1. A Direct Numerical Imaging Method for Point and Extended Targets
Wang Hongxia
Department of Mathematics and Systems Science, NUDT

2. Outline 1. Introduction and motivation
2. Direct imaging for point/extended targets
Algorithm
Numerical results
3. Multiple attenuation in seismic signal
4. Future work

3. 1. Introduction
Probing a medium using waves to detect and image targets is useful in
Medical applications, nondestructive testing, mine detection, target detection … 3 3

4. 1. Introduction Imaging is to resolve an inverse problem Methods
nonlinear, ill-posed problem
Methods
Iteration + regularization
Direct imaging algorithm (MUSIC)

5. 2. Direct imaging for point targets
The response matrix
The reflected signal
i.e.

6. 2. Direct imaging for point targets
From Born approximation, the reflected signal at the j-th transducer is
where
the Greens function
the reflectivity of the k-th scatters located at xk.
the location of the i-th transducers

7. 2. Direct imaging for point targets
We have

8. 2. Direct imaging for point targets
Define the time reversal matrix
The point spread function
In the well-resolved case
Right singular vector
Singular value

9. 2. Direct imaging for point targets
VS: the signal space
SVD
VN: the noise space
The imaging function

10. 2. Direct imaging for point targets
An improved imaging function
where

11. Direct imaging for extended targets
(1) Sound-soft target (Dirichlet boundary condition)

12. Direct imaging for extended targets
From Greens formula we have
The scattered wave received at xj is
The response matrix can be written as

13. Direct imaging for extended targets
Imaging function
where
the shape space VS = span{v1,v2,…,vM}(the leading SVs)
the noise space VN: the orthogonal complement of VS

14. Direct imaging for extended targets
(2) Sound-hard target (Neumann boundary condition)
The scattered field is
The scattered wave received at xj is
The response matrix is

15. Direct imaging for extended targets
Imaging function
where
the normal direction at the boundaries is unknown, we use a set of fixed search directions at each point and take the maximum among them.

16. 2.2 Numerical Results
λ= 0.5m, s = 10m, L = 200m.

17. 2.2 Numerical Results
Fig.1 Imaging results for one-point target

18. 2.2 Numerical Results
Fig.1 Imaging results for one-point target

19. 2.2 Numerical Results
Fig. Imaging results for a target in the shape of circle, r = 20m.

20. 2.2 Numerical Results
Fig. Imaging results for a target in the shape of a five- leaves object.

21. 3. Numerical Results
Fig. Imaging results for a target in the shape of a kite shaped object.

22. 3. Multiples

23. 3. Multiples

24. 3. Multiples Seismic data acquistion by high density Noise attenuation
random noise
Temporal-space filtering
wavelet, …
coherent noise
region filtering
Radon transform
beamforming …

25. 3. Multiples
(1) f-k domain
(2) kx-ky domain

26. The constrain matrix C

27. The constrain matrix C

28. 3. Multiples
where
Output of beamforming is

29. The constrain matrix C
where
Let

30. The beamforming algorithm
the trace of R;
two inverse matrices;
three matrix – products and one matrix-vector production.

31. Numerical results

32. Numerical results

33. Numerical results
Beamformed for level 1.

34. Numerical results
Beamformed for level 1.

35. Numerical results
Beamformed for level 2.

36. Numerical results
Beamformed for level 2.

37. Numerical results
Beamformed for level 3.

38. Numerical results
Beamformed for level 3.

39. Numerical results
Beamformed for level 3.

40. Numerical results

41. Numerical results

42. 4. Future work Further consideration of direct imaging method
Noise
The estimate of the number of singular vectors
Multiple – attenuation or utilization
Iteration imaging method:
point and extended targets
seismic structure
sparse-conserve iteration

43. Research Progress Project: Paper: NSF grand 61072118;
National 863 plan (partially supported).
Paper:
[1] Reweighted minimization model for MR image reconstruction with split Bregman method, Science in China, to appear.
[2] A well resolved condition for point targets in MUSIC imaging algorithm, ICMSEEC2011.
[3] An adaptive MUSIC method for resolution enhancing using an improved imaging function, submitted.
[4] A modified seismic noise attenuation by beamforming algorithm, prepare.

44. Thanks!