Normal Delay Time Refraction

In normal shooting, we work with shot gathers rather than geophone gathers in the reciprocal approach 8.4.6.4. Shot profiles k008.seg and k009.seg are used in this example. Offsets from 50 to 250 meters are used. The BREF command is

bref 0809 2 50. 250. 0 0 k008.seg k009.seg

The output files are:

An observant reader will ask, why not include the split spread, k004.seg? That would be better, and one would normally do that. However, by taking only two reverse profiles, I can show you how to add constraint equations when needed. With only the two lines, the system matrix, G0809, will be singular. The problem is a lack of reverse profiles in the near offset ranges. Receivers in the offsets 50 meters and beyond all receive signals from both sources at the end of the lines, so they are OK. To get a solution, we need to add a couple of extra lines at the bottom of the G0809 matrix.


Constraint Make the shot and nearest geophone for that shot have the same delay time.


The first two columns of the G matrix correspond to the shots, k008 and k009 (columns 1 and 2 respectively). The first shot, k008, has a near geophone in column 13. We create a new row at the bottom of the matrix by placing a 1 in column 1 (for the shot) and a -1 in column 13 (for the nearest geophone with a refraction). In the last column, we put a zero instead of a distance since this is a constraint equation, $T_{shot} - T_{geophone} = 0 $. To get the zero, we add a row to the bottom of the data vector, D0809. We put two zeros in this last row (one for the time column, one for the station number). Again, this is a constraint equation, not a data equation.

We do this again, adding one more row to the G matrix. This time, a 1 in column 2 (for shot k009). The negative one (-1) goes in column 39 corresponding to the nearest geophone with a refraction for shot k009. We then edit D0809 data vector with a pair of zeros as above. Note: We leave E0809 alone, no need to change it.

The rest of the matrix above the constraint equations are simply delay time equations.

$\displaystyle T_{shot} + T_{geophone} + \dfrac{X_{sg}}{V_2} = T{obs} ~~,$ (14)

where $T_{shot}$ is the shot delay time, $T_{geophone}$ is the geophone delay time, $X_{sg}$ is the distance on the surface of the ground between shot and geophone, and $T_{obs}$ is the observed arrival time from the first break pick. The refractor velocity is given by $V_2$. Details on this approach are given in Michaels (1995). The solution is found by a weighted least squares (weighting minimizes the roughness of the solution, ie. makes it smoother at the slight cost of a poorer fit).

 RUNNING delaytm.m 
 Start octave, type
	delaytm	
GUI, change G001 to G0809, etc
GUI, number of shots = 2
GUI, smoothness weight 0.1 
GUI, shows refractor velocity =4122 m/s and shot delay times of 10.3 and 12.4 msec, OK
Plot showing delay times for geophones
GUI, overburden velocity 923
Plot shows ground elevation and refractor indicating a variable soil thickness.
Alternative solution, default to 10 meters, Plot shows how an alternative extreme
solution of constant soil thickness with overburden
velocity varying.
GUI, 2 constraint equations, OK
Plot shows fit of solution to observed times.
Preference is for variable soil thickness solution based on geologic context.

Figure 40: OCTAVE DELAYTM: Structure solution for shots k008 and k009. Ground surface in blue, top of bedrock in red. Soil velocity 923 m/s between blue and red. Bedrock velocity 4121 m/s.
\includegraphics[scale=0.47]{DTstruct.pdf}

Figure 41: OCTAVE DELAYTM: Computed solution and observed times for k008 and k009.
\includegraphics[scale=0.47]{DTtimes.pdf}