let dj , j=1,  ,m , mn , be bounded and open subsets of rn , n=2,3 , such that rndj are connected and assume that they are disjoint .
,m , of dj is of class c2 and the unit normal vector  is directed into the exterior of dj . finally , we set dj=1mdj .
we denote by  the density function such that =1 in rnd , continuous inside d and has a discontinuity across d .
we also denote by  and  the lam coefficients and we assume that those coefficients are constant in rn and satisfy the conditions >0 and 2+3>0 .
we are concerned with the scattering problem of elastic waves by the obstacle d at a fixed frequency . precisely , if ui , which is a vector field satisfying ui+(+)divui+2ui=0 in rn , is the incident field , then the total field utui+u , with uc2(rnd ) as the scattered field , is the solution to the following inhomogeneous problem associated with the lam system ( 1){ut+(+)divut+2ut=0,in  rnlim|x||x|n12(up|x|ikpup)=0,andlim|x||x|n12(us|x|iksus)=0 , where the last two limits are uniform in all directions xx|x|sn1  the unit sphere in rn .
here , we denoted upkp2divu to be the longitudinal ( or the pressure ) part of the field u and usks2curl  curl  u to be the transversal ( or the shear ) part of the field u corresponding to the helmholtz decomposition u = up+us .
the constants kp2+ and ks are known as the longitudinal and the transversal wavenumbers , respectively .
it is well known that the scattering problem ( 1 ) is well posed ; see for instance  [ 13 ] .
the scattered field u satisfies the following asymptotic expansion at infinity ( 2)u(x)eip|x||x|n12up(x)+eis|x||x|n12us(x)+o(1|x|n+12),|x| uniformly in all directions xsn1 ; see   for instance .
the fields up(x ) and us(x ) defined on sn1 are called the longitudinal and transversal parts of the far field pattern , respectively .
the longitudinal part up(x ) is normal to sn1 while the transversal part us(x ) is tangential to sn1 . due to this property
note that it is not necessarily true for near field measurements . in this case , see   for an approximate separation of these two components .
now , we specify the type of incident waves used in this work . as usual in the scattering problems , we use plane waves as incident waves . for the lam system , they have the analytic forms ( 3)ui , p(x,)eipxandui , s(x,)eisx , where  is any vector in sn1 orthogonal to . remark that ui , p(, ) is normal to sn1 and ui , s(, ) is tangential to sn1 .
hence , we can define the matrix ( 4)(ui , p , ui , s)f(ui , p , ui , s)[up,p(,)up,s(,)us,p(,)us,s(, ) , ] where 1.(up,p(,),us,p(, ) ) is the far field pattern associated with the pressure incident field ui , p(,).2.(up,s(,),us,s(, ) ) is the far field pattern associated with the shear incident field ui , s(, ) . ( up,p(,),us,p(, ) )
is the far field pattern associated with the pressure incident field ui , p(, ) . ( up,s(,),us,s(, ) )
is the far field pattern associated with the shear incident field ui , s(, ) . in this paper , we are interested in the following inverse scattering problem . from the knowledge of the matrix  ( 4 )  for all directions 
 x 
 and 
  
 in 
 sn1 ,
determine 
 d. several works have been published regarding this inverse problem , see for instance  [ 68 ] using the full matrix ( 4 ) for all directions x and  in sn1 . for near field measurements ,
we also mention the works [ 1417 ] regarding small obstacles and   for imaging extended obstacles .
we consider now the cavity problem ( 5){u+(+)divu+2u=0,in  rnd(u)=(ui),on  dlim|x||x|n12(up|x|ikpup)=0,andlim|x||x|n12(us|x|iksus)=0 , where (u)(2+div+curl)u .
this problem is well posed , see , and we have a similar asymptotic behavior as ( 2 ) .
with this at hand , we state the similar inverse problem as for the inhomogeneous medium case .
