Make data



In [41]:

    
%%file data.scons

Flow('trace',None,'spike n1=2001 d1=0.001 k1=1001 | ricker1 frequency=30')
Flow('gather','trace','spray axis=2 n=49 d=25 o=0 label=Offset unit=m | nmostretch inv=y half=n v0=2000')

Result('gather','window f1=888 n1=392 | grey title=Gather')









    



Overwriting data.scons



In [1]:

    
from m8r import view
view('gather')









    Out[1]:

Display the same with Python (matplotlib)



In [43]:

    
import m8r
gather = m8r.File('gather.rsf')



In [44]:

    
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
plt.imshow(np.transpose(gather[:,888:1280]),aspect='auto')









    Out[44]:





<matplotlib.image.AxesImage at 0x7f3da2350950>

Apply moveout with incorrect velocity



In [47]:

    
%%file nmo.scons

Flow('nmo','gather','nmostretch half=n v0=1800')

Result('nmo','window f1=888 n1=200 | grey title=NMO')









    



Overwriting nmo.scons



In [48]:

    
view('nmo')









    Out[48]:

Slope estimation



In [51]:

    
%%file slope.scons

Flow('slope','nmo','dip rect1=100 rect2=5 order=2')

Result('slope','grey color=linearlfb mean=y scalebar=y title=Slope')









    



Overwriting slope.scons



In [52]:

    
view('slope')









    Out[52]:

Non-physical flattening by predictive painting



In [59]:

    
%%file flat.scons

Flow('paint','slope','pwpaint order=2')

Result('paint','window f1=888 n1=200 | contour title=Painting')

Flow('flat','nmo paint','iwarp warp=${SOURCES[1]}')

Result('flat','window f1=888 n1=200 | grey title=Flattening')









    



Overwriting flat.scons



In [58]:

    
view('paint')









    Out[58]:



In [62]:

    
view('flat')









    Out[62]:

Velocity estimation by time warping

Predictive painting produces $t_0(t,x)$. Time warping converts it into $t(t_0,x)$.



In [19]:

    
%%file twarp.scons

Flow('twarp','paint','math output=x1 | iwarp warp=$SOURCE')

Result('twarp','window j1=20 | transp | graph yreverse=y min2=0.888 max2=1.088 pad=n title="Time Warping" ')









    



Overwriting twarp.scons



In [20]:

    
view('twarp')









    Out[20]:

We now want to fit

$t^2(t_0,x)-t_0^2 \approx \Delta S\,x^2$,

where $\Delta S = \frac{1}{v^2} - \frac{1}{v_0^2}$.

The least-squares fit is

$\Delta S = \displaystyle \frac{\int x^2\left[t^2(t_0,x)-t_0^2\right]\,dx}{\int x^4\,dx}$.

The velocity estimate is

$v = \displaystyle \frac{v_0}{\sqrt{\Delta S\,v_0^2 + 1}}$.



In [37]:

    
%%file lsfit.scons

Flow('num','twarp','math output="(input*input-x1*x1)*x2^2" | stack norm=n')
Flow('den','twarp','math output="x2^4" | stack norm=n')
Flow('vel','num den','div ${SOURCES[1]} | math output="1800/sqrt(1800*1800*input+1)" ')

Result('vel',
       '''
       window f1=888 n1=200 | 
       graph yreverse=y transp=y title="Estimated Velocity" label2=Velocity unit2=m/s grid2=y pad=n min2=1950 max2=2050
       ''')









    



Overwriting lsfit.scons



In [38]:

    
view('vel')









    Out[38]:

Last step - physical flattening



In [39]:

    
%%file nmo2.scons

Flow('nmo2','gather vel','nmo half=n velocity=${SOURCES[1]}')

Result('nmo2','window f1=888 n1=200 | grey title="Physical Flattening" ')









    



Writing nmo2.scons



In [40]:

    
view('nmo2')









    Out[40]:



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