A Direct Numerical Imaging Method for Point and Extended Targets Wang Hongxia Department of Mathematics and Systems Science, NUDT 2011-06-21

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

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

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

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

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

2. Direct imaging for point targets We have

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

3. Multiples

3. Multiples

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

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

The constrain matrix C

The constrain matrix C

3. Multiples where Output of beamforming is

The constrain matrix C where Let

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

Numerical results

Numerical results

Numerical results Beamformed for level 1.

Numerical results Beamformed for level 1.

Numerical results Beamformed for level 2.

Numerical results Beamformed for level 2.

Numerical results Beamformed for level 3.

Numerical results Beamformed for level 3.

Numerical results Beamformed for level 3.

Numerical results

Numerical results

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

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.

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