Guide to programming with madagascar

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This guide demonstrates a simple time-domain finite-differences modeling code in RSF.

Introduction

This section presents time-domain finite-difference modeling written with the RSF library. The program is demonstrated with the C, C++ and Fortran 90 interfaces. The acoustic wave-equation

$\Delta U-{\frac {1}{v^{2}}}{\frac {\partial ^{2}U}{\partial t^{2}}}=f(t)$

can be written as

$\left[\Delta U-f(t)\right]v^{2}={\frac {\partial ^{2}U}{\partial t^{2}}}\;.$

$\Delta$ is the Laplacian symbol, $f(t)$ is the source wavelet, $v$ is the velocity, and $U$ is a scalar wavefield. A discrete time-step involves the following computations:

$U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}\;,$

where $U_{i-1}$, $U_{i}$ and $U_{i+1}$ represent the propagating wavefield at various time steps.

C program

/* time-domain acoustic FD modeling */
#include <rsf.h>
int main(int argc, char* argv[])
{
    /* Laplacian coefficients */
    float c0=-30./12.,c1=+16./12.,c2=- 1./12.;

    bool verb;           /* verbose flag */
    sf_file Fw=NULL,Fv=NULL,Fr=NULL,Fo=NULL; /* I/O files */
    sf_axis at,az,ax;    /* cube axes */
    int it,iz,ix;        /* index variables */
    int nt,nz,nx;
    float dt,dz,dx,idx,idz,dt2;

    float  *ww,**vv,**rr;     /* I/O arrays*/
    float **um,**uo,**up,**ud;/* tmp arrays */

    sf_init(argc,argv);
    if(! sf_getbool("verb",&verb)) verb=0;

    /* setup I/O files */
    Fw = sf_input ("in" );
    Fo = sf_output("out");
    Fv = sf_input ("vel");
    Fr = sf_input ("ref");

    /* Read/Write axes */
    at = sf_iaxa(Fw,1); nt = sf_n(at); dt = sf_d(at);
    az = sf_iaxa(Fv,1); nz = sf_n(az); dz = sf_d(az);
    ax = sf_iaxa(Fv,2); nx = sf_n(ax); dx = sf_d(ax);

    sf_oaxa(Fo,az,1); 
    sf_oaxa(Fo,ax,2); 
    sf_oaxa(Fo,at,3);

    dt2 =    dt*dt;
    idz = 1/(dz*dz);
    idx = 1/(dx*dx);

    /* read wavelet, velocity & reflectivity */
    ww=sf_floatalloc(nt);     sf_floatread(ww   ,nt   ,Fw);
    vv=sf_floatalloc2(nz,nx); sf_floatread(vv[0],nz*nx,Fv);
    rr=sf_floatalloc2(nz,nx); sf_floatread(rr[0],nz*nx,Fr);

    /* allocate temporary arrays */
    um=sf_floatalloc2(nz,nx);
    uo=sf_floatalloc2(nz,nx);
    up=sf_floatalloc2(nz,nx);
    ud=sf_floatalloc2(nz,nx);
    
    for (iz=0; iz<nz; iz++) {
	for (ix=0; ix<nx; ix++) {
	    um[ix][iz]=0;
	    uo[ix][iz]=0;
	    up[ix][iz]=0;
	    ud[ix][iz]=0;
	}
    }
    
    /* MAIN LOOP */
    if(verb) fprintf(stderr,"\n");
    for (it=0; it<nt; it++) {
	if(verb) fprintf(stderr,"\b\b\b\b\b%d",it);
	
	/* 4th order laplacian */
	for (iz=2; iz<nz-2; iz++) {
	    for (ix=2; ix<nx-2; ix++) {
		ud[ix][iz] = 
		    c0* uo[ix  ][iz  ] * (idx+idz) + 
		    c1*(uo[ix-1][iz  ] + uo[ix+1][iz  ])*idx +
		    c2*(uo[ix-2][iz  ] + uo[ix+2][iz  ])*idx +
		    c1*(uo[ix  ][iz-1] + uo[ix  ][iz+1])*idz +
		    c2*(uo[ix  ][iz-2] + uo[ix  ][iz+2])*idz;	  
	    }
	}

