Taylor Series derivation of nonstationary wavefield extrapolators
Gary F. Margrave, Robert James Ferguson
Taylor series expansion of extrapolated wavefields leads directly to the elementary nonstationary wavefield extrapolators combination and convolution. Other more commonly implemented, extrapolators are derived in this way, and a comparison between them and nonstationary extrapolators is made. Nonstationary combination is found to be equivalent to infinite series implementations of recursive explicit extrapolators (often called -x methods), and thus more correctly approximates one-way extrapolation. No existing -x analogue is found for nonstationary convolution.
The relationship between nonstationary extrapolators and pseudo-differential operators provides a basis for error analysis. The errors corresponding to the combination and convolution operators are found to be complimentary. That is, any composition of these operators, resulting in an averaging of their vertical wavnumbers, tends to increase the order of the resulting error and cancels complex values. A new symmetric extrapolator suggested by this analysis, and an existing one (symmetric nonstationary phase shift), are shown to be more accurate and more stable than the elementary extrapolators.