Direct Wave Method

Before we analyze refractions, we can examine the direct wave to determine the overburden velocity. The program, direct.m packaged with BSU uses the least squares method to simultaneously solve for an overburden velocity using several shot records and arrival time picks on near offsets (less than 30 meters in our data). Figure 30 shows the general setup in a simplified case. Only data from near the shots are used. The basic travel time equation for the direct wave between shot A and geophone 1 is

**Figure 30:** Direct wave raypaths used by program direct.m
$\includegraphics[scale=.7]{FigureQ}$

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

(20)

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

$\displaystyle G \centerdot m = d$

(21)

which expands to

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

(22)

The ordinary least squares (OLS) solution is given by (Menke (7)

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

(23)

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