	/* inject wavelet */
	for (iz=0; iz<nz; iz++) {
	    for (ix=0; ix<nx; ix++) {
		ud[ix][iz] -= ww[it] * rr[ix][iz];
	    }
	}

	/* scale by velocity */
	for (iz=0; iz<nz; iz++) {
	    for (ix=0; ix<nx; ix++) {
		ud[ix][iz] *= vv[ix][iz]*vv[ix][iz];
	    }
	}
	
	/* time step */
	for (iz=0; iz<nz; iz++) {
	    for (ix=0; ix<nx; ix++) {
		up[ix][iz] = 
		    2*uo[ix][iz] 
		    - um[ix][iz] 
		    + ud[ix][iz] * dt2; 
		
		um[ix][iz] = uo[ix][iz];
		uo[ix][iz] = up[ix][iz];
	    }
	}
	
	/* write wavefield to output */
	sf_floatwrite(uo[0],nz*nx,Fo);
    }
    if(verb) fprintf(stderr,"\n");    
    sf_close()
    exit (0);
}

• Declare input, output and auxiliary file tags: Fw for input wavelet, Fv for velocity, Fr for reflectivity, and Fo for output wavefield.

    sf_file Fw,Fv,Fr,Fo; /* I/O files */

• Declare RSF cube axes: at time axis, ax space axis, az depth axis.

    sf_axis at,az,ax;    /* cube axes */

• Declare multi-dimensional arrays for input, output and computations.

    float  *ww,**vv,**rr;     /* I/O arrays*/

• Open files for input/output.

    Fw = sf_input ("in" );
    Fo = sf_output("out");
    Fv = sf_input ("vel");
    Fr = sf_input ("ref");

• Read axes from input files; write axes to output file.

    at = sf_iaxa(Fw,1); nt = sf_n(at); dt = sf_d(at);
    az = sf_iaxa(Fv,1); nz = sf_n(az); dz = sf_d(az);
    ax = sf_iaxa(Fv,2); nx = sf_n(ax); dx = sf_d(ax);

    sf_oaxa(Fo,az,1); 
    sf_oaxa(Fo,ax,2); 
    sf_oaxa(Fo,at,3);

• Allocate arrays and read wavelet, velocity and reflectivity.

    ww=sf_floatalloc(nt);     sf_floatread(ww   ,nt   ,Fw);
    vv=sf_floatalloc2(nz,nx); sf_floatread(vv[0],nz*nx,Fv);
    rr=sf_floatalloc2(nz,nx); sf_floatread(rr[0],nz*nx,Fr);

• Allocate temporary arrays.

    um=sf_floatalloc2(nz,nx);
    uo=sf_floatalloc2(nz,nx);
    up=sf_floatalloc2(nz,nx);
    ud=sf_floatalloc2(nz,nx);

• Loop over time.

    for (it=0; it<nt; it++) {

• Compute Laplacian: $\Delta U$.

	for (iz=2; iz<nz-2; iz++) {
	    for (ix=2; ix<nx-2; ix++) {
		ud[ix][iz] = 
		    c0* uo[ix  ][iz  ] * (idx+idz) + 
		    c1*(uo[ix-1][iz  ] + uo[ix+1][iz  ])*idx +
		    c2*(uo[ix-2][iz  ] + uo[ix+2][iz  ])*idx +
		    c1*(uo[ix  ][iz-1] + uo[ix  ][iz+1])*idz +
		    c2*(uo[ix  ][iz-2] + uo[ix  ][iz+2])*idz;	  
	    }
	}

• Inject source wavelet: $\left[\Delta U-f(t)\right]$

	for (iz=0; iz<nz; iz++) {
	    for (ix=0; ix<nx; ix++) {
		ud[ix][iz] -= ww[it] * rr[ix][iz];
	    }
	}

• Scale by velocity: $\left[\Delta U-f(t)\right]v^{2}$

	for (iz=0; iz<nz; iz++) {
	    for (ix=0; ix<nx; ix++) {
		ud[ix][iz] *= vv[ix][iz]*vv[ix][iz];
	    }
	}

