Guide to programming with madagascar: Difference between revisions

Latest revision as of 02:29, 6 May 2021

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

Introduction[edit]

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}+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[edit]

C program[edit]

/* 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; /* verbose flag */

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

    /* MAIN LOOP */
    if(verb) fprintf(stderr,"
");
    for (it=0; it<nt; it++) {
	if(verb) fprintf(stderr,"\b\b\b\b\b %d",it);
	
	/* 4th order laplacian */
	for (ix=2; ix<nx-2; ix++) {
	    for (iz=2; iz<nz-2; iz++) {
		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 (ix=0; ix<nx; ix++) {
	    for (iz=0; iz<nz; iz++) {
		ud[ix][iz] -= ww[it] * rr[ix][iz];
	    }
	}

	/* scale by velocity */
	for (ix=0; ix<nx; ix++) {
	    for (iz=0; iz<nz; iz++) {
		ud[ix][iz] *= vv[ix][iz]*vv[ix][iz];
	    }
	}
	
	/* time step */
	for (ix=0; ix<nx; ix++) {
	    for (iz=0; iz<nz; iz++) {
		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");

    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=NULL,Fv=NULL,Fr=NULL,Fo=NULL; /* I/O files */
```
Declare RSF cube axes: at time axis, ax space axis, az depth axis. \tiny
```
    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 (ix=2; ix<nx-2; ix++) {
	    for (iz=2; iz<nz-2; iz++) {
		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 (ix=0; ix<nx; ix++) {
	    for (iz=0; iz<nz; iz++) {
		ud[ix][iz] -= ww[it] * rr[ix][iz];
	    }
	}

Scale by velocity:

\left[\Delta U-f(t)\right]v^{2}

	for (ix=0; ix<nx; ix++) {
	    for (iz=0; iz<nz; iz++) {
		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 (ix=0; ix<nx; ix++) {
	    for (iz=0; iz<nz; iz++) {
		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];
	    }
	}

\newpage

C++ program[edit]

// 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 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); 

    // 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 ix=2; ix<nx-2; ix++) {
	    for (int iz=2; iz<nz-2; iz++) {
		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 Fv("vel","i"); Fv.headin(); //Fv.report();
    CUB Fr("ref","i"); Fr.headin(); //Fr.report();
    CUB Fo("out","o"); Fo.setup(3);

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);

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 ix=2; ix<nx-2; ix++) {
	    for (int iz=2; iz<nz-2; iz++) {
		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: $\left[\Delta U-f(t)\right]$ $\left[\Delta U-f(t)\right]$
```
	ud -= ww[it] * rr;
```
Scale by velocity: $\left[\Delta U-f(t)\right]v^{2}$ $\left[\Delta U-f(t)\right]v^{2}$
```
	ud *= vv*vv;
```
Time step: $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$ $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$
```
	up=(float)2 * uo - um + ud * dt2;
	um =   uo;
	uo =   up;
```

Fortran 90 program[edit]

! 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
  integer    :: nt,nz,nx
  real       :: dt,dz,dx
  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); nt = at%n; dt = at%d
  call iaxa(Fv,az,1); nz = az%n; dz = az%d
  call iaxa(Fv,ax,2); nx = ax%n; dx = ax%d
  
  call oaxa(Fo,az,1)
  call oaxa(Fo,ax,2)
  call oaxa(Fo,at,3)

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

  ! read wavelet, velocity & reflectivity
  allocate(ww(nt));    call rsf_read(Fw,ww)
  allocate(vv(nz,nx)); call rsf_read(Fv,vv)
  allocate(rr(nz,nx)); call rsf_read(Fr,rr)

  ! allocate temporary arrays
  allocate(um(nz,nx)); um=0.
  allocate(uo(nz,nx)); uo=0.
  allocate(up(nz,nx)); up=0.
  allocate(ud(nz,nx)); ud=0.

