Appendix B. Error Analysis

Assume a crosswell experiment with one source and one receiver, and an interwell model parameterization of one cell as shown below. For a single ray that passes through the single slowness cell of width l, the direct wave traveltime error dt is related to the model error ds by

ds = dt/l. (2.17)

In this case the error diminishes as l becomes large, i.e., the longer we sample the cell the smaller our slowness error for a given traveltime error. Repeating the same experiment N times will lead to different trav eltime picking errors with the same raypath the same raypath but different slowness errors. Assuming a zero-mean traveltime error will allow for an estimate of the slowness variance as:

[ ds² ] = [ dt² ]/ [ N l² ] , (2.18)

where the brackets indicate an averaging over the different outcomes (or random variables) of the experiment. This equation says that slowness variance is inversely proportional to the squared segment length of the ray that passes through the cell times the number of rays that pass through the cell. Thus the model error can be assessed as a function of data errors and raypath segment lengths. For arbitrary source-receiver distributions and model parameterizations, the covariance matrix [L^TL]^-1 can be used to determine model errors in terms of the data errors.