• Time step: $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$

	for (iz=0; iz<nz; iz++) {
	    for (ix=0; ix<nx; ix++) {
		up[ix][iz] = 
		    2*uo[ix][iz] 
		    - um[ix][iz] 
		    + ud[ix][iz] * dt2; 
		
		um[ix][iz] = uo[ix][iz];
		uo[ix][iz] = up[ix][iz];
	    }
	}

C++ program

<cpp> // time-domain acoustic FD modeling

1. include <valarray>
2. include <iostream>
3. include <rsf.hh>
4. include <cub.hh>
5. include <vai.hh>

using namespace std;

int main(int argc, char* argv[]) {

   // Laplacian coefficients
   float c0=-30./12.,c1=+16./12.,c2=- 1./12.;

   sf_init(argc,argv);// init RSF
   bool verb;         // vebose flag
   if(! sf_getbool("verb",&verb)) verb=0;

   // setup I/O files
   CUB Fw("in", "i"); Fw.headin(); //Fw.report();
   CUB Fv("vel","i"); Fv.headin(); //Fv.report();
   CUB Fr("ref","i"); Fr.headin(); //Fr.report();
   CUB Fo("out","o"); Fo.setup(3,Fv.esize()); 

   // Read/Write axes
   sf_axis at = Fw.getax(0); int nt = sf_n(at); float dt = sf_d(at);
   sf_axis az = Fv.getax(0); int nz = sf_n(az); float dz = sf_d(az);
   sf_axis ax = Fv.getax(1); int nx = sf_n(ax); float dx = sf_d(ax);

   Fo.putax(0,az); 
   Fo.putax(1,ax); 
   Fo.putax(2,at);
   Fo.headou();

   float dt2 =    dt*dt;
   float idz = 1/(dz*dz);
   float idx = 1/(dx*dx);

   // read wavelet, velocity and reflectivity
   valarray<float> ww( nt    ); ww=0; Fw >> ww;
   valarray<float> vv( nz*nx ); vv=0; Fv >> vv;
   valarray<float> rr( nz*nx ); rr=0; Fr >> rr;
  
   // allocate temporary arrays
   valarray<float> um(nz*nx); um=0;
   valarray<float> uo(nz*nx); uo=0;
   valarray<float> up(nz*nx); up=0;
   valarray<float> ud(nz*nx); ud=0;

   // init ValArray Index counter
   VAI k(nz,nx);

   // MAIN LOOP
   if(verb) cerr << endl;
   for (int it=0; it<nt; it++) {

if(verb) cerr << "\b\b\b\b\b" << it;

// 4th order laplacian for (int iz=2; iz<nz-2; iz++) { for (int ix=2; ix<nx-2; ix++) { ud[k(iz,ix)] = c0* uo[ k(iz ,ix )] * (idx+idz) + c1*(uo[ k(iz ,ix-1)]+uo[ k(iz ,ix+1)]) * idx + c1*(uo[ k(iz-1,ix )]+uo[ k(iz+1,ix )]) * idz + c2*(uo[ k(iz ,ix-2)]+uo[ k(iz ,ix+2)]) * idx + c2*(uo[ k(iz-2,ix )]+uo[ k(iz+2,ix )]) * idz; } }

// inject wavelet ud -= ww[it] * rr;

// scale by velocity ud *= vv*vv;

// time step up=(float)2 * uo - um + ud * dt2; um = uo; uo = up;

// write wavefield to output output Fo << uo;

   }
   if(verb) cerr << endl;

   exit(0);

}

</cpp>

• Declare input, output and auxiliary file cubes (of type CUB).

<cpp>

   CUB Fw("in", "i"); Fw.headin(); //Fw.report();
   CUB Fv("vel","i"); Fv.headin(); //Fv.report();
   CUB Fr("ref","i"); Fr.headin(); //Fr.report();
   CUB Fo("out","o"); Fo.setup(3,Fv.esize()); 

</cpp>

• Declare, read and write RSF cube axes: at time axis, ax space axis, az depth axis.

