Reciprocal Delaytime Refraction

The difference between conventional refraction shooting 8.4.6.3 and reciprocal refraction shooting is that the former employs shot gathers and in this case, geophone gathers. The analysis is essentially the same. Reciprocal shooting is applied when crossing a river. Placing geophone in the river would subject the geophones to a noisy environment, particularly with a strong current bouncing the phones around. When an existing bridge needs replacement, one can deploy an airgun source from the bridge at stations of about 5 meters and record into geophone arrays on the river banks. An example of this approach is shown in Figure 42.

Figure 42: OCTAVE DELAYTMR: Reciprocal shooting across a river. Airgun source deployed at stations across bridge (Michaels, 2001a).

The geophone arrays can be summed to cancel traffic noise, beam forming to receive signals from the air guns. Recorded data are sorted into 3 northern geophone gathers, and 3 southern geophone gathers using the BEXT program (N000.seg, N001.seg, N002.seg, S001.seg, S002.seg, S003.seg). BREF is run with the following command:

bref 001 6 10. 110. 0 1 N000.seg N001.seg N002.seg S000.seg S001.seg S002.seg

The BREF program produces output files G001, D001, and E001 as in the normal shooting example (8.4.6.3 ). Some editing is required. The BREF code detected one constraint equation on this run. Further, a need for 2 more constraints was found so that all delaytimes were made equal at stations 90, 95, 100. These constraints were strongly weighted (factor of 50), and the G001 matrix was edited as shown below (relevant tail of the matrix):

  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0      45.363 
  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0      40.364 
  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0      35.382 
  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0      30.329 
  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0      20.336 
  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0      15.075 
  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0      10.564 
  0  0  0 50  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 -50 0  0       0.000  
  0  0  0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 -50 50  0       0.000 
  0  0  0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 -50 0  50       0.000

The first 6 columns are for the 6 geophone gathers. The D001 matrix tail is shown here:

    0.03300     55
    0.03120     60
    0.02840     65
    0.02660     70
    0.02390     80
    0.01880     85
    0.01720     90
    0.00000      0 
    0.00000      0 
    0.00000      0

One extra file is required when shooting this way. One needs a water depth file which gives the depth to the river bottom below the surface shots. This is a two column ASCII text file (depth, shot#). One starts an octave session and then runs delaytmR.m:

 delaytmR
 GUI enter names of the files G001, D001, E001 and wds.data
 GUI number of geophones on shore, 6
 GUI smoothness weight, 0.1 a good value
 GUI displays refractor velocity (2216 m/s) and geophone delay
 times below geophones (7.8, 9.9, 11.0, 2.5, 8.7, 8.6 ms), OK
 PLOT shows delay times under the trans-river sources
 GUI enter an overburden velocity in m/s, for example 1500 m/s
 PLOT showing structure if overburden velocity is as assumed
 (ground or water surface, bottom of river, refractor structure)
 GUI enter alternative limiting case of constant depth refractor
 PLOT of alternative, constant depth refractor, V1 varies
 GUI enter number of constraint equations, 3 OK
 PLOT of the observed data and fit to the equations.

The resulting structural plot is shown in Figure 43 and the arrival time fit in Figure 44. Delay times, $T_{dt}$, are related to the distance from the surface to the refractor, $H$ by the critical angle, $\theta_c$:

$$T_{dt} = \dfrac{H cos(\theta_c)}{V_1}$$ (15)

In normal shooting, one solves for $H$ using the critical angle $\theta_c= sin^{-1}(V_1/V_2)$, the refractor velocity, $V_2$, being the last unknown in the G matrix setup, and a result of obtaining a solution. This is straight forward for the delay times on the river banks, under the geophones. Under the shots, the water layer creates some additional complexity to the problem (see the code, delaytmR.m). In short, if the refractor velocity, $V_2$, is greater than water velocity, then it is simple ray optics. Velocity $V_1$ is the soil layer between the bottom of the river and the top of the refractor. To keep it simple in this example, I made $V_1=1500$, water velocity, and that would mean no ray bending from water to overburden soil. Recall that the ray parameter, p, is a constant, $p=sin(\theta_j)/V_j$, and to get a critical refraction at bedrock, we need to know $V_2$ relative to water velocity.

Figure 43: OCTAVE DELAYTMR: Structure assuming an overburden velocity of 1500 m/s. River water surface and bottom of river bottom in blue. Refractor structure in red.

Figure 44: OCTAVE DELAYTMR: Observed arrival times and fit assuming an overburden velocity of 1500 m/s.