it says that every column of the matrix ( 4 ) for all directions x and  in sn1 , determines d. sampling type methods for solving this obstacle inverse scattering problem have been developed by several authors , see   using the full matrix ( 4 ) for all directions x and  in sn1 .
we remark that in the above works , not only the information over all directions of incidence and observation , but also both pressure and shear parts of the far field pattern are needed . in 
, we proved that it is possible to reduce the amount of data for detecting d as follows .
theorem  1.1every component of the matrix  ( 4 )  known for all directions 
 x 
 and 
  
 in 
 sn1 , determines 
 d 
 for the model  ( 5 ) .
every component of the matrix  ( 4 )  known for all directions 
 x 
 and 
  
 in 
 sn1 , determines 
 d 
 for the model  ( 5 ) .
remark that in theorem  1.1 , we need only the longitudinal part ( or only the transverse part ) of the far field pattern if we use longitudinal incident waves or transversal incident waves .
the result in theorem  1.1 is also valid for the inhomogeneous medium model ( 1 ) .
the proof is based on the asymptotic expansion of the singular solutions of the models in ( 1 ) and ( 5 ) , see   for the impenetrable case and   for the penetrable case related to the scalar equations .
first , we would like to propose numerical methods corresponding to the uniqueness result in theorem  1.1 .
second , we would like to see whether the choice of the type of incident field is relevant or not .
for this , we restrict ourselves to the case of point - like obstacles for which more explicit calculations can be done .
note that none of the known methods ( iterative , sampling , probe , etc . ) have been applied for detection by elastic waves using the reduced amount of data mentioned in theorem  1.1 . to our knowledge ,
the only result considering the use of one type of elastic scattered waves for the detection is the one in   who used p incident waves and the p part of the scattered waves to detect point - like obstacles .
however , no mathematical justification , as in theorem  1.1 or in theorem  3.1 , was given there .
using a music type algorithm , we show that indeed one type of waves is enough for the reconstruction .
in addition , using s incident waves we obtain better resolution than when using p incident waves in the presence of noise .
this can be explained by the fact that the s incident waves have shorter wavelengths than the p incident waves .
we note that , since we make use of a weak scattering model to simulate the measured data , it is not physically meaningful to apply this model to the case of close scatterers .
therefore , the notion of resolution in this paper should be understood as the minimum distance between two point - like scatterers that can be resolved by the algorithm in the presence of measurement noise .
however , we should remark that this weak scattering assumption is merely for the simplicity of the forward modeling . in a future work
, we will investigate the resolution of the music type algorithms using a more physically meaningful model which can be used also for close scatterers .
the rest of the paper is organized as follows . in section  2 , we describe briefly the scattering of point - like obstacles including weak ( born ) approximation . section  3 is devoted to the music algorithms for scalar and elastic waves . finally , section  4 shows numerical examples of the music algorithms and to confirm our discussions on the resolution limits .
suppose that they are illuminated by an incident plane elastic wave ui(x,),xrn,sn1 . as described in the introduction , here ui(x,)=ui , p(x, ) or ui(x,)=ui , s(x, ) . as it is shown in  [ 24 ,  section 8.4 ] , the total scalar field ut corresponding to the scalar model ( acoustic model for instance ) is written as follows(6)ut(x)=ui(x)+m=1mmut(ym)(x , ym ) , where ui is the incident scalar field and  is the fundamental solution of the associated helmholtz model .
schwinger equation by replacing the source , given by the density in each dm , m=1, 
,m , by m(ym ) . here ,  is the dirac measure . following this approach , using the lippmann 
schwinger equation corresponding to problem ( 1 ) , under the assumption that the lam coefficients  and  are constant in rn , the total vector field corresponding to the lam system can be described as follows(7)ut(x)=ui(x)+m=1mmg(x , ym)ut(ym ) , where ui is the incident vector field and g is the fundamental tensor associated with the lam system .