<cpp>

   sf_axis at = Fw.getax(0); int nt = sf_n(at); float dt = sf_d(at);
   sf_axis az = Fv.getax(0); int nz = sf_n(az); float dz = sf_d(az);
   sf_axis ax = Fv.getax(1); int nx = sf_n(ax); float dx = sf_d(ax);

   Fo.putax(0,az); 
   Fo.putax(1,ax); 
   Fo.putax(2,at);
   Fo.headou();

</cpp>

• Declare multi-dimensional valarrays for input, output and read data.

<cpp>

   valarray<float> ww( nt    ); ww=0; Fw >> ww;
   valarray<float> vv( nz*nx ); vv=0; Fv >> vv;
   valarray<float> rr( nz*nx ); rr=0; Fr >> rr;

</cpp>

• Declare multi-dimensional valarrays for temporary storage.

<cpp>

   valarray<float> um(nz*nx); um=0;
   valarray<float> uo(nz*nx); uo=0;
   valarray<float> up(nz*nx); up=0;
   valarray<float> ud(nz*nx); ud=0;

</cpp>

• Initialize multidimensional valarray index counter (of type VAI).

<cpp>

   VAI k(nz,nx);

</cpp>

• Loop over time.

<cpp>

   for (int it=0; it<nt; it++) {

</cpp>

• Compute Laplacian: $\Delta U$.

<cpp> for (int iz=2; iz<nz-2; iz++) { for (int ix=2; ix<nx-2; ix++) { ud[k(iz,ix)] = c0* uo[ k(iz ,ix )] * (idx+idz) + c1*(uo[ k(iz ,ix-1)]+uo[ k(iz ,ix+1)]) * idx + c1*(uo[ k(iz-1,ix )]+uo[ k(iz+1,ix )]) * idz + c2*(uo[ k(iz ,ix-2)]+uo[ k(iz ,ix+2)]) * idx + c2*(uo[ k(iz-2,ix )]+uo[ k(iz+2,ix )]) * idz; } } </cpp>

• Inject source wavelet: $\left[\Delta U-f(t)\right]$

<cpp> ud -= ww[it] * rr; </cpp>

• Scale by velocity: $\left[\Delta U-f(t)\right]v^{2}$

<cpp> ud *= vv*vv; </cpp>

• Time step: $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$

<cpp> up=(float)2 * uo - um + ud * dt2; um = uo; uo = up; </cpp>

Fortran 90 program

<fortran> ! time-domain acoustic FD modeling program AFDMf90

 use rsf

 implicit none

 ! Laplacian coefficients
 real :: c0=-30./12.,c1=+16./12.,c2=- 1./12.

 logical    :: verb         ! verbose flag
 type(file) :: Fw,Fv,Fr,Fo  ! I/O files
 type(axa)  :: at,az,ax     ! cube axes
 integer    :: it,iz,ix     ! index variables
 real       :: idx,idz,dt2

 real, allocatable :: vv(:,:),rr(:,:),ww(:)           ! I/O arrays
 real, allocatable :: um(:,:),uo(:,:),up(:,:),ud(:,:) ! tmp arrays

 call sf_init() ! init RSF
 call from_par("verb",verb,.false.)

 ! setup I/O files
 Fw=rsf_input ("in")
 Fv=rsf_input ("vel")
 Fr=rsf_input ("ref")
 Fo=rsf_output("out")

 ! Read/Write axes
 call iaxa(Fw,at,1); call iaxa(Fv,az,1); call iaxa(Fv,ax,2)
 call oaxa(Fo,az,1); call oaxa(Fo,ax,2); call oaxa(Fo,at,3)

 dt2 =    at%d*at%d
 idz = 1/(az%d*az%d)
 idx = 1/(ax%d*ax%d) 

 ! read wavelet, velocity & reflectivity
 allocate(ww(at%n     )); ww=0.; call rsf_read(Fw,ww)
 allocate(vv(az%n,ax%n)); vv=0.; call rsf_read(Fv,vv)
 allocate(rr(az%n,ax%n)); rr=0.; call rsf_read(Fr,rr)

 ! allocate temporary arrays
 allocate(um(az%n,ax%n)); um=0.
 allocate(uo(az%n,ax%n)); uo=0.
 allocate(up(az%n,ax%n)); up=0.
 allocate(ud(az%n,ax%n)); ud=0.

