Editing the Gxxxx and Dxxxx files

Suggested constraint equations are weighted by a factor of 9. If you inserted my picks into the *.seg data trace headers, then you will note that bref has returned a Gxxxx matrix with 4 rows at the bottom that need editing. One strategy is to set the delay time for an unpicked station exactly equal to the nearest neighbor which does have a pick. One does this by replacing a “0” with a “-9” at the station which has the constraining value. The idea is to form an equation which sums to zero. Thus, to constrain the delay time for geophone station 8 to equal that at station 7, we edit the file, placing a -9 in column 7 of the row with a bref provided 9 in column 8. The corresponding entry in the same row of Dxxxx will be zero. Thus, the product Gm=d produces a constraint equation of

$$-9T_{g_{7}}+9T_{g_{8}}=0.$$ (35)

where $T_{g_{7}}$ is the delay time at geophone 7, and $T_{g_{8}}$ is the delay time at geophone 8. In the least squares solution, this is given a weight of 9, which makes the constraint fairly strong.

In our example problem, one needs additional constraints, beyond what is required for unpickable data. First, to tie the line 1 solution, we need to constrain the shot delay times to match the values obtained from the line 1 solution. In this case, the decision is to give these constraints a weight of 10 (the math is simple, move the decimal over one place). Thus, one adds two rows in the G0003 file, one has a 10 in column 1, and the other has a 10 in column 2. Corresponding rows need to be added to the D0003 file. In these rows (first column) we place the delay times from the line 1 solution (scaled by a factor of 10). Thus, for the row with a 10 in column 1 of the G0003 file, we enter a value of 0.0890 in the D0003 file, which is 10 times the delay time at station 24 of line 1, the same location where the hammer was placed for shot 10, file k010.seg. For shot 11 (column 2 has a 10 in G0003), we place a value of 0.1500 which is 10 times the delay time at station 1 of line 1. Thus, the product Gm=d produces a constraint equations

$$\begin{array}{c} 10T_{s_{1}}=.0890\\ 10T_{s_{2}}=0.150\end{array},$$ (36)

where $T_{s_{1}}$ is the delay time for shot 10 (first column), and $T_{s_{2}}$ is the delay time for shot 11 (second column).

One last constraint equation must be added. This is to constrain the refractor velocity. We can not solve for this since both source positions are basically measuring a single apparent velocity on the refractor. We draw on the line 1 solution again (it had reverse profiles and was able to resolve the refractor velocity). The refractor velocity was found to be 4187 m/s for the granite. One way to do this is to add one more row with the only non-zero value being the refractor velocity in the last column of G0003. Normally, the source-receiver distance goes in this column, but here, we make it a constraint by adding a last row to D0003, and inserting a “data” value of 1.000 in the first column. Thus, the product GM=d produces a constraint equation

$$\frac{4187}{V_{2}}=1.000$$ (37)

from this last row. The files G0003.mod, D0003.mod and E0003.mod included with the data sample are modified as described above.