the constant mc,m0 , represents the scattering strength of the m - th scatterer dm . the main difficulty in using the model ( 7 ) to generate the far field is the calculation of u(yj ) .
this is due to the singularities of g on the points yj , see   for more details . to avoid this
however , we should note that the music type algorithms are applicable for the nonlinear model ( 7 ) since the proofs of theorem 4.1 of   and theorem  3.1 are also valid for this case . for results using the scalar model ( 6 ) ,
a current work is being carried out for the elastic model ( 7 ) and we will discuss this in a future work .
assume that there is no multiple scattering between the scatterers ( born approximation ) , then the scattered wave can be written in the form(8)u(x,)=m=1mmg(x , ym)ui(ym, ) , by replacing in the right hand side of ( 7)ut by ui .
the asymptotic behavior of the green tensor at infinity is given as follows(9)g(x , ym)=apxxeikp|x||x|n12eikpxym+as(ixx)eiks|x||x|n12eiksxym+o(|x|n+12 ) , with x=x|x| and i being the identity matrix in rn , ap = kp242 and as = ks242 , see for instance  .
it follows from ( 8) and ( 9 ) that the p and s parts of the far field pattern associated with the p incident wave ui , p are given by(10)up,p(x,)=apm=1mm(xx)eikpym(x),(11)us,p(x,)=asm=1mm(ixx)eikpymeiksymx. similarly , the p and s parts of the far field pattern associated with the s incident wave ui , s can be written as(12)us,p(x,)=apm=1mm(xx)eiksymeikpymx,(13)us,s(x,)=asm=1mm(ixx)eiksym(x ) .
here we have used the subscripts p and s to represent the p and s parts of the far field pattern and the superscripts p and s to represent the p and s incident waves , respectively .
the main difficulty in using the model ( 7 ) to generate the far field is the calculation of u(yj ) .
this is due to the singularities of g on the points yj , see   for more details . to avoid this
however , we should note that the music type algorithms are applicable for the nonlinear model ( 7 ) since the proofs of theorem 4.1 of   and theorem  3.1 are also valid for this case . for results using the scalar model ( 6 ) ,
a current work is being carried out for the elastic model ( 7 ) and we will discuss this in a future work .
assume that there is no multiple scattering between the scatterers ( born approximation ) , then the scattered wave can be written in the form(8)u(x,)=m=1mmg(x , ym)ui(ym, ) , by replacing in the right hand side of ( 7)ut by ui .
the asymptotic behavior of the green tensor at infinity is given as follows(9)g(x , ym)=apxxeikp|x||x|n12eikpxym+as(ixx)eiks|x||x|n12eiksxym+o(|x|n+12 ) , with x=x|x| and i being the identity matrix in rn , ap = kp242 and as = ks242 , see for instance  .
it follows from ( 8) and ( 9 ) that the p and s parts of the far field pattern associated with the p incident wave ui , p are given by(10)up,p(x,)=apm=1mm(xx)eikpym(x),(11)us,p(x,)=asm=1mm(ixx)eikpymeiksymx. similarly , the p and s parts of the far field pattern associated with the s incident wave ui , s can be written as(12)us,p(x,)=apm=1mm(xx)eiksymeikpymx,(13)us,s(x,)=asm=1mm(ixx)eiksym(x ) .
here we have used the subscripts p and s to represent the p and s parts of the far field pattern and the superscripts p and s to represent the p and s incident waves , respectively .
the first music algorithm for determining the locations of point - like scatterers was first developed in   in 2000 using near field measurements of electromagnetic waves .
so far , several works have studied this type of algorithms for both near field and far field measurements and for different types of waves . for the elasticity , ammari et  al .  
used the music algorithm with full green s matrix as the measurements to reconstruct the locations of small inclusions and simonetti   showed some numerical results using a music algorithm for only one part ( s or p ) of the scattered waves . in this paper
, we also use the music type algorithms for reconstructing the locations of the scatterers but using only one part of the far field patterns and one type of incident plane waves as described in the previous section .
the idea is to convert the vector - type far field pattern to scalar one and make use of the music algorithm for scalar waves with some modifications .
we first briefly recall the classical music algorithm for scalar waves with far field measurements in the next subsection .
consider the scattering of acoustic wave by point - like scatterers associated with incident plane wave ui(x,)=eikx , where k is the wavenumber and sn1 is the direction of incidence . then under the assumption of weak scattering
, it follows from ( 6 ) that the far field pattern can be given in  ( 14)u(x,)=m=1mmeikym(x),xsn1 .