  ! MAIN LOOP
  do it=1,nt
     if(verb) write (0,*) it

     ud(3:nz-2,3:nx-2) = &
          c0* uo(3:nz-2,3:nx-2) * (idx + idz)            + &
          c1*(uo(3:nz-2,2:nx-3) + uo(3:nz-2,4:nx-1))*idx + &
          c2*(uo(3:nz-2,1:nx-4) + uo(3:nz-2,5:nx  ))*idx + &
          c1*(uo(2:nz-3,3:nx-2) + uo(4:nz-1,3:nx-2))*idz + &
          c2*(uo(1:nz-4,3:nx-2) + uo(5:nz  ,3:nx-2))*idz

     ! 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); nt = at%n; dt = at%d
  call iaxa(Fv,az,1); nz = az%n; dz = az%d
  call iaxa(Fv,ax,2); nx = ax%n; dx = ax%d
  
  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(nt));    call rsf_read(Fw,ww)
  allocate(vv(nz,nx)); call rsf_read(Fv,vv)
  allocate(rr(nz,nx)); call rsf_read(Fr,rr)

Allocate temporary arrays.

  allocate(um(nz,nx)); um=0.
  allocate(uo(nz,nx)); uo=0.
  allocate(up(nz,nx)); up=0.
  allocate(ud(nz,nx)); ud=0.

Loop over time.
```
  do it=1,nt
```

Compute Laplacian:

\Delta U

.

     ud(3:nz-2,3:nx-2) = &
          c0* uo(3:nz-2,3:nx-2) * (idx + idz)            + &
          c1*(uo(3:nz-2,2:nx-3) + uo(3:nz-2,4:nx-1))*idx + &
          c2*(uo(3:nz-2,1:nx-4) + uo(3:nz-2,5:nx  ))*idx + &
          c1*(uo(2:nz-3,3:nx-2) + uo(4:nz-1,3:nx-2))*idz + &
          c2*(uo(1:nz-4,3:nx-2) + uo(5:nz  ,3:nx-2))*idz

Inject source wavelet: $\left[\Delta U-f(t)\right]$ $\left[\Delta U-f(t)\right]$
```
     ud = ud - ww(it) * rr
```
Scale by velocity: $\left[\Delta U-f(t)\right]v^{2}$ $\left[\Delta U-f(t)\right]v^{2}$
```
     ud= ud *vv*vv
```
Time step: $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$ $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$
```
     up = 2*uo - um + ud * dt2
     um =   uo
     uo =   up
```

Python program[edit]

#!/usr/bin/env python

import sys
import numpy
import m8r

c0=-30./12.
c1=+16./12.
c2=- 1./12.

par = m8r.Par()
verb = par.bool("verb",False) # verbosity

# setup I/O files
Fw=m8r.Input()
Fv=m8r.Input ("vel")
Fr=m8r.Input ("ref")
Fo=m8r.Output()

# Read/Write axes
at = Fw.axis(1); nt = at['n']; dt = at['d']
az = Fv.axis(1); nz = az['n']; dz = az['d']
ax = Fv.axis(2); nx = ax['n']; dx = ax['d']

Fo.putaxis(az,1)
Fo.putaxis(ax,2)
Fo.putaxis(at,3)

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

# read wavelet, velocity & reflectivity
ww = numpy.zeros(nt,'f');      Fw.read(ww)
vv = numpy.zeros([nz,nx],'f'); Fv.read(vv)
rr = numpy.zeros([nz,nx],'f'); Fr.read(rr)

# allocate temporary arrays
um = numpy.zeros([nz,nx],'f')
uo = numpy.zeros([nz,nx],'f')
up = numpy.zeros([nz,nx],'f')
ud = numpy.zeros([nz,nx],'f')

# MAIN LOOP
for it in range(nt):
    if verb:
        sys.stderr.write("\b\b\b\b\b %d" % it)

    ud[2:-2,2:-2] = \
    c0* uo[2:-2,2:-2] * (idx + idz)        + \
    c1*(uo[2:-2,1:-3] + uo[2:-2,3:-1])*idx + \
    c2*(uo[2:-2, :-4] + uo[2:-2,4:  ])*idx + \
    c1*(uo[1:-3,2:-2] + uo[3:-1,2:-2])*idz + \
    c2*(uo[ :-4,2:-2] + uo[4:  ,2:-2])*idz

    # 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

if verb:
    sys.stderr.write("\n")
    Fo.write(uo)

sys.exit(0)

Open files for input/output.

