least squares kirchho depth migration with preconditioningkstanton/files/lsmig.pdf · running head:...

Least squares Kirchhoff depth migration with

preconditioning

Aaron Stanton∗

∗University of Alberta,

Department of Physics,

4-183 CCIS,

Edmonton AB T6G 2E1

(April 2, 2013)

Running head: Least Squares Migration

ABSTRACT

In this article the Kirchhoff operator is used for least squares migration (LSM) of 2D seismic

data. The algorithm produces Common Image Gathers (CIGs) that when passed to the

forward operator fits the input data in a least squares sense. The inverse problem is shown

to be underdetermined. Preconditioning is added to the algorithm to constrain the solution

to be smooth. This is achieved via a simple five-point triangle filter to smooth the data

along the offset dimension of the model, along with a band-pass filter which aims to sup-

press low and high-frequency artifacts. A modified Conjugate Gradients algorithm is shown

that allows for preconditioning operators to be easily added and allows for their contribu-

tion to the model update to be easily measured. This makes setting the hyper-parameter

straightforward. Two examples are shown using synthetic data, the first is a simple linear

velocity model with a single cup-shaped reflector. In this case the LSM provides superior

results to the adjoint operator, while the LSM with preconditioning offers a slight improve-

1

ment to the smoothness of the solution. The second example uses acoustic finite difference

synthetic data from the Marmousi model. This model is more complex than the first. The

least squares approach to migration offers an improved result compared to using the adjoint

operator. In this example preconditioning offers a very minor improvement, suggesting a

more advanced migration operator and preconditioning operator are required.

2

INTRODUCTION

Seismic migration is a key step in the processing of seismic data. It is responsible for trans-

forming reflected energy into a meaningful coordinate system for structural interpretation

and parameter inversion (Tarantola, 1984). Migration can also be thought of in terms of

downward continuation– a survey consisting of source and receivers on the earths surface is

successively re-datumed to deeper recording surfaces (Schneider, 1978).

Kirchhoff migration, using the adjoint Kirchhoff operator, aims to solve the migration

problem by integration. Because we record discrete data with a finite aperture and aim to

solve for a discrete earth model the integrals are replaced with finite discrete summations.

This is the primary source of error for these methods. Another problem with Kirchhoff

methods relates to the artifacts created by spreading amplitudes over travel time curves.

While typically the bulk of the energy constructively sums in place of true reflectors, a great

deal of energy persists in the form of ”smiles” or migration artifacts.

Least squares migration (LSM) can attempt to deal with the above problems. Instead

of simply using the adjoint operator, the inverse is sought by attempting to match the data

predicted by the model with the observed data. Significant work has been done in the field

of LSM. Nemeth et al. (1999) use a Kirchhoff operator to perform LSM of incomplete surface

seismic and Vertical Seismic Profile (VSP) data. They find that the migration results are

more focussed and suffer less from acquisition footprint compared with the adjoint operation.

Kuehl and Sacchi (2003) use LSM with a Double Square Root (DSR) operator to express

reflectivity as a function of subsurface position and ray parameter to be used for further

Amplitude Versus reflection Angle (AVA) inversion. Guitton et al. (2007) used LSM to

attenuate low frequency Reverse Time Migration (RTM) artifacts due to crosscorrelation of

3

diving-waves, head-waves, or backscattered waves. The attenuation of artifacts is achieved

via preconditioning using prediction-error filters.

This paper considers least squares prestack depth migration using a Kirchhoff operator.

Two synthetic data examples are provided, the first involves a single reflector that has been

sampled by a small number of shots modelled using a forward Kirchhoff operator, while the

second involves a more complicated reflectivity structure with input shot gathers modelled

using an acoustic finite difference approach. In both cases it is clear that LSM offers an

improved result over application of the adjoint operator. A simple preconditioning strategy

is used to impose smoothness across offsets. In the first example the desired CIGs contain

a single reflector with a flat, constant amplitude response with offset. The preconditioning

is successful in this scenario. In the second more complicated scenario, the gathers are

not flat, and display amplitude variations with offset. In this case the preconditioning is

less successful in attenuating migration artifacts, although a computational advantage is

achieved, as the misfit is lowered more quickly than the case when no preconditioning is

used.

