approximate inverse for the common offset …rieder/media/slideslinz2017.pdf · approximate inverse...

Approximate inverse for the common offset acquisition geom-etry in 2D seismic imaging

Andreas Rieder Christine Grathwohl Peer Kunstmann Todd Quinto

KIT The Research University in the Helmholtz Association

FAKULTAT FUR MATHEMATIK INSTITUT FUR ANGEWANDTE UND NUMERISCHE MATHEMATIK

www.kit.edu

CRC 1173

Organization of the material

2 cAndreas Rieder Approximate Inverse for the common offset acquisition geometry in 2D seismic imaging 100 Years of The Radon transform, March 27, 2017

The Problem

The Method

The Experiments

The Future

The Problem

The ProblemThe Method

The Experiments

The Future


An inverse problem for the acoustic wave equation


u(t;x,xs) acoustic potential in x R2 at time t 0

1

22t uxu = (x xs)(t)

= (x) speed of sound, xs excitation (source) point.

Seismic imaging

Recover from the backscattered (reflected) fields

u(t;xr,xs), t [0, Tmax], (xr,xs) R Swhere

S/R sets of source/receiver points, and

Tmax observation period.

Generalized Radon transform


Consider the ansatz1

2(x)=

1 + n(x)

c2(x),

c = c(x) smooth and known background velocity.Determine n from

Fw(T ;xr,xs) =

T

0(u u)(t;xr,xs)dt

with the generalized Radon transform

Fw(T ;xr,xs) =

w(x)

c2(x)a(x,xs)a(x,xr)

(T (x,xs) (x,xr)

)dx

which integrates w over reflection isochrones: T = (,xs) + (,xr).Travel-time and amplitude a can be computed from

|x | = c1 and div(a2x) = 0.

Historical note: Kirchhoff migration


Since the 1950s Kirchhoff migration is the standard technique to ap-

proximately solve the integral equation.

Beylkin (1984, 1985) showed that there is a convolution type operator

K and a dual transform F# such that

F#KF = I +

where is compact. Further, Kirchhoff migration is the direct applica-tion of F#K to the measured data.

Historical note: Kirchhoff migration


Since the 1950s Kirchhoff migration is the standard technique to ap-

proximately solve the integral equation.

Beylkin (1984, 1985) showed that there is a convolution type operator

K and a dual transform F# such that

F#KF = I +

where is compact. Further, Kirchhoff migration is the direct applica-tion of F#K to the measured data.

We advocate a different approach which we think

is more flexible,

allows a better control of the involved parameters, and

gives a better understanding of the propagation of singularities

which hopefully results in better reconstructions.

The Method

The Problem

The MethodThe Experiments

The Future


The elliptic Radon transform in 2D


background velocity c = 1: (x,y) = |x y| and a(x,y) = 1/|x y|

n is compactly supported in the lower half space x2 > 0 (x2 > 0 pointsdownwards),

common offset scanning geometry:

xs(s) = (s , 0) and xr(s) = (s+ , 0)

where > 0 is the common offset.

b bx1

x2

(s1, 0) (s2, 0)rS rS rS rS

xs xr xs xr

The imaging operator


As an inversion formula for F is unknown we define the reconstructionoperator

= F F

where

= (s, t) is a smooth compactly supported cutoff function,

F is a formal (weighted) L2-adjoint of F , and

is the Laplacian.

From the elliptic means g = Fn we can recover

n = F g.

Why this choice of ?


is a do of order 1 and n emphasizes singularities (e.g., jumpsalong curves) of n which are tangent to ellipses being integrated over.(follows from results by Guillemin & Sternberg, 1977, and by Krishnan et al., 2012)

b bx1

x2

(s1, 0) (s2, 0)rS rS rS rS

xs xr xs xr

Inversion scheme: approximate inverse


Instead of n(p) we try to compute

n(p) := n, ep,L2(R2) = n e0,(p)

where ep, , > 0, is a mollifier:

supp ep, = B(p),

ep,(x)dx = 1, ep,

0 ( p).

We use

ep,,k(x) = Ck,

{(2 2)k : < ,

0 : , = |x p|,

with k > 0 and

Ck, =k + 1

2(k+1).