,m , of the scatterers from the measured far field pattern u(x, ) for a finite set of incidence and scattered directions , i.e. ,  x,{j , j=1,  ,n}sn1 . here
we assume that the number of scatterers is not larger than the number of incidence ( and observation ) directions , i.e. ,  nm .
given the measured far field pattern , we define the multistatic response matrix fcnn by ( 15)fj , l = u(j,l)=m=1mmeikym(lj ) . in order to determine the locations ym
for each point z , we define the vector zcn by ( 16)z=(eikz1,eikz2,  ,eikzn)t .
the use of the music algorithm is based on the property that z is in the range r(f ) of f iff z is at one of the locations of the scatterers .
that is , z{y1,y2,  ,ym } iff pz=0 , where p is the projection onto the null space n(f)=r(f) of the adjoint matrix f of f[25 ,  chapter 4 ] . in applying the music algorithm for elastic waves
, we have noticed that care must be taken in designing measurement setups as well as some modifications are needed in forming the multistatic response matrix .
for example , if we use the p part of the far field patterns of the p incident plane waves , i.e. ,  up,p , it is clear that the measured data vanishes in the directions orthogonal to the incidence direction . that is , the measured data in these directions are useless .
more generally , the information contained in the far field patterns is proportional to |x|  the cosine of the angle between the incidence and observation directions .
therefore , to obtain usable data , the measurement system should be set up in such a way that |x|>0 . with this system setup , given the p part of the far field patterns , we can calculate the scalar far field pattern ( 17)u(x,)=up,p(x,)ap(x)2=m=1mmeikpym(x ) . in this case
, we can use the same algorithm as in the scalar case to find the locations of the scatterers , with z in ( 16 ) being replaced by the test vector ( 18)zp=(eikpz1,eikpz2,  ,eikpzn)t which corresponds to the longitudinal far field of the p part of a point source located at z. the case of s incident waves and s part of the far field patterns is treated in the same way by using the test vector ( 19)zs=(eiksz1,eiksz2,  ,eikszn)t , which represents the transversal far field of the s part of the point source . now consider the mixed cases , i.e. ,  s incident waves and p part of the far field patterns or p incident waves and s part of the far field patterns . by similar arguments as above ,
we note that the observation directions should not be parallel or anti - parallel to the incidence directions . in these cases ,
modifications are needed in applying the music algorithm since we have the presence of both s and p wavenumbers ks and kp .
similar to the above case , we assume that |x|>0 . under this assumption , we can calculate from the p far field pattern ( 12 ) the following modified multistatic response matrix fcnn by ( 20)fj , l=m=1mmeiksymleikpymxj with l , l=1, 
,n , the observation directions . note that this modified multistatic response matrix is different from the scalar one due to the presence of two different s wavenumber ks and p wavenumber kp .
following the same arguments as in , we factorize the matrix f as ( 21)f=hpths , where hp and hs are matrices in cmn defined by ( 22)hmjp=|m|eikpymxj , hmjs=|m|eiksymj , m=1,  ,m;j=1,  ,n .
now for each sampling point zrn , we also make use of the test vector zs defined by ( 19 ) . as in the scalar case ,
the key properties of the music algorithm are ( i ) the vector zs belongs to r(hs ) iff z{y1,  ,ym } and ( ii ) the range r(ff ) of ff coincides the range r(hs ) of hs. they are proved in the following theorem .