Fw=m8r.Input()
Fv=m8r.Input ("vel")
Fr=m8r.Input ("ref")
Fo=m8r.Output()

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

at = Fw.axis(1); nt = at['n']; dt = at['d']
az = Fv.axis(1); nz = az['n']; dz = az['d']
ax = Fv.axis(2); nx = ax['n']; dx = ax['d']

Fo.putaxis(az,1)
Fo.putaxis(ax,2)
Fo.putaxis(at,3)

Allocate arrays and read wavelet, velocity and reflectivity.

ww = numpy.zeros(nt,'f');      Fw.read(ww)
vv = numpy.zeros([nz,nx],'f'); Fv.read(vv)
rr = numpy.zeros([nz,nx],'f'); Fr.read(rr)

Allocate temporary arrays.

um = numpy.zeros([nz,nx],'f')
uo = numpy.zeros([nz,nx],'f')
up = numpy.zeros([nz,nx],'f')
ud = numpy.zeros([nz,nx],'f')

Loop over time.
```
for it in range(nt):
```

Compute Laplacian:

\Delta U

.

    ud[2:-2,2:-2] = \
    c0* uo[2:-2,2:-2] * (idx + idz)        + \
    c1*(uo[2:-2,1:-3] + uo[2:-2,3:-1])*idx + \
    c2*(uo[2:-2, :-4] + uo[2:-2,4:  ])*idx + \
    c1*(uo[1:-3,2:-2] + uo[3:-1,2:-2])*idz + \
    c2*(uo[ :-4,2:-2] + uo[4:  ,2:-2])*idz

Inject source wavelet: $\left[\Delta U-f(t)\right]$ $\left[\Delta U-f(t)\right]$
```
    ud = ud - ww[it] * rr
```
Scale by velocity: $\left[\Delta U-f(t)\right]v^{2}$ $\left[\Delta U-f(t)\right]v^{2}$
```
    ud= ud *vv*vv
```
Time step: $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$ $U_{i+1}=\left[\Delta U-f(t)\right]v^{2}\Delta t^{2}+2U_{i}-U_{i-1}$
```
    up = 2*uo - um + ud * dt2
    um =   uo
    uo =   up
```

References[edit]

↑ "Hello world" of seismic imaging.

[1] "Hello world" of seismic imaging.

[1]

Guide to programming with madagascar: Difference between revisions

Latest revision as of 02:29, 6 May 2021

Contents

Introduction[edit]

C program[edit]

C program[edit]

C++ program[edit]

Fortran 90 program[edit]

Python program[edit]

References[edit]