Inversion scheme: reconstruction kernel


Lemma: For k 3 we have that

n(p) = Fn, p,,kL2(R]2,[,t2 dtds)

with the reconstruction kernel

p,,k(s, t) = 4k Ck,

((k 1)F

(| p|2 ep,,k2

)(s, t) F ep,,k1(s, t)

)

with ep,,k = ep,,k/Ck, .

Proof: By duality, n(p) = F Fn, ep,,k = Fn, p,,k with

p,,k = Fep,,k = Ck,Fep,,k

and ep,,k = 4k(k 1) | p|2 ep,,k2 4k ep,,k1 yields the result. X

Plotting the kernel


10 5 0 5 10

midpoint s

5

10

15

20

travel time (diameter) t

p, , 3, = 1.00, = 0.80, p= (0.00, 3.00)

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

4 6 8 10 12 14

travel time (diameter) t

1.0

0.5

0.0

0.5

1.0p, , 3(0, : ), = 1.00, = 0.80, p= (0.00, 3.00)

The Experiments

The Problem

The Method

The ExperimentsThe Future


Discretization


We compute

n(p) = Fn, p,,3L2(R]2,[,t2 dtds)

from the discrete data

g(i, j) = (si, tj)Fn(si, tj), i = 1, . . . , Ns, j = 1, . . . , Nt,

where{si} [smax, smax] and {tj} [tmin, tmax], tmin > 2,

are uniformly distributed with step sizes hs and ht, respectively.

n(p) n(p) := hshtNs

i=1

tjTi(p)

g(i, j)p,,3(si, tj) t2j

with |Ti(p)| .

The phantom n and its transform Fn


x10 2

4

2

x2

n

15 10 5 0 5 10 15

midpoint s

15

20

25

30

35

40tr

avel tim

e (dia

mete

r) t

Elliptic 2D-Radon transform (sinogram), = 5.00

0.000

0.015

0.030

0.045

0.060

0.075

0.090

0.105

0.120

Ns = Nt = 600

Reconstructed images n


2 1 0 1 2 3 4 5

p1

2

3

4

5

6

7

p2

= 2.00, = 0.20, smin = -15.00, tmax = 34.90

32

24

16

8

0

8

16

24

2 1 0 1 2 3 4 5

p1

2

3

4

5

6

7

p2

= 5.00, = 0.20, smin = -15.00, tmax = 40.50

45

30

15

0

15

30

45

Reconstructed images n: limited data


2 1 0 1 2 3 4 5

p1

2

3

4

5

6

7

p2

= 5.00, = 0.20, smin = -7.50, tmax = 25.50

45

30

15

0

15

30

45

si [7.5, 7.5]

Data from the wave equation


Fw(T ;xr,xs) =

T

0(u u)(t;xr,xs)dt

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.2

0.3

0.4

0.5

0.6

0.7

0.80.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

[0.1, 1] [0.1, 0.8] with absorbing bc using PML. Step size 0.01.

17 source/receiver pairs, = 0.05, positioned at (s, 0.1), s {0.15+0.05i : i = 0, . . . , 16}, to record u at the receivers.

Temporal source signal: scaled Gaussian.

u was computed with constant sound speed c = 1.

= 1

= 1.5

Wavefields


Sine profile Cosine profile

PySIT Seismic Imaging Toolbox for Python

by L. Demanet & R. Hewitt

Preprocessed seismograms


0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

t

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

s

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

t

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

s

y(s, t) =

T

0(u u)(t;xr(s),xs(s))dt

Reconstructed images 0.06n


The Future

The Problem

The Method

The Experiments

The Future


Next steps


Generalization to 3D

Symbol calculation for

n(y)=

n(x)(x,y, p)eip(xy)dxdp

New reconstruction kernels

Non-constant background velocity

2 1 0 1 2 3 4 5

p1

2

3

4

5

6

7

p2

= 5.00, = 0.20, smin = -15.00, tmax = 40.50

45

30

15

0

15

30

45

Next steps


Generalization to 3D

Symbol calculation for

n(y)=

n(x)(x,y, p)eip(xy)dxdp

New reconstruction kernels

Non-constant background velocity

2 1 0 1 2 3 4 5

p1

2

3

4

5

6

7

p2

= 5.00, = 0.20, smin = -15.00, tmax = 40.50

45

30

15

0

15

30

45

Thank you for your attention!

