# Guide to programming with madagascar

This page was created from the LaTeX source in book/rsf/rsf/demo.tex using latex2wiki

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

## Introduction

This section presents time-domain finite-difference modeling [1] 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:

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

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");

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 */

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

// time-domain acoustic FD modeling
#include <valarray>
#include <iostream>
#include <rsf.hh>
#include <cub.hh>
#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 Fo("out","o"); Fo.setup(3,Fv.esize());

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);

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);
}
• Declare input, output and auxiliary file cubes (of type CUB).
    CUB Fw("in", "i"); Fw.headin(); //Fw.report();
CUB Fo("out","o"); Fo.setup(3,Fv.esize());
• Declare, read and write RSF cube axes: at time axis, ax space axis, az depth axis.
    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();
• Declare multi-dimensional valarrays for input, output and read data.
    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;
• Declare multi-dimensional valarrays for temporary storage.
    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;
• Initialize multidimensional valarray index counter (of type VAI).
    VAI k(nz,nx);
• Loop over time.
    for (int it=0; it<nt; it++) {
• Compute Laplacian: $\Delta U$.
	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 source wavelet: $\left[ \Delta U - f(t) \right]$
	ud -= ww[it] * rr;
• Scale by velocity: $\left[ \Delta U - f(t) \right] v^2$
	ud *= vv*vv;
• Time step: $U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}$
	up=(float)2 * uo - um + ud * dt2;
um =   uo;
uo =   up;

## Fortran 90 program

! 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")

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 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
• Declare input, output and auxiliary file tags.
  type(file) :: Fw,Fv,Fr,Fo  ! I/O files
• Declare RSF cube axes: at time axis, ax space axis, az depth axis.
  type(axa)  :: at,az,ax     ! cube axes
• Declare multi-dimensional arrays for input, output and computations.
  real, allocatable :: vv(:,:),rr(:,:),ww(:)           ! I/O arrays
real, allocatable :: um(:,:),uo(:,:),up(:,:),ud(:,:) ! tmp arrays
• Open files for input/output.
  Fw=rsf_input ("in")
Fv=rsf_input ("vel")
Fr=rsf_input ("ref")
Fo=rsf_output("out")
• Read axes from input files; write axes to output file.
  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)
• Allocate arrays and read wavelet, velocity and reflectivity.
  allocate(ww(at%n     )); ww=0.; call rsf_read(Fw,ww)
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.
• Loop over time.
  do it=1,at%n
• Compute Laplacian: $\Delta U$.
     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 source wavelet: $\left[ \Delta U - f(t) \right]$
     ud = ud - ww(it) * rr
• Scale by velocity: $\left[ \Delta U - f(t) \right] v^2$
     ud= ud *vv*vv
• Time step: $U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}$
     up = 2*uo - um + ud * dt2
um =   uo
uo =   up

## References

1. "Hello world" of seismic imaging.