THEORY

The Kirchhoff operator

The Kirchhoff operator is currently the most widely used migration operator. One of

the main reasons for the algorithm’s popularity lies in its simplicity. Kirchhoff migration

involves integrating trace amplitudes over a reflectivity model. After travel times and

Kirchhoff weights have been calculated, the migration process can be written as a trace by

trace process (Audebert, 2001). This means that a tiny amount of memory is required for

4

this process, and perhaps more importantly the task is highly parallelizable. The speed

of this algorithm also makes it possible to perform iterative velocity analysis. For these

reasons the adjoint Kirchhoff operator is currently the most popular migration algorithm

in commercial processing flows.

In the case of LSM the forward and adjoint migration operators are required. Given

the simplicity of the Kirchhoff adjoint operator, the forward operator is straightforward to

define.

In addition to its simplicity, the Kirchhoff operator also suffers from artifacts that could

benefit from LS migration with preconditioning that aims to mitigate these artifacts as will

be shown later in this paper.

The basis of Kirchhoff migration is the Huygen-Fresnel principle which states when a

disturbance reaches a point, this point behaves as a secondary source, and that summing

the waves from this source can be used to determine the form of the wave at a later time.

The forward Kirchhoff operator can be written

d(s, g, t) =∑Nx

∑Nz

m(x, z)K(s, g, x, z, t)

where d(s, g, t) is the data, m(x, z) is the model, and K(s, g, x, z, t) are the Kirchhoff

weights. Here t is the travel time that is obtained via ray tracing through the velocity

model. The adjoint, or migration operator can be written

m(x, z) =∑

Ns∗Ngd(s, g, t)K(s, g, x, z, t).

Algorithm 1 shows an implementation of the Kirchhoff operator that was used for the

synthetic data experiments in this paper.

The main elements of the 2D Kirchhoff depth migration operator described in Algo-

5

Algorithm 1 Kirchhoff operatorprocedure Kirchhoff(d,m, v, sx, gx, dt, Ntrace, Nx, Nz , fwd, adj) . fwd diffracts, adj migrates

for itrace = 1 : Ntrace do

t← ray tracing(sx(itrace), gx(itrace), V )

it = bt/dtc

b = (t− it ∗ dt)/it ∗ dt

a = 1− b

for ix = 1 : Nx do

for iz = 1 : Nz do

K ← kirchoff weights(sx(itrace), gx(itrace), V, ix, iz)

if fwd then

d(itrace,itime) = d(itrace,itime) + a*m(ix,iz)*K

d(itrace+1,itime) = d(itrace,itime) + b*m(ix,iz)*K

end if

if adj then

m(ix,iz) = m(ix,iz) + (a*d(itrace,itime) + b*d(itrace,itime + 1))*K

end if

end for

end for

end for

end procedure

rithm 1, functions ”ray tracing” and ”kirchhoff weights” are taken from the open source

SeismicUnix code sukdmig2d.c (Liu, 1995).

Least squares migration

Given the forward Kirchhoff operator, L, data can be generated from a reflectivity model.

Given recorded data, we may want to collapse diffractions to the position where reflection

occurred. To do this we should use the inverse of the forward operator, L−1. Typically

for imaging the inverse is approximated by the adjoint operator, LT . While the adjoint

can produce reasonable structural images, the produced model is not true amplitude in the

sense that applying the forward operator will not recover the recorded data to a reasonable

degree, especially if the acquisition geometry contains significant holes, or irregular geometry

(Nemeth et al., 1999). LSM aims to recover the model, m, such that the application of

6

the forward operator reproduces the recorded data in a least squares sense. The system of

equations for LSM is

Lm = d + n

where d is the observed data and n is noise. The minimum norm solution comes from

minimizing the cost function

J = ||Lm− d||22 + λ2||m||22,

where J is the cost to be minimized, m is the reflectivity model to be recovered, and λ is

a trade-off parameter which controls the under-fitting or over-fitting to the observed data,

d.