Why this choice of ?


is a do of order 1 and n emphasizes singularities (e.g., jumpsalong curves) of n which are tangent to ellipses being integrated over

Proof:

Under the Bolker assumption any hypersurface Radon transform R on Rd

and its (formal, smoothly weighted) L2-adjoint R are FIOs of order (d 1)/2. (Guillemin & Sternberg, 1977)

If they can be composed, then RR is a do.

Our F on R2 satisfies the Bolker assumption (Krishnan et al., 2012), that is,F F is of order 1.

Reconstructed images n: erroneous offset


2 1 0 1 2 3 4 5

p1

2

3

4

5

6

7

p2

data = 2.00, recon = 2.50

50

40

30

20

10

0

10

20

2 1 0 1 2 3 4 5

p1

2

3

4

5

6

7

p2

data = 2.00, recon = 1.50

30

24

18

12

6

0

6

12

18

Computing the kernel


Let be the indicator function of Br(p) which is in the lower half-space.To evaluate

F(s, t) =

A(s,x)(x)

(t (s,x)

)dx, t > 2,

we transform the integral by elliptic coordinates x(s, t, ) = (x1, x2),

x1 = s+t

2cos and x2 =

t2

4 2 sin.

Note: E(s, t) ={x(s, t, ) : [0, 2]

}ellipse wrt xs(s), xr(s), and t.

Thus,

F(s, t) =1

t2 42

0(x(s, t, )

)d.

Computing the kernel (continued)


To evaluate F(s, t) further we provide the following quantities

T/+ = T/+(s, r,p) = min /max{(s,x) : x Br(p)

}.

b

b

b

x1

x2

(s, 0)

T

T+

rS rSxs xr

E(s, t) Br(p) 6= T < t < T+

For t ]T, T+[ :

E(s, t)Br(p) ={x(s, t, ) : [1, 2]

}

F(s, t) =

0 : t 6 ]T, T+[2 1t2 42

: t ]T, T+[



Remaining tasks: Compute T/+, 1/2.

T/+ = min /max{() : [0, 2[

}

where

() := (s,p+ r(cos, sin)

).

attains exactly one minimum and one maximum in [0, 2[.

As both extrema are clearly separated, we can apply Newtons method

to get the two zeros of .



Having T we solve

r2 = |p x(s, t, )|2 for .For t ]T, T+[, s R we have exactly the two solutions 1 and 2.

We substitute

z = cos, b = (s p1) t,

c = (p1 s)2 + p22 +t2

4 2 r2, d =

t2 42 p2,

to obtain the equation

d

1 z2 = c+ b z + 2 z2

which has exactly two solutions 1 z2 < z1 1.

By Newtons method again,

i = arccos zi, i = 1, 2.



The kernel p,,k = Fep,,k can be computed just as F.

Indeed, let k = 3, then

ep,,3(x) = C3,( 36 |x p|4 + 482 |x p|2 124

)B(p)(x).

Now F can be applied to each of the components of ep,,3, e.g.,

F(|p|4B(p)

)(s, t) =

0 : t 6 ]T, T+[,1

t2 42 2

1

|x(s, t, ) p|4 d : t ]T, T+[.Here,

|x(s, t, ) p|4 =((

s p1 +t

2cos

)2+( t2

4 2 sin p2

)2)2

is a trigonometric polynomial which can be integrated analytically.

The ProblemAn inverse problem for the acoustic wave equationGeneralized Radon transformHistorical note: Kirchhoff migration

The MethodThe elliptic Radon transform in 2DThe elliptic Radon transform (continued)The imaging operatorWhy this choice of ?Inversion scheme: approximate inverseInversion scheme: reconstruction kernelPlotting the kernel

The ExperimentsDiscretizationThe phantom n and its transform FnReconstructed images "0365nReconstructed images "0365n: limited dataData from the wave equationWavefieldsPreprocessed seismogramsReconstructed images "03650.06 n

The FutureNext stepsWhy this choice of ?Reconstructed images "0365n: erroneous offsetComputing the kernelComputing the kernel (continued)Computing the kernel (continued)Computing the kernel (continued)Computing the kernel (continued)

fd@rm@0: fd@rm@1: fd@rm@2:

approximate inverse for the common offset …rieder/media/slideslinz2017.pdf · approximate inverse...

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