Navigation menu

@@ Line 33: / Line 33: @@
 [[Image:wavec.png|frame|center|Wave propagation snapshot.]]
+==C program==
 <syntaxhighlight lang="c">
@@ Line 95: / Line 97: @@
      /* MAIN LOOP */
-     if(verb) fprintf(stderr,"\n");
+     if(verb) fprintf(stderr,"
+");
      for (it=0; it<nt; it++) {
-	if(verb) fprintf(stderr,"\b\b\b\b\b%d",it);
+	if(verb) fprintf(stderr,"\b\b\b\b\b %d",it);
 	/* 4th order laplacian */
@@ Line 147: / Line 150: @@
 </syntaxhighlight>
-#Declare input, output and auxiliary file tags: <tt>Fw</tt> for input wavelet,  <tt>Fv</tt> for velocity, <tt>Fr</tt> for reflectivity, and <tt>Fo</tt> for output wavefield.
-<syntaxhighlight lang="c">
+#Declare input, output and auxiliary file tags: <tt>Fw</tt> for input wavelet,  <tt>Fv</tt> for velocity, <tt>Fr</tt> for reflectivity, and <tt>Fo</tt> for output wavefield.  <syntaxhighlight lang="c">
      sf_file Fw=NULL,Fv=NULL,Fr=NULL,Fo=NULL; /* I/O files */
 </syntaxhighlight>
-#Declare RSF cube axes: <tt>at</tt> time axis, <tt>ax</tt> space axis, <tt>az</tt> depth axis.
+#Declare RSF cube axes: <tt>at</tt> time axis, <tt>ax</tt> space axis, <tt>az</tt> depth axis. \tiny <syntaxhighlight lang="c">
-<syntaxhighlight lang="c">
      sf_axis at,az,ax;    /* cube axes */
 </syntaxhighlight>
-#Declare multi-dimensional arrays for input, output and computations. <syntaxhighlight lang="c">
+#Declare multi-dimensional arrays for input, output and computations.  <syntaxhighlight lang="c">
      float  *ww,**vv,**rr;     /* I/O arrays*/
 </syntaxhighlight>
-#Open files for input/output.
+#Open files for input/output.  <syntaxhighlight lang="c">
-<syntaxhighlight lang="c">
      Fw = sf_input ("in" );
      Fo = sf_output("out");
      Fv = sf_input ("vel");
      Fr = sf_input ("ref");
 </syntaxhighlight>
-#Read axes from input files; write axes to output file. \tiny <syntaxhighlight lang="c">
+#Read axes from input files; write axes to output file. <syntaxhighlight lang="c">
      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);
@@ Line 173: / Line 175: @@
      sf_oaxa(Fo,ax,2);
      sf_oaxa(Fo,at,3);
 </syntaxhighlight>
-#Allocate arrays and read wavelet, velocity and reflectivity. \tiny <syntaxhighlight lang="c">
+#Allocate arrays and read wavelet, velocity and reflectivity. <syntaxhighlight lang="c">
      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);
 </syntaxhighlight>
-#Allocate temporary arrays.
+#Allocate temporary arrays. <syntaxhighlight lang="c">
-<syntaxhighlight lang="c">
      um=sf_floatalloc2(nz,nx);
      uo=sf_floatalloc2(nz,nx);
@@ Line 186: / Line 187: @@
      ud=sf_floatalloc2(nz,nx);
 </syntaxhighlight>
-#Loop over time.
+#Loop over time. <syntaxhighlight lang="c">
-<syntaxhighlight lang="c">
      for (it=0; it<nt; it++) {
 </syntaxhighlight>
-#Compute Laplacian: <math>\Delta U</math>.
+#Compute Laplacian: <math>\Delta U</math>. <syntaxhighlight lang="c">
-<syntaxhighlight lang="c">
 	for (ix=2; ix<nx-2; ix++) {
 	    for (iz=2; iz<nz-2; iz++) {
@@ Line 217: / Line 216: @@
 	}
 </syntaxhighlight>
-#Time step:  <math>U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}</math>
+#Time step:  <math>U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}</math>  <syntaxhighlight lang="c">
-<syntaxhighlight lang="c">
 	for (ix=0; ix<nx; ix++) {
 	    for (iz=0; iz<nz; iz++) {
@@ Line 230: / Line 228: @@
 	    }
 	}