The minimum norm solution to this cost function is

m̂ = (LTL + λ2I)−1LTd.

Typically the size of the model is greater than the size of the data which makes this an

underdetermined problem. The next section discusses methods to precondition the problem

in order to narrow down the range of solutions.

Preconditioning

To obtain a smooth solution a derivative operator can be applied to the model. Sharp

changes in the model are then penalized in the cost function resulting in a smoother set of

solutions. This is written:

J = ||Lm− d||22 + λ2||Dm||22,

where D can be referred to as a ”bad pass” operator, such as a derivative across the

offset dimension of Common Image Gathers (CIGs).

7

An alternative way to write this is via a change of variables:

z = Dm

giving

J = ||LD−1z− d||22 + λ2||z||22.

If the derivative operator, D, acts as a high-pass operator then its inverse, D−1 acts as

a low-pass operator (Wang, 2005). It becomes clear that operators placed into the right

hand side (model-space part) of the cost function are ”bad-pass” operators that emphasize

parts of the model we wish to penalize, while terms on the left hand side (data-space) are

”good-pass” operators that emphasize parts of the model we wish to reinforce.

This cost function can be incorporated into a Conjugate Gradients framework such as

given by Scales (1987), by simply concatenating the preconditioning operator, D, with the

migration operator, L, to form the new operator L̃ = LD−1, with adjoint L̃T = (D−1)TLT .

An alternative formulation is to incorporate one or more regularization terms directly

into the Conjugate Gradients algorithm∗. Algorithm 2 shows a typical implementation of

Conjugate Gradients, but with two operators, A, and B. A clear advantage to writing the

algorithm in this way is that the contribution of each operator to the model update step

∆ =< As,As > +p < Bs,Bs > can be controlled by the hyper-parameter p. Considering

B to be a ”good-pass” operator such as a smoothing operator, setting p = 0 provides the

minimum-norm solution, while setting p to a value that makes the contribution of the two

inner products of comparable magnitude provides a regularized solution. Finally, setting p

to be a very large value makes the contribution of < Bs,Bs > larger than < As,As >,

restricting the update to be mainly controlled by the smoothness of the model.

∗via personal communication with M. D. Sacchi

8

The good pass operator used in this study is a five-point triangle filter, P, which is

applied across offsets within each Common Image Gather. Unlike a rectangular filter, a

triangle filter gives higher weight to the central point on the filter compared to the edges.

When smoothing a signal of length 8 the filter can be written in matrix form as

P =

1 0 0 0 0 0 0 0

14

12

14 0 0 0 0 0

19

29

13

29

19 0 0 0

0 19

29

13

29

19 0 0

0 0 19

29

13

29

19 0

0 0 0 19

29

13

29

19

0 0 0 0 0 14

12

14

0 0 0 0 0 0 0 1

, with adjoint

PT =

1 14

19 0 0 0 0 0

0 12

29

19 0 0 0 0

0 14

13

29

19 0 0 0

0 0 29

13

29

19 0 0

0 0 19

29

13

29 0 0

0 0 0 19

29

13

14 0

0 0 0 0 19

29

12 0

0 0 0 0 0 19

14 1

.

This operator has very few non-zero entries, so writing the smoothing operator in oper-

ator form makes this forward adjoint pair computationally inexpensive.

