Estimating and Handling Random Data Errors

In this paper, we assume zero-offset VSP (ZVSP) data. We derive a simple measure of ZVSP data uncertainty below. The uncertainty in surface seismic data depends on velocity and raypath effects in a more complex manner, although Clapp (2001) has made encouraging progress in bounding the uncertainty.

Somewhat counter to intuition, we adopt the convention that traveltime is the independent variable in a time/depth pair, i.e., z = f(t). Bad first break picks and ray bending introduce errors into the traveltimes of ZVSP data, but depth in the borehole to the receiver is well known. To obtain an equivalent depth error, we need only scale the traveltime error in ZVSP data by the average overburden velocity. By definition, the traveltime t (along a straight ray) is related to depth z in the following way:

 

z = t v_avg, (8)

where v_avg is the average overburden velocity. If the traveltime is perturbed with error $\Delta t$ it follows that the corresponding depth error, $\Delta z$ is simply the traveltime error scaled by v_avg:  

$$\Delta z = \Delta t v_{avg}.$$ (9)

If the data errors are independent and follow a Gaussian distribution, least squares theory prescribes Strang (1986) that the data residual of equation (5) be weighted by the inverse variance of the data.

Assuming that we have translated a priori data uncertainty into an estimate of data variance, we can define a diagonal matrix $\bf W$ where the diagonal elements $w_{ii} = \sigma_i^{-1}$: the inverse of the variance of the i^th datum. This diagonal operator is applied to the data residual of equation (7):  

$$\left[\begin{array} {c} \bf WA \\ \epsilon \bold C \end{a... ...y} {c} \bf W \boldsymbol \zeta \\ \bold 0 \end{array}\right]$$ (10)