Theory

The basic travel time equation for the direct wave between shot A and geophone 1 is

$\displaystyle X_{a1} \centerdot \dfrac{1}{V_1}=t_{a1}$

(10)

where $X_{a1}$ is the distance between the shot A and geophone 1. The overburden velocity is given by $V_1$

and the observed first arrival time is $t_{a1}$ .

We set up a matrix problem in the form

$\displaystyle G \centerdot m = d$

(11)

which expands to

$\begin{displaymath}\left[ \begin{array}{c} X_{a1}\\ X_{a2}\\ X_{b8}\\ X_{b9} ... ...ray}{c} t_{a1}\\ t_{a2}\\ t_{b8}\\ t_{b9} \end{array}\right]\end{displaymath}$

(12)

The ordinary least squares (OLS) solution is given by Menke (1989)

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

(13)

It follows that the overburden velocity determination is $V_1=\frac{1}{m}$ .

**Figure 39:** BREF: Output plot.ps for direct wave analysis. Title shows the least squares solution for the overburden velocity, $923 \pm 35 m/s$ . Range of offsets 0 -> 30 m.
$\includegraphics[scale=0.5]{FigureDirect.pdf}$