To avoid low and high frequency artifacts in the solution the smoothing operator is

concatenated with a band-pass filter. This operator can be written Q = F−1TF. Where

9

F and F−1 are the forward and inverse Fourier transforms respectively and T is a diagonal

matrix (TT = T) that weights spectral coefficients outside of the pass-band to zero. Since

the Fourier transform is orthogonal (FT = F−1), the adjoint of this operator is written

QT = (F−1TF)T = FTTT(F−1)T = F−1TF. This implies that the band-pass operator is

self adjoint. The concatenation of the smoothing operator, P, and the band-pass operator,

Q, is written B = PQ, with adjoint BT = QPT. To ensure that the forward adjoint

pair are not written incorrectly, the dot product test is performed. Two vectors of random

numbers are created, one with the dimension of the data, d1, and one with the dimension

of the model, m1. d1 is passed through the concatenation of Adjoint operators to generate

m2, while m1 is passed through the concatenation of Forward operators to generate d2.

The inner products < d1,d2 > and < m1,m2 > are compared and found to agree within

machine precision.

SYNTHETIC DATA EXAMPLES

Simple synthetic

The algorithm is first tested on a simple reflectivity model consisting of a cup shaped

reflector in a simple background velocity shown on the right in Figure 1. Using the known

reflectivity the forward model is used to generate 6 shot gathers, shown on the left in

Figure 1. Adjoint, LSM, and LSM with preconditioning are applied to the input data. In

the case of preconditioning the hyper-parameter, p, is selected by first allowing it to be

equal to one, then monitoring the difference in magnitude between the two inner products

that contribute to the model update. p = 1000 was then chosen to equalize the contribution

of these two terms, and the code was restarted. The adjoint operator is applied to these

10

gathers resulting in the selected Common Image Gathers (CIGs) shown on the left in Figure

2. Here 8 gathers are shown with increasing offset to the right within each gather. The

minimum norm solution after 20 iterations of Conjugate Gradients is shown in the middle

of the figure, while the right figure shows the gathers after LSM with preconditioning

(again with 20 iterations). It is apparent that the amplitudes of the artifacts in the adjoint

are of the same level as the flat reflector amplitudes, while the LSM results have higher

amplitudes corresponding to the flat reflector. Between the two LSM results it is apparent

that the use of preconditioning provides slightly less artifacts compared with the minimum

norm solution. Figure 3 shows the results of adjoint (left), LSM (middle), and LSM with

preconditioning (right) after stacking CIGs. It is clear that LSM mitigates much of the

migration artifacts seen in the adjoint stack (left), while preconditioning (right) provides a

slightly clearer result compared with the minimum norm solution (middle). Figure 4 shows

the data predicted from the migrated model. The data predicted from the adjoint (left)

does not come close to modelling the original data, while the minimum norm LSM result

(middle) matches the original data quite closely. The LSM with preconditioning (right) also

matches the original data closely, but appears to be slightly noisier than that obtained using

the minimum norm solution. Figure 5 shows the misfit versus iteration number for both

LSM (a) and LSM with preconditioning (b). Here the curves are similar in both these cases,

but it is clear that the minimum norm solution reaches a lower misfit than the least squares

solution with preconditioning. Figure 6 shows the amplitude spectrum of the adjoint stack

(a) and the coefficients of the band-pass operator that was applied in the preconditioning

(b).

11

Marmousi finite difference synthetic

This example follows the advice of Gray et al. (2001), namely, that a more accurate forward

modelling operator should be used compared to the migration operator being tested. The

shot gathers are modelled using an acoustic finite difference algorithm and the Marmousi

subsurface model. The velocity model is shown in Figure 7. Both LSM and LSM with pre-

conditioning were run with 20 iterations. For LSM with preconditioning a hyper-parameter

of 20000 was chosen to equalize the action of the migration and regularization operators on

the model update. Figure 8 shows selected CIGs after adjoint (left), LSM (middle), and

LSM with preconditioning (right). All three figures appear similar, but with slightly less

noise from artifacts on the results after LSM and LSM with preconditioning. What these

CIGs also show is that the velocity structure is likely too complicated for ray-based imaging

methods, and a wave-equation based method would be better suited for this dataset. This

is further discussed in Gray et al. (2001). Figure 9 shows stacked data after adjoint (top),

