In this chapter I assume that the multiple model has already been estimated by some method and concentrate on the adaptive subtraction step to get the estimated-matched primaries. I present a new adaptive matching algorithm that simultaneously matches estimates of the primaries and the multiples to the data. In contrast, the most standard algorithms adaptively match the multiples only. Matching also the estimate of the primaries help constrain the matching of the multiples thus reducing the leak of residual multiples (so-called cross-talk) on the estimated primaries.
The new adaptive-matching algorithm estimates, in the least-squares sense, non-stationary filters (, ) that simultaneously match both the estimates of the primaries and the multiples to the data. These filters act on micro-patches i.e small, overlapping pieces of data (, ) and can handle inaccuracies in the estimated multiples in terms of both amplitudes and kinematics. Once the solution to the least-squares problem is computed, I iteratively re-estimate the multiple and primary models until the residual (the sum of the matched primaries and multiples minus the data) is close to zero. In my experience, as few as three to five iterations of the least-squares inversion (``outer'' iterations) seem sufficient.
I apply this new method to two synthetic datasets contaminated with multiples. In the first test, I match kinematically perfect estimates of primaries and multiples contaminated with 40 of cross-talk and show that for this simple case the method produces a cross-talk-free result. Then I apply the method to an inaccurate estimate of both the primaries and the multiples obtained via migration-demigration as described in (). Even with a poor initial estimate of both primaries and multiples, with strong cross-talk on both, the matched results are very good, with little cross-talk. To illustrate the method with stacked data, I apply it to a migrated section of the Sigsbee model. Here the multiples were estimated with an image space version of SRME (, ). The results show that the method attenuated most of the multiples and produced a largely multiple-free estimate of the primaries.
The method performs well with real data, as I demonstrate by applying it to match the estimated multiples computed in the previous chapter. I adaptively matched and subtracted the multiple estimate of each individual ADCIGs and then stacked the estimated primaries to form an angle stack of primaries only. The method performed very well and the multiples were nicely attenuated in the angle stack.
Finally, to illustrate that the method may have applications beyond the matching of primaries and multiples, I apply it to a different problem, namely the separation of ground-roll and body-waves. I use a shot gather from a land dataset contaminated with strong, spatially-aliased, ground-roll and show that most of the residual ground-roll can be attenuated in the final estimate of the body waves. In a way, this is a more challenging problem because the non-stationarity characteristics of the ground-roll and the body waves are different. The requirements of filter lengths and patch sizes to match the data are therefore different for the ground-roll and the body-waves. I chose to preserve the body waves even if that meant allowing some residual ground-roll.