Spatial versus ensemble averaging for modeling wave propagation in finely layered media

Abstract

The shapes of plane-wave pulses propagating normally through a discretely layered medium are predicted using three methods. The first uses the method of averaging to predict the pulse traveling through a particular layered medium. This method shows that the pulse spreads solely due to admittance variations from layer to layer. The slowness variations affect only the arrival time of the pulse. The second method applies ensemble averaging. The predicted pulse suffers spreading due to impedance variations, and velocity variations. Ensemble averaging over pulses which have different arrival times (due to velocity variations) causes spreading in excess of that in any realization. The third method applies ensemble averaging after straightening the characteristics (i.e., transforming the spatial variable from distance to travel time). This gives the same result as the first method, although the significance is different in that the first method applies to any realization, but the third only to the ensemble. Thus ensemble averages must be applied with care if there is interest in a single realization, i.e., a particular layered medium; and method one is preferable when it applies.