LSM (middle), and LSM with preconditioning (bottom). The amplitudes of reflectors are

more balanced on the LSM result compared with the adjoint result, while the result with

preconditioning appears to have very slightly less noise, especially in the deeper 2-3km of

the section. What is also apparent is that the uppermost reflector at depth ≈ 0km has

been attenuated in the preconditioned result. This is likely due to the fact that the pre-

conditioning is a smoothing operator applied across offsets. Because the reflector has an

average fold of approximately 2 at this depth (see Figure 8), the preconditioning is asking

for solutions that are more continuous across offset, thereby lowering the amplitude of this

uppermost reflector. This suggests that either a more accurate velocity field is required to

flatten this event, or a more accurate migration operator should be used for such a compli-

cated velocity structure. Yet another alternative is to use a more complex preconditioning

12

strategy that can smooth along the local dips in the data (see for example Rebollo and

Sacchi (2010)). Figure 10 shows selected input shot gathers (upper left), residual after

using the adjoint migration operator (upper right), residual after LSM (bottom left), and

residual after LSM with preconditioning (bottom right). It is clear that the adjoint does

a poor job in recovering the input data, while LSM and LSM with preconditioning have

a much lower amplitude residual. Figure 11 shows the misfit versus iteration number for

both LSM (a) and LSM with preconditioning (b). A positive effect of preconditioning in

this example is the steepness of the curve for early iterations. The preconditioning has the

effect of clustering eigenvalues in the LTL term which stabilizes the inversion (Wang, 2005).

Figure 12 shows the amplitude spectrum of the adjoint stack (a) and the coefficients of the

band-pass operator that was applied in the preconditioning (b).

CONCLUSIONS

The two synthetic examples considered in this paper demonstrate that a least squares

formulation of Kirchhoff depth migration is effective in providing an improved image of

the subsurface. Preconditioning is added to the LSM using a 5 point triangle smoothing

operator across offsets along with a band-pass operator that attempts to limit low and

high frequency artifacts in the solution. In the first example a significant improvement is

achieved through the use of preconditioning, while in the second example the improvement

due to preconditioning is less. This was likely due to the complexity of the velocity model

used, suggesting a wave-equation migration operator would be better suited to migrate such

data. A drawback of this method is that the entire dataset must be loaded to memory. For

extremely large datasets this would be impractical. A topic of further research could be how

to implement an inversion procedure that can utilize ”warm starts” from a shot-by-shot LS

13

migration procedure.

ACKNOWLEDGMENTS

I would like to thank Zhenyue Liu for contributing the 2d Kirchhoff depth migration code

in the open-source SeismicUnix package. This code provided the basic framework for de-

veloping the LSM code.

14

REFERENCES

Audebert, F., 2001, 3-D prestack depth migration: why Kirchhoff?: Technical Report 80,

Stanford Exploration Project.

Gray, S. H., J. Etgen, J. Dellinger, and D. Whitmore, 2001, Seismic migration problems

and solutions: Geophysics, 66, 1640.

Guitton, A., B. Kaelin, and B. Biondi, 2007, Least-squares attenuation of reverse-time-

migration artifacts: GEOPHYSICS, 72, S19–S23.

Kuehl, H., and M. D. Sacchi, 2003, Least-squares wave-equation migration for avp/ava

inversion: Geophysics, 68, 262–273.

Liu, Z., http://www.seismicunix.com/w/Sukdmig2d.

Nemeth, T., C. Wu, and G. T. Schuster, 1999, Least-squares migration of incomplete re-

flection data: Geophysics, 64, 208–221.

Rebollo, R. C., and M. D. Sacchi, 2010, Time domain least-squares prestack migration:

GeoCanada Expanded Abstracts, 4.

Scales, J. A., 1987, Tomographic inversion via the conjugate gradient method: Geophysics,

52, 179–185.