 </syntaxhighlight>
+\newpage
 ==C++ program==
@@ Line 240: / Line 240: @@
 #include <rsf.hh>
 #include <cub.hh>
-#include <vai.hh>
+#include "vai.hh"
 using namespace std;
@@ Line 256: / Line 258: @@
      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());
+     CUB Fo("out","o"); Fo.setup(3);
      // Read/Write axes
@@ Line 292: / Line 294: @@
 	// 4th order laplacian
-	for (int iz=2; iz<nz-2; iz++) {
+	for (int ix=2; ix<nx-2; ix++) {
-	    for (int ix=2; ix<nx-2; ix++) {
+	    for (int iz=2; iz<nz-2; iz++) {
 		ud[k(iz,ix)] =
 		    c0* uo[ k(iz  ,ix  )] * (idx+idz) +
@@ Line 321: / Line 323: @@
      exit(0);
 }
 </syntaxhighlight>
-*Declare input, output and auxiliary file cubes  (of type <tt>CUB</tt>).
+#Declare input, output and auxiliary file cubes  (of type <tt>CUB</tt>). <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="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());
+     CUB Fo("out","o"); Fo.setup(3);
 </syntaxhighlight>
-*Declare, read and write RSF cube axes: <tt>at</tt> time axis, <tt>ax</tt> space axis, <tt>az</tt> depth axis.
+#Declare, read and write RSF cube axes: <tt>at</tt> time axis, <tt>ax</tt> space axis, <tt>az</tt> depth axis.  <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="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);
+</syntaxhighlight>
-    Fo.putax(0,az);
+#Declare multi-dimensional  <tt>valarrays</tt> for input, output and read data. <syntaxhighlight lang="cpp">
-    Fo.putax(1,ax);
-    Fo.putax(2,at);
-    Fo.headou();
-</syntaxhighlight>
-*Declare multi-dimensional  <tt>valarrays</tt> for input, output and read data.
-<syntaxhighlight lang="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;
 </syntaxhighlight>
-*Declare multi-dimensional  <tt>valarrays</tt> for temporary storage.
+#Declare multi-dimensional  <tt>valarrays</tt> for temporary storage. <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="cpp">
      valarray<float> um(nz*nx); um=0;
      valarray<float> uo(nz*nx); uo=0;
@@ Line 354: / Line 348: @@
      valarray<float> ud(nz*nx); ud=0;
 </syntaxhighlight>
-*Initialize multidimensional <tt>valarray</tt>  index counter (of type <tt>VAI</tt>).
+#Initialize multidimensional <tt>valarray</tt>  index counter (of type <tt>VAI</tt>). <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="cpp">
      VAI k(nz,nx);
 </syntaxhighlight>
-*Loop over time.
+#Loop over time. <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="cpp">
      for (int it=0; it<nt; it++) {
 </syntaxhighlight>
-*Compute Laplacian: <math>\Delta U</math>.
+#Compute Laplacian: <math>\Delta U</math>. <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="cpp">
+	for (int ix=2; ix<nx-2; ix++) {
-	for (int iz=2; iz<nz-2; iz++) {
+	    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) +
@@ Line 374: / Line 365: @@
 	    }
 	}
 </syntaxhighlight>
-*Inject source wavelet: <math>\left[ \Delta U - f(t) \right]</math>
+#Inject source wavelet: <math>\left[ \Delta U - f(t) \right]</math> <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="cpp">
 	ud -= ww[it] * rr;
 </syntaxhighlight>
-*Scale by velocity: <math>\left[ \Delta U - f(t) \right] v^2</math>
+#Scale by velocity: <math>\left[ \Delta U - f(t) \right] v^2</math> <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="cpp">
 	ud *= vv*vv;
 </syntaxhighlight>
-*Time step:  <math>U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}</math>