theorem  3.1suppose that the sets 
 { n , nn}sn1 
 and 
 { xn , nn}sn1 
 are dense on 
 sn1 
 in the sense that any analytic function on 
 sn1that vanishes on one of these sets vanishes on the whole 
 sn1 
 .
let 
 k 
 be a compact set of 
 rn 
 containing 
 { ym , m=1,  ,m } 
 .
then there exists a number 
 n0n 
 such that for all 
 nn0 , the following properties are satisfied(i)z{y1,y2,  ,ym } 
 iff 
 zsr(hs ) 
 for 
 zk.(ii)r(ff)r(hs).therefore , 
 z{y1,y2,  ,ym } 
 iff 
 zsr(ff ) 
 or equivalently , 
 pzs=0 , where 
 p 
 is the projection onto the null space of the self - adjoint matrix 
 ff. suppose that the sets 
 { n , nn}sn1 
 and 
 { xn , nn}sn1 
 are dense on 
 sn1 
 in the sense that any analytic function on 
 sn1that vanishes on one of these sets vanishes on the whole 
 sn1 
 . let 
 k 
 be a compact set of 
 rn 
 containing 
 { ym , m=1,  ,m } 
 . then there exists a number 
 n0n 
 such that for all 
 nn0 , the following properties are satisfied(i)z{y1,y2,  ,ym } 
 iff 
 zsr(hs ) 
 for 
 zk.(ii)r(ff)r(hs).therefore , 
 z{y1,y2,  ,ym } 
 iff 
 zsr(ff ) 
 or equivalently , 
 pzs=0 , where 
 p 
 is the projection onto the null space of the self - adjoint matrix 
 ff. z{y1,y2,  ,ym } 
 iff 
 zsr(hs ) 
 for 
 zk .
the only difference is that in this case we make use of two different sets of incidence and observation directions . using the same arguments as of the mentioned theorem
, we can prove first that there exists a number n1n such that the vectors y1s,  ,yms,zs are linearly independent for nn1 and zk{y1,  ,ym } and the point ( i ) of the theorem exactly follows from the proof of theorem 4.1 of  .now consider the point ( ii ) .
equivalently , hs is surjective from cn to cm.concerning the matrix hp , we define the following vector ( 24)zp=(eikpzx1,eikpzx2,  ,eikpzxn)t . following the same arguments of the point ( i ) , we also can prove that there exists a number n2n such that the vectors y1p,  ,ymp are linearly independent for nn1 .
now for nn0=max{n1,n2 } , there exists ycn such that x = thsy as hs is surjective . that is y = hsthsyr(ff ) .
the only difference is that in this case we make use of two different sets of incidence and observation directions . using the same arguments as of the mentioned theorem
, we can prove first that there exists a number n1n such that the vectors y1s,  ,yms,zs are linearly independent for nn1 and zk{y1,  ,ym } and the point ( i ) of the theorem exactly follows from the proof of theorem 4.1 of  .
concerning the matrix hp , we define the following vector ( 24)zp=(eikpzx1,eikpzx2,  ,eikpzxn)t . following the same arguments of the point ( i )
, we also can prove that there exists a number n2n such that the vectors y1p,  ,ymp are linearly independent for nn1 .
now for nn0=max{n1,n2 } , there exists ycn such that x = thsy as hs is surjective . that is y = hsthsyr(ff ) .
 remark  3.1instead of the test vector zs , we can also use the vector zp for the music algorithm .
however , in this case , the matrix ff must be replaced by ff. instead of the test vector zs , we can also use the vector zp for the music algorithm .
consider the scattering of acoustic wave by point - like scatterers associated with incident plane wave ui(x,)=eikx , where k is the wavenumber and sn1 is the direction of incidence . then under the assumption of weak scattering
, it follows from ( 6 ) that the far field pattern can be given in  ( 14)u(x,)=m=1mmeikym(x),xsn1 .
