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Finite Difference Modeling

Finite difference (FD) shot and data modeling can be performed on the Sigsbee models using Madagascar. This example will use the Sigsbee2A model but but it could be easily extended to perform modeling on Sigsbee2B.

For the purposes of this example a shot will be fired at 10 km along the horizontal coordinate and at a depth of 10 meters. Receivers are spread at a depth of 0 meters every 7.62 m (25 ft) along the entire scope of the model. This long receiver cable is impractical but useful for these purposes. Data is recorded on every receiver at time increments of 1 ms 3000 times resulting in 3 seconds of data.

An SConstruct file located within sigsbee/fdmod2A/ properly formats the model and inputs necessary parameters to perform a shot on the Sigsbee model. This file is reproduced below in Table 9.

from rsf.proj import *
import fdmod
 # Sigsbee 2A
         segyread tape=$SOURCE tfile=${TARGETS[1]} |
         o1=0  d1=25     label1=z unit1=ft
         o2=10000 d2=25  label2=x unit2=ft
# ------------------------------
par = {
    'nt':7000, 'dt':0.001,'ot':0,    'lt':'t','ut':'s',
    'kt':100,    # wavelet delay
    'nx':3201, 'ox':0, 'dx':0.00762, 'lx':'x','ux':'km',
    'nz':1201, 'oz':0, 'dz':0.00762, 'lz':'z','uz':'km',
# add F-D modeling parameters
# ------------------------------
# wavelet
         spike nsp=1 mag=1 n1=%(nt)d d1=%(dt)g o1=%(ot)g k1=%(kt)d |
         ricker1 frequency=15 |
         scale axis=123 |
         put label1=t label2=x label3=y |
         ''' % par)    
       'transp | window n1=200 | graph title="" label1="t" label2= unit2=')
# ------------------------------
# experiment setup
Flow('r_',None,'math n1=%(nx)d d1=%(dx)g o1=%(ox)g output=0' % par)
Flow('s_',None,'math n1=1      d1=0      o1=0      output=0' % par)
# receiver positions
Flow('zr','r_','math output="0" ')
Flow('xr','r_','math output="x1"')
     cat axis=2 space=n
     ${SOURCES[0]} ${SOURCES[1]} | transp
     ''', stdin=0)
# source positions
Flow('zs','s_','math output=.01')
Flow('xs','s_','math output=10.0')
Flow('rs','s_','math output=1')
     cat axis=2 space=n
     ${SOURCES[0]} ${SOURCES[1]} ${SOURCES[2]} | transp
     ''', stdin=0)

# ------------------------------

# Velocity
     scale rscale=.0003048 |
     put o1=%(oz)g d1=%(dz)g  o2=%(oz)g d2=%(dz)g
     ''' % par)

allpos=y bias=1.5 pclip=100 color=j title=Survey Design labelsz=4 titlesz=6 wheretitle=t

# ------------------------------

# density
Flow('den','vel','math output=1')

# ------------------------------
# finite-differences modeling
fdmod.awefd('dat','wfl','wav','vel','den','ss','rr','free=y dens=y',par)

Plot('wfl',fdmod.wgrey('pclip=99 title=Wavefield Movie labelsz=4 titlesz=6 wheretitle=t',par),view=1)

for item in ['9','19','29','39']:
           window f3=%s n3=1 min1=0 min2=0 | grey gainpanel=a
           pclip=99 wantframenum=y title=Wavefield at %s s labelsz=4
           titlesz=6 screenratio=.375 screenht=2 wheretitle=t
           label1=z label2=x unit1=kft unit2=kft
           ''' % (item,times[cntr]))
    cntr = cntr + 1
Result('dat','window j2=4 j1=2 | transp |' + fdmod.dgrey('pclip=99',par))

Table 9. Scons script that performs a finite difference synthetic shot on Sigsbee2A.

Typing Command 4 within the sigsbee/fdmod2A/ directory runs the FD modeling script.

\texttt{bash-3.1\$\ scons\ view} \end{displaymath} (4)

This script first constructs the survey acquisition geometry as was previously mentioned. An image of the survey is created and presented in Figure 9.

Figure 9.
FD model geometry as performed on the Sigsbee 2A velocity model. The X represents the shot while the smaller * symbols represent receivers. The receivers extent along the right hand side although it is not clear in this image.
[pdf] [png] [scons]

Firing the shot results the propagation of a wavefield which can be seen in the movie wfl.vpl that is generated. Typing Command 5 within the sigsbee/fdmod2A directory displays the wavefield movie.

\texttt{bash-3.1\$\ scons\ wfl.vpl} \end{displaymath} (5)

Four frames from this movie are presented in Figure 10 illustrating the propagation of the wavefield in the model.

time9 time19 time29 time39
Figure 10.
Images of the propagating wavefield in the Sigsbee model generated by a finite difference model.
[pdf] [pdf] [pdf] [pdf] [png] [png] [png] [png] [scons]

The resulting data is then presented in the file dat.vpl. This plot is reproduced here in Figure 11.

Figure 11.
Data gathered by the receivers in the FD model survey.
[pdf] [png] [scons]

FD models can be performed on the Sigsbee2B model in a similar fashion. The primary change would be in appending line six, the model input file, in the SConstruct file shown in Table 9.

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