full waveform inversion via matched source · pdf fileoverviewmatched source waveform...

Overview Matched Source Waveform Inversion Analysis of Gradient and Hessian Numerical Examples Conclusion and Discussion

Full Waveform Inversion via Matched SourceExtension

Guanghui Huang and William W. Symes

TRIP, Department of Computational and Applied Mathematics

May 1, 2015, TRIP Annual Meeting

G. Huang and W. W. Symes MSWI 1 / 31


Outline

1 Overview

2 Matched Source Waveform Inversion

3 Analysis of Gradient and Hessian

4 Numerical Examples

5 Conclusion and Discussion



Outline

1 Overview







Overview

Assume that the received pressure field p(xr, t;xs) generated by acausal isotropic point radiator at source position x = xs satisfiesthe constant density acoustic wave equation,

1

v2∂2p

∂t2−∆p = δ(x− xs)f(t) in R2. (1)

p|t=0 =∂p

∂t

∣∣∣t=0

= 0 (2)

Let’s introduce the forward modeling operator S[v] to relate thevelocity v(x, z) and wavelet function f(t) to the scattering field atthe receiver xr,

S[v, f ](xr, t;xs) = p(xr, t;xs). (3)



Full Waveform Inversion

Given recorded traces d(xr, t;xs), find velocity v and waveletfunction f such that S[v, f ] = d.

FWI via data fitting,

JFWI[v, f ] =1

2

∑xr,xs

∫|S[v, f ](xr, t;xs)− d(xr, t;xs)|2dt. (4)

FWI objective function is quadratic with respect to f , buthighly nonlinear and nonconvex in velocity v (frequencydependent).

Cycle skipping problem (eg. Symes, 1994).



Outline

1 Overview







Extended Modeling & Null Space

Extended Modeling:

Let f̄(xr,xs, t) be the extended model of f(t), define the extendedmodeling operator

S̄[v, f̄ ] = p̄(xr, t;xs)

where p̄ is the solution of (1)-(2) with source function being f̄ .

Null Space (Annihilator):

Differential semblance operator A = ∂zs (see Symes (1994));

t-moment operator after source signature deconvolutionA = tf−1(t) (eg. deconvolution-based by Luo and Sava(2011), AWI by Warner (2014)).



MSWI

Matched Source Waveform Inversion (MSWI) is stated as follows,

JMS[v] =1

2

∑xr,xs

∫|Af̄ |2dt (5)

s.t. S̄[v, f̄ ](xr, t;xs) = d(xr, t;xs). (6)

Key feature: Even given wrong velocity, data fitting is perfect,hence no cycle skipping problem!


Overview Matched Source Waveform Inversion Analysis of Gradient and Hessian Numerical Examples Conclusion and DiscussionAnalysis of Gradient Local Convexity of Hessian Relation with DSO formulation

Outline

1 Overview


3 Analysis of Gradient and HessianAnalysis of GradientLocal Convexity of HessianRelation with DSO formulation





1 Overview







Factorization of Tangent Operator

Under single arrival approximation, we have

S̄[v, f̄ ](xr, t;xs) ≈ a(xr,xs)f̄(xr,xs, t− τ(xr,xs))

Then taking the first order variation of S̄[v, f̄ ] formally gives us thedesired factorization of operator

(DS̄[v]δv)f̄(xr, t;xs)≈a(xr,xs)∂

∂tf̄(xr,xs, t− τ(xr,xs)

)(−Dτ [v]δv)

, S̄[v,Q[v, δv]f̄ ].

where Dτ [v] is the tangent operator of traveltime function, andQ[v, δv]f̄ is bilinear operator with respect to δv and f with

Q[v, δv] = −(Dτ [v]δv)∂

∂t.



Backprojection of Traveltime Differences

Taking the first order perturbation of JMS, we have

DJMS[v]δv ≈∑xr,xs

∫ATAf̄(Dτ [v]δvf̄t)dt

Hence the gradient is given by,

g ≈∑xr,xs

Dτ [v]T(∫

(ATAf̄)f̄tdt)

Here Dτ [v]T is the adjoint operator of Dτ [v], which backprojectsits arguments along rays.



Adjoint Source in the Gradient

Assume that data is noise-free and well approximated by geometricoptics,

d(xr, t;xs) ≈ a∗(xr,xs)f∗(t− τ [v∗](xr,xs)). (7)

Denote by ∆τ(xr,xs) = τ [v∗](xr,xs)− τ [v](xr,xs), then

For differential semblance operator A = ∂zs ,∫(ATAf̄)f̄tdt ≈ −

(a∗a

)2 ∫ (∂f∗∂t

)2dt(

∂

∂zs)T( ∂

∂zs∆τ [v](zr, zs)

).

For t-moment operator, we have∫(ATAf̄)f̄tdt ≈ −(a∗/a)2∆τ [v](zr, zs).