,m , of the scatterers from the measured far field pattern u(x, ) for a finite set of incidence and scattered directions , i.e. ,  x,{j , j=1,  ,n}sn1 . here
we assume that the number of scatterers is not larger than the number of incidence ( and observation ) directions , i.e. ,  nm .
given the measured far field pattern , we define the multistatic response matrix fcnn by ( 15)fj , l = u(j,l)=m=1mmeikym(lj ) . in order to determine the locations ym
for each point z , we define the vector zcn by ( 16)z=(eikz1,eikz2,  ,eikzn)t .
the use of the music algorithm is based on the property that z is in the range r(f ) of f iff z is at one of the locations of the scatterers .
that is , z{y1,y2,  ,ym } iff pz=0 , where p is the projection onto the null space n(f)=r(f) of the adjoint matrix f of f[25 ,  chapter 4 ] .
in applying the music algorithm for elastic waves , we have noticed that care must be taken in designing measurement setups as well as some modifications are needed in forming the multistatic response matrix .
for example , if we use the p part of the far field patterns of the p incident plane waves , i.e. ,  up,p , it is clear that the measured data vanishes in the directions orthogonal to the incidence direction . that is , the measured data in these directions are useless . more generally , the information contained in the far field patterns is proportional to |x|  the cosine of the angle between the incidence and observation directions .
therefore , to obtain usable data , the measurement system should be set up in such a way that |x|>0 . with this system setup ,
given the p part of the far field patterns , we can calculate the scalar far field pattern ( 17)u(x,)=up,p(x,)ap(x)2=m=1mmeikpym(x ) . in this case
, we can use the same algorithm as in the scalar case to find the locations of the scatterers , with z in ( 16 ) being replaced by the test vector ( 18)zp=(eikpz1,eikpz2,  ,eikpzn)t which corresponds to the longitudinal far field of the p part of a point source located at z. the case of s incident waves and s part of the far field patterns is treated in the same way by using the test vector ( 19)zs=(eiksz1,eiksz2,  ,eikszn)t , which represents the transversal far field of the s part of the point source . now consider the mixed cases , i.e. ,  s incident waves and p part of the far field patterns or p incident waves and s part of the far field patterns . by similar arguments as above , we note that the observation directions should not be parallel or anti - parallel to the incidence directions . in these cases ,
modifications are needed in applying the music algorithm since we have the presence of both s and p wavenumbers ks and kp .
similar to the above case , we assume that |x|>0 . under this assumption , we can calculate from the p far field pattern ( 12 ) the following modified multistatic response matrix fcnn by ( 20)fj , l=m=1mmeiksymleikpymxj with l , l=1, 
note that this modified multistatic response matrix is different from the scalar one due to the presence of two different s wavenumber ks and p wavenumber kp .
following the same arguments as in , we factorize the matrix f as ( 21)f=hpths , where hp and hs are matrices in cmn defined by ( 22)hmjp=|m|eikpymxj , hmjs=|m|eiksymj , m=1,  ,m;j=1,  ,n .
now for each sampling point zrn , we also make use of the test vector zs defined by ( 19 ) . as in the scalar case ,
the key properties of the music algorithm are ( i ) the vector zs belongs to r(hs ) iff z{y1,  ,ym } and ( ii ) the range r(ff ) of ff coincides the range r(hs ) of hs. they are proved in the following theorem .
theorem  3.1suppose that the sets 
 { n , nn}sn1 
 and 
 { xn , nn}sn1 
 are dense on 
 sn1 
 in the sense that any analytic function on 
 sn1that vanishes on one of these sets vanishes on the whole 
 sn1 
 .
let 
 k 
 be a compact set of 
 rn 
 containing 
 { ym , m=1,  ,m } 
 .