 ! MAIN LOOP
 do it=1,at%n
    if(verb) write (0,*) it

    ! 4th order laplacian
    do iz=2,az%n-2
       do ix=2,ax%n-2
          ud(iz,ix) = &
               c0* uo(iz,  ix  ) * (idx + idz)        + &
               c1*(uo(iz  ,ix-1) + uo(iz  ,ix+1))*idx + &
               c2*(uo(iz  ,ix-2) + uo(iz  ,ix+2))*idx + &
               c1*(uo(iz-1,ix  ) + uo(iz+1,ix  ))*idz + &
               c2*(uo(iz-2,ix  ) + uo(iz+2,ix  ))*idz
       end do
    end do

    ! inject wavelet
    ud = ud - ww(it) * rr

    ! scale by velocity
    ud= ud *vv*vv

    ! time step
    up = 2*uo - um + ud * dt2
    um =   uo
    uo =   up

    ! write wavefield to output
    call rsf_write(Fo,uo)
 end do

 call exit(0)

end program AFDMf90 </fortran>

• Declare input, output and auxiliary file tags.

<fortran>

 type(file) :: Fw,Fv,Fr,Fo  ! I/O files

</fortran>

• Declare RSF cube axes: at time axis, ax space axis, az depth axis.

<fortran>

 type(axa)  :: at,az,ax     ! cube axes

</fortran>

• Declare multi-dimensional arrays for input, output and computations.

<fortran>

 real, allocatable :: vv(:,:),rr(:,:),ww(:)           ! I/O arrays
 real, allocatable :: um(:,:),uo(:,:),up(:,:),ud(:,:) ! tmp arrays

</fortran>

• Open files for input/output.

<fortran>

 Fw=rsf_input ("in")
 Fv=rsf_input ("vel")
 Fr=rsf_input ("ref")
 Fo=rsf_output("out")

</fortran>

• Read axes from input files; write axes to output file.

<fortran>

 call iaxa(Fw,at,1); call iaxa(Fv,az,1); call iaxa(Fv,ax,2)
 call oaxa(Fo,az,1); call oaxa(Fo,ax,2); call oaxa(Fo,at,3)

</fortran>

• Allocate arrays and read wavelet, velocity and reflectivity.

<fortran>

 allocate(ww(at%n     )); ww=0.; call rsf_read(Fw,ww)
 allocate(vv(az%n,ax%n)); vv=0.; call rsf_read(Fv,vv)
 allocate(rr(az%n,ax%n)); rr=0.; call rsf_read(Fr,rr)

</fortran>

• Allocate temporary arrays.

<fortran>

 allocate(um(az%n,ax%n)); um=0.
 allocate(uo(az%n,ax%n)); uo=0.
 allocate(up(az%n,ax%n)); up=0.
 allocate(ud(az%n,ax%n)); ud=0.

</fortran>

• Loop over time.

<fortran>

 do it=1,at%n

</fortran>

• Compute Laplacian: $\Delta U$.

<fortran>

    do iz=2,az%n-2
       do ix=2,ax%n-2
          ud(iz,ix) = &
               c0* uo(iz,  ix  ) * (idx + idz)        + &
               c1*(uo(iz  ,ix-1) + uo(iz  ,ix+1))*idx + &
               c2*(uo(iz  ,ix-2) + uo(iz  ,ix+2))*idx + &
               c1*(uo(iz-1,ix  ) + uo(iz+1,ix  ))*idz + &
               c2*(uo(iz-2,ix  ) + uo(iz+2,ix  ))*idz
       end do
    end do

</fortran>

• Inject source wavelet: $\left[\Delta U-f(t)\right]$

<fortran>

    ud = ud - ww(it) * rr

</fortran>

• Scale by velocity: $\left[\Delta U-f(t)\right]v^{2}$

<fortran>

    ud= ud *vv*vv

</fortran>

• Time step: $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$

<fortran>

    up = 2*uo - um + ud * dt2
    um =   uo
    uo =   up

</fortran>