+#Time step:  <math>U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}</math> <syntaxhighlight lang="cpp">
-<syntaxhighlight lang="cpp">
 	up=(float)2 * uo - um + ud * dt2;
 	um =   uo;
 	uo =   up;
 </syntaxhighlight>
 ==Fortran 90 program==
@@ Line 406: / Line 394: @@
    type(axa)  :: at,az,ax     ! cube axes
    integer    :: it,iz,ix     ! index variables
+  integer    :: nt,nz,nx
+  real       :: dt,dz,dx
    real       :: idx,idz,dt2
@@ Line 421: / Line 411: @@
    ! Read/Write axes
-   call iaxa(Fw,at,1); call iaxa(Fv,az,1); call iaxa(Fv,ax,2)
+   call iaxa(Fw,at,1); nt = at%n; dt = at%d
-   call oaxa(Fo,az,1); call oaxa(Fo,ax,2); call oaxa(Fo,at,3)
+  call iaxa(Fv,az,1); nz = az%n; dz = az%d
+  call iaxa(Fv,ax,2); nx = ax%n; dx = ax%d
+   call oaxa(Fo,az,1)
+  call oaxa(Fo,ax,2)
+  call oaxa(Fo,at,3)
-   dt2 =    at%d*at%d
+   dt2 =    dt*dt
-   idz = 1/(az%d*az%d)
+   idz = 1/(dz*dz)
-   idx = 1/(ax%d*ax%d)
+   idx = 1/(dx*dx)
    ! read wavelet, velocity & reflectivity
-   allocate(ww(at%n     )); ww=0.; call rsf_read(Fw,ww)
+   allocate(ww(nt));    call rsf_read(Fw,ww)
-   allocate(vv(az%n,ax%n)); vv=0.; call rsf_read(Fv,vv)
+   allocate(vv(nz,nx)); call rsf_read(Fv,vv)
-   allocate(rr(az%n,ax%n)); rr=0.; call rsf_read(Fr,rr)
+   allocate(rr(nz,nx)); call rsf_read(Fr,rr)
    ! allocate temporary arrays
-   allocate(um(az%n,ax%n)); um=0.
+   allocate(um(nz,nx)); um=0.
-   allocate(uo(az%n,ax%n)); uo=0.
+   allocate(uo(nz,nx)); uo=0.
-   allocate(up(az%n,ax%n)); up=0.
+   allocate(up(nz,nx)); up=0.
-   allocate(ud(az%n,ax%n)); ud=0.
+   allocate(ud(nz,nx)); ud=0.
    ! MAIN LOOP
-   do it=1,at%n
+   do it=1,nt
       if(verb) write (0,*) it
-      ! 4th order laplacian
+      ud(3:nz-2,3:nx-2) = &
-     do iz=2,az%n-2
+          c0* uo(3:nz-2,3:nx-2) * (idx + idz)            + &
-        do ix=2,ax%n-2
+          c1*(uo(3:nz-2,2:nx-3) + uo(3:nz-2,4:nx-1))*idx + &
-           ud(iz,ix) = &
+          c2*(uo(3:nz-2,1:nx-4) + uo(3:nz-2,5:nx  ))*idx + &
-                c0* uo(iz,  ix  ) * (idx + idz)        + &
+          c1*(uo(2:nz-3,3:nx-2) + uo(4:nz-1,3:nx-2))*idz + &
-                c1*(uo(iz  ,ix-1) + uo(iz  ,ix+1))*idx + &
+          c2*(uo(1:nz-4,3:nx-2) + uo(5:nz  ,3:nx-2))*idz
-                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
@@ Line 473: / Line 463: @@
 end program AFDMf90
 </syntaxhighlight>
-*Declare input, output and auxiliary file tags.
+#Declare input, output and auxiliary file tags. <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
    type(file) :: Fw,Fv,Fr,Fo  ! I/O files
 </syntaxhighlight>
-*Declare RSF cube axes: <tt>at</tt> time axis, <tt>ax</tt> space axis, <tt>az</tt> depth axis.
+#Declare RSF cube axes: <tt>at</tt> time axis, <tt>ax</tt> space axis, <tt>az</tt> depth axis. <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
    type(axa)  :: at,az,ax     ! cube axes
 </syntaxhighlight>
-*Declare multi-dimensional arrays for input, output and computations.
+#Declare multi-dimensional arrays for input, output and computations. <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
    real, allocatable :: vv(:,:),rr(:,:),ww(:)           ! I/O arrays
    real, allocatable :: um(:,:),uo(:,:),up(:,:),ud(:,:) ! tmp arrays
 </syntaxhighlight>
-*Open files for input/output.
+#Open files for input/output. <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
