Theory
The basic travel time equation for the direct wave between shot A and geophone 1 is
 |
(10) |
where
is the distance between the shot A and geophone 1. The overburden velocity
is given by
and the observed first arrival time is
.
We set up a matrix problem in the form
 |
(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}](img96.svg) |
(12) |
The ordinary least squares (OLS) solution is given by Menke (1989)
![$\displaystyle m=\left[G^{T}G\right]^{-1}G^{T}\centerdot d$](img97.svg) |
(13) |
It follows that the overburden velocity determination is
.
Figure 39:
BREF: Output plot.ps for direct wave analysis. Title shows the least squares solution for the overburden velocity,
. Range of offsets 0 -> 30 m.
|