A Direct Numerical Imaging Method for Point and extended Targets Wang Hongxia Department of Mathematics and systems science NUDT 2011-06-21
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 .A 2. Direct imaging for point/extended targets ◆ Algorithm Numerical results 3 Multiple attenuation in seismic signal 今4. Future work (過两科学大季 A Direct Numerical Imaging Method National University of Defense Technology
A Direct Numerical Imaging Method Outline ❖ 1. Introduction and motivation ❖ 2. Direct imaging for point/extended targets ◆ Algorithm ◆ Numerical results ❖ 3. Multiple attenuation in seismic signal ❖ 4. Future work 2
l。 Introduction o Probing a medium using waves to detect and image targets is useful in Medical applications nondestructive testing, mine detection, target detection… 气枪 检波器 防科学时太季 A Direct Numerical Imaging Method National University of Defense Technology
A Direct Numerical Imaging Method 3 1. Introduction ❖ Probing a medium using waves to detect and image targets is useful in ◆ Medical applications, nondestructive testing, mine detection, target detection …
l。 Introduction o Imaging is to resolve an inverse problem nonlinear ill-posed problem 骣 CD 0,Dy+k2y=0 ☆ Methods Iteration regularization Direct imaging algorithm MUSIc (過两科学大季 A Direct Numerical Imaging Method 4 National University of Defense Technology
A Direct Numerical Imaging Method 1. Introduction ❖ Imaging is to resolve an inverse problem ◆ nonlinear, ill-posed problem ❖ Methods ◆ Iteration + regularization ◆ Direct imaging algorithm (MUSIC) 4 2 2 2 1 0; v t y 骣ç ¶ ÷ çD - = ÷ ç ÷ ç桫 ÷ ¶ 2 D + = y y k 0
2. Direct imaging for point targets the response matrix P()=鵝() PML N′N ☆ The reflected signal r(t=P(te(t) e,):()L,() transducers =P()*).2(),L,:、( 防科学时太季 A Direct Numerical Imaging Method 5 National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets ❖ The response matrix ❖ The reflected signal 5 ( ) ( ) ij N N P t p t ´ = 轾犏臌 r t P t e t ( ) ( ) * ( ) = v v i.e. 1 2 1 2 ( ), ( ), , ( ) ( ) * ( ), ( ), , ( ) N N r t r t r t P t e t e t e t 轾犏臌 = 轾犏臌 L L
2. Direct imaging for point targets From Born approximation the reflected signal at the j-th transducer is M N f()=邋G(x,x,)G(x,x() k=1i=1 where G(x,x): the Greens function the reflectivity of the k-th scatters k located at x he location of the i-th transducers (過两科学大季 A Direct Numerical Imaging Method 6 National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets ❖ From Born approximation, the reflected signal at the j-th transducer is 6 0 G x ( , ) : x 0 0 1 1 ˆ ( ) ( , ) ( , ) ( ) M N j j k k i k i k i r G G e w t w = = = 邋 x x where the Greens function : k t the reflectivity of the k-th scatters located at xk . : i x the location of the i-th transducers
2. Direct imaging for point targets We have M P(w)=a gig k=1 s=(xx)G"(x…x)LG"(x Ey Rank(P(w))=M (過两科学大季 A Direct Numerical Imaging Method 7 National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets We have 7 0 0 1 0 0 0 0 1 2 ( ) , ( , ), ( , ), , ( , ) M T k k k k T k k k N k P g g g G G G w t = = = 轾犏臌 å v v v x x x L R ank P M ( ( )) . w =
2. Direct imaging for point targets Define the time reversal matrix R()@Po)p(p)a(点 k,k≠=1 a k, k) The point spread function kke In the well-resolved case Right singular vector Singular value (過学故时大学 A Direct Numerical Imaging Method National University of Defense Technology
A Direct Numerical Imaging Method 2 2 0 0 0 0 0 0 ( ) , ( ) . T T k k k k k k k k P g g g P g g g w t w t = = v v v v v v 2. Direct imaging for point targets 8 Define the time reversal matrix ( ) 0 0 0 0 , 1 , , 1 ( ) ( ) ( )= ( , ) M T T T k k k k k k k k M k k k k k k R P P g g g g w w w t t ⅱ ⅱ ? ¢= ⅱ ¢= × L G å å v v v v @ @ x x In the well-resolved case The point spread function Right singular vector Singular value
2. Direct imaging for point targets : the signal space SVD vs=spanu,u, L, u,i dVN. the noise space The imaging function (x l p. cf al-a kox)? i P k=1 (過两科学大季 A Direct Numerical Imaging Method National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets 9 P( ) w The imaging function V S : the signal space SVD V N: the noise space 2 2 2 2 1 1 1 ( ) ( ) ( ) S ( ) ( ) N V k k I x g P g g g u = = = - - ? å v v v v v x x x x 1 2 { , , , } S N V span = v v v u u u L
2. Direct imaging for point targets An improved imaging function X n?1 x ax Where I(x)=22 g(x u k=1 (過两科学大季 A Direct Numerical Imaging Method 10 National University of Defense Technology
A Direct Numerical Imaging Method 2. Direct imaging for point targets 10 An improved imaging function max ( ) ( ) , 1. ( ) n n I x I x n I x 轾 = ? 犏犏犏臌 where 2 1 ( ) ( ) . M T k k I x g x u = = å