then there exists a number 
 n0n 
 such that for all 
 nn0 , the following properties are satisfied(i)z{y1,y2,  ,ym } 
 iff 
 zsr(hs ) 
 for 
 zk.(ii)r(ff)r(hs).therefore , 
 z{y1,y2,  ,ym } 
 iff 
 zsr(ff ) 
 or equivalently , 
 pzs=0 , where 
 p 
 is the projection onto the null space of the self - adjoint matrix 
 ff. suppose that the sets 
 { n , nn}sn1 
 and 
 { xn , nn}sn1 
 are dense on 
 sn1 
 in the sense that any analytic function on 
 sn1that vanishes on one of these sets vanishes on the whole 
 sn1 
 .
let 
 k 
 be a compact set of 
 rn 
 containing 
 { ym , m=1,  ,m } 
 .
then there exists a number 
 n0n 
 such that for all 
 nn0 , the following properties are satisfied(i)z{y1,y2,  ,ym } 
 iff 
 zsr(hs ) 
 for 
 zk.(ii)r(ff)r(hs).therefore , 
 z{y1,y2,  ,ym } 
 iff 
 zsr(ff ) 
 or equivalently , 
 pzs=0 , where 
 p 
 is the projection onto the null space of the self - adjoint matrix 
 ff. z{y1,y2,  ,ym } 
 iff 
 zsr(hs ) 
 for 
 zk .
the only difference is that in this case we make use of two different sets of incidence and observation directions . using the same arguments as of the mentioned theorem
, we can prove first that there exists a number n1n such that the vectors y1s,  ,yms,zs are linearly independent for nn1 and zk{y1,  ,ym } and the point ( i ) of the theorem exactly follows from the proof of theorem 4.1 of  .now consider the point ( ii ) .
equivalently , hs is surjective from cn to cm.concerning the matrix hp , we define the following vector ( 24)zp=(eikpzx1,eikpzx2,  ,eikpzxn)t . following the same arguments of the point ( i ) , we also can prove that there exists a number n2n such that the vectors y1p,  ,ymp are linearly independent for nn1 .
now for nn0=max{n1,n2 } , there exists ycn such that x = thsy as hs is surjective .
the only difference is that in this case we make use of two different sets of incidence and observation directions . using the same arguments as of the mentioned theorem
, we can prove first that there exists a number n1n such that the vectors y1s,  ,yms,zs are linearly independent for nn1 and zk{y1,  ,ym } and the point ( i ) of the theorem exactly follows from the proof of theorem 4.1 of  .
concerning the matrix hp , we define the following vector ( 24)zp=(eikpzx1,eikpzx2,  ,eikpzxn)t . following the same arguments of the point ( i )
, we also can prove that there exists a number n2n such that the vectors y1p,  ,ymp are linearly independent for nn1 .
now for nn0=max{n1,n2 } , there exists ycn such that x = thsy as hs is surjective . that is y = hsthsyr(ff ) .
 remark  3.1instead of the test vector zs , we can also use the vector zp for the music algorithm .
however , in this case , the matrix ff must be replaced by ff. instead of the test vector zs , we can also use the vector zp for the music algorithm .
in this section , we illustrate the performance of the music algorithm for elasticity using one type of wave and compare the results for the case of s and p incident plane waves .
it is expected that , since they have shorter wavelength , the s incident waves should provide sharper results compared to the p incident waves .
this is confirmed in the following numerical examples . for the convenience in visualizing the results ,
we should mention that the algorithm in two and three dimensional spaces are the same .
but we are aware that the three dimensional case is a bit more complicated since there are two linearly independent directions for the transverse waves . as we have mentioned in the previous section , care
must be taken in setting up the measurements to avoid small values of the measured far field patterns .
for this purpose , we propose the following setups in two situations . for p incident waves and p part of far field patterns ( pp case ) , or s incident waves and s part of far field patterns ( ss case )
denote by nd the number of incidence directions ( angles ) used in a quarter of the unit circle . in the first and the third quarter , we use the following incidence angles ( see fig . 
