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, . 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.
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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.
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