Adding Constraint Equations

In practice, one may not have a geophone at shot B, or there may be geophones without observed first arrival times. In those cases, one must add constraint equations to the bottom of the matrix, G. For example, if we were unable to pick a time for the signal from Shot A to geophone 2, we might add a constraint that sets delay time at geophone 1 equal to that at geophone 2 (ie. $T_1 \simeq T_2$). That system would look like this:

$$\left[ \begin{array}{ccccc} 1 & 0 & 1 & 0 & X_{a1}\\ 1 & 1 &... ... t_{a1}\\ t_{ab}\\ t_{b1}\\ t_{b2}\\ 0.0 \end{array}\right]$$ (27)

In this example, we set $T_1$ equal to $T_2$ with a weight of 9 fold in the least squares solution. This is the last row of matrix $G$ in equation 27. It is like the following equation 28 being repeated 9 times in the matrix equation 27:

$$T_1 -T_2 = 0$$ (28)

The effect is to strongly weight these two delay times equal. The OLS solution to Equation 27 is again given by

$$m=\left[G^{T}G\right]^{-1}G^{T}\centerdot d$$ (29)

where, in the case of Equation 27, the vector, $m$, contains the delay times for the shots, receivers, and the refractor slowness.

$$m= \left[ \begin{array}{c} T_{a}\\ T_{b}\\ T_{1}\\ T_{2}\\ \frac{1}{V_{2}}\end{array}\right]$$ (30)

If we want to constrain the refractor velocity there is a way to do that too. Perhaps we lack reverse profiles or a second shot with at least a different angle between the shots and geophones which would yield more than one apparent refractor velocity. Modifying the above situation to include only shot A with a constrained refractor velocity, and a constrained delay time, $T_{a}$, we could set up a system like this:

$$\left[ \begin{array}{cccc} 1 & 1 & 0 & X_{a1}\\ 1 & 0 & 1 & ... ...n{array}{c} t_{a1}\\ t_{ab}\\ 1.0 \\ .005 \end{array}\right]$$ (31)

Here, we set the velocity of the refractor at $4000$ m/s and constrain the shot A delay time to $.005$ seconds. Both of these values would need to come from some other source, like an intersecting survey that was absent the defects here.