Time residual estimation by adaptive stacking

B.L.N. Kennett and N. Rawlinson

Figure 1: The vertical-component records of the South Vanuatu event at the short-period stations in the TIGGER deployment.

A segment of the original records surrounding the main P arrivals is selected, typically 20-40 s long. The theoretical patterns of arrival are calculated for the AK135 model for the assigned event parameters. The station tsi3 is taken as the reference point for the TIGGER array and the time shifts from the reference value are applied to approximately align the traces. The application of these theoretical time shifts gives a good general alignment (Figure 2) but there are clear systematic patterns in the residuals from the AK135 predictions. The charcter of the records themsleves are used to determine the residual pattern across the stations

Figure 2: The records of the South Vanuatu event aligned with the time corrections for the AK135 model relative to station tsi3. Both the P and pP arrivals are quite well aligned but clear residual patterns remain.

The linear stack of a suitable portion of the aligned traces is used as an estimator of the typical P waveform across the array, and the quadratic stack is also calculated as an indicator of the spread in alignment. These traces are shown at the top of figure 2.

For each moveout-corrected trace the time shift needed to minimise the difference from the linear stack is sought, using an L3 measure of the differences between the actual andseismogram and the shifted stack. The best fitting shift is found by a simple direct search. The L3 measure is intolerant of discrepancies but interestingly the procedure works well even with noisy data because just the minimum for a range of time shifts is sought; good alignment of the main features of the record will give the least value for this L3 measure.

Once the set of time shifts are found using the current estimate of the lienar stack, they are applied to improve the alignment of the full set of traces and the stacks are recalculated. The estimation of the residuals from the AK135 model is then carried out with the improved stacks. Five iterations of this adaptive stacking procedure yields excellent alignment of the corrected traces and stable estimates of the residual pattern. The quadratic stack then displays tight peaks as in figure 3. Excellent alignement of the traces is achieved (even where there is significant noise)

Figure 3: The records of the South Vanuatu event aligned with the time corrections for the AK135 model relative to station tsi3 and the corrections for residuals from AK135. Note the well defined alignment indicating high quality estimnates of the residuals. The upper quadratic stack traces is now strongly peaked.

The stack traces can then be used to try to estimate the absolute onset time corresponding to the reference station. Note that alignment can frequently be most effective on pP and the actual P arrivals can be quite small.