,nd , and in the second and the fourth quarter , we make use of the incidence angles nd+j=2+4nd+(j1)2nd,3nd+j=32+4nd+(j1)2nd , j=1,  ,nd .
, we have |x|sin(4nd ) for all x,{j , j=1,  ,4nd}. to avoid parallel or anti - parallel directions in the case of p incidence waves and s part of far field patterns ( ps case ) or s incidence waves and p part of far field patterns ( sp case ) , we choose the incidence and observation angles as follows ( fig . 
, the minimum angle between the incidence and observation angles is 4nd . in the following examples ,
the parameters are chosen as =2 , =1 and k=2 resulting in kp= and ks=2. let us first consider four point - like scatterers located at y1=(0,0 ) , y2=(0,0.5 ) , y3=(1,1 ) and y4=(1,1 ) .
since the music algorithm is an exact method , see theorem  3.1 ( see also  ) , the reconstruction is very accurate if there is no noise in the measured data . in this paper , we concentrate on the resolution of the algorithm in the case of noisy data . to analyze the effect of noise level on the resolution of the algorithm ,
24 show the pseudo spectrum of the scatterers with 1% , 5% and 10% random noise in the measured far field patterns , respectively .
we should emphasize that , by converting from the vector far field patterns to the scalar one as in ( 17 ) , the noise in the measured far field patterns is amplified in the modified multistatic response matrix resulting worse results than the scalar case .
2 shows good reconstructions for all scatterers in both pp and sp cases even though in the latter case the peaks are a bit sharper at the locations y1 and y2 . in fig .  3 , with 5% noise in the data ,
the two scatterers at y1 and y2 are not well separated anymore in the pp case but they are still very well separated in the sp case .
here the two close scatterers are still clearly visible in the sp case but not anymore distinguishable in the pp case .
these results show that using the s incident waves we can obtain better resolution with the music algorithm than using the p incident waves .
5 shows that , with nine scatterers , the scatterers close to each other ( at the distance of about one fourth of the wavelength ) are hardly or even impossible to be separated in the pp case although the noise level is only 5% , but they are still distinguishable in the sp case .
however , even in the sp case , the result is less accurate than the previous example of four scatterers . actually , the higher the number of scatterers , the lower the accuracy in both cases .
finally , we should mention that the reconstruction results depend on the choice of the signal and noise subspaces of the multistatic response matrix . for small measurement noise ,
these two subspaces are easy to choose since there is a clear cut in the distribution of the singular values of the multistatic response matrix .
however , for large noise , the distribution of the singular values are smooth and it becomes more difficult to separate the singular values of the signal and noise subspaces ; see fig . 
6 for the pp case with four scatterers . in this paper , since we want to compare the pp and sp cases , the singular values were separated manually which is based on the true number of point - like scatterers .
as a conclusion , we can say that using the s incident waves provides more accurate reconstruction of the locations of point - like scatterers using the music algorithm compared to the p incident waves .
moreover , the larger the lam parameter  , the better the reconstruction with the s incident waves compared to the p incident waves since , in this case , the wavelength of the p - incident wave is much larger than the one of the s - incident wave .
it is worth mentioning that from the numerical tests , we observed that the p and s parts of far field patterns , for a given incident wave , provide almost the same resolution .
however , the magnitudes of the peaks may be different using p or s parts of the far field patterns . this may result in a better or worse reconstruction quality
 in this paper , we are concerned with the detection of point - like obstacles using elastic waves . 
 we show that one type of waves , either the p or the s scattered waves , is enough for localizing the points . 
 we also show how the use of s incident waves gives better resolution than the p waves . 
 these affirmations are demonstrated by several numerical examples using a music type algorithm . 
Introduction
Point-like obstacles
Weak scattering approximation
MUSIC algorithms
MUSIC algorithm for scalar waves
MUSIC algorithm for elastic waves
Numerical results and discussions
Conclusion