    Fw=rsf_input ("in")
    Fv=rsf_input ("vel")
    Fr=rsf_input ("ref")
    Fo=rsf_output("out")
+</syntaxhighlight>
+#Read axes from input files; write axes to output file. <syntaxhighlight lang="fortran">
+  call iaxa(Fw,at,1); nt = at%n; dt = at%d
+  call iaxa(Fv,az,1); nz = az%n; dz = az%d
+  call iaxa(Fv,ax,2); nx = ax%n; dx = ax%d
+  call oaxa(Fo,az,1)
+  call oaxa(Fo,ax,2)
+  call oaxa(Fo,at,3)
 </syntaxhighlight>
-*Read axes from input files; write axes to output file.
+#Allocate arrays and read wavelet, velocity and reflectivity. <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
+   allocate(ww(nt));    call rsf_read(Fw,ww)
-   call iaxa(Fw,at,1); call iaxa(Fv,az,1); call iaxa(Fv,ax,2)
+  allocate(vv(nz,nx)); call rsf_read(Fv,vv)
-   call oaxa(Fo,az,1); call oaxa(Fo,ax,2); call oaxa(Fo,at,3)
+   allocate(rr(nz,nx)); call rsf_read(Fr,rr)
 </syntaxhighlight>
-*Allocate arrays and read wavelet, velocity and reflectivity.
+#Allocate temporary arrays. <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
+   allocate(um(nz,nx)); um=0.
-   allocate(ww(at%n     )); ww=0.; call rsf_read(Fw,ww)
+  allocate(uo(nz,nx)); uo=0.
-   allocate(vv(az%n,ax%n)); vv=0.; call rsf_read(Fv,vv)
+   allocate(up(nz,nx)); up=0.
-   allocate(rr(az%n,ax%n)); rr=0.; call rsf_read(Fr,rr)
+   allocate(ud(nz,nx)); ud=0.
 </syntaxhighlight>
-*Allocate temporary arrays.
+#Loop over time. <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
+   do it=1,nt
-  allocate(um(az%n,ax%n)); um=0.
+</syntaxhighlight>
-  allocate(uo(az%n,ax%n)); uo=0.
+#Compute Laplacian: <math>\Delta U</math>. <syntaxhighlight lang="fortran">
-  allocate(up(az%n,ax%n)); up=0.
+      ud(3:nz-2,3:nx-2) = &
-  allocate(ud(az%n,ax%n)); ud=0.
+          c0* uo(3:nz-2,3:nx-2) * (idx + idz)            + &
-</syntaxhighlight>
+          c1*(uo(3:nz-2,2:nx-3) + uo(3:nz-2,4:nx-1))*idx + &
-*Loop over time.
+          c2*(uo(3:nz-2,1:nx-4) + uo(3:nz-2,5:nx  ))*idx + &
-<syntaxhighlight lang="fortran">
+          c1*(uo(2:nz-3,3:nx-2) + uo(4:nz-1,3:nx-2))*idz + &
-   do it=1,at%n
+          c2*(uo(1:nz-4,3:nx-2) + uo(5:nz  ,3:nx-2))*idz
 </syntaxhighlight>
-*Compute Laplacian: <math>\Delta U</math>.
+#Inject source wavelet: <math>\left[ \Delta U - f(t) \right]</math> <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="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
-</syntaxhighlight>
-*Inject source wavelet: <math>\left[ \Delta U - f(t) \right]</math>
-<syntaxhighlight lang="fortran">
       ud = ud - ww(it) * rr
 </syntaxhighlight>
-*Scale by velocity: <math>\left[ \Delta U - f(t) \right] v^2</math>
+#Scale by velocity: <math>\left[ \Delta U - f(t) \right] v^2</math> <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
       ud= ud *vv*vv
 </syntaxhighlight>
-*Time step:  <math>U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}</math>
+#Time step:  <math>U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}</math> <syntaxhighlight lang="fortran">
-<syntaxhighlight lang="fortran">
       up = 2*uo - um + ud * dt2
       um =   uo
       uo =   up
+</syntaxhighlight>
+==Python program==
+<syntaxhighlight lang="python">
+#!/usr/bin/env python
+import sys
+import numpy
+import m8r
+c0=-30./12.
+c1=+16./12.
+c2=- 1./12.
+par = m8r.Par()
+verb = par.bool("verb",False) # verbosity
+# setup I/O files
+Fw=m8r.Input()
+Fv=m8r.Input ("vel")
