puckin
Figure 2 Input synthetic data. |
residual
Figure 3 Residuals, i.e. an evaluation of U_{x} - p U_{t}. |
The residual shows several features: It is small in the central region of the data; it is large where there is spatial aliasing; and it is rough because it is made from derivatives and the synthetic data was made with nearest neighbor interpolation.
puckout
Figure 4 Output values of p are shown by the slope of short line segments. |
The residuals are small. If a weaker dip were present it would be revealed in the residuals. In Figure 3 there is no weaker dip, only artifacts of inadequate sampling of the signals and the transient onset of the signal. Later in chapter 8 several strong dips will be estimated simultaneously.
# slide a window around on a wall of data measuring coherency, dip, residual # subroutine slider( n1,n2, w1,w2, k1,k2, data, coh, pp, res) integer n1,n2, w1,w2, k1,k2, s1,s2, e1,e2, i1,i2 integer p1,p2 # math size of pp(p1,p2), coh(p1,p2) real data(n1,n2) # input real res(n1,n2) # outputs. math size (n1-1,n2-1) real pp(n1,n2), coh(n1,n2) # outputs. temporary real window(n1,n2), tres(n1,n2), count(n1,n2) call zero( n1*n2, res) call zero( n1*n2, count) p2=1; s2=1; e2=s2+w2-1; while( e2<=n2) { p1=1; s1=1; e1=s1+w1-1; while( e1<=n1) { do i1=1,w1 do i2=1,w2 window(i1,i2) = data(s1+i1-1, s2+i2-1) call puck ( n1, w1, w2, window, coh(p1,p2), pp(p1,p2), tres) do i1=1,w1-1 { do i2=1,w2-1 { res( s1+i1-1, s2+i2-1) = res(s1+i1-1, s2+i2-1) + tres(i1,i2) count(s1+i1-1, s2+i2-1) = count(s1+i1-1, s2+i2-1) + 1. }} p1=p1+1; s1=s1+k1; e1=s1+w1-1 } p1=p1-1 p2=p2+1; s2=s2+k2; e2=s2+w2-1 } p2=p2-1 do i2=1,n2-1 do i1=1,n1-1 if( count(i1,i2) > 0. ) res(i1,i2) = res(i1,i2) / count(i1,i2) return; end
A disadvantage of the method is that the finite difference operator is susceptible to spatial aliasing as well as distortions at spatial frequencies that are high but not yet aliased. This suggests the next step--estimating missing interlaced traces--that is taken up in chapter 8.