Schneider, W., 1978, Integral formulation for migration in two and three dimensions: Geo-

physics, 43, 49–76.

Tarantola, A., 1984, Linearized inversion of seismic reflection data: Geophysical Prospect-

ing, 32, 998–1015.

Wang, J., 2005, 3-d least-squares wave-equation avp/ava migration of common- azimuth

data: PhD thesis, University of Alberta, least squares migration.

15

APPENDIX A

CONJUGATE GRADIENTS WITH MORE THAN ONE OPERATOR

Algorithm 2 Conjugate Gradients with more than one operatorprocedure CG(d,Niter, p) . solves ||Ax− d||22 + p||B−1x||22

x = 0

r1 = d

g = AT r1 + BT r2

s = g

γ =< g, g >

γold = γ

while k ≤ Niter do

ss1 = As

ss2 = Bs

∆ =< ss1, ss1 > +p < ss2, ss2 >

α = γ/∆

x = x + αs

r1 = r1 − αss1

r2 = r2 − αss2

g = AT r1 + BT r2

γ =< g, g >

β = γ/γold

γold = γ

s = g + βs

k + +

end while

return x . The solution is x

end procedure

16

LIST OF FIGURES

1 Data and velocity model used for the first example.

2 A selection of Common Image Gathers (CIGs) after adjoint (left), LSM (middle),

and LSM with preconditioning (right).

3 Stack after adjoint (left), LSM (middle), and LSM with preconditioning (right).

4 Predicted data after application of adjoint (left), LSM (middle), and LSM with

preconditioning (right).

5 Misfit vs. Iteration number for LSM (a), and LSM with preconditioning (b).

6 Amplitude spectrum for stacked data (a) and band-pass filter applied as part of

the preconditioning operator (b). Note that the band-pass filtering operator is self adjoint.

7 Smooth velocity model used for the migration.

8 A selection of Common Image Gathers (CIGs) after adjoint (left), LSM (middle),


9 Stack after adjoint (top), LSM (middle), and LSM with preconditioning (bottom).

10 A selection of shot gathers representing input data (upper left), residual after

predicting data using the adjoint (upper right), residual after using the LSM result to

predict data (bottom left), and residual data after using LSM with preconditioning to

predict data (bottom right).

11 Misfit vs. Iteration number for LSM (a), and LSM with preconditioning (b).

12 Amplitude spectrum for stacked data (a) and band-pass filter applied as part of


17

0

0.5

1.0

1.5

2.0

2.5

Tim

e (

s)

2 4 6 8

0

500

1000

1500

2000

2500

De

pth

(m

)

0 1000 2000 3000Distance (m)

Figure 1: Data and velocity model used for the first example.Stanton & Sacchi –

18

0

500

1000

1500

2000

2500

De

pth

(m

)

0.2 0.4 0.6 0.8 1.0x104

0

500

1000

1500

2000

2500

De

pth

(m

)

0.2 0.4 0.6 0.8 1.0x104

0

500

1000

1500

2000

2500

De

pth

(m

)

0.2 0.4 0.6 0.8 1.0x104

Figure 2: A selection of Common Image Gathers (CIGs) after adjoint (left), LSM (middle),


Stanton & Sacchi –

19

0

500

1000

1500

2000

2500

De

pth

(m

)

1000 2000 3000Midpoint (m)

0

500

1000

1500

2000

2500

De

pth

(m

)

1000 2000 3000Midpoint (m)

0

500

1000

1500

2000

2500

De

pth

(m

)

1000 2000 3000Midpoint (m)

Figure 3: Stack after adjoint (left), LSM (middle), and LSM with preconditioning (right).


20

0

0.5

1.0

1.5

2.0

2.5

Tim

e (

s)

2 4 6 8

0

0.5

1.0

1.5

2.0

2.5

Tim

e (

s)

2 4 6 8

0

0.5

1.0

1.5

2.0

2.5T

ime

(s)

2 4 6 8

Figure 4: Predicted data after application of adjoint (left), LSM (middle), and LSM with

preconditioning (right).