Traveltime Tomography

Travel tomography attempts to minimize the difference betweenthe computed traveltime and picked traveltime from data,

JTT =1

2

∑xr,xs

‖τ [v](xr,xs)− τ [v∗](xr,xs)‖2,

The gradient of JTT is given by,

∇JTT = −∑xr,xs

Dτ [v]T (∆τ [v](zr, zs)).



1 Overview







Local Convexity of Hessian

The Hessian of JMS at consistent data Af̄ = 0 is given by

D2J [v∗](δv, δv) ≈∑xr,xs

∫|[A,Q](v∗, δv)f̄ |2dt

where [A,Q] = AQ−QA.

for A = ∂zs ,

D2J [v∗](δv, δv) ≈∫ (∂f∗

∂t

)2dt( ∑

xr,xs

∣∣∣ ∂∂zs

Dτ [v∗]δv∣∣∣2)

for A = tf−1(t),

D2J [v∗](δv, δv) ≈∑xr,xs

|Dτ [v∗]δv|2

Hessian of JMS is proportional to the Hessian of a traveltimeobjective function, and is as convex as tomographic objective.



1 Overview







Relation with DSO formulation

DSO formulation seeks the balance between data fitting andannihilator term, i.e.

Jα[v, f̄ ] =1

2

∑xr,xs

∫|S̄[v, f̄ ]− d|2dt+

α

2

∑xr,xs

∫|Af̄ |2dt.

As α→ 0, the gradient and Hessian of 1αJα is the same as

JMS.

As α→ +∞, Af̄ = 0, then it’s equivalent to minimize JFWI



Outline

1 Overview







Example 1: Velocity Scan of JMS

0.3 0.4 0.5 0.6 0.70

2

4

6

8

10

Slowness (s/km)

Obj

ectiv

e fu

nctio

n va

lue

Figure: JMS[v] for homogeneous velocity v, 0.25 s/km ≤ v−1 ≤ 0.75s/km. Correct velocity is 2 km/s.



Example 2: Gaussian Lens

In this example, the target velocity consists of two Gaussianvelocity anomalies embedded in a v = 2km/s background:

v(x, z) = 2− 0.6e− (x−0.25)2+(z−0.3)2

(0.2)2 − 0.6e− (x−0.25)2+(z−0.7)2

(0.1)2 ,

where x ∈ [0, 0.5]km, z ∈ [0, 1]km. The initial model is given bythe constant velocity v0 = 2 km/s. Data is consisted of 50 shotsand 99 receivers for each shot, which are uniformly distributed.




Distance (m)

Tim

e (s

)

0 1000 2000 3000 4000

0

0.2

0.4

0.6

0.8

1 −400

−200

0

200

400

600

800

Distance (m)

Tim

e (s

)

0 1000 2000 3000 4000

0

0.2

0.4

0.6

0.8

1 −400

−200

0

200

400

600

800




Distance (m)

Dep

th (

m)

0 100 200 300 400 500

0

200

400

600

800

1000 1400

1500

1600

1700

1800

1900

2000

Distance (m)

Dep

th (

m)

0 100200300400500

0

200

400

600

800

1000

1600

1700

1800

1900

2000




Distance (m)

Tim

e (s

)

0 1000

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1 −40

−20

0

20

40

60

80

Figure: Extended source functions



Example 3: Big Gaussian

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

200

400

600

800

1000 1400

1500

1600

1700

1800

1900

Figure: Low velocity Gaussian anomaly model with radius 250m× 150membedded in the constant background velocity v0 = 2000m/s




0 50 100 150 200 250 300 350

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 −200

−100

0

100

200

300

Figure: Receivers are uniformly distributed long xr = 1990m fromzr = 10m to xr = 990m for each shots, shots interval is 20m. Thezero-phase ricker wavelet with main frequency f = 10Hz for generatingthe data.




0 500 1000 1500 2000

0

100

200

300

400

500

600

700

800

900

1000 1700

1750

1800

1850

1900

1950

2000

2050

0 500 1000 1500 2000

0

100

200

300

400

500

600

700

800

900

1000

1500

2000

2500

3000

3500

Figure: Inverted velocity after 50 iterations by MSWI (top) and FWI(bottom)



Outline

1 Overview







Conclusion and Discussion

Conclusion

Establish the relation between waveform inversion andtraveltime tomography;

Convexity of objective function is independent of frequencycontent.

MSWI still get stuck in local minima due to the strongmultipaths.

Discussion

Does perfect data fitting (no cycle skipping) mean avoidinglocal minima;

How do we choose the extended model space and the relatedannihilator.



To be cont’d

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

200

400

600

800

1000 1400

1500

1600

1700

1800

1900

2000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

200

400

600

800

1000 1400

1500

1600

1700

1800

1900

2000

Figure: Inverted velocity after 100 iterations and 200 iterations



Thanks

Total E&P USA, for funding me

The Rice Inversion Project, for supporting me

TRIP members, for helping me

All of you, for listening to me


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