Computing error bars for velocity.

The computation of error bars is based on the scatter of the points plotted in the bvasqc.ps file. Consider an alternative X-Y plot of points about a linear trend, where the “x” values are the depths, $z_{i}$ , of each geophone station, and the “y” values are pseudo arrival times computed from the bvas solution velocity and relative time shifts, $t_{s}$ , about the mean shift, $\overline{t}$ ,

$$y_{i}=\frac{z_{i}}{V}+\left(t_{s}-\overline{t}\right).$$ (8)

The slope of a least squares linear fit to these pseudo times, $y_{i}$ , would correspond to the slowness, 1/V. The problem is to then estimate the variance in the reciprocal of the slope of linear solution. That is, if $y=mx+b$ , then we solve for the variance, $\sigma_{1/m}^{2}$ , assuming a least squares solution to the problem. For N pairs of (x,y), we can write the velocity as the reciprocal of the least squares solution for the slope of the line as

$$V=\frac{1}{m}=\frac{\left[\left(\sum x_{i}\right)^{2}-N\sum\left(... ...(\sum x_{i}\right)\left(\sum y_{i}\right)-N\sum\left(x_{i}y_{i}\right)\right]}.$$ (9)

The variance of the velocity, $\sigma_{V}^{2}=\sigma_{\frac{1}{m}}^{2}$ , is given by

$$\sigma_{V}^{2}=\sigma_{\frac{1}{m}}^{2}=\sum\left(\frac{\partial V}{\partial y_{i}}\right)^{2}\sigma_{y_{i}}^{2},$$ (10)

where $\sigma_{y}^{2}$ is assumed a constant for all $y_{i}$ and is estimated by the scatter around the mean $t_{s}$ . In other words, $\sigma_{y}^{2}$ is given by

$$\sigma_{y}^{2}=\sigma_{t_{s}}^{2}=\frac{\sum\left(t_{s}-\overline{t}\right)^{2}}{N-1}.$$ (11)

After some algebra, we find that,

$$\sigma_{V}^{2}=\sigma_{\frac{1}{m}}^{2}=\sigma_{t_{s}}^{2}\cdot\f... ...m x_{i}\right)\left(\sum y_{i}\right)-N\sum\left(x_{i}y_{i}\right)\right]^{4}}.$$ (12)

This permits us to treat the semblance determined velocity, V, as though it were the result of a least squares fit to picked arrival times, and thus obtain an estimate of the uncertainty in the phase velocity determination. The velocity error bars are computed as the square root of $\sigma_{V}^{2}$ (units of m/s). These error bars may be scaled by 1.96 to obtain an estimate of the 95% confidence interval (assuming normally distributed errors). The unscaled values are output to the file bvas.his, and then later used in the Octave joint inversion, cainv3.m, to obtain confidence limits on both stiffness, $C_{1}$ and damping, $C_{2}$ .