21

0

2000

4000

6000

8000

10000

12000

14000

16000

0 2 4 6 8 10 12 14 16 18 20

Mis

fit

Iteration number

(a)

10000

20000

30000

40000

50000

60000

70000

80000

0 2 4 6 8 10 12 14 16 18 20

Mis

fit

Iteration number

(b)

Figure 5: Misfit vs. Iteration number for LSM (a), and LSM with preconditioning (b).


22

0 0.02 0.04 0.06 0.08 0.10 0.12(1/m)

0.2

0.4

0.6

0.8

1.0

1.2

x10 -3

Am

plit

ude

(a)

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Filt

er

Am

plit

ud

e

Frequency (1/m)

(b)

Figure 6: Amplitude spectrum for stacked data (a) and band-pass filter applied as part of



23

0

1000

2000

3000

Depth

(m

)

0 2000 4000 6000 8000Distance (m)

Figure 7: Smooth velocity model used for the migration.Stanton & Sacchi –

24

0

500

1000

1500

2000

2500

3000

De

pth

(m

)

2000 2200 2400 2600 2800 3000

0

500

1000

1500

2000

2500

3000

De

pth

(m

)

2000 2200 2400 2600 2800 3000

0

500

1000

1500

2000

2500

3000

De

pth

(m

)

2000 2200 2400 2600 2800 3000

Figure 8: A selection of Common Image Gathers (CIGs) after adjoint (left), LSM (middle),



25

0

1000

2000

3000

Depth

(m

)

2000 3000 4000 5000 6000 7000 8000 9000Midpoint (m)

0

1000

2000

3000

Depth

(m

)

2000 3000 4000 5000 6000 7000 8000 9000Midpoint (m)

0

1000

2000

3000

Depth

(m

)

2000 3000 4000 5000 6000 7000 8000 9000Midpoint (m)

Figure 9: Stack after adjoint (top), LSM (middle), and LSM with preconditioning (bottom).

Stanton & Sacchi –26

0

0.5

1.0

1.5

2.0

2.5

Tim

e (

s)

0.2 0.4 0.6 0.8 1.0x104

0

0.5

1.0

1.5

2.0

2.5

Tim

e (

s)

0.2 0.4 0.6 0.8 1.0x104

0

0.5

1.0

1.5

2.0

2.5

Tim

e (

s)

0.2 0.4 0.6 0.8 1.0x104

0

0.5

1.0

1.5

2.0

2.5

Tim

e (

s)

0.2 0.4 0.6 0.8 1.0x104

Figure 10: A selection of shot gathers representing input data (upper left), residual after

predicting data using the adjoint (upper right), residual after using the LSM result to predict

data (bottom left), and residual data after using LSM with preconditioning to predict data

(bottom right).


27

1.4e+11

1.6e+11

1.8e+11

2e+11

2.2e+11

2.4e+11

2.6e+11

2.8e+11

3e+11

3.2e+11

3.4e+11

0 2 4 6 8 10 12 14 16 18 20

Mis

fit

Iteration number

(a)

2e+11

2.5e+11

3e+11

3.5e+11

4e+11

4.5e+11

5e+11

5.5e+11

0 2 4 6 8 10 12 14 16 18 20

Mis

fit

Iteration number

(b)

Figure 11: Misfit vs. Iteration number for LSM (a), and LSM with preconditioning (b).


28

0 0.02 0.04 0.06 0.08 0.10 0.12(1/m)

0.5

1.0

1.5

Am

plit

ude

(a)

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Filt

er

Am

plit

ud

e

Frequency (1/m)

(b)

Figure 12: Amplitude spectrum for stacked data (a) and band-pass filter applied as part of



29

least squares kirchho depth migration with preconditioningkstanton/files/lsmig.pdf · running head:...

Documents