric to seismic signals. arXiv preprint arXiv:1311.4581,2013.

Bjorn Engquist, Kui Ren, and Yunan Yang. The quadraticWasserstein metric for inverse data matching. Inverse Problems,36(5):055001, May 2020.

References

Gaspard Monge. Memoire sur la theorie des deblais et des remblais.Histoire de l’Academie Royale des Sciences de Paris, 1781.

Yunan Yang. Analysis and Application of Optimal Transport ForChallenging Seismic Inverse Problems, 2019.

velocity inversion using the quadratic wasserstein metric...oct 03, 2020 · -8 -6 -4 -2 0 2 4 6...

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