+Fr=m8r.Input ("ref")
+Fo=m8r.Output()
+# Read/Write axes
+at = Fw.axis(1); nt = at['n']; dt = at['d']
+az = Fv.axis(1); nz = az['n']; dz = az['d']
+ax = Fv.axis(2); nx = ax['n']; dx = ax['d']
+Fo.putaxis(az,1)
+Fo.putaxis(ax,2)
+Fo.putaxis(at,3)
+dt2 =    dt*dt
+idz = 1/(dz*dz)
+idx = 1/(dx*dx)
+# read wavelet, velocity & reflectivity
+ww = numpy.zeros(nt,'f');      Fw.read(ww)
+vv = numpy.zeros([nz,nx],'f'); Fv.read(vv)
+rr = numpy.zeros([nz,nx],'f'); Fr.read(rr)
+# allocate temporary arrays
+um = numpy.zeros([nz,nx],'f')
+uo = numpy.zeros([nz,nx],'f')
+up = numpy.zeros([nz,nx],'f')
+ud = numpy.zeros([nz,nx],'f')
+# MAIN LOOP
+for it in range(nt):
+    if verb:
+        sys.stderr.write("\b\b\b\b\b %d" % it)
+    ud[2:-2,2:-2] = \
+    c0* uo[2:-2,2:-2] * (idx + idz)        + \
+    c1*(uo[2:-2,1:-3] + uo[2:-2,3:-1])*idx + \
+    c2*(uo[2:-2, :-4] + uo[2:-2,4:  ])*idx + \
+    c1*(uo[1:-3,2:-2] + uo[3:-1,2:-2])*idz + \
+    c2*(uo[ :-4,2:-2] + uo[4:  ,2:-2])*idz
+    # 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
+if verb:
+    sys.stderr.write("\n")
+    Fo.write(uo)
+sys.exit(0)
 </syntaxhighlight>
+[[Image:wavepython.png|frame|center|Wave propagation snapshot.]]
+#Open files for input/output. <syntaxhighlight lang="python">
+Fw=m8r.Input()
+Fv=m8r.Input ("vel")
+Fr=m8r.Input ("ref")
+Fo=m8r.Output()
+</syntaxhighlight>
+#Read axes from input files; write axes to output file. <syntaxhighlight lang="python">
+at = Fw.axis(1); nt = at['n']; dt = at['d']
+az = Fv.axis(1); nz = az['n']; dz = az['d']
+ax = Fv.axis(2); nx = ax['n']; dx = ax['d']
+Fo.putaxis(az,1)
+Fo.putaxis(ax,2)
+Fo.putaxis(at,3)
+</syntaxhighlight>
+#Allocate arrays and read wavelet, velocity and reflectivity. <syntaxhighlight lang="python">
+ww = numpy.zeros(nt,'f');      Fw.read(ww)
+vv = numpy.zeros([nz,nx],'f'); Fv.read(vv)
+rr = numpy.zeros([nz,nx],'f'); Fr.read(rr)
+</syntaxhighlight>
+#Allocate temporary arrays. <syntaxhighlight lang="python">
+um = numpy.zeros([nz,nx],'f')
+uo = numpy.zeros([nz,nx],'f')
+up = numpy.zeros([nz,nx],'f')
+ud = numpy.zeros([nz,nx],'f')
+</syntaxhighlight>
+#Loop over time. <syntaxhighlight lang="python">
+for it in range(nt):
+</syntaxhighlight>
+#Compute Laplacian: <math>\Delta U</math>. <syntaxhighlight lang="python">
+    ud[2:-2,2:-2] = \
+    c0* uo[2:-2,2:-2] * (idx + idz)        + \
+    c1*(uo[2:-2,1:-3] + uo[2:-2,3:-1])*idx + \
+    c2*(uo[2:-2, :-4] + uo[2:-2,4:  ])*idx + \
+    c1*(uo[1:-3,2:-2] + uo[3:-1,2:-2])*idz + \
+    c2*(uo[ :-4,2:-2] + uo[4:  ,2:-2])*idz
+</syntaxhighlight>
+#Inject source wavelet: <math>\left[ \Delta U - f(t) \right]</math> <syntaxhighlight lang="python">
+    ud = ud - ww[it] * rr
+</syntaxhighlight>
+#Scale by velocity: <math>\left[ \Delta U - f(t) \right] v^2</math> <syntaxhighlight lang="python">
+    ud= ud *vv*vv
+</syntaxhighlight>
+#Time step:  <math>U_{i+1} = \left[ \Delta U -f(t) \right] v^2 \Delta t^2 + 2 U_{i} - U_{i-1}</math>  <syntaxhighlight lang="python">
+    up = 2*uo - um + ud * dt2
+    um =   uo
+    uo =   up
+</syntaxhighlight>
 ==References==
 <references/>

Guide to programming with madagascar: Difference between revisions

Latest revision as of 02:29, 6 May 2021

Introduction[edit]

C program[edit]

C program[edit]

C++ program[edit]

Fortran 90 program[edit]

Python program[edit]

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