Image Processing Reference

In-Depth Information

so much a function of either sources or receivers by themselves, but an expres-

sion involving a combination of the two.

The first intuitive response could be that it may be simply a product of the

two or
N
s
×
N
r
, which would simply be
N
2
. This could be tested quickly using

the model previously used. For this scenario
N
s
/
N
r
is calculated to be 4.9, so

for these conditions we could declare that
N
= 5. If in fact
N
2
is the relation-

ship, we should be able to satisfy this by setting
N
s
= 6 and
N
r
= 5. The results

are shown in Figure 11.4 and compared with the case for
N
= 12,960.

Reviewing the results in Figure 11.4, it does not appear that the “overall”

number of degrees of freedom has been satisfied The combined requirement

for the sources and receivers, if this is in fact the case, may be as simple as

applying a Nyquist-Shannon theorem type approach which might look some-

thing like
2N
s
× 2
N
r
= 4
N
2
. Applying this criterion to the square model used

before would mean that the minimum number of degrees of freedom or data

points required would be
N
= 4(5)
2
= 100. This can be achieved (or exceeded)

in the square model by using 12 sources and 10 receivers randomly spaced.

This too was implemented with the results shown in Figure 11.5, again along

with the case for
N
= 12,960.

The image obtained from using 12 sources and 10 receivers shown in

Figure 11.5b does seem to confirm that the minimum degrees of freedom have

been met when compared with the image in Figure 11.5d for the case with 36

(a)

×10
-4

(b)

×10
-3

0

30

9

8

7

6

5

4

3

2

1

4

20

50

3.5

10

3

100

0

2.5

-10

150

2

1.5

-20

200

-30

1

-40

250

-40

-30

-20

-10

0

10

20

30

40

0

50

100

150

200

250

(c)

(d)

×10
-5

40

0

4

0.04

30

3.5

50

0.035

20

3

0.03

10

0

100

2.5

2

0.025

150

-10

1.5

1

0.5

0.02

-20

-30

200

0.015

0.01

250

-40

-40

-20

0

20

40

0

50

100

150

200

250

Figure 11.4
(a) Ewald circle using six sources and five receivers randomly spaced. (b) Resulting reconstructed

image from (a). (c) Ewald circle using 36 source and 360 receivers equally spaced for comparison with (a) and (b).

(d) Resulting image from (c).

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